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Peter Russell may refer to: Peter H. Russell, Canadian political scientist and writer Peter Nicol Russell 18161905, Australian philanthropist and university benefactor Peter Russell cricketer born 1939, English cricketer Peter Russell politician 17331808, Canadian gambler, government official, politician and judge Peter Russell poet 19212003, British poet, translator and critic Peter Russell footballer born 1935, English footballer active in the 1950s Peter Russell fashion designer 18861966, IncSoc founder member and designer active in Britain from 193153 Peter Russell ice hockey born 1974, British ice hockey player and coach See also Russell Peters born 1970, Canadian comedian Russ Peterson disambiguation

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circumstellar discs are a natural by - product of the star formation process ( e.g. shu , adams & lizano 1987 ) . lada et al . ( 2000 ) found that 97 percent of the optical proto - planetary discs in the trapezium cluster exhibit excess in the _ jhkl _ colour - colour diagram , indicating that the most likely origin of the observed ir excesses are the circumstellar discs . the _ disc lifetime _ derived by examining the fraction of ir excess stars ( the _ disc fraction _ ) of young stellar populations as a function of age provides an empirical limit on the duration of the disc accretion phase . this is critical for understanding the evolutionary paths followed by pms stars in the hr diagram , their angular momentum histories , and the timescales available for planet building ( e.g. hartmann et al . 1998 ; hillenbrand et al . 1998 ; telesco et al . 2000 ) . considerable debate has waged concerning the longevity of discs around low mass pms stars . the effort is difficult because samples are often incomplete and biased : ctt stars are typically located by h@xmath4 and near - ir excess surveys , while wtt stars are mostly found through x - ray surveys ( feigelson & montmerle 1999 ) . an influential early study on the timescale for disc dissipation by strom et al . ( 1989 ) , based on several dozen pms stars in the taurus - auriga complex , reported the ctt / wtt transition occurs around an age @xmath8 myr . this result is supported by a recent _ jhkl _ survey of 7 clusters with mean ages from 0.5 to 5 myr that shows half the stars lose their discs within 3 myr and essentially all lose their discs in 6 myr ( haisch , lada & lada 2001 ) . at a later age of @xmath9 myr , only 1/110 sco - cen late - type stars show spectroscopic ctt emission lines and @xmath10-band excesses ( mamajek , meyer & liebert 2002 ) . however , other studies suggest discs are more enduring . no evolution in disc fraction is found in the stellar populations of the orion nebula cluster from @xmath11 myr ( hillenbrand et al . 1998 ) , and in ngc 2264 from @xmath12 myr ( rebull et al . 2002 ) . the chamaeleon i cloud population , where the sample is enhanced through x - ray and _ iso _ surveys , shows no difference in the age distribution of ctt and wtt stars from @xmath13 myr ( lawson , feigelson & huenemoerder 1996 ) . spectroscopic study of the @xmath7 myr - old tw hydrae association members found active accretion in 2 stars ; tw hya and hen 3 - 600a ( muzerolle et al . 2000 ) , albeit at mass accretion rates @xmath14 magnitudes lower than is derived for younger accreting systems . the measurement of disc lifetimes is muddied by several other issues . first , despite the broad overlap in ctt and wtt age distributions , few intermediate cases are known suggesting that the transition between the two states is rapid ( wolk & walter 1996 ) . secondly , some studies suggest that two disc lifetimes must be considered ; one associated with a rapid decline in accretion onto the star and another associated with a slower dissipation of the outer disc ( clarke , gendrin & sotomayor 2001 ) . third , efforts to reconcile rotational distributions of both pms and zero - age main sequence stellar clusters assuming star - disc rotational coupling during the accretion phase have deduced relatively long @xmath15 myr disc lifetimes for slowly rotating stars ( bouvier , forestini & allain 1997 ; tinker , pinsonneault & terndrup 2002 ) , but @xmath16 myr disc lifetimes may suffice if the stellar interiors have sufficient radial differential rotation ( barnes , sofia & pinsonneault 2001 ) . finally , the astrophysical processes leading to disc dissipation are highly uncertain . possible mechanisms include accretion onto the star , bipolar outflow , stellar winds , photoevaporation , close gravitational encounters and incorporation of disc material into planets ( see the review by hollenbach , yorke & johnstone 2000 ) . the recently discovered @xmath1 chamaeleontis cluster has the potential of clarifying some of the observational issues . it is a nearby ( @xmath17 pc ) , compact ( extent @xmath181 pc ) and coeval ( @xmath19 myr ; lawson & feigelson 2001 ) system of pms stars with a small ( 15 known primaries ) population of stars spanning a relatively large range in mass ( @xmath20 m@xmath21 ; mamajek , lawson & feigelson 1999 , 2000 ; lawson et al . 2001 , 2002 ) . the cluster includes @xmath1 cha ( spectral type b8 ) , hd 75505 ( a5 ) , the a7+a8 binary and @xmath22 scuti system rs cha , 11 wtt stars ( k5@xmath23m4 ) and 1 ctt star ( m2 ) . importantly , this census is known to be virtually complete in the inner region from a combination of a deep @xmath24 exposure , optical photometry and proper motion study of the field . unlike most other pms populations , these membership criteria are independent of the presence or absence of circumstellar discs . we can thus use the @xmath1 cha population as an unbiased laboratory to investigate the fraction of ir excess and accretion at a critical intermediate - age phase of disc evolution . a @xmath25-band ( 3.5 @xmath26 m ) map of the @xmath1 cha cluster was made during 1999 july @xmath27 with the 0.6-m south pole infrared explorer ( spirex ) telescope , using the abu camera which had a 1k @xmath28 1k insb detector array . the abu camera had a plate scale of 0.6 arcsec pixel@xmath29 , giving a field - of - view ( fov ) of 100 arcmin@xmath30 . our observations were conducted in good conditions with below - average background levels . the raw data frames were pipeline - reduced at the rochester institute of technology before being delivered to us . using custom iraf routines written for the spirex / abu system , we then merged the multiple dithered images made of each field into single frames for further analysis . _ jhk@xmath31_-band images of the cluster members were obtained during 2002 march @xmath32 with the cryogenic array spectrometer / imager ( caspir ) on the 2.3-m telescope operated by mount stromlo and siding spring observatories ( mssso ) . caspir uses a 256 @xmath28 256 insb detector array and we selected a fov of 4.5 arcmin@xmath30 giving a resolution of 0.5 arcsec pixel@xmath29 . our observations were obtained under photometric conditions in @xmath33 arcsec seeing . the caspir images were linearized , sky - subtracted and flat - fielded using customized iraf routines based on , e.g. ccdproc . the spirex and caspir images were analysed using photometric routines ( such as phot ) running within iraf . fluxes were extracted and calibrated by comparison to standard stars listed on the spirex homepage and iris photometric standards ( carter & meadows 1995 ) , respectively . examination of the image profiles for the early - type cluster members ( @xmath1 cha , rs cha and hd 75505 ) suggested the onset of saturation in the caspir @xmath34 and @xmath35 frames . for this reason we obtained _ jhkl _ photometry ( in the saao system ; carter 1995 ) for these 3 stars ( _ jhk _ only for hd 75505 ) using the 0.75-m telescope and mark ii ir photometer at the south african astronomical observatory ( saao ) . these data , along with measurements of ir standard stars , were obtained during the week of 2002 april 30 to may 6 . the saao _ kl _ data shows close agreement ( @xmath36 mag ) to the caspir @xmath10 and spirex @xmath25 data except for the @xmath25-band measurement of rs cha , which we discuss below . table 1 lists the _ photometry of the @xmath1 cha cluster members . we adopt 1@xmath3 uncertainties of 0.03 mag for the saao _ jhk _ and the caspir _ jhk@xmath31 _ data . for the brighter @xmath25-band sources ( @xmath37 ) we adopt a 1@xmath3 uncertainty of 0.05 mag . fainter @xmath25-band magnitudes are uncertain by 0.1 mag . for several recx stars we can compare our observations to on - line denis _ jk _ observations and to _ jhkl _ data published by alcal et al . in most cases , the magnitudes differ by @xmath38 mag , which we consider to be insignificant given differences between ir photometric systems , and the intrinsic variability of these stars ( lawson et al . 2001 ) . a special case is the eclipsing binary rs cha . using the ephemeris of clausen & nordstrm ( 1978 ) we find the spirex @xmath25-band measurement was obtained during the secondary eclipse , whereas the caspir and saao magnitudes were obtained at maximum light . in the following sections we make use of the saao photometry for the 3 early - type stars ; otherwise we adopt our caspir and spirex observations . colours derived from the individual magnitudes were transformed to the ` homogenized ' ir system of bessell & brett ( 1988 ) making use of equations provided by bessell & brett ( 1988 ) and mcgregor ( 1994 , 1997 ) . for ( @xmath39 ) and ( @xmath40 ) colours derived from the caspir photometry , the correction is small ( @xmath41 mag ) . no correction was applied to the ( caspir @xmath10 spirex @xmath25 ) colour due to the mix of photometric systems , and any correction is likely swamped by the @xmath42 mag uncertainty in the @xmath25-band photometry . for colours derived for the early - type stars from saao photometry , the effect of transforming the colours from the saao system to the ` homogenized ' system is a correction of @xmath43 mag . @lrrrr@ star & @xmath34 & @xmath35 & @xmath10 & @xmath25 + + + + recx 1 & 8.20 & 7.60 & 7.27 & 6.97 + recx 3 & 10.57 & 9.86 & 9.61 & 9.11 + recx 4 & 9.69 & 8.92 & 8.66 & 8.32 + recx 5 & 10.99 & 10.29 & 9.96 & 9.26 + recx 6 & 10.42 & 9.74 & 9.46 & 8.93 + recx 7 & 8.61 & 7.92 & 7.69 & 7.48 + recx 9 & 10.53 & 9.83 & 9.50 & 8.82 + recx 10 & 9.68 & 8.95 & 8.78 & 8.48 + recx 11 & 8.85 & 8.06 & 7.71 & 7.08 + recx 12 & 9.38 & 8.70 & 8.51 & 8.02 + @xmath1 cha & & & 5.73 & 5.62 + rs cha & & & 5.45 & 5.84 + hd 75505 & & & 6.98 & 6.81 + echa j0841.57853 & 11.90 & 11.30 & 10.94 & + echa j0843.37905 & 10.66 & 9.93 & 9.45 & 8.40 + + + + @xmath1 cha & 5.68 & 5.72 & 5.75 & 5.65 + rs cha & 5.60 & 5.46 & 5.43 & 5.35 + hd 75505 & 7.10 & 7.02 & 7.00 & + colour - colour diagrams , constructed from multi - wavelength ir photometric and imaging surveys , have been shown to be a powerful tool for identifying ir excesses and circumstellar discs around stars in young clusters and star - forming regions . in particular , @xmath25-band data combined with shorter wavelength observations permits evaluation of the fraction of sources with ir excess emission from circumstellar discs the _ disc fraction _ ( see , e.g. haisch , lada & lada 2000 ; kenyon & gmez 2001 ; lada et al . as kenyon & gmez ( 2001 ) convincingly showed in their spirex @xmath25-band study of the cha i molecular cloud , @xmath25-band photometry is nearly essential for a meaningful evaluation of the disc fraction in a young stellar population . _ jhk _ observations alone do not extend to a long enough wavelength range to enable a complete or unambiguous census of circumstellar discs in young clusters . data obtained at 3.5 @xmath26 m provides more contrast relative to photospheric emission from the central star compared to the shorter wavelength observations . figure 1 shows ( a ) _ jhk _ and ( b ) _ jhkl _ colour - colour diagrams of the @xmath1 cha cluster members . all 15 cluster members are shown in figure 1(a ) . only 14 are shown in figure 1(b ) ; the m4 cluster member echa j0841.5 - 7853 was not observed at @xmath25 band . in these diagrams the solid curves are the locus of colours corresponding to main - sequence stars of spectral types b8 m5 ( bessell & brett 1988 ) , which encompasses the range of spectral types of the cluster members . in each figure , the dashed parallel lines define the reddening bands derived from relationships given by bessell & brett ( 1988 ) , where @xmath44/@xmath45 and @xmath44/@xmath46 , respectively . stars which lie in the right of the reddening band , after due consideration of uncertainties in the reddening law and in the photometry , are ir excess objects and therefore circumstellar disc candidates . as kenyon & gmez ( 2001 ) found in their study of cha i , only those stars with the largest ir excesses fall to the right of the reddening band in the _ jhk _ plane , whereas many more stars fall to the right of the reddening band once @xmath25-band data is available . if photometric errors were negligible , we could count all late - type stars with ( @xmath47 ) immediately to the right of the reddening band ; however with @xmath42 mag uncertainties in the @xmath25-band data , we count only those late - type stars with ( @xmath47 ) colours 0.1 mag redder than the reddening band as ir - excess objects . this criterion also largely eliminates uncertainty in the reddening law as a contributor to the disc fraction , as the reddening line is almost constant in ( @xmath47 ) colour over the narrow range of ( @xmath39 ) colours for the late - type stars . from figure 1(b ) , the ` gap ' in the ( @xmath47 ) colours between 0.34 and 0.49 allows us to count 7 stars as the most likely number of ir excess objects in the low - mass population ) = 0.4 delineated the disced and the discless populations . ] . to estimate the uncertainty in this number , we calculated the variation in the number of stars if the photometric uncertainties are considered at the 2@xmath3 level , i.e. twice the level of the adopted photometric errors ( see section 2 ) . now we find @xmath48 could be counted as ir excess objects . use of standard ir colour - colour diagrams such as figure 1(b ) could lead to an under - estimate of the disc fraction since k - type stars need to have a colour excess @xmath49 to be counted , whereas late - m stars need only @xmath50 . an alternative ir colour - colour diagram which largely eliminates this problem is shown in figure 2 . in this plane the reddening band is narrow , and the reddening vector is parallel to the main - sequence , enabling stars to be counted using the ( @xmath47 ) excess as the criterion . for @xmath51 above the main - sequence line of bessell & brett ( 1988 ) , we count the same 7 stars as the most - likely ir excess objects . if we adopt the same 2@xmath3 error budget as above , we again count @xmath48 as the range of ir excess stars . also , as the faint cluster member echa j0841.57853 has _ jhk _ colours consistent with its m4 spectral type , it is unlikely to have a large ir excess , and so we conclude @xmath48 out of the 12 known late - type members are ir excess objects . for the 3 early type stars , we may be justified in defining a lower colour excess as being significant . @xmath1 cha is a @xmath52 pic - disc source , and with 0.05 mag uncertainty in the saao @xmath25-band flux the star has a ( @xmath47 ) excess that is significant at the 3@xmath3 level . for hd 75505 , a 0.05 mag uncertainty in the spirex @xmath25-band magnitude gives rise to a 4@xmath3 excess . thus with @xmath53 certainty , 2 of the 3 early - type cluster members are ir excess objects . since an ir excess most likely indicates the presence of a circumstellar disc , we might then expect to find a correlation between the ir excess and accretion indicators such as enhanced h@xmath4 emission , which is believed to originate in magnetospheric columns allowing transport of disc material to the stellar surface . two recent studies have demonstrated a correlation between the h@xmath4 equivalent width ( @xmath54 ) and the ( @xmath47 ) colour in cha i pms stars ( kenyon & gmez 2001 ) , or the ir colour excess in the ngc 2264 population ( rebull et al . 2002 ) . comparison between the h@xmath4 @xmath54 and the _ colour excess _ is desirable to eliminate redundancy caused by the spread in _ colour _ across a stellar population . determination of the colour excess for individual stars requires knowledge of the spectral type . for @xmath1 cha cluster members , spectral types have been determined from high resolution studies for several recx stars that are also rosat all - sky survey stars ( alcal et al . 1995 ) , medium resolution spectra of all cluster members ( mamajek et al . 1999 , lawson et al . 2002 ) and optical photometric study ( lawson et al . 2001 , 2002 ) . for stars common to 2 or 3 studies , comparison of the spectral types suggests a typical uncertainty of @xmath55 subtype , a value comparable to the uncertainties present in the individual studies . in figure 3 , we find a strong correlation between the h@xmath4 @xmath54 and the ( @xmath47 ) excess for the late - type members . adopting the same colour excess as in figure 2 , we again count the same @xmath48 stars as having an ir excess when photometric errors are considered . merging the above results , we conclude @xmath56 out of the 15 known stellar systems in the cluster , or a fraction of @xmath2 ( 2@xmath3 ) have ir excesses . this number also defines the fraction of stars with _ dust discs _ , or the _ disc fraction _ of the population . however , a more important parameter in pms star evolution is the fraction of stars possessing _ accretion discs _ in a young stellar population . by combining studies of populations of different ages it is possible to determine the timescale for the end of significant star - disc activity and probably also the timescale for jovian planet building ( e.g. hillenbrand & meyer 1999 ) . ) /(@xmath40 ) colour - colour diagram for members of the @xmath1 cha cluster surveyed at 3.5 @xmath26 m . see the caption to figure 1 for the meaning of the solid and dashed lines and other symbols.,height=321 ] in other recent studies the number of accreting stars has been determined from the ( @xmath40 ) colour excess ( e.g. rebull et al . 2002 ) . this has been a useful technique since the ( @xmath40 ) colour is sensitive to warm inner discs , and also since _ jhk _ detector arrays have allowed entire star formation regions to be surveyed . with the introduction of @xmath25-band imagers , sensitive to cooler dust and to lower luminosity discs due to the increased contrast provided by the @xmath25-band photometry , we expect the disc fraction to increase as we demonstrate in figure 1 . but with the availability of @xmath25-band data , do we continue to detect accretion discs or are the @xmath25-band measurements increasingly sensitive to remnant dust discs in systems where there is no longer star - disc coupling ? the ( @xmath47 ) excesses measured for the @xmath1 cha cluster stars ( figure 3 ) suggest a continuum of behaviour ; from a ctt star ( echa j0843.37905 ) showing strong star - disc interaction , stars that are ctt wtt transition objects still showing evidence for on - going accretion ( recx 5 , 9 and 11 ) , wtt stars with weak ir excesses ( recx 3 , 6 and 12 ) , and wtt stars with little or no ir excess ( recx 1 , 4 , 7 and 10 , and echa j0841.5 - 7853 ) . of the 15 known cluster members , the 4 stars with ir excesses @xmath5 are all likely to be experiencing on - going accretion ( the uncertainty in this number is @xmath57 stars ; 2@xmath3 ) . we have spectroscopically confirmed that accretion is present in two cases : echa j0843.3 - 7905 ( lawson et al . 2002 ) and recx 11 ( see below ) . for most of the late - type stars in the cluster , @xmath58 , so our accretion criterion is little different from that adopted by rebull et al . ( 2002 ) , where @xmath59 . these stars allow us to define an accretion fraction for the cluster of @xmath6 ( @xmath60 ) . we further consider the issue of accretion in section 4 . in figure 4 we show spectral energy distributions for 4 of the cluster stars . we plot the measured fluxes of the ctt star echa j0843.3 - 7905 , and offset the fluxes of the other stars to illustrate the range of ir signatures that are present in these stars . echa j0843.3 - 7905 shows a flat spectrum at near - ir wavelengths with high colour excess . on - going accretion in this star is supported by its rich optical emission spectrum , with a h@xmath4 @xmath61 . the star is likely associated with iras f084507854 . iras _ faint source catalogue ( _ iras _ fsc ) indicates high - quality 25-@xmath26 m and 60-@xmath26 m fluxes of 0.30 jy and 0.28 jy , respectively ; approximately twice the @xmath25-band flux . ( the _ iras _ fsc indicates an upper limit 12-@xmath26 m flux of 0.27 jy . ) recx 11 is one of 3 recx stars with @xmath62 , and with a h@xmath4 @xmath63 that places them near the traditional ctt - wtt star ` boundary ' of activity . mssso 1.9-m / coud spectroscopy of recx 11 shows the star is still accreting from its circumstellar disc , with broad ( width @xmath64 kms@xmath29 ) and variable infall signatures at h@xmath4 that are phased to the 3.95-d rotation period of the star ( lyo et al . , in preparation ) . recx 11 appears associated with iras f084877848 , with 12-@xmath26 m , 25-@xmath26 m and 60-@xmath26 m fluxes of 0.29 jy , 0.32 jy and 0.27 jy , respectively . recx 6 is representative of wtt stars in the cluster with ir excesses of @xmath65 , and recx 10 is a wtt star with little or no ir excess ( see figure 3 ) . emission - line _ ew _ versus @xmath66 excess for the late - type stars in the @xmath1 cha cluster . the individual stars are identified by their recx number , except for the ctt star echa j0843.37905 . stars with @xmath51 are identified as ir excess objects with circumstellar discs . those stars with @xmath5 are identified as stars with accretion discs , see section 3.3.,width=317 ] in addition to the a7+a8 dual - lined eclipsing binary and @xmath22 scuti system rs cha ab , on - going study of the cluster population has found several confirmed or probable binary systems . mamajek et al . ( 1999 ) noted that the h@xmath4 emission profiles of recx 7 and 9 were double , indicating that these stars may be spectroscopic binaries . lawson et al . ( 2001 ) noted several of recx stars had elevated @xmath67 magnitudes compared to other cluster members of similar spectral type ; recx 9 and 12 are elevated by @xmath68 mag ( suggesting near - equal mass systems ) and 2 of the k - type stars ( recx 1 and 7 ) are @xmath69 mag brighter than the third k - type cluster member recx 11 . speckle @xmath10-band imaging of recx 1 and 9 by khler ( 2001 ) found both stars have companions at separations of @xmath70 arcsec . observations of recx 1 made during 1996 and 2000 showed motion that might indicate a decade - long orbit . consideration of the stellar background density suggests these nearby stars are likely to be physically related to the primaries . if future observations confirm these systems , recx 1ab and recx 9ab have @xmath10-band brightness ratios of @xmath71 and 0.5 , respectively . observations of recx 7 made by us using the 1.9-m telescope and coud spectrograph at mssso during 2002 february showed recx 7 to be a dual - lined spectroscopic binary with a period of 2.6 d ( the same as the photometric period ; lawson et al . 2001 ) and a mass ratio of @xmath72 2.3:1 . ( recx 7 has also been observed to be a spectroscopy binary by donati , private communication . ) if recx 7a is a near - solar mass star , then recx 7b is a @xmath73 m@xmath21 early - m star . recx 12 remains a candidate binary . however , lawson et al . ( 2001 ) found 2 periods ( 1.3 and 8.6 d ) in the @xmath67-band light curve of the star in observations made in 1999 and 2000 . one of these periods may be the binary period . -band fluxes and assuming no reddening . the adopted spectral types are uncertain by @xmath55 subtype ; see section 3.2 . the worst - case photometric errors ( the @xmath74 percent uncertainties in the @xmath25-band magnitudes ) translate to roughly the plotted size of the points . the broad band magnitudes were converted to fluxes using the conversions of bessell ( 1979 ) , bessell & brett ( 1988 ) and koornneef ( 1983).,height=321 ] we have considered the effect of a binary companion on the ir excesses determined from our _ jhkl _ photometry . for a system such as recx 7 with a mass ratio of @xmath75:1 we calculate , using the pms models of siess et al . ( 2000 ) , that the near - ir colours will appear redder due to the presence of the secondary by @xmath76 mag . binaries with higher mass ratios show less distortion of the primary star colours . due to the luminosity ratio of such systems , any ir excess present is likely associated with the primary . for distant near - equal mass systems such as recx 1 the available data does not allow us to determine which member of the binary contains the ir excess , or if the excess is shared . for close binaries , we would envisage a circumbinary disc if one were present ; the 2 confirmed spectroscopic binaries in the cluster ( rs cha and recx 7 ) have no ir excess ( see figure 2 ) . since our criterion for the presence of an ir excess is @xmath51 , we conclude that binarity will not significantly distort our results . photometry of the cluster members indicates reddening is absent or low . for the early - type stars , westin ( 1985 ) found @xmath77(@xmath78 ) @xmath79 for @xmath1 cha , and the light curve of the binary rs cha has been modelled assuming @xmath77(@xmath80 ) = 0.0 ( e.g. , clausen & nordstrm 1978 ) . for the late - type stars , our ir photometry indicates @xmath81 , whereas the reddening vector in this plane has a gradient of 2.47 ( see figure 1b ) . we conclude that reddening is unimportant in our determination of the disc fraction . comparison of the ir colour excesses indicates the 3 k - type stars ( recx 1 , 7 and 11 ) have @xmath10-band excesses of @xmath82 mag , which has the effect of increasing the ( @xmath40 ) colours and decreasing the ( @xmath47 ) colours . this effect is not seen in the m - type stars , and can not be attributed to reddening . without 2-@xmath26 m spectroscopy we can not confirm if the excess is due to @xmath10-band activity , e.g. greene & lada ( 1996 ) detected br@xmath83 emission in half the ctt stars in a sample of @xmath84 oph pms stars , although in none of the wtt stars . for recx 1 , an increase in the ( @xmath47 ) colour of @xmath82 mag would make the ir excess significant . however , the star is already included in our estimate of ir excess objects once we account for observational uncertainties . considerable uncertainty reigns concerning the longevity , or more likely , the distribution of longevities , of circumstellar discs ( section 1 ) . despite its modest population , the @xmath1 cha cluster provides a rare opportunity to examine at high sensitivity disc properties of pms stars with intermediate ages whose selection is unbiased with respect to disc existence cha cluster in this respect is recognized by its inclusion in the first year guaranteed time programme of the _ space infrared telescope facility_. ] . if discs decay rapidly as indicated by some past studies , then no discs at all are expected in this cluster . we find , however , that 9/15 or 60 percent of @xmath1 cha primaries show ir excesses in the @xmath85 diagram ( the late - type stars identified in figure 2 , plus @xmath1 cha and hd 75505 ) . the excess can not be attributed to errors in photometry ( section 2 ) , binarity ( section 3.4.1 ) or reddening ( section 3.4.2 ) . long - lived circumstellar discs are the only plausible explanation . one of these stars , echa j0843.3 - 7905 , is a ctt star with active accretion ( lawson et al . 2002 ) , and the h@xmath86 correlation seen in the late - type population suggests that up to 3 other stars may be accreting ( see figure 3 ) . high - resolution spectroscopic study now underway will address this issue . why do we find a high disc fraction at @xmath87 myr when some other studies find discs largely disappear by @xmath88 myr ? we first recognize that , except for echa j0843.3 - 7905 , the @xmath1 cha discs would have been mostly missed from _ colours alone ( see figure 1a ) . sensitive @xmath89band surveys are essential for the detection of aging pms discs . the principal discrepancy among @xmath89band studies lies between our high disc fraction ( 9/15 or 0.60 ) in @xmath1 cha and the low disc fraction ( 9/75 or 0.12 ) for the 5 myr old cluster ngc 2362 found by haisch et al . we suggest several explanations for this difference . first , the assigned age of ngc 2362 pms stars relies solely on the turnoff age of the o9ib supergiant @xmath90 cma ( balona & laney 1996 ) and the assumption that all stars in the cluster are coeval . secondly , ngc 2362 is 1480 pc distant compared to 97 pc for @xmath1 cha . the distance ratio alone degrades the @xmath25-band sensitivity by a factor of 200 . because of this , faint discs in ngc 2362 may not have been detected . also , the limiting mass of the haisch et al . ( 2001 ) study of ngc 2362 is @xmath91 m@xmath21 ( spectral type mid - k ) compared to @xmath92 m@xmath21 ( spectral type m4 ) in our study of @xmath1 cha . as we discuss in section 3.1 , it is easier to detect an ir excess in a late - m star , compared to a k - type star , using standard ir colour - colour plane analysis . also , it is possible that disc lifetimes are shorter in higher mass stars . third , there might be variance in the disc destruction rate amongst clusters due to different rates of close encounters or photoevaporation due to massive stars , e.g. the discs in ngc 2362 might have been stripped by the uv / wind of @xmath90 cma ( see hollenbach et al . 2000 for theory ) . a combination of the above factors can explain why there is a @xmath93 dispersion in the observed disc fraction for pms star clusters of a similar age ( hillenbrand & meyer 1999 ) . while noting these differences , our results seen together with studies of other older nearby pms stars ( e.g. study of the tw hydrae association members by muzerolle et al . 2000 ) indicate that ir - detected discs can be present in @xmath94 percent , and accretion discs can be present in @xmath95 percent , of @xmath7 myr - old pms stars . we thank c. kaminski for obtaining our spirex / abu observations , and j. kastner and colleagues at the rochester institute of technology for performing the pipeline analysis of the raw observations . we also thank m. burton ( unsw ) for coordinating the australian allocation of spirex observing time , s. james for installing the spirex / abu reduction package at adfa , and n. zarate ( noao ) at the iraf help desk for on - line assistance . spirex was a facility operated by the center for astrophysical research in antartica . we thank the mssso and saao time allocation committees for telescope time during 2002 which enabled us to complete this project . mssso is operated by the research school of astronomy and astrophysics , australian national university . saao is a national facility operated by the national research foundation of south africa . arl acknowledges the support of a unsw / adfa international postgraduate research scholarship . wal acknowledges support from the unsw research support programme and unsw / adfa special research grants . eem thanks the sirtf legacy science program for support . edf s research is supported in - part by nasa contracts nas8 - 38252 and nag5 - 8422 . lac is supported by a national research foundation post - graduate scholarship . alcal j. m. , krautter j. , schmitt j. h. m. m. , covino e. , wichmann r. , mundt r. , 1995 , a&as , 114 , 109 balona l.a . , laney c.d . , 1996 , mnras , 281 , 1341 barnes s. , sofia s. , pinsonneault m. , 2001 , apj , 548 , 1071 bessell m. s. , 1979 , pasp , 91 , 589 bessell m. s. , brett j. m. , 1988 , pasp , 100 , 1134 bouvier j. , forestini m. , allain s. , 1997 , a&a , 326 , 1023 carter b. s. , 1995 , ap&ss , 230 , 163 carter b. s. , meadows v. s. , 1995 , mnras , 276 , 734 clarke c. j. , gendrin a. , sotomayor m. , 2001 , mnras , 328 , 485 clausen j. v. , nordstrm b. , 1978 , a&as , 33 , 87 feigelson e. d. , montmerle t. , 1999 , ara&a , 37 , 363 greene t.p . , lada c. j. , 1996 , aj , 112 , 2184 haisch k. e. jr . , lada e. a. , lada c. j. , 2000 , aj , 120 , 1396 haisch k. e. jr . , lada e. a. , lada c. j. , 2001 , apj , 553 , l153 hartmann l. , calvet n. , gullbring e. , dalessio p. , 1998 , apj , 495 , 385 hillenbrand l. a. , meyer m. r. , 1999 , am . soc . , 195 , 209 hillenbrand l. a. , strom s. e. , calvet n. , merrill k. m. , gatley i. , makidon r. b. , meyer m. r. , skrutskie m. f. , 1998 , aj , 116 , 1816 hollenbach d. j. , yorke h. w. , johnstone d. , 2000 , in mannings v. , boss a. p. , russell s. s. , eds , protostars and planets iv , university of arizona , p. 401 kenyon s. j. , gmez m. , 2001 , aj , 121 , 2673 kenyon s. j. , hartmann l. , 1995 , apjs , 101 , 117 khler r. , 2001 , in jayawardhana r. , greene t. p. , eds , asp conf . 244 , young stars near earth . , san francisco , p. 277 koornneef j. , 1983 , a&a , 128 , 84 lada c. j. , muench a. a. , haisch k. e. jr . , lada e. a. , alves j. f. , tollestrup e v. , willner s. p. , 2000 , aj , 120 , 3162 lawson w. a. , crause l. a. , mamajek e. e. , feigelson e. d. , 2001 , mnras , 321 , 57 lawson w. a. , crause l. a. , mamajek e. e. , feigelson e. d. , 2002 , mnras , 329 , l29 lawson w. a. , feigelson e. d. , huenemoerder d. p. , 1996 , mnras , 280 , 1071 lawson w. a. , feigelson e. d. , 2001 , in montmerle t. , andr p , eds , asp conf . series vol . 243 , from darkness to light , astron . pac . , san francisco , p. 591 mamajek e. e. , lawson w. a. , feigelson e. d. , 1999 , apj , 516 , l77 mamajek e. e. , lawson w. a. , feigelson e. d. , 2000 , apj , 544 , 356 mamajek e. e. , meyer m. r. , liebert j. w. , 2002 , aj , in press mcgregor p. j. , 1994 , pasp , 106 , 508 mcgregor p. j. , 1997 , caspir manual , http://www.mso.anu.edu muzerolle j. , calvet n. , briceo c. , hartmann l. , hillenbrand l. , 2000 , apj , 535 , l47 rebull l. m. , et al . , 2002 , aj , 123 , 1528 shu f. h. , adams f. c. , lizano s. , 1987 , ara&a , 25 , 23 siess l. , dufour e. , forestini m. , 2000 , a&a , 358 , 593 strom k. m. , strom s. e. , edwards s. , cabrit s. , skrutskie m. f. , 1989 , aj , 97 , 1451 telesco c. m. , et al . , 2000 , apj , 530 , 329 tinker j. , pinsonneault m. , terndrup d. , 2002 , apj , 564 , 877 westin t. n. g. , 1985 , a&as , 60 , 99 wolk s. j. , walter f. m. , 1996 , aj , 111 , 2066

arxiv

a frequently used model for stochastically evolving multitype populations is the fleming viot diffusion . in the neutral case the corresponding genealogy at a fixed time @xmath1 is described by the kingman coalescent which was introduced some 30 years ago as the random genealogy relating the individuals of a population of large constant size in equilibrium ( @xcite ) . as a genealogical tree with infinitely many leaves , the kingman coalescent exhibits some distinct geometric properties . in particular , s. evans studied the random tree as a metric space , where the distance of two leaves is given by the time to their most recent common ancestor ( @xcite ; see also the fine - properties of the metric space derived in @xcite ) . using this picture , the kingman coalescent close to its leaves has a nice shape : roughly speaking , in the limit @xmath3 , approximately @xmath6 balls of radius @xmath2 are needed to cover the whole tree ; see section 4.2 in d. aldous review article ( @xcite ) . equivalently , there are @xmath6 families whose individuals have a common ancestor not further than @xmath7 in the past . moreover , these @xmath8 families have sizes of order @xmath7 . more precisely , the size of a typical family is exponential with parameter @xmath8 ( see eq ( 35 ) in@xcite ) , and the empirical distribution of the family sizes converges to this exponential distribution . however , these results have been proved only for the genealogy of a population at a fixed time . in a series of papers of the authors , in part with a. winter , @xcite the kingman coalescent was extended to a tree - valued process @xmath0 , where @xmath5 gives the genealogy of an evolving population at time @xmath1 . the resulting process , the tree - valued fleming viot process , is connected to the fleming viot measure - valued diffusion , which describes the evolution of type - frequencies in a large ( i.e.infinite ) population of constant size . in the simplest case of neutral evolution all individuals have the same chance to produce viable offspring , i.e. , the frequency of offspring of any subset of individuals is a martingale . however , biologically most interesting is the _ selective case _ where the evolutionary success of an individual depends on its ( allelic ) type and where also mutation ( i.e. random changes in types ) may occur . this case including mutation and selection was studied in @xcite . we note that rather than studying the full - tree valued process in the infinite population limit , it is possible to obtain limits of its functionals directly as well . for the neutral tree - valued fleming - viot process , this has been done for the height @xcite and the length @xcite . in addition , functionals of other tree - valued processes have been studied , e.g. for the height of the tree in branching processes @xcite and for the height and length of a population with the bolthausen - sznitman coalescent as long - time limit @xcite . * goals : * the construction of the tree - valued fleming viot process allows one to ask if the above mentioned properties of the geometry of the kingman coalescent trees are almost sure path properties of the tree - valued fleming - viot process . furthermore , while we gave a construction of the tree - valued fleming viot process under neutrality in @xcite and under mutation and selection in @xcite , some questions about path behavior remained open . we will carry over some ( not all ) of the geometric properties of the fixed random trees to the evolving paths of trees in theorems [ t1 ] [ t4 ] of this work . in the next section , we explain in detail how we model _ genealogical trees_. in order to formulate open questions let us briefly mention here that we use a _ marked metric measure space ( mmm - space ) _ , that is , a triple @xmath9 where @xmath10 is a complete metric space describing genealogical distances between individuals and @xmath11 is a probability measure on the borel-@xmath12 algebra of @xmath13 , where @xmath14 is the set of possible ( allelic ) types . in particular , the tree - valued fleming viot process @xmath0 takes values in the space of ( continuous ) paths in the space of mmm - spaces . to state two open questions from earlier work ( see remark 3.11 in @xcite ) , let @xmath15 be the state of the tree - valued fleming viot process at time @xmath16 . first , we ask if the measure @xmath17 has atoms for some @xmath4 . to understand what this means , recall that the state of the measure - valued fleming viot process is purely atomic for all @xmath4 , almost surely . however , in the tree - valued case , existence of an atom in the measure @xmath18 implies that there exists a set of positive @xmath17-mass such that individuals belonging to this set have zero genealogical distance to each other . as we will see in theorem [ t5 ] , this is not possible , and the tree - valued fleming viot process is non - atomic for all @xmath4 , almost surely . second , we ask if every individual in @xmath19 can uniquely be assigned a type which is of course the case for the moran model , but does not automatically carry over to the ( infinite population ) diffusion limit . this is the case iff the support of @xmath17 is given by @xmath20 for a function @xmath21 . in theorem [ t6 ] , we will see that this is indeed the case and every individual can be assigned a type for all @xmath4 , almost surely . * methods : * since the tree - valued fleming viot process was constructed using a well - posed martingale problem , we will frequently use martingale techniques in our proofs . these allow us to study the _ sample laplace - transform _ for the distance of two points of the tree as a semi - martingale . in addition , population models have specific features that will also be useful . for example all individuals have unique ancestors even though not all individuals have descendants and if an individual has a descendant , she might as well have many . this simple structure can be used for finite population models ( e.g. the moran model ) or the tree - valued fleming viot process , since this infinite model arises as a large - population limit from finite moran models ( for the neutral case see theorem 2 of @xcite and for the selective case theorem 3 of @xcite ) to derive properties of the family structure . an important point of the proofs is that we can transfer properties from the neutral case since for most forms of selection ( which are determined by the interacting fitness functions , which gives the dependence of the offspring distribution depends on the allelic type ) , the resulting process is _ absolutely continuous _ to the _ neutral case _ ( which comes with no dependency between allelic type and offspring distribution ) via a _ girsanov transform_. * outline : * the paper is organized as follows : in section [ s.tvfv ] we recall the definition of the state space of the tree - valued fleming viot process , its construction by a well - posed martingale problem and some of its properties . in section [ s : res ] , we give our main results . theorem [ t1 ] states that the law of large numbers for the number of ancestors of kingman s coalescent holds along the whole path of the tree - valued fleming viot process . moreover , we discover a brownian motion within the tree - valued fleming viot process based on the fluctuations of the number of ancestors ; see theorem [ t2 ] . another law of large numbers is obtained for a statistic concerning the family sizes and we make a big step towards this result in theorem [ t3 ] . another brownian motion is discovered within the tree - valued fleming viot process based on family sizes in theorem [ t4 ] . finally we show the non - atomicity along the path in theorem [ t5 ] and obtain existence of a mark function in theorem [ t6 ] . in section [ s.proofpcover ] we prove theorem [ t1 ] and after some preparatory moment computations in section [ s : prep ] , we give in the subsequent sections the remaining proofs of the main results . we note that various proofs have been carried out using mathematica and can be reproduced by the reader via the accompanying mathematica - file . in this section , we recall the tree - valued fleming viot process given as the unique solution of a _ martingale problem _ on the space of _ marked metric measure spaces_. the material presented here is a condensed version of results from @xcite and @xcite . we only recall notions needed to follow our arguments in the present paper . let us fix some notation first . + for [ not : aux ] a polish space @xmath22 the set of all bounded measurable functions is denoted by @xmath23 , its subset containing the bounded and continuous functions by @xmath24 , the set of cdlg function @xmath25 by @xmath26 ( which is equipped with the skorohod topology ) and the subset of continuous functions by @xmath27 . the set of probability measures on ( the borel @xmath12-algebra of ) @xmath22 is denoted by @xmath28 and @xmath29 denotes either weak convergence of probability measures or convergence in distribution of random variables . if @xmath30 for some polish space @xmath31 then the image measure of @xmath32 under @xmath33 is denoted by @xmath34 . for functions @xmath35 and @xmath36 , we write @xmath37 if there is @xmath38 such that @xmath39 uniformly for all @xmath40 . furthermore for @xmath41 we write @xmath42 if @xmath43 and @xmath44 are asymptotically equivalent as @xmath45 , i.e. if @xmath46 as @xmath47 . for product spaces @xmath48 we denote the projection operators by @xmath49 . when there is no chance of ambiguity we use the shorter notation @xmath50 . at any time @xmath51 the state of the neutral tree - valued fleming - viot process without types is a genealogical tree describing the ancestral relations among individuals alive at time @xmath1 . such trees can be encoded by ultrametric spaces and vice versa where the distance of two individuals is given by the time back to their most recent common ancestor . adding selection and mutation to the process requires that we not only keep track of the genealogical distances between individuals but also of the type of each individual . this leads to the concept of marked metric measure spaces which we recall here . for more details and interpretation of the state space we refer to section 2.3 in @xcite and to remark [ rem : int - st - sp ] below . throughout , we fix a _ compact metric space @xmath14 _ which we refer to as the _ ( allelic ) type space_. an _ @xmath14-marked ( ultra-)metric measure space _ , abbreviated as @xmath14-mmm space or just mmm - space in the following , is a triple @xmath9 , where @xmath10 is an ultra - metric space and @xmath52 is a probability measure on @xmath13 . the state space of the tree - valued fleming viot process is @xmath53 where @xmath54 is the equivalence class of the @xmath14-mmm space @xmath9 , and two mmm - spaces @xmath55 and @xmath56 are called _ equivalent _ if there exists an isometry ( here @xmath57 of a measure denotes its support ) @xmath58 with @xmath59 . the subspace of compact mmm - spaces @xmath60 will play an important role . + 1 . in [ rem : int - st - sp ] our presentation , only _ ultra_-metric spaces @xmath10 will appear . the reason is that we only consider stochastic processes whose state at time @xmath1 describes the genealogy of the population alive at time @xmath1 , which makes @xmath61 an ultra - metric . 2 . there are several reasons why we consider equivalence classes of marked metric spaces instead of the marked metric spaces themselves . the most important is that we view a genealogical tree as a metric space on its set of leaves . since in population genetic models the individuals are regarded as exchangeable ( at least among individuals carrying the same allelic type ) , reordering of leaves does not change ( in this view ) the tree . in order to construct a stochastic process with cdlg paths and state space @xmath62 , we have to introduce a topology . to this end , we need to introduce test functions with domain @xmath62 . + we set @xmath63 . a function @xmath64 is a _ polynomial _ , if there is a measurable function @xmath65 depending only on finitely many coordinates such that @xmath66 where @xmath67 is the infinite product measure , i.e. the law of a sequence sampled independently with sampling measure @xmath11 . let us remark that functions of the form are actually monomials . however , products and sums of such monomials are again monomials , and hence we may in fact speak of polynomials ; cf . the example below . + assume that @xmath33 only depends on the first @xmath68 coordinates in @xmath69 and the first @xmath70 in @xmath71 . then , we view a function of the form as taking a sample of size @xmath70 according to @xmath11 from the population , observing the value under @xmath33 of this sample and then taking the @xmath11-sample mean over the population . + some [ rem : intphi ] functions of the form will appear frequently in this paper , for example @xmath72 , @xmath73 this function arises from sampling two leaves , @xmath74 and @xmath75 , from the genealogy @xmath10 according to @xmath76 and averaging over the test function @xmath77 of this sample . then @xmath78 is again of the form and @xmath79 another function that will be used and which also depends on types is given by @xmath80 in this function @xmath74 and @xmath75 contribute to the integral only if their types , @xmath81 and @xmath82 agree . since we use polynomials as the domain of the generator for the tree - valued fleming viot process , we need to restrict this class to smooth functions . + we denote by @xmath83 the set of _ smooth ( in the first coordinate ) polynomials_. furthermore we denote by @xmath84 the subset of @xmath85 consisting of all @xmath86 for which @xmath87 depends at most on the first @xmath68 coordinates of @xmath88 and the first @xmath70 of @xmath89 and hence have degree at most @xmath70 . + the _ marked gromov - weak topology _ on @xmath62 is the coarsest topology such that all @xmath90 with ( in both variables ) continuous @xmath33 are continuous . the following is from theorems 2 and 5 in @xcite : + the following properties hold : 1 . the space @xmath62 equipped with the marked gromov - weak topology is polish . 2 . the set @xmath85 is a convergence determining algebra of functions , i.e. for random @xmath62-valued variables @xmath91 we have @xmath92 \xrightarrow{n\to\infty } \mathbf e[\phi(x ) ] \ ; \mbox { for all } \ ; \phi\in\pi^1.\ ] ] the tree - valued fleming viot process will be defined via a well - posed martingale problem . let us briefly recall the concept of a martingale problem . + let [ d:01 ] @xmath22 be a polish space , @xmath93 , @xmath94 and @xmath95 a linear operator on @xmath96 with domain @xmath97 . the law @xmath98 of an @xmath22-valued stochastic process @xmath99 is a _ solution of the @xmath100-martingale problem _ if @xmath101 has distribution @xmath102 , @xmath103 has paths in the space @xmath104 , almost surely , and for all @xmath105 , @xmath106 is a @xmath98-martingale with respect to the canonical filtration . the @xmath107-martingale problem is said to be _ well - posed _ if there is a unique solution @xmath98 . let us first specify the _ generator _ of the tree - valued fleming viot process . it is a _ linear operator with domain @xmath85 _ , given by @xmath108 here , for @xmath109 , the linear operators @xmath110 are defined as follows : 1 . we define the _ growth operator _ by @xmath111 with @xmath112 2 . we define the _ resampling operator _ by @xmath113 with @xmath114 , where @xmath115 3 . for the _ mutation operator _ , let @xmath116 and @xmath117 be a markov transition kernel from @xmath14 to the _ borel sets _ of @xmath14 and set @xmath118 where @xmath119 we say that mutation has a parent - independent component if @xmath120 is of the form @xmath121 for some @xmath122 $ ] , @xmath123 and a probability transition kernel @xmath124 on @xmath14 . 4 . for _ selection _ , consider the _ fitness function _ @xmath125 \end{aligned}\ ] ] with @xmath126 for all @xmath127 . we require that @xmath128 , i.e. @xmath129 is continuous and continuously differentiable with respect to its third coordinate . then , with @xmath130 ( the selection intensity ) and @xmath131 we set @xmath132 + if @xmath133 does not depend on @xmath61 , and if there is @xmath134 $ ] such that @xmath135 we say that selection is additive and conclude that with @xmath136 we have @xmath137 + the growth , resampling , mutation and selection generator terms are interpreted as follows : 1 . _ growth : _ the distance of any pair of individuals is given by the time to the most recent common ancestor ( mrca ) . when time passes this distance grows at speed 1 . note that in @xcite and @xcite the corresponding distance was twice the time to mrca the reason for this change were some simplifications of the terms in the computations that we will see later . 2 . _ resampling : _ the term @xmath138 describes the action of an event where an offspring of individual @xmath139 replaces individual @xmath140 in the sample corresponding to the polynomial @xmath86 . this term is analogous to the measure - valued case ; see e.g. ( 3.21 ) in @xcite , but acts on both , the genealogy and the types . mutation : _ it is important to note that mutation only affects types , but not genealogical distances . hence , the mutation operator agrees with the measure - valued case ; see e.g.(3.16 ) in @xcite . note that here we consider only jump operators @xmath141 . selection : _ this term is best understood when considering a finite population . consider for simplicity the case of additive selection ( i.e. holds ) in particular covering haploid models . here , the offspring of an individual of type @xmath142 replaces some randomly chosen individual at rate @xmath143 due to selection . in the large population limit , we only consider a sample of @xmath70 individuals and this sample changes only if some offspring of an individual outside the sample , e.g. the @xmath144st individual by exchangeability , replaces an individual within the sample , the @xmath139th say , due to selection . after this selection event , the fitness of the @xmath139th individuals is @xmath145 which is also seen from the generator term . in the case of selection acting on diploids , the situation is similar , but one has to build diploids from haploids first and then apply the fitness function . in @xcite the tree - valued fleming viot processes were constructed via well - posed martingale problems . the following proposition summarizes theorems 1 , 2 and 4 from @xcite . + for [ p : main ] @xmath146 the @xmath147-martingale problem is well - posed . its solution @xmath148 defines a feller semigroup , i.e. @xmath149 $ ] is continuous for all @xmath150 , and hence , @xmath103 is a strong markov process . furthermore , the process @xmath151 satisfies the following properties : 1 . 2 . @xmath153 . let @xmath109 such that @xmath33 is symmetric under permutations . then , the quadratic variation of the semi - martingale @xmath154 is given by @xmath155_t & = \int_0^t \big\langle \mu_s , \big ( \rho_s - \langle \mu_s , \rho_s\rangle\big)^2\big\rangle\ , ds , \\ \label{eq : qv2 } \rho_s(u_1 ) & \coloneqq \int \mu_s^{\otimes { \mathbb{n } } } ( d(u_2,u_3,\dots ) ) \phi((r_s(u_i , u_j))_{1\leq i < j } ) . \end{aligned}\ ] ] 4 . let @xmath156 be the distribution of @xmath103 with selection intensity @xmath157 . then , for all @xmath158 , the laws @xmath159 and @xmath160 are absolutely continuous with respect to each other . 5 . if either ( i ) @xmath161 and the process with generator @xmath162 has a unique equilibrium or ( ii ) @xmath163 and mutation has a parent - independent component , then the process @xmath103 is ergodic . that is , there is an @xmath164-valued random variable @xmath165 , depending on the model parameters but not the initial law , such that @xmath166 . using [ def : treev ] the same notation as in proposition [ p : main ] , we call the process @xmath103 the _ tree - valued fleming viot process _ and in the case @xmath161 its ergodic limit @xmath167 is called _ kingman marked measure tree_. + the random variable @xmath165 arises from the marked ultrametric measure space which is associated with the partition - valued entrance law of the kingman coalescent @xcite . + in [ rem : qv ] some of the proofs , we will need to compute the quadratic variation of @xmath154 for specific @xmath168 via ( [ eq : qv1 ] ) explicitly . for @xmath169 as in example [ rem : intphi ] , we have by @xmath170_t & = \int_0^t \bigl(\psi^{12,23}_\lambda(x_s ) - \psi^{12,34}_\lambda(x_s)\bigr)\ , ds , \end{aligned}\ ] ] with ( cf . definition [ def : psiphi ] ) @xmath171 our main goal is to establish almost sure properties of the paths of the tree - valued fleming viot process , beyond continuity of paths and the property that the states are compact marked metric measure space for every @xmath4 , almost surely . we start by studying the geometry of the marked metric measure tree at time @xmath1 of the tree - valued fleming viot process . first we recall in section [ ss.geoprop ] some well - known facts concerning the geometry of the kingman coalescent and then extend them in section [ sss.pptv ] to the tree - valued fleming viot process . in section [ ss : respath ] we take advantage of our results and techniques and state some further path properties of the tree - valued fleming viot process answering two open questions . we focus on the kingman marked measure tree @xmath165 introduced in proposition [ p : main].5 , but for most assertions in this subsection we can ignore the marks ( i.e. think of @xmath14 consisting of only one element ) . since the introduction of the partition - valued kingman coalescent in @xcite , this random tree has been studied extensively for instance in @xcite and @xcite see also @xcite . in our present formalism ( using metric measure spaces ) , @xmath165 appeared first in @xcite . in this section , we mostly reformulate known results , but also add a new one in proposition [ p.fluc ] . the kingman measure tree , @xmath165 , arises from the partition - valued kingman coalescent , but can also be realized as a discrete graph tree using the following construction ( see also figure [ fig : not ] ) . let @xmath172 be independent exponentially distributed random variables with parameter 1 . starting with two lines from the root the tree stays with these two lines for time @xmath173 . at time @xmath174 one of the two lines chosen at random splits in two , such that three lines are present . in general after the jump from @xmath175 to @xmath139 lines the tree stays with that @xmath139 lines for a period of time @xmath176 and then one of the @xmath139 lines chosen at random splits , such that there are @xmath177 lines . the total tree height is thus @xmath178 , where @xmath179 , i.e. @xmath180 is the time it takes the coalescent to go from @xmath70 to infinitely many lines . the time of the root is called the time of the most recent common ancestor ( mrca ) and @xmath181 is the present time of the population . in order to derive the kingman _ marked _ metric measure tree , consider the uniform distribution on the branches and construct a tree - indexed markov process , by using a collection of independent mutation processes as follows . start with an equilibrium value of the mutation processes at the root up to the next splitting time where we continue with two independent mutation processes both starting from the type in the vertex , etc . running from the root to the leaves and letting time approach @xmath181 we finally obtain @xmath165 . without marks . in the `` dashed region '' the tree comes down from infinitely many lines at the treetop ( time @xmath182 ) to six lines at time @xmath183 . we have @xmath184 . the thick grey sub - tree is the closed and open ball of radius @xmath185 around @xmath186 and around @xmath187 . the balls coincide because @xmath188 is an ultrametric.,scaledwidth=85.0% ] at time @xmath7 ( counted from the top of the tree , for @xmath189 ) , a random number @xmath190 of lines are present . equivalently , @xmath190 is the minimal number of @xmath7-balls needed to cover ( the leaves of ) the random tree @xmath165 . it is a well - known fact using de finetti s theorem that the frequency of the family descending from every of the @xmath190 lines can be defined for all @xmath191 . in addition , these frequencies are distributed as the spacings between @xmath190 on @xmath192 $ ] uniformly distributed random variables @xcite . > from these considerations several results on the geometry of @xmath165 near the leaves can be derived . we briefly recall and extend some of them and reprove them later in our setting . roughly we will show that there are @xmath193-many families in which the genealogical distance between the individuals is at most @xmath2 . furthermore , each of the families has mass of order @xmath2 , as @xmath3 . more precisely , the distribution of ( by @xmath2 rescaled ) family sizes is exponential with rate 2 . we split the above picture in two parts . first we study the number of families and then their size in both cases looking at a lln and then at a clt . we begin with a law of large numbers and a central limit theorem for @xmath190 ( see ( 35 ) in @xcite ) . our proofs are given in sections [ ss.proofpcover ] and [ s.proofpcover2 ] . + let [ p : cover ] @xmath194 be the kingman measure tree . moreover , let @xmath190 be the ( minimal ) number of @xmath7-balls needed to cover @xmath195 . then @xmath196 + with [ p : cover2 ] the same notation as in proposition [ p : cover ] and @xmath197 , @xmath198 \xrightarrow{\varepsilon\to 0 } 1 , & & \mathbb e\big[\big(\frac{n_\varepsilon - 2/\varepsilon}{\sqrt{2/(3\varepsilon)}}\big)^4\big ] \xrightarrow{\varepsilon\to 0 } 1 . \end{aligned}\ ] ] we now come to the _ family structure _ of @xmath199 close to the leaves . for @xmath191 , we define @xmath200 as the disjoint balls of radius @xmath7 that cover @xmath201 and the corresponding frequencies by @xmath202 recall that in an ultrametric space two balls of the same radius are either equal or disjoint ( see also figure [ fig : not ] ) . therefore , the vectors @xmath203 above are defined in a unique way . it can be viewed as the frequency vector of a sequence of exchangeable random variables and we can ask for the law of the empirical distribution of the scaled masses in the limit @xmath3 , where the underlying sequence , even if scaled , becomes i.i.d . and we should get the scaled law of a single scaled @xmath204 . it turns out , a first step ( cf . remark [ r.refine ] ) is to see that the following law of large numbers holds , the proof of which ( together with the proof of lemma [ l.reform ] ) appears in section [ s : prooft3 ] . + for [ p : smallballs ] @xmath205 as above , @xmath206 almost surely . the classical proof of proposition [ p : smallballs ] uses the fact that the random vector @xmath207 has the same distribution as the vector of spacings between @xmath190 random variables uniformly distributed on @xmath192 $ ] . this vector in turn has the same distribution as @xmath208 , where @xmath209 are i.i.d . exp@xmath210 random variables . then , using a moment computation , can be proved . for details we refer to section 2 in @xcite . we will use a different route for which we need the following auxiliary tauberian result . + the assertions [ l.reform ] @xmath211 with @xmath169 from example [ rem : intphi ] are equivalent . moreover , the equivalence remains true if we replace @xmath7 by @xmath212 , @xmath40 by @xmath213 with @xmath214 and let @xmath215 . + actually , @xcite contains [ r.refine ] refinements of proposition [ p : smallballs ] . 1 . in equation ( 35 ) of @xcite it is claimed that ( correcting a typo in aldous equation ) @xmath216 this means that the kingman coalescent at distance @xmath2 from the tree top consists of approximately @xmath6 families , and the size of a randomly sampled family has an exponentially distributed size with expectation @xmath217 , in particular the rescaled empirical measure of the family sizes converges weakly to the exponential distribution with mean @xmath218 , denoted by @xmath219 . + in order to show this assertion using moments of @xmath220 , it is necessary and sufficient that for @xmath221 @xmath222 the sufficiency follows since the moment problem for the exponential distribution is well posed , while for the necessity , we assume that holds , and then one concludes ( recall the notation @xmath223 from remark [ not : aux ] ) @xmath224 as well as , for @xmath225 , @xmath226 2 . the statement raises the issue to determine the fluctuations in that lln , i.e. to derive a clt . here , ( 36 ) in @xcite states that @xmath227 where @xmath228 is a brownian bridge . another , for us more suitable formulation is to consider the sum multiplied by @xmath229 instead of @xmath230 , so that @xmath231 disappears on the right hand side . in this case one would consider the fluctuations of the empirical measure of masses of the @xmath232-balls that cover the kingman coalescent tree . we have so far investigated the behavior near the treetop looking at the family sizes with respect to fixed degree @xmath2 of kinship for @xmath3 . this picture can be refined by obtaining fluctuation results in ( or ) . we obtain a partial result by considering a degree of kinship @xmath233 for @xmath1 varying in @xmath234 and letting @xmath3 . this gives a _ profile _ of the _ family sizes _ of varying degrees of kinship and their correlation structure close to the leaves , if we view the scaling limit as a function of @xmath4 . this profile should be the deterministic flow of distributions @xmath235 which are the limits of @xmath236 as @xmath237 . again we consider the laplace transform given through @xmath169 and obtain the following fluctuation result proved in section [ ss : prooft3 ] . + let [ p.fluc ] @xmath165 be the kingman measure tree . define the process @xmath238 by @xmath239 then every sequence @xmath240 with @xmath241 has a convergent subsequence @xmath242 with @xmath243 for some process @xmath244 with continuous paths . furthermore all limit points satisfy @xmath245 & = 0,\\ \mathbf { var}[z_t ] & = \frac 2 t,\\ \mathbf{cov}(z_s , z_t ) & = \frac{4st}{(s+t)^3},\\ \mathbf e[z_t^3 ] & = 0,\\ \mathbf e[z_t^4 ] & = \frac{3}{4t^2}. \end{aligned}\ ] ] + we conjecture that there is a unique limit process @xmath246 in proposition [ p.fluc ] . moreover , we note that @xmath247 $ ] and @xmath248 $ ] are in the relation if @xmath249 , which raises the question whether @xmath250 is a gaussian process . although the kingman measure tree , @xmath165 , only arises as the long - time limit of the neutral tree - valued fleming viot process , @xmath251 , near the leaves , @xmath5 ( for @xmath4 ) and @xmath165 have similar geometry . the reason is that the structure near the leaves of @xmath165 or @xmath5 only depends on resampling events in the ( very ) recent past . hence , we expect that the properties of @xmath165 from propositions [ p : cover ] and [ p : smallballs ] hold along the paths of @xmath103 . this will be shown in theorem [ t1 ] and theorem [ t3 ] , respectively . furthermore we conjecture ( but do nt have a proof ) that the more ambitious refinements described in remark [ r.refine ] ( see ) also hold along the paths . in addition , in the stationary regime @xmath252 , theorems [ t2 ] and [ t4 ] give two results on convergence to a brownian motion along the tree - valued fleming viot process . the following theorem is proved in section [ ss.prooft1 ] . + let [ t1 ] @xmath251 with @xmath253 be the tree - valued fleming viot process ( started in some @xmath254 ) and selection coefficient @xmath130 . moreover , let @xmath255 be the number of @xmath7-balls needed to cover @xmath256 . then , @xmath257 while the fluctuations in proposition [ p : cover2 ] are dealing with a fixed - time genealogy , we can view the fluctuations of the path @xmath258 arising in theorem [ t1 ] in the limit @xmath3 . this program is now carried out along the tree - valued fleming viot process . in order to obtain a meaningful limit object , we consider time integrals . it is important to understand that the part of the time-@xmath1 tree @xmath5 which is at most @xmath7 apart from the treetop is independent of @xmath259 as long as @xmath260 . the following is proved in section [ ss.prooft2 ] . + let [ t2 ] @xmath251 with @xmath261 be the neutral tree - valued fleming viot process ( i.e. @xmath161 ) started in equilibrium , @xmath252 , and @xmath262 given by @xmath263\big ) ds . \end{aligned}\ ] ] then , @xmath264 where @xmath265 is a brownian motion started in @xmath266 . + in one would rather like to replace @xmath267 $ ] by @xmath6 to measure the fluctuations around the limit profile , i.e. to consider @xmath268 defined by @xmath269 instead of @xmath270 . we will see that @xmath271 converges as @xmath237 to a brownian motion @xmath272 with _ drift _ , but unfortunately we can not identify the latter . indeed , from proposition [ p : cover2 ] , in particular using boundedness of second moments , we see that , approximately , @xmath273 \approx \frac 2 \varepsilon$ ] in the sense that @xmath274 \xrightarrow{\varepsilon\to 0 } 2 $ ] . however , this only implies @xmath275=2/\varepsilon + o(1/\varepsilon)$ ] and the error term can be large . in order to sharpen this expansion to @xmath275 = 2/{\varepsilon}+ \mathcal o(1)$ ] , we use results from @xcite . his section 5.4 ( with @xmath276 and @xmath277 ) yields @xmath278 & = \sum_{k=1}^\infty \rho_k(\varepsilon)(2k-1 ) \frac{(k-1)\cdots ( k - j+1)\cdot k\cdots(k+j-2)}{(j-1 ) ! } , \end{aligned}\ ] ] with @xmath279 . from this , writing @xmath280 we also see that @xmath281 & = \frac{2}{\delta^2}\sum_{x\in\delta\mathbb n } \exp(-x(x-\delta)/2 ) ( x-\delta/2)\delta \\ & = \frac{2}{\delta^2}\sum_{x\in\delta\mathbb n } \exp(-x^2/2)(1 + x\delta/2 + o(\delta^2))(x-\delta/2)\delta \\ & = \frac{2}{\delta^2 } \int_0^\infty xe^{-x^2/2 } dx + \frac{1}{\delta } \int_0^x ( x^2 - 1)e^{-x^2/2}dx + \mathcal o(1 ) \\ & = \frac{2}{\varepsilon } + \mathcal o(1 ) \end{aligned}\ ] ] as @xmath237 . this , together with theorem [ t2 ] , implies that @xmath271 is of the form @xmath282 that is , @xmath283 is a brownian motion with drift . now we come to a generalization of proposition [ p : smallballs ] to the tree - valued fleming viot process . together with lemma [ l.reform ] , we obtain the following result on the laplace transform of two randomly sampled points . the proof is based on martingale arguments which will also be useful in the proof of theorem [ t6 ] . theorem [ t3 ] is proved in section [ ss : prooft3 ] . + let [ t3 ] @xmath251 with @xmath253 be the tree - valued fleming viot process with selection coefficient @xmath284 , started in some @xmath254 , and let @xmath169 be as in remark [ rem : intphi ] . then @xmath285 + denote by @xmath286 the sizes of the @xmath287 balls of radius @xmath2 needed to cover @xmath256 . if we could show that @xmath288 , we could use lemma [ l.reform ] in order to see that @xmath289 however , our proof of theorem [ t3 ] is based on a computation involving the evolution of fourth moments of @xmath169 in order to show tightness of @xmath290 . based on these computations , we can only claim convergence in probability rather than almost sure convergence . + as an ultimate goal one would want to prove that ( compare with ) @xmath291 this would mean that the assertion that roughly the tree consists of @xmath8 families of mean @xmath292 exponentially distributed sizes holds at all times . using our conclusions from remark [ r.refine ] , this goal can be achieved if we show that holds for @xmath221 uniformly at all times . ( while the case @xmath293 is trivial , note that a combination of theorem [ t3 ] and lemma [ l.reform ] gives for @xmath294 . ) in principle , the technique of our proof of proposition [ p : smallballs ] can be extended in order to obtain for a given but arbitrary @xmath139 which would require controlling higher order moments of @xmath169 . if we could do this for general @xmath139 then we would obtain a proof of . but since we are using mathematica for these calculations the problem remains open . again , we can formulate a result on fluctuations . integrating over time ( to get a process rather than white noise ) the quantity @xmath295 , which appears in theorem [ t3 ] , and using the right scaling , we again obtain a brownian motion as the weak limit . the following result is proved in section [ ss.prooft4 ] . + let [ t4 ] @xmath251 with @xmath261 be the neutral tree - valued fleming viot process ( i.e. @xmath161 ) started in equilibrium , i.e. , @xmath252 and let @xmath296 be given by @xmath297 with @xmath169 as in example [ rem : intphi ] . then , @xmath298 where @xmath299 is a brownian motion started in @xmath300 . + assume that @xmath40 is large . then , @xmath301 depends approximately only on resampling events which happened within an interval @xmath302 $ ] for some large @xmath303 . in particular , on different time intervals ( which are at least of order @xmath304 apart ) , the increments of @xmath305 are approximately independent . thus , it is reasonable to expect that the limiting process is a local martingale . in fact , using some stochastic calculus we can show that the limiting process is continuous ( i.e. the family @xmath306 is tight in the space @xmath307 ) and the limiting object of @xmath308 is a local martingale as well . by lvy s characterization of brownian motion , @xmath309 must be a brownian motion . using the calculus developed for the statements in section [ sss.pptv ] we obtain two further properties of the states of the tree - valued fleming viot process @xmath310 , @xmath311 , namely that the states are _ atom - free _ and admit a _ mark function_. more precisely , theorem [ t5 ] says that at no time it is possible to sample two individuals with distribution @xmath17 with distance zero ; cf.remark [ rem : intt5 ] below . furthermore theorem [ t6 ] says that we can assign marks to all individuals in the sense that @xmath17 has the form @xmath312 for some measurable function @xmath313 . these two theorems are proved in section [ s : prooft5 ] . + let [ t5 ] @xmath251 with @xmath261 be the tree - valued fleming viot process . then , @xmath314 [ rem : intt5 ] 1 . at first glance the fact that @xmath17 is non - atomic for all @xmath315 might seem to contradict the fact that the measure - valued fleming viot diffusion is purely atomic for every @xmath316 . however , both properties are of different kind and the probability measures in question are different objects : @xmath17 is a sampling measure and the state of the measure - valued fleming viot diffusion is a probability measure on the type space . the above theorem implies that randomly sampled individuals from the tree - valued fleming viot process have distance of order @xmath317 , whereas genealogically the atomicity of the measure - valued fleming viot diffusion expresses the fact that at every time @xmath4 one can cover the state with a finite number of balls with radius @xmath1 . the proof is based on a simple observation : for a measure @xmath318 , @xmath319 hence , the proof of is based on a detailed analysis of the laplace transform of the distance of two points , independently sampled with distribution @xmath17 . the next goal is to establish that at any time there is a _ mark function_. briefly , the state @xmath54 of a tree - valued population dynamics admits a mark function @xmath320 iff every individual @xmath321 can be assigned a ( unique ) type @xmath322 . this situation occurs in particular in finite population models , e.g. in the moran model . the question for the tree - valued fleming viot model is whether types in the finite moran model can change at a fast enough scale so that an individual can have several types in the large population limit . such a situation can occur , if the cloud of very close relatives ( as measured in the metric @xmath61 ) is not close in location ( as measured in the type space @xmath14 ) . + we [ def : markfct ] say that @xmath323 admits a mark function if there is a measurable function @xmath324 such that for a random pair @xmath325 with values in @xmath326 and distribution @xmath11 @xmath327 equivalently , @xmath323 admits a mark function if there is @xmath328 and @xmath329 with @xmath330 we set @xmath331 + let us note that admitting a mark function is a property of an equivalence class . assume @xmath332 ( with an isometry @xmath333 as in ) , where @xmath54 admits a mark function @xmath324 , i.e. holds . then , clearly for @xmath334 we have @xmath335 in other words , @xmath336 admits the mark function @xmath337 . + let [ t6 ] @xmath251 , @xmath338 be the tree - valued fleming viot - dynamics . then , @xmath339 + for a series of exchangeable population models it is possible to construct the state of an infinite population via the lookdown construction @xcite . this construction immediately allows us to define a mark function on a countable number of individuals specifying their types at all times , which suggests that should hold . however , the metric space read off from the lookdown process is not complete , and the mark function is not continuous . it seems possible to extend the definition of the mark function to the completion of the corresponding metric space by defining a ( right - continuous ) mark - function on the tree from root to the leaves . however , we do not pursue this direction here . instead , our proof of theorem [ t6 ] in section [ s : prooft6 ] uses again martingale arguments and moment computations . the proofs of our results are of two types . on the one hand , the proofs of propositions [ p : cover ] and [ p : cover2 ] , theorems [ t1 ] and [ t2 ] use as the basic tools the fine properties of coalescent times in kingman s coalescent . this means they are carried out without specific martingale properties of the tree - valued fleming viot process . on the other hand , propositions [ p : smallballs ] and [ p.fluc ] , theorems [ t3 ] , [ t4 ] , [ t5 ] and [ t6 ] are proved by calculating expectations ( moments ) of polynomials , which is possible by using the martingale problem for the tree - valued fleming viot process . the polynomials we have to consider here ( see also remark [ rem : intphi ] ) are either @xmath169 or @xmath340 , i.e. polynomials based on the test functions @xmath341 or @xmath342 and products , powers and linear combinations thereof . for the calculations of the moments of this type we develop some methodology which we explain in section [ s : prep ] . propositions [ p : cover ] and [ p : cover2 ] , theorems [ t1 ] and [ t2 ] are proved in section [ s.proofpcover ] while propositions [ p : smallballs ] and [ p.fluc ] , theorems [ t3 ] and [ t4 ] are proved in section [ s : prooft3 ] . the latter results are then used to prove theorems [ t5 ] and [ t6 ] in section [ s : prooft5 ] . recall from section [ ss.geoprop ] that @xmath343 is the time the kingman coalescent needs to go down to @xmath70 lines , where @xmath344 are i.i.d.exponential random variables with rate 1 . before we begin , we prove some simple results on the times @xmath180 . + let [ l : momentstn ] @xmath180 be the time the kingman coalescent needs to go from infinitely many to @xmath70 lines . then , @xmath345 & = \frac{2}{n},\\ \mathbf e\bigl[(t_n-2/n)^2\bigr ] & = \frac{4}{3n^3}(1 + \mathcal o(1/n)),\\ \mathbf e\bigl[(t_n-2/n)^3\bigr ] & = \frac{16}{5n^5}(1 + \mathcal o(1/n)),\\ \mathbf e\bigl[(t_n-2/n)^4\bigr ] & = \frac{16}{9n^6}(1 + \mathcal o(1/n)),\\ \mathbf e\bigl[(t_n-2/n)^6\bigr ] & = \frac{64}{27n^9}(1 + \mathcal o(1/n)),\\ \mathbf e\bigl[(t_n-2/n)^8\bigr ] & = \frac{4 ^ 4}{3 ^ 4n^{12}}(1 + \mathcal o(1/n)),\\ \mathbf e[e^{-\lambda t_n } ] & \lesssim e^{-\tfrac 43 ( \tfrac \lambda n \wedge \tfrac{\sqrt{\lambda}}2 ) } , \quad \lambda\geq 0 . \end{aligned}\ ] ] recall that @xmath346 = k!\sum_{i=0}^{k } ( -1)^i / i!$ ] . we start by writing @xmath347 & = \sum_{i = n+1}^\infty \frac{\mathbf e[s_i]}{\binom i 2 } = \sum_{i = n+1}^\infty \frac{2}{i(i-1 ) } = 2\sum_{i = n+1}^\infty \frac{1}{i-1 } - \frac{1}{i } = \frac 2 n. \end{aligned}\ ] ] next , @xmath348 & = \mathbf{var}[t_n ] = \sum_{i = n+1}^\infty \frac{4\mathbf { var}[s_i]}{i^2(i-1)^2 } = 4\sum_{i = n+1}^\infty \frac{1}{i^2(i-1)^2 } \\ & = 4 \int_n^\infty \frac{1}{x^4 } dx + \mathcal o(1/n^4 ) = \frac{4}{3n^3}(1+\mathcal o(1/n ) ) . \end{aligned}\ ] ] for third moments , using @xmath349 = 2 $ ] @xmath350 & = \mathbf e\big[\big(\sum_{i = n+1}^\infty \frac{2}{i(i-1)}(s_i - 1)\big)^3\big ] \\ & = \sum_{i = n+1}^\infty \frac{2 ^ 3}{i^3(i-1)^3}\mathbf e[(s_i-1)^3 ] \\ & = \frac{16}{5n^5}(1+\mathcal o(1/n)),\\ \end{aligned}\ ] ] for fourth moments , @xmath351 & = \mathbf e\big[\big(\sum_{i = n+1}^\infty \frac{2}{i(i-1)}(s_i - 1)\big)^4\big ] \\ & = \sum_{i = n+1}^\infty \frac{2 ^ 4}{i^4(i-1)^4}\mathbf e[(s_i-1)^4 ] \\ & \qquad + \sum_{i , j = n+1 \atop i\neq j}^\infty \frac{4}{i^2(i-1)^2 } \frac{4}{j^2(j-1)^2 } \mathbf e[(s_i-1)^2]\cdot \mathbf e[(s_j-1)^2 ] \\ & = \big(\sum_{i = n+1}^\infty \frac{4}{i^2(i-1)^2}\mathbf e[(s_i-1)^2]\big)^2 + \mathcal o(1/n^7 ) \\ & = \frac{16}{9n^6}(1+\mathcal o(1/n ) ) \end{aligned}\ ] ] for sixth moments , @xmath352 & = \mathbf e\big[\big(\sum_{i = n+1}^\infty \frac{2}{i(i-1)}(s_i - 1)\big)^6\big ] \\ & = \big(\sum_{i = n+1}^\infty \frac{4}{i^2(i-1)^2}\mathbf e[(s_i-1)^2]\big)^3 + \mathcal o(1/n^{10 } ) \\ & = \frac{64}{27n^9}(1+\mathcal o(1/n ) ) . \end{aligned}\ ] ] with analogous calculations , the results for the 8th moment follows . finally for the exponential moments , we compute for any @xmath353 @xmath354 & = \prod_{i = n+1}^\infty \frac{\binom i 2}{\binom i 2 + \lambda } = \exp\big(\sum_{i = n+1}^\infty \log\big(1 - \frac{\lambda}{\binom i 2 + \lambda}\big)\big ) \\ & \leq \exp\big ( -\sum_{i = n+1}^\infty \frac{\lambda}{\binom i 2 + \lambda}\big ) \leq \exp\big ( -\sum_{i = ( n+1)\vee \lceil \sqrt{4\lambda } + 1 \rceil}^\infty \frac{\lambda}{\binom i 2 + \lambda}\big ) \\ & \leq \exp\big ( -\sum_{i = ( n+1)\vee \lceil \sqrt{4\lambda } + 1\rceil}^\infty \frac{\tfrac 23 \lambda}{\binom i 2}\big ) = \exp\big(-\frac 43 \cdot\frac\lambda{n\vee \big\lceil \sqrt{4\lambda}\big\rceil } \big ) \lesssim e^{-\tfrac 43 ( \tfrac \lambda n \wedge \tfrac{\sqrt{\lambda}}2)}. \end{aligned}\ ] ] let @xmath180 be as in the last subsection and recall @xmath190 from proposition [ p : cover ] . then , is equivalent to @xmath355 in order to see this , note that @xmath356 by definition of @xmath180 and @xmath357 since @xmath358 as @xmath215 ( and @xmath359 as @xmath237 ) , the equivalence of and follows . for , it suffices to note that @xmath360}{(\varepsilon / n)^4 } \leq \frac{16}{9\varepsilon^4 n^2}(1+\mathcal o(1/n ) ) . \end{aligned}\ ] ] by lemma [ l : momentstn ] . since the right hand side is summable , @xmath361 almost surely for all @xmath191 . in other words , @xmath362 almost surely . by the lindeberg feller central limit theorem , we see from the moment computations of lemma [ l : momentstn ] that @xmath363 recalling , we set @xmath364 such that for every @xmath365 : @xmath366 which implies . however , we also need to show convergence of moments up to fourth order . we write @xmath367 & = 2\int_0^\infty x\mathbf p\big(\big|\frac{n_\varepsilon - 2/\varepsilon}{\sqrt{2/(3\varepsilon)}}\big|>x\big ) dx,\\ \label{eq : int10 } \mathbf e\big[\big(\frac{n_\varepsilon - 2/\varepsilon}{\sqrt{2/(3\varepsilon)}}\big)^4\big ] & = 4\int_0^\infty x^3\mathbf p\big(\big|\frac{n_\varepsilon - 2/\varepsilon}{\sqrt{2/(3\varepsilon)}}\big|>x\big ) dx.\end{aligned}\ ] ] to estimate the right hand side of we first show that for a suitably chosen @xmath368 and @xmath369 we have @xmath370 where the equality follows because for @xmath371 the integrand is identically @xmath182 . using the exponential chebyshev inequality , we obtain for all @xmath372 @xmath373\,dy , \\ \intertext{now taking the lower bound for $ c$ , setting $ \delta'=2\delta/3 $ and using \eqref{eq:1679 } we get } & \leq \frac 94 \int_{\sqrt{4-\delta'}}^{2 } \frac{y^3}{\varepsilon^2 } e^{\lambda_{y,\varepsilon } \varepsilon}\mathbb e\big[e^{-\lambda_{y,\varepsilon}t_{\lfloor \frac{2-y}{\varepsilon } \rfloor}}\big]\ , dy \\ & \lesssim \frac 94 \int_{\sqrt{4-\delta'}}^{2 } \frac{y^3}{\varepsilon^2}\exp\big\{\lambda_{y,\varepsilon } \varepsilon - \frac43 \big(\frac{\lambda_{y,\varepsilon } \varepsilon}{2-y } \wedge \frac{\sqrt{\lambda_{y,\varepsilon}}}{2}\big ) \ , \big\ } \,dy . \end{split}\end{aligned}\ ] ] now choose @xmath374 and let @xmath375 be the solution of @xmath376 . for @xmath377 $ ] we have @xmath378 thus , @xmath379 it is easy to see that on the interval @xmath380 $ ] the function @xmath381 is bounded below by @xmath382 ( its value in @xmath383 ) . it follows @xmath384 for a suitable constant @xmath385 , and hence we have shown . in order to show convergence of fourth ( and second ) moments , using , since @xmath386 pointwise , we need to show that there is an integrable function dominating @xmath387 for some @xmath388 and @xmath389 . for this , we get , by the markov inequality and @xmath390}{(\varepsilon - 2/a_\varepsilon(x))^6 } + \frac{\mathbf e[(t_{a_\varepsilon(-x ) } - 2/a_\varepsilon(-x))^6]}{(\varepsilon - 2/a_\varepsilon(-x))^6 } \\ & = \frac{64(1+\mathcal o(1/a_\varepsilon(x))}{27 a_\varepsilon(x)^9(x\sqrt{\varepsilon^3/6}(1+\mathcal o(\sqrt{\varepsilon})))^6 } + \frac{64(1+\mathcal o(1/a_\varepsilon(x))}{27 a_\varepsilon(-x)^6(-x\sqrt{\varepsilon^3/6}(1+\mathcal o(\sqrt{\varepsilon})))^6 } \\ & = \frac{2(1+\mathcal o(1/a_\varepsilon(x))}{x^6(1+\mathcal o(\sqrt\varepsilon ) ) } \leq \frac{2}{x^6(1+\mathcal o(\varepsilon ) ) } + \mathcal o\big(\frac{\varepsilon}{x^6(1+x\sqrt{\varepsilon/6})}\big ) , \end{aligned}\ ] ] since @xmath391 since the area @xmath392 is restricted to @xmath393 in , the @xmath394-term on the right hand side of does not have a pole . it is now easy to obtain an integrable function dominating , leading to . by proposition [ p : main].5 . ( see theorem 2 of @xcite for details ) the tree - valued fleming viot process with selection has a law which is absolutely continuous with respect to the neutral process . therefore it suffices to consider the neutral case , @xmath161 . we observe that for @xmath161 the claim is not affected by mutation . moreover , it suffices to deal with the case @xmath395 . the reason is that is equivalent to the assertion that for all @xmath396 and uniformly for all @xmath397 we have @xmath398 almost surely . then one can use the independence of @xmath255 and @xmath101 for @xmath399 . let @xmath400 i.e. @xmath401 is the minimal time we have to go back from time @xmath1 such that we have @xmath70 ancestral lineages . it suffices to show ( see around ) that @xmath402 . to prove this , we need to extend the proof of proposition [ p : cover ] . it suffices to show that @xmath403 uniformly for @xmath404 ( if @xmath252 ) . first , by lemma [ l : momentstn ] , for all @xmath16 we have @xmath405\bigr)^8\bigr ] \lesssim \frac{1}{n^{12}}.\end{aligned}\ ] ] therefore considering the process along a discrete grid , we have for any @xmath406 , @xmath407\bigr)^8\bigr]}{({\varepsilon}/n)^8 } \lesssim \frac{1}{{\varepsilon}^8 n^2}. \end{aligned}\ ] ] since the right hand side is summable , @xmath408 bounds from above and from below . first observe that @xmath409 for @xmath410 for tree - valued fleming viot process . ( this property holds in every population model , arising as a diffusion limit from an individual based population where we can define ancestors , since the ancestors at time @xmath411 of the population at time @xmath1 must then be ancestors at time @xmath412 of the population at time @xmath413 , for @xmath414 . ) we can now write , for @xmath191 and @xmath415 : @xmath416 hence @xmath417 almost surely , by the borel cantelli lemma . for the other direction of the inequality we use @xmath418 for @xmath419 , to get @xmath420 hence @xmath421 almost surely . combining both , the estimate from above and from below we obtain the assertion of theorem [ t1 ] . we proceed in the following four steps . * * step 0 * : warm up ; computation of the first two moments of @xmath422 . * * step 1 * : computation of the first two conditional moments of @xmath422 . * * step 2 * : the family @xmath423 is tight in @xmath424 . * * step 3 * : for @xmath425 a limit point @xmath425 as well as @xmath426 are martingales . throughout we let @xmath427 be the canonical filtration of @xmath428 . _ step 0 : computation of first two moments of @xmath429 . _ the first moment of @xmath430 equals 0 since by assumption the tree - valued fleming viot process is in equilibrium . for the second moment , we start by noting that ( see proposition [ p : cover2 ] ) @xmath431 for some random variable @xmath432 . note that @xmath433 and @xmath434 are independent given @xmath435 \cap ( s-\varepsilon , s ] = \emptyset$ ] . ( the reason is that in this case @xmath433 depends on resampling events in the time interval @xmath436 $ ] while @xmath434 only depends on resampling events in @xmath435 $ ] , and these two sets of events are independent . ) without loss of generality , we set @xmath437 and compute the variance of @xmath438 as @xmath439 & = 3 \int_0^t \int_0^s \mathbf{cov}(n_\varepsilon^r , n_\varepsilon^s ) dr ds = 3 \int_0^t \int_{0\vee ( s-\varepsilon)}^s\mathbf{cov}(n_\varepsilon^r , n_\varepsilon^s ) dr ds \\ & = 3 \int_0^t \int_{0}^{\varepsilon\wedge t}\mathbf{cov}(n_\varepsilon^{s-\delta } , n_\varepsilon^s ) d\delta ds \stackrel{\varepsilon\to 0}{\approx } 6 t \cdot \int_0^\varepsilon \mathbf{cov}(n_\varepsilon^{0 } , n_\varepsilon^\delta ) d\delta . \end{aligned}\ ] ] in order to compute the integrand of the last expression , we decompose @xmath440 . now @xmath441 is independent of @xmath442 . the former is the number of lines the tree at time @xmath182 looses between times @xmath443 and @xmath7 in the past , and therefore only depends on resampling events between times @xmath444 and @xmath445 . the latter only depends on resampling events between times @xmath445 and @xmath446 . hence , @xmath447 consider now the dual representation of our equilibrium by the kingman coalescent . let @xmath448 be number of lines of a subtree , starting with @xmath449 lines , in a tree starting with @xmath450 many lines , at the time the big tree has @xmath451 lines . in lemma 4 of @xcite it is shown that @xmath452 = \frac{in}{n+i-1}. \end{aligned}\ ] ] hence , for independent @xmath453 , @xmath454 ) \\ & = \mathbf{cov}(n_{\varepsilon-\delta}^{0 } , \mathbf e[k_{n_{\varepsilon-\delta}^{0}}^{n_\delta^\delta}|n_\delta^\delta , n_{\varepsilon-\delta}^0])\\ & = \mathbf{cov}\big(n_{\varepsilon-\delta}^{0 } , \frac{n_{\varepsilon-\delta}^0\cdot n_{\delta}^\delta}{n_{\varepsilon-\delta}^0 + n_{\delta}^\delta -1}\big ) \\ & \stackrel{\varepsilon\to 0}\approx \mathbf{cov}\bigg(\sqrt{2/(3(\varepsilon-\delta))}z+o(1/\sqrt{\varepsilon}),\\ & \qquad \qquad \frac{\big(\frac{2}{\varepsilon-\delta } + \sqrt{2/(3(\varepsilon-\delta))}z + o(\sqrt{1/\varepsilon}\,)\big)\big(\frac{2}{\delta } + \sqrt{2/(3\delta)}z ' + o(\sqrt{1/\varepsilon}\,)\big)}{\frac{2}{\varepsilon-\delta } + \frac{2}{\delta } + \sqrt{2/(3(\varepsilon-\delta))}z + \sqrt{2/(3\delta)}z ' + o(1/\sqrt{\varepsilon}\,)}\bigg ) \\ & = \frac{2}{\varepsilon } \mathbf{cov}\bigg(\sqrt{2/(3(\varepsilon-\delta))}z+o(1/\sqrt{\varepsilon}\,),\\ & \qquad \qquad \frac{\big(1 + \sqrt{(\varepsilon-\delta)/6}z + o(\sqrt{\varepsilon}\,)\big ) \big(1 + \sqrt{\delta/6}z ' + o(\sqrt{\varepsilon}\,))\big)}{1 + \frac{\delta}{\varepsilon } \sqrt{(\varepsilon-\delta)/6)}z + \frac{\varepsilon-\delta}{\varepsilon}\sqrt{\delta/6}z ' + o(\sqrt{\varepsilon}\,)}\bigg ) \\ & = \frac{2}{\varepsilon}\mathbf{cov}\big ( \sqrt{2/(3(\varepsilon-\delta))}z+o(1/\sqrt{\varepsilon}\,),\\ & \qquad\qquad \qquad \big(\big(1 + \sqrt{(\varepsilon-\delta)/6}z + \sqrt{\delta/6}z ' + o(\sqrt{\varepsilon}))\big ) \\ & \qquad \qquad \qquad \qquad \qquad \cdot\big(1 - \frac{\delta}{\varepsilon } \sqrt{(\varepsilon-\delta)/6)}z - \frac{\varepsilon-\delta}{\varepsilon}\sqrt{\delta/6}z ' + o(\sqrt{\varepsilon}\,)\big ) \\ & = \frac{2}{\varepsilon}\mathbf{cov}\big ( \sqrt{2/(3(\varepsilon-\delta))}z+o(1/\sqrt{\varepsilon}\ , ) , \frac{\varepsilon-\delta}{\varepsilon } \sqrt{(\varepsilon-\delta)/6}z + \frac{\delta}{\varepsilon}\sqrt{\delta/6}z ' + o(\sqrt{\varepsilon}\,)\big ) \\ & = \frac{2(\varepsilon-\delta)}{3\varepsilon^2}(1 + o(1 ) ) \end{aligned}\ ] ] leading to @xmath455 & = 3 t \int_0^\varepsilon \mathbf{cov}(n_\varepsilon^0 , n_\varepsilon^\delta)d\delta \stackrel{\varepsilon\to 0}\approx 3 t \int_0^\varepsilon \frac{2\delta}{3\varepsilon^2 } d\delta = t.\end{aligned}\ ] ] _ step 1 : computation of first two conditional moments of @xmath456 . _ we can compute the _ first conditional moment _ as @xmath457 & = \sqrt{\frac 32}\int_s^t \mathbf e[n_\varepsilon^r - \mathbf e[n_\varepsilon^\infty]|\mathcal f_s ] dr = \sqrt{\frac 32}\int_s^{s+\varepsilon } \mathbf e[n_\varepsilon^r - \mathbf e[n_\varepsilon^\infty]|\mathcal f_s ] \ , dr \end{aligned}\ ] ] since we started in equilibrium and @xmath458 is independent of @xmath459 for @xmath460 . so , by proposition [ p : cover2 ] , @xmath461\big)^2\big ] & \lesssim \varepsilon^2 \mathbf e[(n_\varepsilon^\infty - \mathbf e[n_\varepsilon^\infty])^2 ] \xrightarrow{\varepsilon\to 0 } 0 , \end{aligned}\ ] ] which implies @xmath462 \xrightarrow{\varepsilon\to 0 } 0 \qquad \text { in } l^2.\end{aligned}\ ] ] for the _ second conditional moment _ , we extend our calculation from step 0 . here , @xmath463 & = 3 \int_s^t \int_s^{r_1 } \mathbf{e}[(n_\varepsilon^{r_1}-\mathbf e[n_\varepsilon^{\infty}])(n_\varepsilon^{r_2}-\mathbf e[n_\varepsilon^{\infty}])|\mathcal f_s ] dr_1 dr_2 \\ & = 3\int_{s+\varepsilon}^t \int_{s+\varepsilon}^{r_1 } \mathbf{cov}[n_\varepsilon^{r_1 } , n_\varepsilon^{r_2 } ] dr_1 dr_2 + 3a_\varepsilon , \end{aligned}\ ] ] with @xmath464 & = \mathbf e\big[\big|\int_s^t \int_s^{r_1\wedge ( s+\varepsilon)}\mathbf{e}[(n_\varepsilon^{r_1}-\mathbf e[n_\varepsilon^{\infty}])(n_\varepsilon^{r_2}-\mathbf e[n_\varepsilon^{\infty}])|\mathcal f_s ] dr_1 dr_2\big|\big ] \\ & = \mathbf e\big[\big|\int_s^{s+2\varepsilon } \int_s^{r_1\wedge ( s+\varepsilon ) } \mathbf{e}[|(n_\varepsilon^{r_1}-\mathbf e[n_\varepsilon^{\infty}])(n_\varepsilon^{r_2}-\mathbf e[n_\varepsilon^{\infty}])|\mathcal f_s ] dr_1 dr_2\big|\big ] \\ & \leq \int_s^{s+2\varepsilon } \int_s^{r_1\wedge ( s+\varepsilon ) } \mathbf{e}[|(n_\varepsilon^{r_1}-\mathbf e[n_\varepsilon^{\infty}])(n_\varepsilon^{r_2}-\mathbf e[n_\varepsilon^{\infty}])| ] dr_1 dr_2 \\ & \leq \int_s^{s+2\varepsilon } \int_s^{r_1 } \mathbf{e}[|(n_\varepsilon^{r_1}-\mathbf e[n_\varepsilon^{\infty}])(n_\varepsilon^{r_2}-\mathbf e[n_\varepsilon^{\infty}])| ] dr_1 dr_2 \\ & \leq 2\varepsilon^2 \mathbf { var}[n_\varepsilon^\infty ] \xrightarrow{\varepsilon\to 0 } 0 \end{aligned}\ ] ] by proposition [ p : cover2 ] . so , combining the last two displays with , @xmath465 \xrightarrow{\varepsilon\to 0}t - s \qquad \text { in } l^1.\end{aligned}\ ] ] _ step 2 : the family of the laws of @xmath423 on @xmath424 is tight for @xmath3 . _ we use the kolmogorov chentsov criterion ; see e.g. corollary 16.9 in @xcite . we bound the fourth moment of the increment in @xmath466 . we write @xmath467 $ ] such that , again making use of the independence of @xmath468 and @xmath469 if @xmath470 , as well as of proposition [ p : cover2 ] , we estimate for fixed @xmath1 and @xmath3 @xmath471 \lesssim \int_0^t \int_0^{s_1 } \int_0^{s_2 } \int_0^{s_3 } \mathbf e\bigl [ \prod\nolimits_{i=1}^4 ( n_\varepsilon^{s_i}-a_\varepsilon)\bigr ] \ , ds_4 \ , ds_3 \ , ds_2 \ , ds_1 \\ & \lesssim \big(\int_0^t \int_{0\vee(s_1-\varepsilon)}^{s_1 } \mathbf e[(n_\varepsilon^{s_1}-a_\varepsilon)\cdot ( n_\varepsilon^{s_2}-a_\varepsilon ) ] ds_2 ds_1\big)^2 \\ & \quad + \int_0^t \int_{0\vee(s_1-\varepsilon)}^{s_1 } \int_{0\vee(s_2-\varepsilon)}^{s_2 } \int_{0\vee(s_3-\varepsilon)}^{s_3 } \mathbf e\bigl [ \prod\nolimits_{i=1}^4 ( n_\varepsilon^{s_i}-a_\varepsilon)\bigr ] \ , ds_4 \ , ds_3 \ , ds_2 \ , ds_1 \\ & \lesssim \big(t ( t\wedge \varepsilon ) \mathbf { var}[n_\varepsilon^\infty]\big)^2 + t(t\wedge \varepsilon)^3 \mathbf e[(n_\varepsilon^\infty - a_\varepsilon)^4 ] \\ & \lesssim t^2 \end{split}\end{aligned}\ ] ] and the tightness follows . _ step 3 : if @xmath472 is a limit point , then @xmath472 as well as @xmath473 are martingales . let @xmath265 be a weak limit point of @xmath474 , which has continuous paths by step 2 . we know from step 0 that @xmath475 is square integrable which allows for the following calculations . for @xmath476 and some continuous bounded function @xmath477 , by @xmath478\big| = \lim_{\varepsilon\to\infty } \big|\mathbf e[(b_\varepsilon(t ) - b_\varepsilon(s ) ) \cdot f(b_\varepsilon(r_1),\dots , b_\varepsilon(r_n))]\big| \\ & \leq \lim_{\varepsilon\to\infty } \mathbf e\big [ \big|\mathbf e[b_\varepsilon(t ) - b_\varepsilon(s)|\mathcal f_s ] \big|\cdot f(b_\varepsilon(r_1),\dots , b_\varepsilon(r_n))\big ] = 0 . \end{aligned}\ ] ] since @xmath479 and @xmath480 were arbitrary , @xmath481 is a martingale . similarly , by , @xmath482\big| \\ & = \lim_{\varepsilon\to\infty } \big|\mathbf e[((b_\varepsilon(t ) - b_\varepsilon(s))^2 - ( t - s ) ) \cdot f(b_\varepsilon(r_1),\dots , b_\varepsilon(r_n))]\big| \\ & \leq \lim_{\varepsilon\to\infty } \mathbf e\big [ \big|\mathbf e[(b_\varepsilon(t ) - b_\varepsilon(s))^2-(t - s)|\mathcal f_s ] \big|\cdot f(b_\varepsilon(r_1),\dots , b_\varepsilon(r_n))\big ] = 0 , \end{aligned}\ ] ] which shows that @xmath483 is a martingale . then by lvy s characterization of brownian motion , @xmath481 is a brownian motion . the [ s : prep ] key to the proofs of proposition [ p : smallballs ] and theorems [ t3 ] , [ t4 ] , [ t5 ] and [ t6 ] is a proper understanding of the behavior of the functions @xmath486 and @xmath487 from remark [ rem : intphi ] along the paths of the tree - valued fleming viot dynamics . in this section , we provide useful tools for the analysis of these functions . in particular , we are going to compute moments up to fourth order . throughout this section we assume that @xmath488 i.e. with @xmath489 , and denote by @xmath490 the canonical filtration of @xmath103 . furthermore , for the mutation operator , we will assume throughout this section that @xmath491 this assumption implies that every mutation event leads to a new type . in particular , this will be crucial in lemma [ l1 ] . the key is to obtain the power at which @xmath169 vanishes as @xmath493 respectively at which order the increments of @xmath494 vanish as @xmath495 . proofs of the next two lemmata are given in section [ ss.proofls ] . + for [ l2 ] all @xmath16 , @xmath496 in @xmath497 ( and therefore also in probability ) . + there [ l3]exists @xmath38 , which is independent of @xmath40 , such that @xmath498\leq ct,\\\label{eq:31b } & \sup_{\lambda > 0 } ( \lambda+2\vartheta + 1)^2 \mathbf e[(\widehat \psi^{12}_\lambda(x_t ) - \widehat \psi^{12}_\lambda(x_0))^2]\leq ct . \end{aligned}\ ] ] + if the right hand side of would have been @xmath499 for some @xmath191 , then lemma [ l3 ] would imply the needed tightness for theorem [ t3 ] by the kolmogorov chentsov tightness criterion ; see e.g. corollary 16.9 in @xcite . however , since it is only linear in @xmath1 , we will have to use fourth moments to get the desired tightness property . the main goal of this subsection is therefore to compute the fourth moments of the increments of @xmath500 and of @xmath501 . the proof of the next two results are given in sections [ ss.proofsl4l5 ] and [ ss:56 ] . + for all @xmath16 , [ l4b ] as @xmath502 , @xmath503 & = \frac{3}{4\lambda^2 } + \mathcal o\big(\frac{1}{\lambda^3}\big),\\ \mathbf e\bigl[\bigl((\lambda+2\vartheta + 1)\widehat \psi^{12}_\lambda(x_\infty)-1\bigr)^4\bigr ] & = \frac{3}{4\lambda^2 } + \mathcal o\big(\frac{1}{\lambda^3}\big ) . \end{aligned}\ ] ] in particular , the convergences in hold in @xmath504 as well . + there [ l5 ] exists @xmath38 , such that @xmath505\leq ct^2,\\ & \sup_{\lambda > 0 } ( \lambda+2\vartheta+1)^4 \mathbf e[(\widehat\psi^{12}_\lambda(x_t ) - \widehat\psi^{12}_\lambda(x_0))^4]\leq ct^2 . \end{aligned}\ ] ] we start with second moments and afterwards , we provide an automated procedure how to compute higher order moments using mathematica . here , we show that @xmath506 and @xmath507 in @xmath497 ( lemma [ l2 ] ) and study the second moments of the increments of @xmath508 ( lemma [ l3 ] ) . we start by defining some functions which appear in the first and second moment equations for the polynomials @xmath509 and @xmath510 defined in example [ rem : intphi ] . + we [ def : psiphi ] define the following functions in @xmath85 : @xmath511 and centering around the equilibrium expectations @xmath512 + our main object of study are the functions @xmath509 and @xmath513 . note that since @xmath101 has the same law as @xmath165 and since in @xmath165 the time it takes that two randomly chosen lines coalesce is an exponential random variable @xmath514 with parameter @xmath317 , we have @xmath515 & = \mathbf e[\langle \mu^{\otimes { \mathbb{n } } } , e^{-\lambda r(u_1,u_2)}\rangle ] = \mathbf e[e^{-\lambda r } ] = \frac{1}{\lambda+1}. \end{aligned}\ ] ] moreover , if mutations arise at constant rate @xmath516 , consider again two randomly chosen lines which coalesce at time @xmath514 , and the first mutation event at an independent , exponentially distributed time @xmath517 with rate @xmath518 , using ( [ eq : beta - non - atomic ] ) : @xmath519 & = \mathbf e[\langle \mu^{\otimes { \mathbb{n } } } , { \mathbbm{1}_{\{a_1=a_2\}}}e^{-\lambda r(u_1,u_2)}\rangle ] = \mathbf e[e^{-\lambda r } { \mathbbm{1}_{\{r\leq s\ } } } ] \\ & = \mathbf e[e^{-\lambda r } \cdot \mathbf p(s\geq r|r ) ] = \mathbf e[e^{-(\lambda + \vartheta)r } ] = \frac{1}{\lambda+\vartheta+1}. \end{aligned}\ ] ] moreover , we write @xmath520 and analogously for @xmath521 . next we compute the action of the generator of @xmath522 on the functions from definition [ def : psiphi ] . let [ l1 ] @xmath95 be the generator of the tree - valued fleming viot dynamics . then @xmath523 and @xmath524 2 . the following functionals of @xmath525 are martingales : @xmath526 furthermore for all @xmath527 and all @xmath528 @xmath529 = \mathbf e[\widehat\upsilon_{\lambda}^{k}(x_s ) ] = 0 . \end{aligned}\ ] ] + since [ rem : reference]@xmath530 = \mathbf e[\widehat\upsilon^k_\lambda(x_\infty ) ] = 0 $ ] by , it is also possible to compute the equilibrium values for @xmath531 and @xmath532 . we write for later reference , @xmath533 and @xmath534 where the terms after each @xmath531 and @xmath532 are the corresponding equilibrium values . the results in and follow by simple calculations . the stated martingale properties are consequences of general theory ; see lemma 4.3.2 in @xcite . in particular , since the equilibrium distribution of @xmath103 is ergodic , for @xmath528 and @xmath535 , @xmath536 = \lim_{t\to \infty } \mathbf e[\upsilon^k_{\lambda}(x_t ) ] = \lim_{t\to \infty } e^{-\lambda_k ( t - s)}\mathbf e[\upsilon^k_{\lambda}(x_s ) ] = 0\cdot \mathbf e[\upsilon^k_{\lambda}(x_s ) ] , \end{aligned}\ ] ] which can only hold ( note that @xmath537|<\infty$ ] by definition ) if @xmath538 = 0 $ ] . replacing @xmath539 by @xmath540 , @xmath541 gives the corresponding results for the expectations of @xmath542 . we only give the full proof for the first assertion in since the second follows exactly along the same lines . the proof amounts to computing the variance of @xmath509 . therefore , we need to understand the expectation of @xmath543 ( for the last equality , see ) . to this end we express the @xmath544 s in term of the @xmath545 s and obtain @xmath546 together with , this implies @xmath547 & = ( \lambda+1)^2 \cdot \mathbf e [ \psi_\lambda^{12,34}(x_t ) ] - 2(\lambda+1)\cdot \mathbf e [ \psi_\lambda^{12}(x_t ) ] + 1 \\ & = \frac{2 \lambda^2 } { ( \lambda+3 ) ( 2 \lambda+1 ) ( 2 \lambda + 3 ) } \xrightarrow{\lambda\to\infty}0 . \end{aligned}\ ] ] again , we restrict our proof to since is proved analogously . we compute @xmath548 & = \mathbf e\big[\psi^{12,34}_\lambda(x_0 ) - \frac{1}{\lambda+1}\psi^{12}_\lambda(x_0)\big ] \\ & = \frac{2 \lambda^2}{(\lambda+1)^2(\lambda + 3 ) ( 2\lambda + 1 ) ( 2\lambda + 3 ) } \end{aligned}\ ] ] and ( recall that we start in equilibrium ) @xmath549 \\ & = \mathbf e[(\psi^{12}_\lambda(x_t))^2 - ( \psi^{12}_\lambda(x_0))^2 - 2\psi^{12}_\lambda(x_0 ) ( \psi^{12}_\lambda(x_t)-\psi^{12}_\lambda(x_0 ) ) ] \\ & = \mathbf e\big[2\psi^{12}_\lambda(x_0 ) \int_0^t \omega \upsilon^{12}_\lambda(x_s ) ds\big ] \\ & = \int_0^t \mathbf e[2\psi^{12}_\lambda(x_0)(\lambda+1 ) \mathbf e[\upsilon^{12}_\lambda(x_s)|\mathcal f_0 ] ] ds \\ & = 2\int_0^t(\lambda+1)e^{-(\lambda+1)s}ds \cdot \mathbf e[\psi^{12}_\lambda({x_0})\upsilon^{12}_\lambda({x_0 } ) ] \\ & \le \frac{4}{(\lambda+1)^2 } t \end{aligned}\ ] ] and the result follows . for the proofs of lemma [ l4b ] and lemma [ l5 ] we use moment calculations of the increments of @xmath550 and @xmath551 up to fourth order . the calculations are presented in an algorithmic way such that higher moments can be computed along the same lines . the fundamental idea is that all computations that need to be done are _ linear maps _ on the vector spaces : @xmath552 we point out that one advantage of using matrix algebra is that it is possible to use computer algebra software such as mathematica for automated computations . apparently , a _ basis _ of the vector spaces @xmath553 and @xmath554 are given by ( recall from definition [ def : psiphi ] ) @xmath555 respectively . by inspection of e.g. lemma [ l1 ] it is clear , that the generator @xmath95 gives rise to linear operators @xmath556 and @xmath557 . the bases @xmath558 and @xmath559 can be ordered such that the matrices representing the linear maps @xmath95 on @xmath553 and @xmath554 are _ upper triangular _ ; see also lemma [ l1 ] . in the next section , we give the action of @xmath95 on the first 36 basis vectors for each basis . the last basis vectors in these collections are given by @xmath560 where we write @xmath561 , and @xmath562 respectively . because of @xmath563 and @xmath564 both 36 s basis elements play important roles . here , we give the matrix representation of the generator @xmath95 in terms of the basis @xmath566 , i.e. we find matrices @xmath14 and @xmath567 such that @xmath568 the following list contains the action on the first 36 basis vectors from @xmath569 and @xmath559 . in 4 . as an example ( see below ) we are dealing with the function @xmath570{1 cm } { \beginpicture \setcoordinatesystem units < .5cm,.5 cm > \setplotarea x from 0 to 3 , y from 0 to 0.5 \circulararc -180 degrees from 0 0 center at 0.5 0 \circulararc -180 degrees from 1.5 0 center at 2 0 \endpicture}}as abbreviation.}\ ] ] as this symbol indicates , in @xmath571 , two non - overlapping pairs are sampled . we have already seen in that @xmath572 our list only contains the last two terms ( which are equal in both equations ) which arise from by merging of a pair of leaves , i.e. merging for example the marked leaves in to or merging the marked leaves in to . the coefficient of @xmath573 in @xmath574 ( and of @xmath575 in @xmath576 ) is given such that the sum of coefficients in @xmath574 equals @xmath40 times the number of pairs ( i.e. 2 ) in @xmath573 ( and @xmath518 times the number of different indices ( i.e. 4 ) ) . since the table would be the same for the @xmath577 s , we restrict ourselves to the @xmath578 s . 1 . @xmath579 @xmath580 2 . @xmath581 @xmath580 1x 3 . @xmath582 @xmath580 1x 4 . @xmath583 @xmath580 2x 1x 5 . @xmath584 @xmath580 2x 4x 6 . @xmath585 @xmath580 1x 7 . @xmath586 @xmath580 1x 1x 1x 8 . @xmath587 @xmath580 3x 9 . @xmath588 @xmath580 1x 1x 4x 10 . @xmath589 @xmath580 3x 3x 11 . @xmath590 @xmath580 3x 2x 1x 12 . @xmath591 @xmath580 1x 2x 1x 2x + 4x 13 . 3x 12x 14 . @xmath593 @xmath580 1x 15 . @xmath594 @xmath580 1x 1x 1x 16 . @xmath595 @xmath580 2x 1x 17 . @xmath596 @xmath580 1x 2x 18 . @xmath597 @xmath580 1x 2x 1x 1x 1x 19 . @xmath598 @xmath580 1x 2x 2x 1x 20 . 3x 1x 2x 21 . @xmath600 @xmath580 4x 2x 22 . @xmath601 @xmath580 1x 2x 2x 1x 23 . @xmath602 @xmath580 2x 4x 24 . @xmath603 @xmath580 1x 1x 4x 25 . @xmath604 @xmath580 4x 2x 1x + 2x 1x 26 . @xmath605 @xmath580 2x 2x 1x + 2x 2x 1x 27 . @xmath606 @xmath580 4x 6x 28 . @xmath607 @xmath580 1x 2x 4x 2x + 1x 29 . @xmath608 @xmath580 1x 1x 1x 2x + 2x 2x 1x 30 . @xmath609 @xmath580 1x 3x 6x 31 . @xmath610 @xmath580 4x 4x 4x + 1x 2x 32 . @xmath611 @xmath580 1x 2x 4x + 8x 33 . @xmath612 @xmath580 1x 3x 6x + 2x 3x 34 . @xmath613 @xmath580 1x 3x 4x + 4x 2x 1x 35 . @xmath614 @xmath580 2x 2x + 4x 1x 4x + 8x 36 . @xmath615 @xmath580 4x 24x for the matrices @xmath14 ( and @xmath567 ) representing @xmath95 in terms of the basis @xmath558 ( and @xmath559 ) , we give here only the first 13 ( 5 ) rows and columns for @xmath14 ( @xmath567 ) , which are dealing with samples of at most three ( two ) pairs ( i.e. @xmath616 ( @xmath617 ) in ) . for @xmath14 , we get @xmath618 while for @xmath567 we have @xmath619 note that both matrices are diagonalisable . the reason is that the submatrix containing the same eigenvalue ( take @xmath620 in @xmath14 above as an example ) is diagonal . hence , there are three independent eigenvectors for this eigenvalue and so , we find a basis of eigenvectors . in the rest of this section , the calculations for the @xmath531 s and @xmath532 s are completely analogous using the @xmath621 s instead of the @xmath622 s . hence , we only present the calculations concerning @xmath531 s . during our calculations , we need to be able to compute terms like @xmath623 $ ] for @xmath624 . we do this using the following procedure . we have given the following objects : 1 . a basis @xmath625 such that @xmath626 , i.e. @xmath622 is an eigenvector of the generator @xmath95 for the eigenvalue @xmath539 ( where @xmath627 . as argued below , these eigenvectors exist , since @xmath14 is diagonalisable . 2 . a diagonal matrix @xmath628 with diagonal entries @xmath629 in the @xmath139th line . since the @xmath539 s are given by 1 . , this matrix is readily obtained . an invertible matrix @xmath630 , such that @xmath631 and @xmath632 . these two matrices accomplish a change of basis from @xmath566 to @xmath633 and back . + since @xmath634 is the basis of eigenvectors of the matrix @xmath14 , the matrices @xmath635 and @xmath636 are obtained by standard linear algebra . we stress that both matrices are lower triangular since @xmath14 is lower triangular . then , because @xmath637 are martingales , ( compare with lemma [ l1 ] ) , @xmath638 & = m_{\underline\upsilon_\lambda}^{\underline\psi_\lambda}\cdot \mathbf e[\underline \upsilon_\lambda(x_t)|\mathcal f_s]^\top = m_{\underline\upsilon_\lambda}^{\underline\psi_\lambda}\cdot d_{t - s } \cdot \underline \upsilon_\lambda(x_s)^\top \\ & = m_{\underline\upsilon_\lambda}^{\underline\psi_\lambda}\cdot d_{t - s } \cdot m_{\underline\psi_\lambda}^{\underline\upsilon_\lambda}\cdot \underline \psi_\lambda(x_s)^\top . \end{aligned}\ ] ] this means that conditioning linear combinations of elements in @xmath639 on @xmath459 can be represented by a composition of linear maps . we need to compute the action of multiplying an element of @xmath640 with @xmath509 . since @xmath553 is actually an algebra of functions ( see remark [ rem : intphi ] and remark 2.8 in @xcite ) , this can be done . we have that @xmath641 this means that there is a linear matrix @xmath642 such that @xmath643 apparently , @xmath642 is the matrix for a linear map on @xmath553 , with respect to the basis @xmath558 . again , we only give the calculations for the @xmath578 s in detail . we use @xmath644 by construction , we obtain for all @xmath645 @xmath646 = \mathbf e[(\underline\upsilon_\lambda(x_\infty ) m_{{\underline \upsilon_{\lambda}}}^{{\underline \psi_{\lambda}}})_{k } ] = ( m_{{\underline \upsilon_{\lambda}}}^{{\underline \psi_{\lambda}}})_{\emptyset ; k}\end{aligned}\ ] ] since @xmath530 = 0 $ ] for @xmath647 as in . hence , @xmath648 plugging the last expressions in gives the result . now we can compute the fourth moments of the increments of @xmath649 . again , we only give the proof of the first assertion . the second follows by replacing the @xmath578 s by the @xmath577 s . using minkowski s inequality and the fact that we start in equilibrium we have @xmath650 \bigr)^{1/4 } & \le \bigl ( \mathbf e[(\psi^{12}_\lambda(x_t))^4 ] \bigr)^{1/4 } + \bigl ( \mathbf e[(\psi^{12}_\lambda(x_0))^4 ] \bigr)^{1/4 } \\ & = 2 \bigl ( \mathbf e[\psi^{12,34,56,78}_\lambda(x_0 ) ] \bigr)^{1/4 } \end{aligned}\ ] ] and therefore , since @xmath651 = \mathcal o ( \lambda^{-4})$ ] as @xmath493 by , we have @xmath652 & \le 2 ^ 4 ( \lambda+1)^4 \mathbf e[\psi^{12,34,56,78}_\lambda(x_0 ) ] = \mathcal o(1).\end{aligned}\ ] ] thus , the expression on the left hand side of is _ bounded_. now , using @xmath653 we obtain , @xmath654 \\ & = \mathbf e[\psi^{12,34,56,78}_\lambda(x_t ) - \psi^{12,34,56,78}_\lambda(x_0 ) ] \\ & \qquad \qquad - 4\mathbf e\big[\psi^{12}_\lambda(x_0)\big(\psi^{12,34,56}_\lambda(x_t ) - \psi^{12,34,56}_\lambda(x_0)\big)\big ] \\ & \qquad \qquad \qquad \qquad + 6 \mathbf e\big[\psi^{12,34}_\lambda(x_0)\big(\psi^{12,34}_\lambda(x_t ) - \psi^{12,34}_\lambda(x_0)\big)\big ] \\ & \qquad \qquad \qquad \qquad \qquad \qquad - 4 \mathbf e[\psi^{12,34,56}_\lambda(x_0)\big(\psi^{12}_\lambda(x_t ) - \psi^{12}_\lambda(x_0)\big)\big ] \\ & = -4 \int_0^t \mathbf e[\psi^{12}_\lambda(x_0 ) \omega \psi^{12,34,56}_\lambda(x_s)]ds + 6\int_0^t \mathbf e[\psi^{12,34}_\lambda(x_0)\omega\psi^{12,34}_\lambda(x_s)]ds \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad - 4 \int_0^t \mathbf e[\psi^{12,34,56}_\lambda(x_0)\omega\psi^{12}_\lambda(x_s ) ] ds , \end{aligned}\ ] ] because @xmath655 , since we start in equilibrium . let us consider the expectation in the second term on the right hand side . we have @xmath656 & = \mathbf e[\psi^{12,34}_\lambda(x_0)\mathbf e[\omega\psi^{12,34}_\lambda(x_s)|\mathcal f_0 ] ] \\ & = \mathbf e[\psi^{12,34}_\lambda(x_0)\big(a \cdot \mathbf e[\underline{\psi}_\lambda(x_s)|\mathcal f_0]\big)_{12,34 } ] \\ & = \mathbf e[\psi^{12,34}_\lambda(x_0)\big(a \cdot m_{\underline \upsilon_\lambda}^{\underline\psi_\lambda } \mathbf e[\underline{\upsilon}_\lambda(x_s)|\mathcal f_0]\big)_{12,34 } ] \\ & = \mathbf e[\psi^{12,34}_\lambda(x_0)\big(a \cdot m_{\underline \upsilon_\lambda}^{\underline\psi_\lambda } d_s m_{\underline \psi_\lambda}^{\underline\upsilon_\lambda } \underline{\psi}_\lambda(x_0)\big)_{12,34 } ] \\ & = \mathbf e[\psi^{12}_\lambda(x_0)\big(a \cdot m_{\underline \upsilon_\lambda}^{\underline\psi_\lambda } d_s m_{\underline \psi_\lambda}^{\underline\upsilon_\lambda } q\underline{\psi}_\lambda(x_0)\big)_{12,34 } ] \\ & = \mathbf e[\big(a \cdot m_{\underline \upsilon_\lambda}^{\underline\psi_\lambda } d_s m_{\underline \psi_\lambda}^{\underline\upsilon_\lambda } q q \underline{\psi}_\lambda(x_0)\big)_{12,34 } ] . \end{aligned}\ ] ] the expression in the last line can easily be integrated in @xmath413 over @xmath657 $ ] , because only @xmath658 depends on @xmath413 . in addition , @xmath101 is in equilibrium and hence , the expectation can be evaluated using the equilibrium distribution of @xmath103 . all three terms in the right hand side of can be computed analogously . using mathematica , we see that holds . by lemma [ l.reform ] we have to show that holds . recall @xmath509 from . we abbreviate @xmath659 and compute , using lemma [ l4b ] @xmath660 \xrightarrow{\lambda\to\infty } \frac{3}{4}. \end{aligned}\ ] ] therefore , there is @xmath38 , such that @xmath661}{\varepsilon^4 } \leq \frac{c}{\varepsilon^4 \lambda^2}.\end{aligned}\ ] ] it follows that almost surely @xmath662 for at most finitely many @xmath70 by the borel cantelli lemma . thus , @xmath663 holds almost surely and the result is shown . recall that is the short form of @xmath664 furthermore , since @xmath665 is the probability of sampling a leaf from @xmath666 under the sampling measure ( and hence , @xmath667 is the probability of picking two leaves from @xmath666 ) , for every realisation of @xmath668 and @xmath669 we have @xmath670.\end{aligned}\ ] ] here we set @xmath671 , i.e. the distribution of the distance of two randomly chosen leaves . thus , the assertion of lemma [ l.reform ] is the equivalence of @xmath672 and @xmath673 \xrightarrow{\varepsilon \to 0 } 1 $ ] . this equivalence is implied by classical tauberian theorems , as e.g. given in theorem 3 , chapter xiii.5 of @xcite . we proceed in three steps . * * step 1 : * tightness of one - dimensional distributions of @xmath674 . * * step 2 : * moments and covariance structure . * * step 3 : * tightness of @xmath674 in path - space ( in the space of continuous functions ) . _ step 1 : tightness of one - dimensional distributions of @xmath675 . _ we obtain from remark [ rem : reference ] , @xmath676 & = \lambda \mathbf e[(\lambda t+1)^2 \psi_{\lambda t}^{12,34 } - 2(\lambda t+1)\psi^{12}_{\lambda t}+1 ] \xrightarrow { \lambda\to\infty } \frac1{2 t } , \end{aligned}\ ] ] which shows the claimed tightness . _ step 2 : moments and covariance structure . _ for the covariance structure , we have to compute moments of @xmath677 which are not included in the manuscript up to here . we set @xmath678 then @xmath679 now , in order to compute the covariance structure and higher moments , we write , again using mathematica , @xmath680 & = \lambda \mathbf e[(s\lambda + 1)(t\lambda + 1)\psi^{12,34}_{s\lambda , t\lambda}-1 ] \\ & = \frac{4 s t \lambda^3 } { ( ( s+t)\lambda+1 ) ( ( s+t)\lambda+3 ) ( ( s+t)\lambda+6 ) } \\ & \xrightarrow{\lambda\to\infty}\frac{4st}{(s+t)^3}. \end{aligned}\ ] ] for the third moment , @xmath681 = \lambda^{3/2}\frac{16 t^3\lambda^3 ( 5t^2\lambda^2 + 9 t\lambda - 10)}{(t\lambda + 2)(t\lambda + 3 ) ( t\lambda + 5 ) ( 2t\lambda+1 ) ( 2t\lambda+3 ) ( 3t\lambda+1 ) ( 3 t\lambda+10 ) } \xrightarrow{\lambda\to\infty } 0.\end{aligned}\ ] ] the fourth moment was already given in . _ step 3 : tightness of @xmath674 in path - space . _ here , we show that there exists @xmath38 , which is independent of @xmath40 , such that @xmath682\leq c(t - s)^2\end{aligned}\ ] ] for @xmath683 . tightness then follows from step 1 and the kolmogorov chentsov criterion . in order to show , we simplify the notation and suppress the dependency on @xmath165 . from , we read off @xmath684 $ ] . then , the result follows from @xmath685 with an application of mathematica and is shown . as in the proof of theorem [ t1 ] , we only need to show the assertion in the case @xmath161 due to absolute continuity recalled in proposition [ p : main ] . in this case , the proof of the theorem requires the following three steps : * * step 1 : * instead of starting in @xmath686 , it suffices to start in equilibrium and then show . * * step 2 : * assume that @xmath686 is in equilibrium , i.e. @xmath252 . then , the set of processes @xmath687 is tight in @xmath424 . * * step 3 : * the finite - dimensional distributions of @xmath688 converge to 0 as @xmath502 . these steps imply that the object in converges to @xmath182 in distribution , which is ( in the case of convergence to a constant ) equivalent to convergence in probability . _ step 1 : start in equilibrium . _ let @xmath689 with @xmath690 be the tree - valued fleming viot process started in equilibrium . then , @xmath103 and @xmath691 can be coupled such that for @xmath191 @xmath692 for all @xmath693 which depends only on elements in @xmath88 with values at most @xmath1 . recall the moran model approximation . observe that distances evolve with time of rate @xmath1 and may change by resampling events by being reset to zero . the coupling arises in this model for every @xmath449 by taking the same resampling events for @xmath103 and @xmath691 which defines the two processes on a common probability space . for this coupling , @xmath694 taking now on this common probability space the limit @xmath695 results in the coupled laws . hence , it suffices to prove the assertion when started in equilibrium , i.e. 1 . holds for all assertions which concern properties which depend only on distances below a threshold and in particular all limiting properties close to the leaves . _ step 2 : tightness of @xmath696 . _ this is clearly implied by lemma [ l5 ] and the kolmogorov chentsov criterion for tightness in @xmath424 . _ step 3 : convergence of finite - dimensional distributions to 0 . _ this follows from lemma [ l2 ] . we proceed in several steps . * * step 0 * : warm up ; computation of first two moments of @xmath697 . * * step 1 * : the family @xmath698 is tight in @xmath424 . * * step 2 * : if @xmath699 is a limit point , then @xmath699 and @xmath700 are both martingales . throughout we let @xmath427 be the canonical filtration of the process @xmath0 . _ step 0 : computation of first two moments of @xmath701 . _ we start with some basic computations which we will need in the sequel . first , recall that @xmath702 and by lemma [ l1 ] @xmath703 = e^{-\lambda ( t - s)}\cdot ( \lambda+1)\upsilon^{12}_\lambda(x_s ) . \end{aligned}\ ] ] then , by fubini s theorem @xmath704 & = \lambda \int_s^t \mathbf e[(\lambda+1)\upsilon^{12}_\lambda(x_r)| \mathcal f_s]\ , dr \\ & = \lambda \int_s^t e^{-\lambda ( r - s ) } \cdot ( \lambda+1)\upsilon^{12}_\lambda(x_s ) \ , dr \\ & = ( 1-e^{-\lambda ( t - s)})\cdot ( \lambda+1)\upsilon^{12}_\lambda(x_s ) . \end{aligned}\ ] ] we compute @xmath705 which already implies that @xmath706 \xrightarrow{\lambda\to\infty } 0 \quad \text { in } l^2,\end{aligned}\ ] ] since we started in equilibrium . hence @xmath707 = 0 $ ] as in lemma . next , we come to the second moment @xmath708 = \lambda^2 \cdot \mathbf e\big[\big(\int_s^t ( \lambda+1)^2 \upsilon^{12}_\lambda(r)\ , dr\big)^2\big|\mathcal f_s\big ] \\ & = 2\lambda^2 \cdot \int_s^t \int_{r_1}^t\mathbf e\big [ ( \lambda+1)\upsilon^{12}_\lambda(x_{r_1 } ) \mathbf e[(\lambda+1)\upsilon^{12}_\lambda(x_{r_2})|\mathcal f_{r_1 } ] \big| \mathcal f_{s}\big]\ , dr_2\ , dr_1 \\ & = 2\lambda^2 \cdot \int_s^t \int_{r_1}^t\mathbf e\big [ ( \lambda+1)^2\upsilon^{12}_\lambda(x_{r_1 } ) e^{-(\lambda+1)(r_2-r_1 ) } \upsilon^{12}_\lambda(x_{r_1 } ) \big| \mathcal f_{s}\big ] \ , dr_2\ , dr_1 \\ & = 2\frac{\lambda^2}{\lambda+1 } \int_s^t ( 1-e^{-(\lambda+1)(t - r_1 ) } ) \cdot \mathbf e[((\lambda+1)\upsilon^{12}_\lambda(x_{r_1}))^2|\mathcal f_s ] \ , dr_1 \\ & = t - s + a_{\lambda } , \end{aligned}\ ] ] with @xmath709 \ , dr_1 , \end{split}\end{aligned}\ ] ] where @xmath710 since @xmath711 we have @xmath712 & \xrightarrow{\lambda\to\infty } t - s \quad\text { in } l^1.\end{aligned}\ ] ] _ step 1 : the family @xmath698 is tight in @xmath713 . _ again , we use the kolmogorov chentsov criterion . to this end we bound the fourth moments of the increments in @xmath714 by @xmath715 \\ & = \lambda^4 \cdot \mathbf e\big [ \big ( \upsilon^{12}_\lambda(x_t ) - \upsilon^{12}_\lambda(x_0 ) - \int_0^t ( \lambda+1 ) \upsilon^{12}_\lambda(x_s)ds - \psi^{12}_\lambda(x_t ) + \psi^{12}_\lambda(x_0)\big)^4\big ] \\ & \lesssim \lambda^4 \big ( \mathbf e[\big(\upsilon^{12}_\lambda(x_t ) - \upsilon^{12}_\lambda(x_0 ) - \int_0^t ( \lambda+1 ) \upsilon^{12}_\lambda(x_s)ds \big)^4\big ] \\ & + \mathbf e\big[(\psi^{12}_\lambda(x_t ) - \psi^{12}_\lambda(x_0))^4\big]\big ) . \end{aligned}\ ] ] the second term is bounded by @xmath716 for some @xmath38 which is independent of @xmath40 by lemma [ l5 ] . for the first term , we use the burkholder davis gundy inequality and write ( recall lemma [ l1 ] and the quadratic variation of the semimartingale @xmath717 from remark [ rem : qv ] ) by @xmath718 \\ & \lesssim \lambda^4 \mathbf e\big [ [ \psi^{12}_\lambda(\mathcal x)]_t^2\big ] \\ & = \lambda^4 \int_0^t \int_0^s \mathbf e\big [ ( \psi^{12,23}_\lambda(x_r ) - \psi^{12,34}_\lambda(x_r ) ) \cdot \mathbf e [ \psi^{12,23}_\lambda(x_s ) - \psi^{12,34}_\lambda(x_s ) |\mathcal f_r]\big ] \ , dr \ , ds . \end{aligned}\ ] ] we have to show that the integrand is of order @xmath719 , if @xmath101 is in equilibrium . first , we compute the conditional expectation using lemma [ l1].2 using and obtain @xmath720 \\ & = \mathbf e\big[\frac{1}{10 } \upsilon^{12,12}_\lambda(x_s ) -\frac 13 \upsilon^{12,23}_\lambda(x_s ) - \upsilon^{12,34}_\lambda(x_s ) - \frac{2}{(\lambda+2)(\lambda+5 ) } \upsilon^{12}_\lambda(x_s)\big|\mathcal f_r\big ] \\ & \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad + \frac{\lambda^2}{(\lambda+1)(\lambda+3)(2\lambda+1)(2\lambda+3 ) } \\ & = e^{-(2\lambda+1)(s - r ) } \frac{1}{10 } \upsilon^{12,12}_\lambda(x_r ) -e^{-(2\lambda+3)(s - r)}\frac 13 \upsilon^{12,23}_\lambda(x_r ) - e^{-(2\lambda+6)(s - r)}\upsilon^{12,34}_\lambda(x_r ) \\ & \qquad \qquad - e^{-(\lambda+1)(s - r)}\frac{2}{(\lambda+2)(\lambda+5 ) } \upsilon^{12}_\lambda(x_r ) + \frac{\lambda^2}{(\lambda+1)(\lambda+3)(2\lambda+1)(2\lambda+3)}. \end{aligned}\ ] ] abbreviating @xmath721 , note that by and the last display , the integrand in is a linear combination of terms of the form @xmath722)]$ ] for @xmath723 . we compute all these terms : @xmath724 \\ & = \mathbf e[\psi^{12,23,45 } - \psi^{12,34,56 } ] - \frac{1}{\lambda+1}\mathbf e[\psi^{12,23}_\lambda - \psi^{12,34}_\lambda]\\ & = \frac{4 \lambda^3 ( 5 \lambda^2 + 9 \lambda -10)}{(\lambda+1)^2 ( \lambda+3 ) ( \lambda+5 ) ( 4 \lambda^2 + 8 \lambda + 3 ) ( 9 \lambda^3 + 51 \lambda^2 + 76 \lambda + 20 ) } \\ & = \mathcal o\big(\frac{1}{\lambda^4}\big ) , \end{aligned}\ ] ] @xmath725 \\ & = \mathbf e[\psi^{12,12,34,45}_\lambda - \psi^{12,12,34,56 } ] - \frac{1}{2\lambda+1 } \mathbf e[\psi^{12,23}_\lambda - \psi^{12,34}_\lambda ] \\ & = \frac{2 \lambda^3 ( 1576 \lambda^6 + \mathcal o(\lambda^5 ) ) } { 3 ( \lambda + 3 ) ( 2 \lambda + 1)^2 ( 4 \lambda + 15 ) ( 2 \lambda^2 + 5 \lambda + 3)^2 ( 96 \lambda^5 + \mathcal o(\lambda^4 ) ) } \\ & = \mathcal o\big(\frac{1}{\lambda^4}\big ) , \end{aligned}\ ] ] @xmath726 \\ & = \mathbf e\big[\psi^{12,23,45,56}_\lambda - \psi^{12,23,45,67 } ] - \frac{5\lambda+3}{(\lambda+1 ) ( \lambda+3 ) ( 2\lambda+1 ) ( 2\lambda+3)}\mathbf e[\psi^{12,23}_\lambda - \psi^{12,34}_\lambda ] \\ & = \frac{2 \lambda^2 ( 683976 \lambda^9 + \mathcal o(\lambda^8 ) ) } { 9 ( \lambda+3 ) ( 2\lambda+1)^2 ( 2\lambda^2 + 5\lambda+3)^2 ( 4 \lambda^2 + 41 \lambda + 105 ) ( 1152 \lambda^7 + \mathcal o(\lambda^6 ) ) } \\ & = \mathcal o\big(\frac{1}{\lambda^5}\big ) , \end{aligned}\ ] ] @xmath727 \\ & = \mathbf e[\psi^{12,23,45,67}_\lambda - \psi^{12,34,56,78}_\lambda ] - \frac{4 \lambda^2 + 18 \lambda + 9}{(\lambda+1 ) ( \lambda+3 ) ( 2\lambda+1 ) ( 2\lambda+3)}\mathbf e[\psi^{12,23}_\lambda - \psi^{12,34}_\lambda ] \\ & = \frac{4 \lambda^2 ( 20480 \lambda^{11 } + \mathcal o(\lambda^{10 } ) ) } { ( \lambda+3)^2 ( \lambda+7 ) ( 2\lambda+1)^2 ( 2 \lambda^2 + 5 \lambda + 3 ) ^2 ( 4608 \lambda^9 + \mathcal o(\lambda^8 ) ) } \\ & = \mathcal o\big(\frac{1}{\lambda^5}\big ) , \end{aligned}\ ] ] which shows that the integrand on the right hand side of is @xmath728 . hence , we have shown that there is a constant @xmath38 such that @xmath729 \leq c(t - s)^2\end{aligned}\ ] ] and we have shown tightness of @xmath730 . _ step 2 : if @xmath731 is a limit point , @xmath731 as well as @xmath732 are martingales . _ let @xmath299 be a weak limit point of @xmath733 . the claimed martingale properties follow by the same arguments as in step 3 of the proof of theorem [ t2 ] . to get a first idea , let us do a little computation . by fubini s theorem , dominated convergence theorem and lemma [ l1 ] , we get @xmath734 & = \lim_{\lambda\to\infty } \int_0^t \mathbf e\big [ \psi^{12}_\lambda(x_t)\big ] dt \\ & = \lim_{\lambda\to\infty } \frac{1}{\lambda+1}\int_0^t ( 1 - \mathbf e\big [ \omega\psi^{12}_\lambda(x_t)\big])\ , dt \\ & = \lim_{\lambda\to\infty } \frac{1}{\lambda+1}\big ( t - \mathbf e[\psi^{12}_\lambda(x_t ) - \psi^{12}_\lambda(0)]\big ) = 0 , \end{aligned}\ ] ] thus , since @xmath17 has no atom @xmath735 @xmath736 , @xmath737 our task is to remove the _ almost _ in the @xmath738 on the left hand side . as in the proof of theorem [ t3 ] , it suffices to show the assertion if @xmath252 . recall that @xmath739 from lemma [ l1 ] . hence , for all @xmath740 , the process @xmath741 defined as @xmath742 is a continuous martingale with quadratic variation ( recall remark [ rem : qv ] ) @xmath743_t = \int_0^t \bigl(\psi^{12,23}_\lambda(x_s ) - \psi^{12,34}_\lambda(x_s)\bigr ) ds.\end{aligned}\ ] ] using and recalling from lemma [ l1 ] that the @xmath744 s are mean - zero martingales @xmath745 = \lim_{\lambda \to \infty } \int_0^t \mathbf e[\psi^{12,23}_\lambda(x_s ) - \psi^{12,34}_\lambda(x_s)]\ , ds = 0.\end{aligned}\ ] ] this implies for all @xmath191 by doob s maximal inequality : @xmath746 hence , by and , there is a subsequence @xmath213 with @xmath748 almost surely . recall that we can characterize the existence of atoms by the property whether we can draw two points at distance zero or not which we can in turn characterize by the @xmath493 limit of the sample laplace transform . combining with and with ( which allows us to discard the @xmath749 ) gives @xmath750 + an [ l : char ] mmm - space @xmath323 admits a mark function if there is a sequence @xmath212 with @xmath751 where @xmath752 are two independent pairs distributed according to @xmath11 . equivalently , @xmath753 _ step 1 : proof of lemma [ l : char]_. since @xmath11 is a probability measure on @xmath13 , we can write @xmath754 for some probability kernel @xmath755 from @xmath756 to @xmath14 . we have to show that @xmath757 only has a single atom for @xmath758-almost every @xmath759 . then , for all sequences @xmath212 @xmath763 applying this to the pairs @xmath764 and @xmath765 gives @xmath766 \\ & \leq ( 1-\delta ' ) \cdot \mathbf p(\mathfrak u_1 , \mathfrak u_2\in u ' ) + \mathbf p(\mathfrak u_1\notin u ' \vee \mathfrak u_2\notin u ' ) < 1 . \end{aligned}\ ] ] hence , can not hold if @xmath767 . this quantity however is by assumption bounded below by @xmath768 and we have shown lemma [ l : char ] . in order to see this , it suffices analogously to the proof of theorem [ t3 ] to consider @xmath161 and to start in equilibrium @xmath489 . first , we assume that @xmath771 is non - atomic , i.e. from section [ s : prep ] holds . in this case , we proceed as in steps 2 and 3 from section [ ss : prooft3 ] . these steps rely on lemmata [ l2 ] and [ l5 ] , and the second assertions of these lemmata imply . second , consider the general case , i.e. @xmath772 is not necessarily non - atomic . it is straight - forward to construct a coupling @xmath773 such that @xmath0 and @xmath774 use the same resampling and mutation events , where mutant types in @xmath775 are chosen according to @xmath776 , but mutant types in @xmath774 are chosen ( at the same rate ) according to some non - atomic @xmath777 . if @xmath778 , it is clear that mutant types in @xmath774 lead to new types in any case and thus , the inequality @xmath779 holds for all @xmath16 , almost surely . recall that we have already shown in theorem [ t3 ] that holds if @xmath780 is replaced by @xmath781 , and since @xmath782 is non - atomic , it also holds for @xmath783 by our arguments above . hence , by , it also holds for @xmath784 , i.e. we have shown . _ step 3 : combination of steps 1 and 2 gives theorem [ t6]_. fix @xmath785 . it suffices to show that @xmath786 for all @xmath787 . using theorem [ t3 ] and step 2 above , take a sequence @xmath788 and set @xmath789 such that @xmath790 we use lemma [ l : char ] and the tauberian result from lemma [ l.reform ] with its obvious extension to @xmath513 to write : @xmath791 by . this concludes the proof of theorem [ t6 ] . s. evans . kingman s coalescent as a random metric space . in _ stochastic models : proceedings of the international conference on stochastic models in honour of professor donald a. dawson , ottawa , canada , june 10 - 13 , 1998 ( l.g gorostiza and b.g . ivanoff eds . ) _ , canad . soc . , 2000 . exchangeability and the evolution of large populations . in _ proceedings of the international conference on exchangeability in probability and statistics , rome , 6th-9th april , 1981 , in honour of professor bruno de finetti _ , pages 97112 . north - holland elsevier , amsterdam , 1982 .

arxiv

Rangail is a village in the Saharanpur district in Uttar Pradesh, India. It is situated about 16 km away from Saharanpur proper at the Highway Deoband. The population includes many communities e.g. Tyagi, Kashyap, Pal, Vishvakarma, Harijan, Salmani, Telli, Jhojje And Shekh. In this village, the main occupation is farming and Government Service. There are many villages near Rangail. These are Sunehati kharkhari, Roopri, Feraheri, Kolkie, Pathed, Hariyabas, Itava, khajoori etc. References Category:Villages in Saharanpur district

wikipedia

prevalence of diabetes is rising rapidly in developing countries , and the global number of diabetes is estimated to reach 366 million in 2030 among adults aged 20 years [ 1 , 2 ] . both type 1 and type 2 diabetes are powerful and independent risk factors for coronary artery disease ( cad ) , stroke , and peripheral arterial disease . the global rise in diabetes burden has led to significant increase in health care expenditure . studies have found that thyroid dysfunction is much common in diabetic population compared to nondiabetic population , and diabetes and thyroid disorders have been shown to mutually influence each other [ 5 , 6 ] . most often thyroid dysfunction and type 1 diabetes are due to an autoimmune condition , whereas type 2 diabetes is mainly due to insulin resistance [ 7 , 8 ] . thus , it seems that diabetes , especially type 1 , has potential link with thyroid dysfunction or vice versa . besides the effects due to high blood glucose in diabetics , low thyroid hormones independently increase the risk for cardiovascular diseases in both diabetic and nondiabetic patients [ 3 , 9 ] . hence , assessment of thyroid function in the uprising diabetic patient number may be helpful in identifying cases of clinical and subclinical thyroid dysfunction thereby assisting in mitigating the harmful effects due to low thyroid hormones . thus , we designed a study by selecting confirmed diabetic patients from eastern nepal and performed clinical and laboratory evaluation to investigate thyroid function status and associated risk factors . a cross - sectional study was carried out among the diabetes mellitus patients attending biochemistry laboratory of b. p. koirala institute of health sciences , dharan , nepal , from february 2013 to january 2014 . a total of 419 patients with diabetes mellitus were selected during the study period based on our inclusion criteria of having diabetes for at least 6 months . those diabetic patients with known thyroid disorders , complications of diabetes mellitus , history of other illnesses , and hyperlipidemia as well as on corticosteroids therapy and on medications affecting thyroid function were excluded from the study . diabetes mellitus was diagnosed on the basis of american diabetes association ( ada ) criteria and classification of type 1 diabetes from type 2 diabetes was done on basis of physician classifications based on the age of onset of diabetes and dependence on insulin therapy alone to achieve normal plasma glucose concentrations . consent was taken from each patient and the study protocol was approved by the institute review board of b. p. koirala institute of health sciences , dharan . each subject demographic ( age , sex ) and anthropometric measurements ( height , weight , and bmi ) including blood pressure ( systolic and diastolic ) , duration of diabetes , family history of diabetes mellitus and thyroid disease , and alcohol intake habit and smoking habit was recorded . about 5 ml overnight fasting venous blood samples were collected at morning time from each patient . blood glucose , glycated hemoglobin ( hba1c ) , total cholesterol , high density lipoprotein ( hdl ) cholesterol , low density lipoprotein ( ldl ) cholesterol , triglyceride , free triiodothyronine ( t3 ) , free tetraiodothyronine ( t4 ) , and thyroid stimulating hormone ( tsh ) were estimated . blood glucose and hba1c were estimated using enzymatic method and by nycocard reader ii based on immunochromatographic principle , respectively . serum free t3 , free t4 , and tsh were measured by using fluorescent immunoassay ( vidas , biomeriux sa , france ) . similarly , total cholesterol and triglycerides were measured by enzymatic method ( kits from agappe diagnostics by biolyzer 100 ) . ldl cholesterol and hdl cholesterol were estimated by homogeneous , direct method ( kits from gesan by biolyzer 100 ) . thyroid dysfunction was said to occur if patients thyroid hormones fall outside the reference range ( free t3 ( 4.08.3 pmol / l ) , free t4 ( 9.020.0 pmol / l ) , and tsh level ( 0.255 miu / l ) ) . the data generated from study was entered into ms excel and analyzed using spss version 11.0 . continuous variables were expressed as mean sd values and categorical variables were expressed as percentage ( number ) . one - way anova test was applied for continuous variables and chi square test was applied for categorical variables at 95% confidence interval . relative risk with 95% confidence interval ( ci the study population comprised 53.22% males and 46.77% females with mean age of 51.27 15.33 years . the mean age of males and females was 50.93 15.44 years and 51.66 15.24 years , respectively . type 1 and type 2 diabetes mellitus patients were 4.3% ( n = 18 ) with mean age of 19.28 1.07 years and 95.7% ( n = 401 ) with mean age of 52.7 14.05 years , respectively . the bmi among type 1 and type 2 diabetics was significantly different ( 22.64 1.58 kg / m versus 25.28 2.81 kg / m , p < 0.001 ) . euthyroidism , subclinical hypothyroidism , overt hypothyroidism , and subclinical hyperthyroidism among study population were 64% ( n = 268 ) , 26.5% ( n = 111 ) , 5.5% ( n = 23 ) and 4.1% ( n = 17 ) , respectively . significant difference in thyroid function status was observed among genders ( p = 0.024 ) . significantly higher number of diabetic patients without thyroid dysfunction ( n = 39 , 9.3% ) had hypertension history compared to those with hypothyroidism ( both overt and subclinical ) ( n = 33 , 7.87% ) and subclinical hyperthyroidism ( n = 3 , 0.71% ) ( p = 0.046 ) . parameters of lipid profile were not significantly different among diabetic patients with thyroid dysfunction ( overt and subclinical hypothyroidism and subclinical hyperthyroidism ) and without thyroid dysfunction ( euthyroidism ) . free t3 ( p < 0.001 ) , free t4 ( p < 0.001 ) , and tsh ( p < demographic , anthropometric , and clinical parameters in the study population according to thyroid status , hypothyroidism ( both overt and subclinical ) , subclinical hyperthyroidism , and euthyroidism are shown in table 1 . similarly , the level of different biochemical parameters in the study population according to thyroid function status is shown in table 2 . kg / m versus 24.59 2.73 kg / m , p < 0.001 ) , systolic blood pressure ( 121.84 7.14 mmhg versus 119.72 6.45 mmhg , p = 0.001 ) , and fasting blood glucose ( 179.99 54.72 mg / dl versus 172.5 60.44 mg / dl , p = 0.005 ) than diabetic females , respectively . however , diabetic females had significantly higher hdl cholesterol ( 41.2 4.17 mg / dl versus 39.46 2.78 mg / dl , p < 0.001 ) and triglycerides ( 191.38 105.82 mg / dl versus 172.32 101.02 mg / dl , p = 0.024 ) than diabetic males . diabetic males and females had no significant difference in diastolic blood pressure ( 81.41 5.66 mmhg versus 80.45 6.26 mmhg , p = 0.124 ) , hba1c ( 7.23 1.99% versus 7.19 1.75% , p = 0.411 ) , total cholesterol ( 188.44 54.91 mg / dl versus 192.44 50.97 mg / dl , p = 0.187 ) , ldl cholesterol ( 112.11 47.61 mg / dl versus 110.77 41.72 mg / dl , p = 0.704 ) , free t3 ( 5.03 1.21 pmol / l versus 5.04 1.39 pmol / l , p = 0.587 ) , free t4 ( 11.78 3.46 pmol / l versus 11.78 3.61 pmol / l , p = 0.914 ) , and tsh ( 4.02 3.27 miu / l versus 5.29 5.03 miu / l , p = 0.135 ) , respectively . thyroid dysfunction was more common in type 1 diabetic patients ( 50% , n = 9 ) as compared to type 2 diabetic patients ( 35.41% , n = 142 ) and in diabetic females ( 42.85% , n = 84 ) than in diabetic males ( 30.04% , n = 67 ) . hypothyroidism was the only thyroid dysfunction in type 1 diabetes patients , whereas it was the major thyroid dysfunction in type 2 significant risk factors for thyroid dysfunction were smoking ( relative risk 2.32 with 95% ci ( 1.852.91 , p < 0.001 ) ) , family history of thyroid disease ( relative risk 2.31 with 95% ci ( 1.842.91 , p < 0.001 ) ) , and female gender ( relative risk 1.42 with 95% ci ( 1.11.84 , p = 0.006 ) ) . when separately analyzed , significant risk factors for hypothyroidism included smoking ( relative risk of 2.56 with 95% ci ( 1.993.29 , p < 0.001 ) ) , family history of thyroid disease ( relative risk of 2.57 with 95% ci ( 2.03.31 , p < 0.001 ) ) , and female gender ( relative risk of 1.44 with 95% ci ( 1.091.91 , p = 0.01 ) ) . however , smoking , family history of thyroid disease , and female gender did not appear as significant risk factors for subclinical hyperthyroidism . age 60 years and duration of diabetes 5 years had relative risk of 1.03 with 95% ci ( 0.791.35 , p = 0.79 ) and 1.3 with 95% ci ( 1.01.68 , p = 0.053 ) , respectively , for thyroid dysfunction . diabetic patients have susceptibility to different types of thyroid dysfunction , whether hypothyroidism or hyperthyroidism ; at the same time , patients with thyroid dysfunction are susceptible to suffer from either type 1 diabetes or type 2 diabetes . the present study finds thyroid dysfunction as a common endocrine disorder in diabetic patients , where we reported that 36.03% diabetic patients had thyroid dysfunction . studies have reported high prevalence of thyroid dysfunction in nepal even in the general population . a hospital based study by baral et al . in eastern nepal reported hypothyroidism and hyperthyroidism in 17.19% and 13.68% population , respectively . similarly , in a study in kavre district of central nepal , thyroid dysfunction was observed in 31.84% metabolic syndrome patients . while iodine nutrition has been rapidly improving in the past years in nepal , excess iodine intake , as indicated by recent studies in school children of eastern nepal , could also be a responsible factor for the high prevalence of thyroid dysfunction specifically hypothyroidism in our study population . the most common thyroid dysfunction in the present study was subclinical hypothyroidism ( 26.5% ) followed by overt hypothyroidism ( 5.5% ) and subclinical hyperthyroidism ( 4.1% ) . we reported higher prevalence of thyroid dysfunction in females ( 42.85% ) than in males ( 30.04% ) , which has been well observed in many studies . our results are in agreement with previous studies in india , saudi arabia , and greece [ 5 , 14 , 15 ] . a retrospective study in india reported thyroid dysfunction in 31.2% of type 2 diabetic patients with 27.7% having hypothyroidism . similarly , a study in saudi arabia found thyroid dysfunction in 28.5% of type 2 diabetic patients , with 25.3% having hypothyroidism . prevalence of thyroid dysfunction was much lower than in our study among the type 2 diabetic patients of greece where thyroid dysfunction prevalence was 12.3% ( 5.53% in males and 18.48% in females ) . we observed that thyroid dysfunction is more common in type 1 diabetic patients ( 50% ) than in type 2 diabetic patients ( 35.41% ) . various studies have revealed that autoimmune thyroid disease is the commonest autoimmune disorder associated with type 1 diabetes . type 1 diabetes and autoimmune thyroid disease share an autoimmune disposition , and recent studies have shown a shared genetic susceptibility to both conditions . kordonouri et al . also found thyroid dysfunction to be more common in subjects with type 1 diabetes compared to those with type 2 diabetes and a 3.5-fold increased risk of autoimmune thyroiditis was noticed in gada positive patients . thyroid hormones have prominent roles in regulating a number of metabolic pathways including lipoprotein metabolism . we found that diabetic patients with thyroid disorders , hypothyroidism , or subclinical hyperthyroidism had higher total cholesterol and ldl cholesterol than those without thyroid dysfunction . however , in the study in greece , patients with thyroid dysfunction had higher hdl cholesterol levels and lower values of ldl cholesterol levels in comparison with patients without thyroid dysfunction . the present study reveals that hypertension ( 17.9% ) , smoking habit ( 19.3% ) , alcohol intake ( 31.5% ) , family history of diabetes mellitus ( 32.9% ) , and family history of thyroid disease ( 20.8% ) are common among nepalese patients with diabetes . among the various risk factors for thyroid dysfunction we studied , smoking , family history of thyroid disease , and female gender had significant risk for thyroid dysfunction , similar result was observed by papazafiropoulou et al . , who reported that presence of thyroid dysfunction was related with gender and ldl cholesterol levels in type 2 diabetic patients . similarly , al - geffari et al . found family history of thyroid disease , female gender , and duration of diabetes of > 10 years as significant risk factor for thyroid dysfunction in type 2 diabetic patients . in a study by chubb et al . subclinical hypothyroidism was associated with anti - tpo status and age , but there were no independent associations with serum cholesterol , history of coronary heart disease , hba1c , or hypoglycaemic therapy in diabetic women . the nocturnal tsh peak is blunted or abolished , and the tsh response to trh is impaired . reduced t3 levels have been observed in uncontrolled diabetic patients , which is due to impairment in peripheral conversion of t4 to t3 that normalizes with improvement in glycemic control . higher levels of circulating insulin associated with insulin resistance have shown a proliferative effect on thyroid tissue resulting in larger thyroid size with increased formation of nodules . diabetic patients have strong chance of developing thyroid dysfunction over the time course . a study by chubb et al . , in women with type 2 diabetes without known thyroid disease at baseline , revealed that diabetic women develop subclinical hypothyroidism ( 8.6% ) over time as found after 5 years . furthermore , it seems that unidentified thyroid dysfunction could negatively impact diabetes and its complications . a higher frequency of retinopathy and nephropathy was observed in diabetic patients with subclinical hypothyroidism , and more severe retinopathy was noted as well . thyroid function test is not a commonly recommended investigation in patients with diabetes especially in undeveloped countries like nepal . based on our findings and the reported harmful effects of low thyroid hormones on cardiovascular health , we recommend the frequent screening ( 1 - 2 years ) of thyroid function in patients with diabetes . our findings provide an evidence for the importance and necessity of thyroid function screening in patients with diabetes the prevalence of anti - tpo antibodies in the study population could not be assessed . assessment of anti - tpo antibodies would have provided clues for the thyroid autoimmunity status in the study population , which could have helped to explain the high rate of thyroid dysfunction . autoimmunity against the thyroid gland is one of the most important causes for thyroid dysfunction . anti - tpo antibody estimation helps in establishing the etiological diagnosis of autoimmune thyroid diseases . anti - tpo antibodies are the most prevalent and present in 8090% of hashimoto 's thyroiditis , 6575% of grave 's disease , and 1020% of nodular goiter or thyroid carcinoma . another limitation of the present study was that thyroid function was tested only once , which may have resulted in the inclusion of subjects with nonthyroidal illness . the other limitation of the present study was that the iodine status of the study population was not assessed . iodine excess and deficiency both are found to be associated with thyroid dysfunction and thus the concurrent prevalence of such iodine nutrition conditions in the study population may have some contributions to the high prevalence of thyroid dysfunction . in conclusion , the present study identifies thyroid dysfunction , as well as prominently subclinical hypothyroidism , as a common disorder in nepalese patients with diabetes . diabetic patients with thyroid dysfunction had higher lipid levels ( total cholesterol and ldl cholesterol ) than patients without thyroid dysfunction , so diabetic patients with thyroid dysfunction are at higher risk for cardiovascular diseases than those with normal thyroid function . similarly , smoking , family history of thyroid disease , and female gender were associated with thyroid dysfunction ( mainly hypothyroidism ) in the study population . thus , frequent screening for thyroid dysfunction especially in diabetic patients with family history of thyroid disease , female gender , and smoking habit needs to be done .

