AN EXPOSITION OF GÖTZE S ESTIMATION OF THE RATE OF CONVERGENCE IN THE MULTIVARIATE CENTRAL LIMIT THEOREM. Rabi Bhattacharya Susan Holmes

Transcription

1 AN EXPOSITION OF GÖTZE S ESTIMATION OF THE RATE OF CONVERGENCE IN THE MULTIVARIATE CENTRAL LIMIT THEOREM By Rabi Bhaacharya Susa Holmes Techical Repor No. 2-2 March 2 Deparme of Saisics STANFORD UNIVERSITY Saford, Califoria

2 AN EXPOSITION OF GÖTZE S ESTIMATION OF THE RATE OF CONVERGENCE IN THE MULTIVARIATE CENTRAL LIMIT THEOREM By Rabi Bhaacharya Deparme of Mahemaics Uiversiy of Arizoa Susa Holmes Deparme of Saisics Saford Uiversiy Techical Repor No. 2-2 March 2 This research was suppored i par by Naioal Sciece Foudaio gra DMS ad by Naioal Isiues of Healh gra R GM Deparme of Saisics STANFORD UNIVERSITY Saford, Califoria hp://saisics.saford.edu

3 A Exposiio of Göze s Esimaio of he Rae of Covergece i he Mulivariae Ceral Limi Theorem Rabi Bhaacharya ad Susa Holmes Deparme of Mahemaics Mah 63 Uiversiy of Arizoa rabi@mah.arizoa.edu Deparme of Saisics Sequoia Hall Saford Uiversiy CA susa@sa.saford.edu url: hp://www-sa.saford.edu/~susa Absrac: We provide a explaaio of he mai ideas uderlyig Göze s mai resul i [9] usig Sei s mehod. We also provide deailed derivaios of various iermediae esimaes. Curiously, we are led o a differe dimesioal depedece of he cosa ha ha give i [9]. We would lie o dedicae his o Charles Sei o he occasio of his 9h birhday. AMS 2 subec classificaios: Primary 6F5, 6B; secodary 62H5. Keywords ad phrases: Mulivariae CLT, Sei s Mehod. Suppored by NSF DMS-86 Suppored by NSF DMS ad NIH RGM

4 Bhaacharya ad Holmes/Sei s Mehod for Mulivariae CLT 2. Iroducio I his aricle Göze[9] used Sei s mehod o provide a igeious derivaio of he Berry-Essee ype boud for he class of Borel covex subses of R i he coex of he classical mulivariae ceral limi heorem (CLT). This approach has proved fruiful i derivig error bouds for he CLT uder cerai srucures of depedece as well (see Rio ad Roar []). Our view ad elaboraio of Göze s proof resuled from a collaboraio bewee he auhors ad were firs preseed i a semiar a Saford give by he firs auhor i he summer of 2. The auhors wish o ha Persi Diacois for poiig ou he eed for a more readable accou of Göze s resul ha give i his origial wor. Afer a explaaio of he geeral mehod i Secio, deailed derivaios of various esimaes are give i Secios 2-4 i erms ha would be reasoably familiar o probabiliss. Excep for he smoohig iequaliy i Secio 4, which is fairly sadard, complee proofs are give. Recely Raic[] has followed esseially he same roue as Göze, bu i greaer deail, i derivig Göze s boud. I may be poied ou ha we were uable o verify he dimesioal depedece O() i [9],[]. