A Multivariate CLT for Local Dependence with n &12 log n Rate and Applications to Multivariate Graph Related Statistics

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1 joural of mulivariae aalysis 56, (1996) aricle o A Mulivariae CLT for Local Depeece wih &12 log Rae a Applicaios o Mulivariae Graph Relae Saisics Yosef Rio* Uiversiy of Califoria, Sa Diego a Vlaimir Roar Ceral Ecoomic-Mahemaical Isiue, Russia Acaemy of Scieces, Moscow, Russia, a Uiversiy of Califoria, Sa Diego This paper cocers he rae of covergece i he ceral limi heorem for cerai local epeece srucures. The mai goal of he paper is o obai esimaes of he rae i he muliimesioal case. Cerai oe-imesioal resuls are also improve by usig some more flexible characerisics of epeece. Assumig he summas are boue, we obai raes close o hose for iepee variables. As a applicaio we suy he rae of he ormal approximaio of cerai graph relae saisics which arise i esig equaliy of several mulivariae isribuios Acaemic Press, Ic. 1. Iroucio This paper cocers ceral limi heorems (CLTs) for sums of epee raom vecors, whe he epeece srucure is escribe i erms of epeecy eighborhoos. This ype of epeece is give by iicaig for every erm i he sum a se of oher erms o which i ``esseially'' epes. Such a srucure ee o be associae wih a liear orerig of he summas. This may be compare wih more classical CLTs i which he epeece is specifie i erms of a orerig (espie he fac ha he sum iself is ivaria uer permuaios), such as Markov chais, marigales or various mixig moels. Receive December 9, MSC 1991 subjec classificaios primary; 60F05, 60B12; secoary; 62H15, 04C80. Key wors a phrases Sei's meho, raom graphs, mulivariae saisics. * This work was suppore i par by NSF Gra DMS X Copyrigh 1996 by Acaemic Press, Ic. All righs of reproucio i ay form reserve.

2 334 RINOTT AND ROTAR Depeecy eighborhoos were irouce by Sei [21, Corollary 2, p. 110], i he suy of ormal approximaios. Esimaes of he rae of such a approximaio for osmooh fucios a boue raom variables were improve i Rio [17], a a mulivariae versio for smooh fucios oly, appears i Golsei a Rio [9]. These resuls prove o be useful i applicaios i which he epeece is ``local'' i he sese ha he epeecy eighborhoos are o oo large. This occurs ypically for various cous o graphs, where pars of he graphs ha are iscoece are almos iepee. For such applicaios, where he prese mehos are releva see, for example, [1, 8, 9] a refereces give here. The epeece srucure i his paper is escribe i erms of wo ypes of ses, referre o as firs-orer a seco-orer epeecy eighborhoos. These ses may be raom. See Theorem 2.2 a is iscussio. I he oe-imesioal case, uer his epeece srucure, we provie a more flexible a somwha more raspare esimae of he covergece rae i compariso o he above meioe resuls, coverig he case of osmooh fucios. The mai goal of his paper is he suy of he approximaio raes for he same epeece srucure i he muliimesioal case. I geeral he covergece rae for epee raom variables may be raher slow. We assume here ha he summas are boue. This assumpio eables us o obai a rae of covergece close o ha for iepee raom variables. The fac ha boueess imiishes he effec of epeece o he covergece rae is well kow (see, e.g., [6]). The proofs i his paper are base o Sei's meho. The exesio of hese mehos o osmooh fucios i he mulivariae case is raher orivial. We procee by he mehoology evelope by Barbour [3] a Go ze [10]. The mai resuls a some iscussios are give i Secio 2. A applicaio o graph relae saisics, moivae by hypohesis esig of equaliy of mulivariae isribuios, is give i Secio 3. The proof of Theorem 2.1 is give i Secio Resuls a Discussio Le 8 eoe he saar ormal isribuio i R, a, he correspoig esiy fucio. Give a marix (or a vecor) A, le A T be he raspose, a le A he be sum of he absolue values of is compoes. Usig his orm raher ha Eucliea orm will simplify some oaios a he formulaio of he resuls. This is oly a maer of coveiece sice he orms are equivale.

3 A CLT FOR LOCAL DEPENDENCE 335 Give a fucio h R R, se h + $ (x)=sup[h(x+y) y $], h& $ h (x; $)=h + $ (x)&h& $ (x). (x)=if[h(x+y) y $], Le H be a class of measurable fucios from R o R, which are uiformly boue by some cosa assume o be 1 wihou loss of geeraliy. Suppose H saisfies he followig properies for ay h # H a ay $>0, he fucios h + (x) a $ h& $ (x) are i H, a for ay _ marix A a ay vecor b # R, he fucio h(ax+b) belogs o H. I aiio we assume ha for all $>0 sup { R h (x; $) 8(x) h # H =a$ (2.1) for some cosa a which epes oly o he class H a he imesio. Obviously, we may assume a1. The class of iicaors of covex ses is kow o be such a class (see, e.g., [19, 5]). Theorem 2.1 below provies a ormal approximaio for a sum of raom vecors i erms of cerai ecomposiios of he sum. This heorem is somewha formal a shoul be rea ogeher wih is aural corollary, Theorem 2.2, which moivaes i a is use laer i he applicaios. Give raom vecors X j akig values i R, we suy he proximiy of he isribuio of he raom vecor W= X j o he ormal isribuio. We assume hroughou ha he summas are boue, ha is, X j B,1jfor some cosa B. I paricular applicaios, he cosa B, a ohers below, may be cosiere o epe o. As a measure of he proximiy o he saar ormal isribuio we cosier sup[ Eh(W)&8h h # H], where 8h= R h(z) 8(z). I he case =1, he covergece rae of our heorem is beer (by log ) ha he rae for >1. For his reaso, we sae a bou i he uivariae case separaely. Theorem 2.1. For each,..., assume ha we have wo represeaios of W, W=U j +V j a W=R j +T j, such ha U j A 1, a R j A 2 for cosas saisfyig A 1 A 2. Defie / 1 = E E(X j V j ), / 2 = / 3 = } I& E E(X j U T )&E(X j ju T j T j ), (2.2) E(X j U T j ) },

4 336 RINOTT AND ROTAR where I eoes he ieiy marix. The for =1 here exiss a uiversal cosa c such ha sup[ Eh(W)&8h h # H] c[aa 2 +(a+- EW 2 ) A 1 A 2 B+/ 1 +/ 2 +/ 3 ]. (2.3) For 1 here exiss a cosa c epeig oly o he imesio such ha sup[ Eh(W)&8h h # H]c[aA 2 +aa 1 A 2 B( log A 2 B +log ) +/ 1 +( log A 1 B +log )(/ 2 +/ 3 )]. (2.4) For iepee X j he aural ecomposiio is U j =R j =X j. I he case of epeece, oher ecomposiios prove o be more effecive as show below. I may seem ha he choice of V j a T j is arbirary. However, if, for example, he T j 's are cosa, he / 2 vaishes, bu he A 2 may be large. Similarly, if EW=0 a Var W=I, he choice V j =0 leas o / 3 =0 bu he A 1 may be large. Useful ecomposiios relae o he epeece srucure are cosiere below i eail. Noe ha i he heorem here is o assumpio o he mea a variace of W. However, sice he isribuio of W is beig compare o a saar ormal isribuio, we obviously have i mi he case where EW vaishes a Var W is he ieiy marix I, or approximaely so. Sice H is close uer affie rasformaios, we ca always saarize W (exacly or approximaely) before applyig he heorem. For =1, whe EW 2 =1, or if EW 2 is boue by a uiversal cosa, we may simply omi his erm from he r.h.s. of (2.3) by absorbig i io c, sice we assume a1. Theorem 2.1 appears raher absrac. I was moivae by he followig heorem which suggess aural ecomposiios i erms of he epeecy srucure. The ecomposiio is give by ses S i a N i efie below, which may be raom. Theorem 2.2. Le S i a N i be subses of [1,..., ], which i geeral may be raom, such ha i # S i N i, i=1,...,. Assume ha here exis cosas D 1 D 2, such ha max[ S i i=1,..., ]D 1, a max[ N i i=1,..., ]D 2, where for ses } eoes carialiy. Deoe U j = k # S j X k, V j =W&U j, R j = k # N j X k, T j =W&R j,,...,. (2.5)

5 A CLT FOR LOCAL DEPENDENCE 337 The for =1 here exiss a uiversal cosa c such ha sup[ Eh(W)&8h h # H] c[ad 2 B+(a+- EW 2 ) D 1 D 2 B 3 +/ 1 +/ 2 +/ 3 ], (2.6) where / i are efie i (2.2). For 1 here exiss a cosa c epeig oly o he imesioal such ha sup[ Eh(W)+8h h # H]c[aD 2 B+aD 1 D 2 B 3 ( log B +log ) +/ 1 +( log B +log )(/ 2 +/ 3 )]. (2.7) Theorem 2.1 follows from Theorem 2.2 simply by observig ha he quaiies efie i (2.5) saisfy he assumpios wih A 1 =D 1 B, a A 2 =D 2 B. Also, sice log A 2 appears i (2.4), i woul seem ha he quaiy log D 2 shoul appear i (2.7); however, i may be omie sice D 2. Remarks. I geeral, he epeece srucure affecs he covergece rae i ceral limi heorems for epee raom variables hrough hree ypes of characerisics or coiios. The firs ype, reflece i / 1 above, cocers coiioal expecaios wih respec o appropriae fiels, which shoul be small or vaish, as for example i marigales or for exchageable raom variables, where his ype of coiio appears i erms of correlaios. The seco ype, reflece by / 2 a / 3, cocers coiioal variaces, which shoul be close i some sese o he ucoiioal oes. The hir a mai ype reflecs special aspecs of he epeece srucure. Here, his erm ivolves he bou B o he summas, a he bous D 1 a D 2 o he sizes of he so-calle epeecy eighborhoos S i a N i. A iscussio of characerisics of his aure may be fou for example i Jaco a Shiryayev [14], Lipser a Shiryayev [16], a Roar [18]. Nex, we cosier he above characerisics i more eail. Formally Theorem 2.2 hols for arbirary ses S i a N i. Clearly, oe shoul choose hem so as o miimize he bou give by he heorem, which meas ha boh he / i 's a he D i 's shoul be small. I orer o clarify he aure of such a choice, assume ow ha we have saarize W, so ha EW=0 a Var W=I, a cosier firs he case where for each i here exiss S i such ha X i is iepee of [X j j S i ]. I his case / 1 =0 a / 3 =0. Suppose also ha here exis ses N i such ha, if k a l are boh i S i, he he collecios of variables [X k, X l ] a [X j j N i ] are iepee. I his case / 2 =0. I may eve happe ha a sroger ki of iepeece akes place; ha is, he collecios [X j j # S i ] a [X j j N i ] are iepee. I ay case, i is aural o view N i as a epeecy eighborhoo of [X j j # S i ], or a seco-orer epeecy

6 338 RINOTT AND ROTAR eighborhoo of X i. I oexoic siuaios oe may expec N i = j # Si S j a he D 2 D 2 1. However, his is o he case i geeral; for he laer N i, he prese assumpios o he S i 's imply ha he above X k is iepee of [X j j N i ], a he same hols for X l, bu his oes o ecessarily imply he iepeece of [X k, X l ] a [X j j N i ]. I geeral, he choice escribe above may o be feasible, a oe may ry o achieve a approximaio o his siuaio, ha is, o fi ses S i a N i, ``esseial epeecy ses,'' such ha he / i 's o o vaish bu are small a he D i 's are sill o large. A epeece srucure similar o ha of Theorem 2.2, wih wo ypes of epeecy ses, was use before; see, for example, Barbour, Hols, a Jaso [4], who cosiere Poisso approximaios, a refereces here. Turig o B, oe ha i may applicaios oe sars wih a sum of raom variables Y i, such ha Var(Y 1 +}}}+Y ) has he oer (or I i he mulivariae case), a he Y i 's are boue. Seig X i =Y i - o apply he heorems, we see ha he bou B for he X i 's will have he orer 1-. IfD 1 a D 2 are boue, he he par of he bou (2.7) o ivolvig he /'s has he orer &12 log. Clearly, if EW=*, a Var W=7, we ca apply he heorems o 1 &12 (W&*), where 1 equals or approximaes 7. By he ivariace of H uer affie rasformaios, we he obai a bou o sup h # H Eh(W)&Eh(1 12 Z+*), where Z is a saar ormal vecor i R. We shall cosier saisical applicaios i eail i Secio 3. Here we briefly illusrae he epeecy srucure by a applicaio (wih =1) ue o Bali a Rio [2]. Some of he raes i he laer paper ca be improve by usig Theorem 2.2. Cosier a raom rakig of he =2 m verices of he hypercube [0, 1] m. Le Y i be he iicaor of he eve ha he rak of he ih verex excees ha of all he m eighborig verices (a local maximum). The oal umber of local maxima is M= Y i=1 i. I ca be show ha if he isace bewee ay wo verices i a j is 3 or more, ha is, if he verices iffer i hree cooriaes or more whe viewe as m-vecors of 0's a 1's, he Y i a Y j are iepee. More geerally ay collecios of [Y j j # A i ], i=1, 2, are iepee if he isace bewee he verex ses A 1 a A 2 is a leas 3. Thus, we may choose S i cosisig of he ( m 2 )+m verices of isace a mos 2 from i. This resuls i D 1 which is of he orer m 2, or (log ) 2. Seig N i = j # Si S j, we have D 2 D 2 a / 1 i=0, i=1, 2, 3. Here _ &1 M a hece B have he orer (log ) 12. For W= i=1 (Y i &EY i )_ M, Theorem 2.2 provies a ormal approximaio wih he rae of (log ) 7.5 &12. I he above example a aural liear orerig of he variables oes o exis a he epeecy eighborhoos are eermie by isaces o a graph. Noe also ha he D i 's are o boue, bu icrease i. This

7 A CLT FOR LOCAL DEPENDENCE 339 suggess a geeral scheme, where he epeece srucure is escribe by a graph whose verices correspo o he raom summas, he epeece bewee wo summas is efie as a fucio of he isace bewee correspoig verices, a aural epeecy ses arise as ``geographical'' eighborhoos of verices. If for example, he epeece as reflece by such a fucio of he isace ecays expoeially, oe may choose epeecy ses wih iameer of he orer c log, a for a appropriae c he characerisics / i will have he orer &12, whereas he cosas D i will have he oer of a power of log, leaig o a covergece rae of (log ) k &12 for some k. 3. Applicaios o Graph Relae Saisics Noraom Graphs. For he applicaios escribe laer we ee o cosier raom graphs; however, we sar wih oraom graphs for ease of exposiio. Cosier a fixe regular graph, wih verices a verex egree m. The regulariy implies ha he umber of eges i he graph is N=m2. Suppose ha each verex is iepeely assige oe of colors c i wih probabiliy? i, i=1,...,, saisfyig? i=1 i=1. Le W=(W 1,..., W ), where W i, i=1,...,, is he umber of eges coecig verices which are boh of color c i ; ha is, W i = N X ji, where X ji is he iicaor of he eve ha boh verices associae wih he ege j have he color c i. Se *=EW=N(? 2,..., 1?2). The eries of 7=(_ p ij), he covariace marix of W, saisfy (see [9], hereafer calle GR) _ ii =Var(W i )=N? 2 i (1&?2 i )+2N(m&1)(?3 i &?4); i (3.1) _ ij =Cov(W i, W j )=&N(2m&1)? 2 i?2 j, for i{ j. Le L=[mi 1i [? 2 (1&? i i)]] &12. Give a marix A, le &A& eoe he maximal absolue value of is eries. I is prove i GR ha &7 &12 &N &12 L, a, hece, we ca apply Theorem 2.2 o 7 &12 (W&*), wih he bou o he summas B=N &12 L. For ay ege j we choose a eighborhoo S j cosisig of all eges which share a verex wih j. Wih he aural choice N i = j # Si S j we have he bous D 1 =2m&1 a D 2 = (2m&1) 2 o he carialiy of S j a N i, respecively, a / 1 =/ 2 =/ 3 =0. For fucios h # H, usig he closure of H uer affie rasformaios a he facs m a L1, we erive from (2.7), sup[ Eh(W)&Eh(7 12 Z+*) h #H] cam 32 L 3 ( log L +log ) &12, (3.2) where Z is a -imesioal saar ormal vecor, c is a cosa epeig o he imesio, a a is efie i (2.1). Such a resul, wih rae &12, bu oly for smooh fucios h was obaie i GR.

8 340 RINOTT AND ROTAR Raom Neares Neighbor Graphs. Cosier a sample of i.i.. pois from a absoluely coiuous isribuio F i R k. Le G eoe he eares eighbor graph whose verices are hese pois. This is a irece graph such ha from each verex here is a irece ege poiig o is eares eighbor (wih respec o Eucliea isace, say). As before each verex is iepeely assige oe of he colors c i wih probabiliy? i, i=1,...,. For a verex j, le N( j) eoe is eares eighbor. Le X ji =1 if he verices j a N( j) are boh assige he color c i, a 0 oherwise,,..., ; i=1,...,. The W i = X ji cous he umber of verices havig he color c i as well as heir eares eighbors (wih muual eares eighbors coue wice, oce for each verex). Se W=(W 1,..., W ), a *=EW=(? 2,..., 1?2 ). We ow calculae he covariace marix of W, which we eoe by 7. For a give realizaio of he graph G, le D( j) eoe he egree of he verex j, ha is, he umber of eges poiig o j plus he oe ege emaaig from j. Le A eoe he umber of pairs of verices which are muual (eares) eighbors. For he coiioal variace of W i give he graph G we claim ha Var(W i G)=? 2 i (1&?2)+ D( j) i \ 2 + (?3 i &?4 i )+2A(?2 i &?3 i ). (3.3) To see his oe ha, if j a N( j) are o muual eighbors, he for he D( j) eges coece o j, say l 1,..., l D( j) we have Cov(X lp i, X lq i)=? 3 i &?4. i If j a N( j) are muual eighbors, he he covariace bewee X ji a X N( j)i is o? 3 i &?4, bu i?2 i &?4 i a, herefore, we have o a 2A imes? 2 i &?4 i &(?3 i &?4 i )=?2 i &?3. Sice i D( j)=2, we ca rewrie (3.3) i he form Var(W i G)=(? 2 i &?3 i )+1 2 (?3 i &?4 i ) Also, D( j) Cov(W i, W k G)=& \ 2 +?2 i? 2 k=? 2 i? 2 k& 1 2? 2 i? 2 k D 2 ( j)+2a(? 2 i &?3 i ). (3.4) D 2 ( j). (3.5) Sice EW i =E(W i G) for ay G, we obai he variace a covariace by akig expecaios of he coiioal oes i (3.4) a (3.5). Le =P(N(N( j))=j), ha is, he probabiliy ha he eares eighbor of N( j) isj, a se ; =ED 2 ( j). Boh a ; epe also o F. Noe ha 2EA=. The Var W i =(? 2 i &?3 i )+1 2 ; (? 3 i &?4 i )+ (? 2 i &?3 i ), (3.6) Cov(W i, W k )=? 2 i?2 k & 1 2 ;? 2 i?2 k, i{k.

9 A CLT FOR LOCAL DEPENDENCE 341 Le H, J, a K be he iagoal _ marices wih ih iagoal eries? 3, i?2 i &?4, a i?2 i &?3 i, respecively; le b be a colum vecor wih ih compoe? 2 ; a le i bt eoe is raspose. We ca wrie (3.6) as 7 = 1 2 (; &2)[H&bb T ]+J+ K. (3.7) Noe ha ED( j)=2 a, herefore, ; 4. I is easy o verify ha [H&bb T ] is oegaive efiie, a herefore so is 7 &J. Deoig he specral raius by \, we see ha &7 &12 &\(7 &12 )\( &12 J &12 ) &12 M, (3.8) where M=[mi 1i [? 2 i &?4 i ]]&12. (See, e.g., Hor a Johso [13] for saar facs o marices use here a below.) I applyig Theorem 2.2 o he vecor 7 &12 (W&*), his shows ha we ca ake he bou B i Theorem 2.2 o be B= &12 M. Usig he fac ha he ses S j i Theorem 2.2 may be raom, we choose hem epeig o he graph G. Specifically, give G, efie S j o cosis of j a all verices which are coece wih j by a ege. Noe ha for ay se A, we have P([X ji,1i] # A G, [X li,1i,ls j ]= P([X ji,1i] # A). Takig expecaios coiioe o [X li,1i, ls j ]we obai iepeece of [X ji,1i] a [X li,1i,ls j ]. Wih similar argumes for N l = j # Sl S j, oe may coclue ha / 1 =/ 2 = / 3 =0. I is well kow ha he egrees i he eares eighbor graph i R k are boue by some cosa K(k) which epes o he imesio k, where K(1)=2, K(2)=6, K(3)=12. Esimaes are kow for all K(k) (see, e.g., [15]). For he ses efie above we have he bous D 1 =K(k) a D 2 =K 2 (k) o he carialiy of S j a N l, respecively. For ay fucios h # H we obai, usig he closure of H uer affie rasformaios, sup[ Eh(W)&Eh(7 12 Z+*) h # H] cak 3 (k) M 3 ( log M +log ) &12, (3.9) where Z is a -imesioal saar ormal vecor, c is a cosa epeig o he imesio, a a is efie i (2.1). Noe ha if he color allocaio is o oe a raom a verices havig he same color c i e o cluser ogeher, we shoul expec large values of W i. This pheomeo was use by Heze [12] (see oher refereces here), who propose a saisic similar o W 1 for esig equaliy of wo isribuios a prove is asympoic ormaliy.

10 342 RINOTT AND ROTAR Our resul provies raes a a aural mulivariae exesio for esig equaliy of several isribuios as we briefly iicae ex. Cosier isribuios F i o R k, or populaios 6 i, i=1,...,. A sample of size is obaie by choosig populaio 6 i wih probabiliy? i, akig a raom observaio from he chose populaio a repeaig his proceure iepeely imes. The eares eighbor graph whose verices are he obaie sample pois i R k is cosruce. We color each poi by c i if i is raw from 6 i. Thus a large W i iicaes ha 6 i es o form clusers, a he vecor W ca be use as a es saisic for esig H 0 F 1 =}}}=F. Tess which reuce W o a uivariae saisic a rejec H 0, for example, whe a i=1 i W i p is large, for some p a weighs a i epeig o he aleraive, or whe max 1i W i is large, may appear aural. More geeral rejecio regios i R may arise i coecio wih specific aleraives. For example, if oe suspecs a priori ha some populaios are more likely o iffer from he res ha ohers, he rejecio of H 0 if W i >b i for a leas oe i, i=1,...,, for suiable criical values b i,isa aural choice for a rejecio regio. The asympoic isribuio of W uer H 0, ha is, whe F 1 =}}}=F =F, is eee o eermie criical values for he ess (a, more geerally, rejecio regios). Calculaios similar o hose of Heze [12] show ha a ; of (3.7) coverge o fiie limis, a herefore here exiss a marix 3 such ha (1)7 &3 0. For a coiuous isribuio F of he verex locaios, all hese limis o o epe o F. Replacig he covariace marix 7 by 3 leas o a oparameric saisic. More specifically, he saisic W = &12 3 &12 (W&*), a is asympoic isribuio o o epe o F. As iicae i (3.11) below, here is a price o pay for he coveiece of usig he oparameric saisic W is ormal approximaio rae seems slower compare o ha of W because of he aiioal approximaio of he covariace marix. We apply Theorem 2.2 o he vecor W. I view of (3.8) we have &3 &12 &M a herefore we ca use he same bou B= &12 M as before. The covariace marix of W is o I, a herefore / 3 oes o vaish. Takig io accou ha isea of X j he summas ow are &12 3 &12 (X j &EX j ), i is easy o verify ha / 3 = I& &12 3 &12 _ 7 &12 3 &12. Saar argumes show ha / 3 = &12 3 &12 (3&7 ) &12 3 &12 &12 3 &12 2 3&7 ( 2 &12 M) 2 3&7 = 4 M 2 } 1 7 &3 }. (3.10)

11 A CLT FOR LOCAL DEPENDENCE 343 For h # H we obai sup[ Eh(W)&Eh( Z+*) h # H] cak 3 (k) M 3 ( log M +log ) {&12 + } 1 7 &3 }=, (3.11) where Z is a -imesioal saar ormal vecor, c is a cosa epee o he imesio, a a is efie i (2.1). The rae i which (1) 7 &3 coverges o zero, whe he pois (verices) are isribue accorig o F i R k, epes o F. A careful examiaio a ajusmes of he calculaios i Heze [2] iicae ha uer reasoable coiios, such as ha F is associae wih a probabiliy esiy havig a boue erivaive, i is impossible o asser ha his rae is of orer &12. Moreover, he rae becomes slower as he imesio k icreases. Thus he rae of ormal approximaio i (3.11) will geerally be eermie by he las erm. 4. Proof of Theorem 2.1 We eoe all cosas by c, eve whe we have i mi iffere cosas i he same equaio, as log as hey epe oly o he imesio. The meho we use is base o he followig iffereial equaio ue o Barbour [3] a Go ze [10], 29(x)&x } {9(x)=h(x)&8h, x # R, (4.1) which allows us o evaluae he expecaio Eh(W)&8h. I (4.1), 2 is he Laplacia; ha is 29(z)=Tr 9 (2) (x), where 9 (2) (x) is he Hessia marix of seco erivaives, a { eoes he graie. For =1, his approach is ue o Sei [20, 21]. I his case (4.1) reuces o f $(x)&xf (x)=h(x)&8h, x # R, (4.2) wih f =9$. The uique boue soluio of (4.2) is give by f (x)=,(x) 1 x (h(u)&8h),(u) u. (4.3) & For >1, a soluio for a smoohe versio of h is give below, bu i boh cases we are able o work oly wih smooh fucios. To his e efie he followig smoohig of h h s (x)= h(s 12 y+(1&s) 12 x) 8(y), R 0<s<1. I is worh oig ha 8h s =8h for ay s.

12 344 RINOTT AND ROTAR A bou o he error which arises from his smoohig is provie by he followig versio ue o Go ze [10] of a smoohig lemma of Bhaacharya a Raga Rao [5]. Lemma 4.1. Le Q be a probabiliy measure o R. The here exiss a cosa c>0 which epes oly o he imesio such ha for ay 0<<1, sup {} h(q&8) } h # H R = where a is efie i (2.1). c {} _sup (h&8h) Q } h # H R = +a - &, Le us reur o (4.1). If h is replace by h, oe may verify ha Eq. (4.1) has he soluio 9 (x)=& [h s(x)&8h](s(1&s)) [10]. Noe ha 9 (x) epes also o h, bu his is suppresse i he aioal. Le 9 (1) ={9, a le 9 (2) (x) eoe he _ Hessia marix whose pqh ery, eoe by 9 (2) ( pq)(x), is 2 9 (x)(x p x q ). I was show i Go ze [10] ha for h 1 here exiss a uiversal cosa c such ha 9 (1) (}) <c, 9 2) (}) <clog(&1 ). (4.4) Seig K j =X j U T j,a_raom marix, we have by (4.1) Eh (W)&8h=E[29 (W)&W } {9 (W)]=A&B&C+D, (4.5) where A=ETr _9(2) (W) \I& K j+&, B= C= D= E[X j } {9 (V j )], E[X j }[{9 (W)&{9 (V j )&9 (2) (V j ) U T j ]], ETr[K j [9 (2) (W)&9 (2) (V j )]]. (4.6) The ex Lemma is require i orer o bou Taylor series remaiers arisig i he evaluaio of he quaiies of (4.6).

13 A CLT FOR LOCAL DEPENDENCE 345 Lemma 4.2. Le W, V, a U be ay raom vecors i R saisfyig W=V+U, a le Y be ay raom variable. Suppose U C 1 a Y C 2, where C 1 a C 2 are umbers. Se #=sup[ Eh(W)&8h h # H], a le 9 (3) eoe he hir parial erivaive of 9 (pqr) wih respec o he iicae iices p, q, a r. The here exiss a cosa c which epes oly o he imesio, such ha for ay 0{1 a h # H, EY9 (3) (pqr)(v+{u) cc 2 (#- +ac 1 - +a log ), (4.7) where a is efie i (2.1). Proof of Lemma 4.2. Subsiuio a iffereiaio yiel he formula 9 (3) (pqr)(x)=c 1 (1&s) 12 s R s 32 h(- sz+-1&sx), (3) (pqr)(z) z, where here c= 1 2, bu i he sequel c will sa for ay posiive cosa which may epe o he imesio. Observe ha R, (3) (pqr)(z) z=0, sice his iegral ca be wrie as ( 3 x p x q x r ) R,(z+x) z a x=0, a he las iegral equals he cosa 1. This fac will be use i (4.8) below i he hir equaliy, where he ae erm vaishes. Abbreviaig, (3) for, (3) (pqr), we have EY9 (3) (pqr) (V+{U) = 1 (1&s) 12 }c = } c 1 s 32 s R EYh(- sz+-1&s (V+{U)), (3) (z) z } (1&s) 12 s s 1&s W&- 1&s(1&{)U+- sz), 32 R EYh(- (3) (z) z } (1&s) 12 s s EY[h(- 1&s W&-1&s (1&{)U+- sz) 32 R &h(-1&s W&-1&s (1&{)U)], (3) (z) z } = } c 1 c 1 1 s s C 32 2 R & if u C 1 +-s z E[ sup h(-1&s W+u) u C 1 +- s z h(-1&s W+u)], (3) (z) z =cc s s Eh (- 1&sW; C s z ), (3) (z) z. (4.8) R

14 346 RINOTT AND ROTAR Le Z iicae a iepee -variae saar ormal variable. By aig a subracig he same erm, he las quaiy of (4.8) equals cc s s E[h (-1&s W; C s z )&h (-1&s Z; C 1 +- s z ) R +h (-1&s Z; C 1 +- s z )], (3) (z) z. (4.9) I is easy o see ha by he efiiio of h, for ay =>0, E[h (- 1&s W; =)&h (-1&s Z; =)] E[h + (- 1&s = W)&E[h+ = (- 1&s Z)] + E[h & (- 1&s = W)&E[h& = (- 1&s Z)]. (4.10) By he assumpios o he class H a he efiiio of # we see ha for ay =>0 he expressio i (4.10) is boue by 2#. As 1 (1s32 ) s c-, we coclue ha for some c, 1 1 s s E[h (- 1&s W; C s z ) R &h (-1&s Z; C 1 +- s z )], (3) (z))], (3) (z) zc#-. (4.11) Recorig his fac we ow suy he las erm of (4.9). Noe ha by (2.1) Therefore, 1 Eh (- 1&s Z; C 1 +- s z )a(c 1 +- s z ). 1 s s Eh (- 1&s Z; C s z ), (3) (z) z R a 1 1 s s (C s z ), (3) (z) z R ca(c log ). (4.12) Lemma 4.2 ow follows. We ow reur o (4.6) a sar wih he erm C. Le X jp a U jp eoe he ph compoes of X j a U j, respecively. For,...,, Taylor expasio of {9 (W) ceere a V j shows ha C is equal o E 1 0 (1&{) q=1 r=1 9 (3) (pqr) (V j+{u j ) X jp U jq U jr {. (4.13)

15 A CLT FOR LOCAL DEPENDENCE 347 We apply Lemma 4.2 for each j, wih U=U j a Y=U jp U jq X jr, a obai C ca 2 B(#- +aa a log ), (4.14) where agai #=sup[ Eh(W)&8h h # H]. Nex cosier he erm D i (4.6). Wih he oaio efie above, a firs-orer Taylor expasio yiels 9 (2) (pq) (W)&9(2) (V (pq) j)= r= (3) (pqr) (V j+{u j ) U jr {. (4.15) The erm D is obaie from (4.15) by muliplyig by he eries of K j, a i is easy o see from he efiiio of K j ha his leas o a erm which is similar o he erm of (4.13) a hus D is boue by he r.h.s. of (4.14), possibly wih a iffere cosa. Nex oe ha B of (4.6) saisfies B= E[{9 (V j )}E(X j V j )]. By (4.4) he compoes of {9 (V j ) are uiformly boue, implyig ha for some c>0 B c E E(X jp V j ). (4.16) Fially, we cosier he erm A from (4.6). Wih $ pq =1 if p=q a 0 oherwise, we have Tr _9(2) (W) \I& K j+& = = + q=1 q=1 (W) (pq) \ $ pq& X jq U jp+ 9 (2) 9 (2) (W) (pq) _ $ qp& E(X jq U jp )& E(X jq U jp ) X jq U jp&. (4.17) By (4.4), 9 (2) (pq) (W) <c log(&1 ) for all p, q=1,...,, a 0<<1. Therefore we obai for he firs wo erms o he r.h.s. of (4.17) E } q=1 9 (2) c log (W) (pq) _ $ qp& q=1} $ pq& E(X jq U jp ) &} E(X jp U jp ) }. (4.18)

16 348 RINOTT AND ROTAR We rewrie he expressio ivolvig he las wo erms i (4.17) i he form q=1 [9 (2) (pq) (W)&9(2) (T (pq) j)+9 (2) (T (pq) j)][e(x jq U jp )&X jq U jp ]. (4.19) Taylor expasio of 9 (2) ( pq) (W)&9(2) (pq) (T j) a Lemma 4.2 applie for each j wih U=R j a Y=R jr X jq U jp imply q=1 E[[9 (2) (pq) (W)&9(2) (pq) (T j)][e(x jq U jp )&X jq U jp ]] ca 1 A 2 B(#- +aa 2 - +a log ). (4.20) Reurig o (4.19), we use (4.4) o bou he las erm as q=1 c log E9 (2) (pq) (T j)[e(x jq U jp )&X jp U jp ] q=1 E E(X jp U jq )&E(X jp U jp T j ). (4.21) Combiig Lemma 4.1, a (4.6), (4.14), (4.16), (4.18), (4.20), a (4.21), a oig ha sice A 1 A 2, he erm i he r.h.s. of (4.14) may be igore, beig smaller ha ha of (4.20), we obai #ca 1 A 2 B#- +caa 1 A 2 B(A log ) +c +c log { + E E(X jp V j ) q=1 q=1} $ pq& E(X jp U jq ) } E E(X jp U jq )&E(X jp U jq T j ) =+ca -. (4.22) The choice - =2cA 1 A 2 B, provie i is less ha 1, a simple maipulaios yiel (2.4), afer observig ha he las erm i (4.22) is of lower orer ha he seco erm a may be igore. If for he above choice >1, he he heorem is rivial.

17 A CLT FOR LOCAL DEPENDENCE 349 For he case =1, a beer bou, ha is (2.3), may be obaie wih some exra work which we ow skech. The improveme is achieve by elimiaig he erms log i (4.22) for =1. Iee, we may use (4.2) wih h=h, a he soluio f =9$ has he explici represeaio (4.3). By Lemma 3 of Sei [21, p. 25], f a f $ are boue, a we immeiaely see ha he erm log, arisig i (4.18) a (4.21) is avoie i he prese case sice ow 9 (2) = f$. The erm log arises also i Lemma 4.2 a, cosequely, i (4.20). We ow show ha i ca be avoie i he case =1. Usig f " for 9 (3) a he relaio f "(x)=f(x)+xf $(x)+h$ (x), we recalculae he l.h.s. of (4.7) a obai EY9 (3) (V+{U)=EY[ f (V+{U)+(V+{U) f $(V+{U)+h$ (V+{U)]. (4.23) Sarig wih he firs a las erms i (4.23), we have EY[h$ (V+{U)+f(V+{U)] = } EY { (1&)12 12 R h(- z+-(1&)(v+{u)),$(z) z+ f (V+{U)] } E } Y h(- z+-(1&)(v+{u)),$(z) z } - +cc 2, (4.24) R where he las iequaliy uses he fac ha f is boue. To esimae he iegral o he r.h.s. of (4.24) we follow he same logic as i (4.8)(4.12); he calculaios are simpler because we o o iegrae wih respec o s. We coclue ha EY[h$ (V+{U)+f(V+{U)] cc 2 (#- +ac 1 - +a+1). (4.25) Reurig o he mile erm i (4.23), we have, usig he boueess of f $ a he CauchySchwarz iequaliy, EY(V+{U) f $(V+{U) = EY(W&(1&{)U) f $(V+{U) c[-ey 2 - EW 2 +E YU ]cc 2 (- EW 2 +C 1 ). (4.26) This leas o he oe-imesioal versio of Lemma 4.2 EY9 (3) (V+{U) cc 2 (#- +ac 1 - +a+1+-ew 2 +C 1 ). (4.27) Sice a1 a <1, he r.h.s. of (4.27) may be reuce o cc 2 (#- +ac 1 - +a+- EW 2 ). As his versio of Lemma 4.2 iffers from he origial lemma oly i he absese of log a he aiio of - EW 2, we obai (2.3) by repeaig he above argumes.

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