pubmed

interpolation , which deals with the construction of a function in continuum from its availability in a finite set of points , is a rather old problem dating back to ancient babylon and greece . in view of its increasing relevance in this age of ever - increasing digitization , it is quite natural that the subject of interpolation is receiving more and more attention . consequently , a multitude of different interpolation schemes are being developed and reported in various places in the literature . however , in times where all efforts are directed towards construction of smooth functions , the fact that many experimental and natural signals are rough " having a dense set of nondifferentiable points or even nowhere differentiable was easily ignored . in 1986 , barnsley @xcite introduced the idea of fractal interpolation function ( fif ) aiming at data that require a continuous representation with `` high irregularity '' . in words of barnsley : ..... ( these functions ) appear ideally suited for approximation of naturally occurring functions which display some kind of geometrical self - similarity under magnification , for instance , profiles of mountain ranges , tops of clouds and horizons over forests , temperatures in flames as a function of time , electroencephalograph pen traces and the minute by minute stock market index . in 1989 , barnsley and harrington @xcite provided a seminal result on differentiability of fractal functions , one which initiated a striking relationship between fractal functions and traditional nonrecursive interpolants , and lead to the creation of a subject which is now called fractal splines . since then , many researchers have contributed to the theory of fractal functions by constructing various types of fifs , including hermite or spline fifs ( see , for instance , @xcite ) , hidden variable fifs ( see , e.g. , @xcite ) , multivariate fifs ( see , e.g. , @xcite ) . an operator theoretic formalism for fractal functions which enabled them to find extensive applications in various classical branches of mathematics was introduced and popularized by navascus @xcite . in spite of about a quarter century of its first pronouncement , fifs were extensively investigated only in terms of their algebraic and analytical properties such as fractal dimension , hlder exponent , differentiability , integrability , stability , and perturbation errors . thus , these studies completely ignore one of the important properties of an interpolation scheme , namely , preserving shape inherent in the data . constrained control of interpolating curve is a fundamental task and is an important subject that we face in applications including computer aided geometric design , data visualization , image analysis , cartography . however , fractal functions are not well explored in the field of constrained interpolation . motivated by theoretical and practical needs , the authors have initiated the study of shape preserving interpolation and approximation using fractal functions ; see , for instance , @xcite . however , these researches concern about preserving the three basic shape properties , namely , positivity , monotonicity , and convexity . on the other hand , there are practical situations wherein interpolating curves that lie completely above or below a prefixed curve , for instance , a polygonal ( piecewise linear function ) or a quadratic spline are sought - after . the current article can be viewed as a contribution in this vein . the impetus for the research direction reported here grew out of the works of qi duan et al . @xcite . the structure of this article is as follows . we assemble some relevant definitions and facts concerning fractal interpolation in section [ icafw2sec1 ] . in section [ icafw2sec2 ] , we briefly recall the rational cubic fractal spline studied in @xcite . section [ icafw2sec3 ] is devoted to establish convergence of the rational cubic fractal spline . strategies to select the free parameters involved in the rational cubic fifs so that they render interpolating curves that lie below or above a prescribed spline ( piecewise defined function ) are enunciated in section [ icafw2sec4 ] . in section [ icafw2sec5 ] , we provide some numerical examples . this section targets to a constellation of a few rudiments of fractal interpolation theory . for a detailed exposition , the reader may consult the references @xcite . + the following notation and terminologies will be used throughout the article . the set of real numbers will be denoted by @xmath0 , whilst set of natural numbers by @xmath1 . for a fixed @xmath2 , we shall write @xmath3 for the set of first @xmath4 natural numbers . given real numbers @xmath5 and @xmath6 with @xmath7 , we define @xmath8 $ ] to be the space of all real - valued functions on @xmath9 $ ] that are @xmath10-times differentiable with continuous @xmath10-th derivative . + let @xmath11 and @xmath12 denote the cartesian coordinates of a finite set of points in the euclidean plane with strictly increasing abscissae . set @xmath13 $ ] and @xmath14 $ ] for @xmath15 . in fractal interpolation , a continuous function @xmath16 satisfying @xmath17 for all @xmath18 and whose graph @xmath19 is a fractal ( self - referential set ) in the sense that @xmath19 is a union of transformed copies of itself is sought for . there are two approaches for constructing fractal interpolation functions . first method introduced by barnsley characterizes the graph of fif as an attractor of specifically chosen iterated function system , hence the graph of the fractal interpolant can be approximated by the chaos game " algorithm @xcite . in what follows , we supply the second approach , to wit , the construction of a fractal function as the unique fixed point of read - bajraktarevi operator defined on a suitable function space , which was popularized by massopust @xcite . + suppose @xmath20 , @xmath15 be affine maps satisfying @xmath21 for @xmath22 , let @xmath23 and @xmath24 be functions that are continuous in first argument and fulfill @xmath25 define @xmath26 it is valuable to note that both @xmath27 and @xmath28 are closed ( metric ) subspaces of the banach space @xmath29 . define @xmath30 via @xmath31 the mapping @xmath32 is a contraction with contraction factor @xmath33 , and the fixed point @xmath34 of @xmath32 interpolates the data @xmath35 . consequently , @xmath34 enjoys the functional equation @xmath36 by the banach fixed point theorem , it follows that @xmath34 can be evaluated by using @xmath37 where @xmath38 denotes the @xmath10-fold composition of @xmath32 , and @xmath39 is arbitrary . further , by the collage theorem , for any @xmath40 @xmath41 if @xmath42 is defined by @xmath43 @xmath44 , then @xmath45 next let us point out the well - known fact that the notion of fractal interpolation can be applied to associate a family of fractal functions with a prescribed function @xmath46 . this was observed originally by barnsley and explored in detail by navascus . consider the maps @xmath47 where @xmath48 is a continuous function interpolating @xmath49 at the extremes of the interval and @xmath50 are real parameters satisfying @xmath51 , termed scaling factors . the corresponding fif denoted by @xmath52 is referred to as @xmath53-fractal function for @xmath49 ( fractal perturbation of @xmath49 ) with scale vector @xmath53 , base function @xmath54 and partition @xmath55 of @xmath56 with increasing abscissae . note that here the interpolation points are @xmath57 . the function @xmath58 may be nondifferentiable and its fractal dimension ( minkowski dimension or hausdorff dimension ) depends on the parameter @xmath59 . further , @xmath58 satisfies the functional equation @xmath60 with the notation @xmath61 , the following inequality points towards the approximation of @xmath49 with its fractal perturbation @xmath58 . @xmath62 let @xmath63 be a prescribed set of interpolation data with strictly increasing abscissae . let @xmath64 be the derivative value at the knot point @xmath65 which is given or estimated with some standard procedures . for @xmath15 , denote by @xmath66 , the local mesh spacing @xmath67 , and let @xmath68 and @xmath69 be positive parameters referred to as shape parameters for obvious reasons . a @xmath70-continuous rational cubic spline with linear denominator was introduced in @xcite as follows . @xmath71 where @xmath72 in reference @xcite , the authors observe that for the construction of fractal perturbation @xmath58 of the rational spline @xmath49 involving shape parameters , it is more advantageous to work with a family of base functions @xmath73 instead of a single base function used in the traditional setting ( see eq . ( [ icafweq4 ] ) ) . furthermore , for the fractal perturbation @xmath58 to be @xmath70- continuous , it suffices to choose the scaling parameters @xmath50 such that @xmath74 and base functions @xmath73 so that each @xmath75 agrees with @xmath49 at the extremes of the interpolation interval up to the first derivative . among various possibilities , the following choice is motivated by the simplicity it offers for the final expression of the desired fractal analogue of @xmath49 . @xmath76 where the coefficients @xmath77 , @xmath78 , @xmath79 , and @xmath80 are prescribed as @xmath81 therefore , in view of ( [ icafweq4 ] ) , the desired @xmath70-continuous rational cubic fractal spline is given by the expression @xmath82\big\}\theta(1-\theta)^2+\big\{(r_i+ 2t_i)y_{i+1 } \nonumber \\&&-t_i h_i d_{i+1 } -\alpha_i[(r_i + 2t_i)y_n - t_i(x_n - x_1)d_n]\big\}\theta^2(1-\theta ) , \nonumber\\ q_i(x)=q_i^*(\theta)&=&(1-\theta ) r_i+ \theta t_i,~~ i \in \mathbb{n}_{n-1},~ \theta= \frac{x - x_1}{x_n - x_1}.\end{aligned}\ ] ] a rational cubic spline with linear denominator which is based only on the function values is introduced in @xcite , pointing out that in some manufacturing processes the derivatives are difficult to obtain . however , as far as we know , it differs from ( [ 11a ] ) and its construction only in the following way . consider the given set of data points @xmath83 and the subset @xmath84 of interpolation points . treating @xmath64 to be equal to the chord slope @xmath85 for @xmath86 , the rational cubic spline involving only the function values discussed in detail in @xcite can be deduced at once from ( [ 11a ] ) . consequently , on similar lines , the following expression for the fractal rational cubic splines involving only function values can be obtained . @xmath87\}\theta(1-\theta)^2+\nonumber\\ & & \{(2 t_i+r_i)y_{i+1}- h_i t_i \delta_{i+1}-\alpha_i[(2t_i + r_i)y_n - t_i ( x_n - x_1)\delta_n]\ } \theta^2(1-\theta),\nonumber\\ \hat{q_i}(x)&=&r_i ( 1-\theta ) + t_i \theta,~ i \in \mathbb{n}_{n-1},~ \theta= \frac{x - x_1}{x_n - x_1}.\end{aligned}\ ] ] hence , in principle , the analysis on the fractal spline structure ( [ 15 ] ) set about to do in the subsequent sections can be easily modified and adapted to the rational cubic spline fif given in eq . ( [ 22 ] ) . the rate at which an interpolant @xmath34 approaches @xmath88 , the unknown function generating the data @xmath35 , is perhaps the most important factor deciding the efficacy of an interpolation scheme . in @xcite , it has been established that the rational cubic fractal spline @xmath58 converges to @xmath89 with respect to the @xmath90-norm . since order of continuity of @xmath58 is @xmath70 , it is natural to ask whether @xmath58 possesses uniform convergence as the norm of the partition approaches zero , if the data generating function is assumed to have a reduced order of continuity , namely @xmath70 . in this section , we answer this in the affirmative . + note that the implicit and recursive nature inherent in the description of the fractal spline @xmath58 make it impossible or at least difficult to apply the standard tools for error analysis available in the classical interpolation theory for establishing its convergence . on the other hand , we can base our analysis on the trustworthy triangle inequality @xmath91 following ( [ icafweq5 ] ) , it can be deduced at once that @xmath92 it can be read from @xcite that @xmath93 where @xmath94 , @xmath95 , @xmath96 , and @xmath97 . observing that @xmath98 for all @xmath15 , we obtain @xmath99 hence from ( [ new2 ] ) , the perturbation error obeys @xmath100 as an interlude to our analysis , we have the following theorem which deals with the first summand of the inequality ( [ new1 ] ) , namely , the interpolation error of the traditional rational cubic spline @xmath49 ( cf . ( [ 11a ] ) ) with respect to the uniform norm . [ errorthm1 ] let @xmath35 be a prescribed set of interpolation data generated by a function @xmath101 and @xmath49 be the corresponding traditional nonrecursive rational cubic spline with linear denominator . assume that the derivatives at the knots are given or estimated by some linear approximation methods ( for instance , arithmetic mean method ) . then the local error of the interpolation is given by @xmath102 where for the local variable @xmath103 , the constant @xmath104 is given by @xmath105 consider the error function @xmath106 as a linear functional which operates on @xmath88 . direct calculations confirm that the linear functional annihilates polynomials of degree strictly less than one . by the peano kernel theorem @xcite : @xmath107=e(\phi;x)= \phi(x)-f(x)=\int_{x_i}^{x_{i+1}}\phi'(\tau)l_{x}[(x-\tau)^0_{+ } ] ~d\tau,\ ] ] where the kernel function is given by @xmath108 = \left\ { \begin{array}{lr } r(\tau , x ) ~~ \text{if}~~ x_{i}<\tau < x,\\ s(\tau , x ) ~~ \text{if}~~ x<\tau < x_{i+1}. \end{array } \right.\ ] ] here the notation @xmath109 is used to emphasize that the functional @xmath110 is applied to the truncated power function @xmath111 considered as a function of @xmath112 . denoting @xmath113 , a rigorous calculation yields @xmath114 using the above expression for the kernel function in ( [ e1 ] ) , we obtain the following chain of relations : @xmath115~d\tau \big|,\\ & \le & \|\phi'\| \int_{x_i}^{x_{i+1 } } \big|l_x [ ( x-\tau)^0_{+ } ] \big|~d\tau , \\ & \le & \|\phi'\| \big\{\int_{x_i}^{x } r(\tau , x)~d\tau -\int_{x}^{x_{i+1 } } s(\tau , x)~d \tau \big \},\\ & = & \|\phi'\|\big\ { \frac{r_i ( 1-\varphi)(1-\varphi^2)+t_i\varphi(1-\varphi)^2}{(1-\varphi)r_i+\varphi t_i}(x - x_i)\\&&+\frac{r_i \varphi^2(1-\varphi)+t_i\varphi^2(2-\varphi)}{(1-\varphi)r_i+\varphi t_i}(x_{i+1}-x)\big\},\\ & = & \|\phi'\|h_i c_i,\end{aligned}\ ] ] where @xmath116 denotes the uniform norm on the subinterval @xmath117 $ ] . this offers the promised result . moving now to the crux of this section , we have the following theorem , whose proof is immediate from the foregoing discussion and the theorem . let @xmath35 be a prescribed set of interpolation data generated by a function @xmath101 . assume further that the derivatives at the knots are obtained by some linear approximation methods . let @xmath58 be the corresponding rational cubic spline fif . then @xmath118\\ & & + c h\|\phi'\|_\infty,\end{aligned}\ ] ] where @xmath119 the following remark highlights the important fact to which the preceding theorem sheds light . if @xmath120 is the data generating function and @xmath58 is the rational cubic fractal spline corresponding to the data set , then @xmath121 ergo , @xmath58 converges uniformly to the original function , as the norm of the partition tends to zero . let us point out , at least for the sake of good bookkeeping , that the application of triangle inequality does not imply that the error in approximating a continuously differentiable function @xmath88 with the fractal spline @xmath58 is always greater than or equal to the error in approximation by its classical counterpart @xmath49 . we employed the triangle inequality just to demonstrate that the fractal spline @xmath58 has the same order of convergence as that of the classical rational spline @xmath49 . finding appropriate @xmath53 for which @xmath58 is close to @xmath88 is an entirely different problem , which can be shown to be a constrained convex optimization problem with the aid of collage theorem @xcite . there is no reason to believe that the solution would always be @xmath122 , which corresponds to the classical interpolant @xmath49 . this section features a systematic discussion of selection of parameters involved in the rational cubic spline fif , focusing on its applicability in a more general constrained interpolation problem . to be precise , given a data set @xmath123 and a function @xmath124 ( piecewise linear or piecewise quadratic with joints at the knots @xmath65 ) satisfying @xmath125 , the problem is to construct a rational cubic spline fif @xmath58 such that @xmath126 for all @xmath127 . the reader is bound to have noticed that the values of @xmath58 in the subinterval @xmath128 $ ] depends on its values on other subintervals as well . hence , obtaining conditions for which @xmath58 lies above a piecewise defined function is relatively harder than that in the corresponding classical counterpart . however , this can be remedied by connecting @xmath58 with its classical counterpart @xmath129 , and performing the analysis on @xmath49 rather than on @xmath58 itself . let us commence by noting that @xmath130 where @xmath131 for brevity , we denote @xmath132 it follows that for @xmath133 , it suffices to make @xmath134 recall that on @xmath135 $ ] , @xmath136 , where with local variable @xmath103 , @xmath137 + \varphi^2 ( 1-\varphi ) [ ( r_i+2t_i ) y_{i+1}\\&&-t_ih_id_{i+1}]+\varphi^3 t_i y_{i+1},\\ s_i(x)&= & ( 1-\varphi ) r_i + \varphi t_i.\end{aligned}\ ] ] let @xmath124 be a piecewise linear function with joints at @xmath65 , @xmath18 . if @xmath138 and @xmath139 , then we have @xmath140.\ ] ] therefore @xmath141 for all @xmath127 is satisfied if @xmath142 s_i(x ) \ge 0 , ~ \text{for}~ x \in i_i , ~i \in \mathbb{n}_{n-1}.\ ] ] by using the technique of degree elevation , we assert that @xmath143 s_i(x)&= & r_i p_i ( 1-\varphi)^3 + [ r_i(p_i+p_{i+1})+p_it_i ] \varphi(1-\varphi)^2\\&&+[r_i p_{i+1}+t_i(p_i+p_{i+1 } ) ] \varphi^2(1-\phi)+ \varphi^3 t_i p_{i+1},\\ s_i(x)=(1-\varphi)r_i + \varphi t_i & = & r_i(1-\varphi)^3 + ( 2r_i+t_i)(1-\varphi)^2 \varphi+ ( r_i+2t_i)\varphi^2 ( 1-\varphi)\\&&+ \varphi^3 t_i.\end{aligned}\ ] ] in view of the aforementioned pair of equations , the condition ( [ eqnconstr1 ] ) can be recast as a problem of positivity of a cubic polynomial , namely , @xmath144\varphi(1-\varphi)^2\\&&+ \big[r_i(y_{i+1}-p_{i+1}+k)+t_i(2y_{i+1}-h_id_{i+1}-p_i - p_{i+1}+2k)\big ] \varphi^2 ( 1-\varphi)\\&&+ t_i ( y_{i+1}-p_{i+1}+k ) \varphi^3 \ge 0 , ~\text{for all}~i \in \mathbb{n}_{n-1}. \end{aligned}\ ] ] conditions for positivity ( nonnegativity ) of a cubic polynomial is well - studied in the literature ( see , for instance , @xcite ) . however , to keep our analysis simple enough , we shall impose conditions on parameters so that each coefficient of the cubic polynomial appearing in the above inequality is nonnegative . noting that @xmath145 for all @xmath18 and @xmath146 , we obtain @xmath147 the discussion we had until now is epitomized in the following theorem . [ abovelinecon ] suppose that a data set @xmath35 , where @xmath148 and @xmath124 is a piecewise linear function with joints at knots @xmath65 is prescribed . then a sufficient condition for the rational cubic spline fif @xmath58 to lie above @xmath124 is that the parameters satisfy the following inequalities : a. @xmath74 for @xmath15 , and @xmath149 , + b. @xmath150 , + c. @xmath151 , where @xmath152 a couple of remarks that supplement the stated theorem are in order . on similar lines , conditions for @xmath58 to lie below a piecewise linear function @xmath153 with joints at @xmath65 satisfying @xmath154 can be derived . by coupling these conditions with those prescribed in theorem [ abovelinecon ] , we obtain strategies for selecting the scaling factors and shape parameters so that @xmath155 for all @xmath127 . to keep the article at a reasonable length , we avoid the computational details here . [ exsrem ] having selected the scaling parameters according to the prescription in theorem [ abovelinecon ] , the existence of rational cubic fractal spline @xmath58 that lie above a piecewise linear function @xmath124 points naturally to the existence of positive shape parameters @xmath68 and @xmath69 satisfying the inequalities therein . we shall hint on the existence in the following , see also @xcite . let us rewrite the inequalities involving @xmath68 and @xmath69 ( see items ( ii)-(iii ) in theorem [ abovelinecon ] ) as follows . @xmath156 where @xmath157 , @xmath158 , @xmath159 , @xmath160 , and @xmath161 . if @xmath162 for all @xmath18 and the scaling factors are chosen such that @xmath163 ( that is , @xmath164 , @xmath165 ) , then a positive @xmath166 ( that is , a set of positive @xmath68 and @xmath69 ) satisfying the inequalities in ( [ exstineq ] ) exists except when @xmath167 , @xmath168 , and @xmath169 . if we replace @xmath64 with the slopes @xmath170 , then the coefficient of @xmath68 in item ( ii ) of theorem [ abovelinecon ] reduces to @xmath171 . hence , in view of item ( i ) , the condition in item ( ii ) is trivially satisfied . consequently , we obtain the following conditions for the rational cubic spline fif @xmath172 ( cf . ( [ 22 ] ) ) to lie above a piecewise linear function @xmath124 . @xmath173 next let us consider @xmath124 to be a piecewise defined quadratic polynomial with joints at @xmath65 , @xmath18 . if @xmath138 , @xmath139 , and @xmath174 , then we have @xmath175.\ ] ] analysis similar to the case of piecewise linear @xmath124 establishes the following theorem . [ fifabvqf ] suppose that a data set @xmath35 , where @xmath148 and @xmath124 is a piecewise quadratic function with knots at @xmath65 is given . then a sufficient condition for the rational cubic spline fif @xmath58 to lie above @xmath124 is that the parameters satisfy the following inequalities : a. @xmath74 for @xmath15 , and @xmath149 , + b. @xmath176 , + c. @xmath177 , where @xmath152 note that if @xmath178 is very large , then admissible value for @xmath179 will be close to zero . consequently , the fractal perturbation @xmath58 will almost " coincide with @xmath49 . this is an unpleasant situation as far as construction of a @xmath70-continuous constrained interpolant with fractality " in its derivative is concerned , owing to the fact that larger the value of @xmath179 ( with respect to the interpolation step ) , more pronounced is the irregularity in the derivative of the smooth fractal interpolant . furthermore , in light of remark [ exsrem ] , it can be deduced that there are some cases in which the constrained interpolation can not be solved with the rational cubic spline fif @xmath58 . these bring us motivating influence to seek an alternative approach . we shall now provide a quite brief deliberation of this alternative approach for the constrained interpolation . the following is the key theorem . [ rrafifbelthm ] let @xmath180 and @xmath181 be a partition of @xmath56 satisfying @xmath182 . let the scaling factors be chosen such that @xmath183 and the base functions be chosen fulfilling the conditions @xmath184 , @xmath185 for @xmath186 and @xmath187 for all @xmath15 , @xmath127 . then the corresponding fractal function @xmath188 and @xmath189 for all @xmath127 . recall that fractal function @xmath58 corresponding to @xmath49 satisfies the functional equation @xmath190,~ x \in i.\ ] ] for @xmath74 and base functions coinciding with @xmath49 at the extremes of the interval up to the first derivative , it follows @xcite that @xmath191 . + further , @xmath192 , i.e. , @xmath193 for all @xmath18 ( in fact , the equality ) . note that @xmath194 and @xmath58 is constructed using an iterated scheme according to . whence , to prove @xmath195 for all @xmath127 it is enough to prove that @xmath196 holds good at the points on @xmath56 obtained at @xmath197-th iteration whenever @xmath196 is satisfied for the points on @xmath56 at @xmath198-th iteration . the aforementioned condition is equivalent to @xmath199 for all @xmath15 whenever @xmath200 . we may rewrite in the form @xmath201 assume that the scaling factors are chosen such that @xmath202 for all @xmath15 . by our assumption @xmath203 , hence an appeal to reveals that @xmath204 is satisfied whenever @xmath187 for all @xmath127 and @xmath15 , offering the proof . the foregoing theorem demonstrates that if a constrained interpolation problem is solved with a classical rational interpolant @xmath49 , then the perturbation process can be so designed that the corresponding fractal spline @xmath58 also provides a solution to the same constrained interpolation problem . for instance , let @xmath49 be the rational cubic spline with linear denominator lying above a piecewise linear function @xmath124 . choose @xmath183 and the base functions @xmath205 , where @xmath206 are positive functions satisfying @xmath207 , @xmath208 for @xmath15 , @xmath186 . since positivity preserving hermite polynomial and rational spline interpolants are well - studied in the literature , it is not hard to find a family of functions @xmath209 satisfying the aforementioned conditions . note that cubic hermite interpolants may not serve as candidates for @xmath206 , as they may reduce to zero functions owing to the imposed conditions . the fractal function @xmath58 corresponding to @xmath49 with these choices of scaling parameters and base functions will lie above @xmath124 . thus , by dint of a more careful choice on base functions @xmath75 , the conditions on @xmath50 can be made relatively weaker , to wit , @xmath183 for all @xmath15 . in this section , the developed conditions on parameters are leveraged to obtain rational cubic spline fifs that lie above a prescribed piecewise linear function ( polygonal ) or quadratic function . consider the data set @xmath210 . note that the prescribed data set lies above the piecewise linear function @xmath124 with nodes at @xmath211 given by @xmath212 let the derivatives at knot points be @xmath213 . suppose that , due to some reasons , perhaps for a valid physical interpretation of the underlying process , a constrained interpolant lying above the following piecewise linear function @xmath124 is required . for brevity , let us represent the parameters , namely , the scaling factors @xmath50 satisfying @xmath214 , and the positive shape parameters @xmath215 and @xmath216 as vectors denoted by @xmath53 , @xmath217 and @xmath218 in the four dimensional euclidean space @xmath219 . the details of the scaling vectors and shape parameters used in the construction of constrained rational fifs are provided in table [ table : data ] for a quick reference . by selecting the scaling and shape parameters according to the conditions prescribed in theorem [ abovelinecon ] ( see table [ table : data ] ) , a rational cubic spline fif ( cf . ( [ 15 ] ) ) lying above the prescribed polygonal is generated in fig . [ figabvl](a ) . we construct rational cubic spline fif in fig . [ figabvl](b ) by changing the scaling vector @xmath220 with respect to the parameters of fig . [ figabvl](a ) . by taking the null scale vector , i.e. , @xmath221 , we retrieve the classical rational cubic spline plotted in fig . [ figabvl](c ) . + with specific choices of parameters ( see table [ table : data ] ) satisfying conditions prescribed in theorem [ fifabvqf ] , we obtain constrained rational cubic fifs displayed in figs . [ figabvq](a)-(c ) lying above the quadratic spline @xmath222 defined as follows . @xmath223 in contrast to the classical rational spline @xmath224 , the rational cubic spline fif @xmath58 has derivative @xmath225 having nondifferentiability in a finite or dense subset of the interpolation interval . further , the irregularity may be quantified in terms of box - counting dimension , which depends mainly on the scaling vector @xmath53 . this may find potential applications in various nonlinear and nonequilibrium phenomena wherein smooth interpolant satisfying a restraint ( for instance , positivity , or more generally lying above or below a prescribed curve ) and possessing irregularity in the derivative of suitable order is required . + * note : * parts of the results of this paper were presented by the third author at the international conference on applications of fractals and wavelets ( icafw-2015 ) held at amrita school of engineering , amrita vishwa vidyapeetham , coimbatore , tamilnadu , india . p. viswanathan and a.k.b . chand , @xmath53-fractal rational splines for constrained interpolation , electron . 41(2014)420 - 442 . p. viswanathan and a.k.b . chand , a @xmath228-rational cubic fractal interpolation function : convergence and associated parameter identification problem , acta appl . 136(2015)19 - 41 .

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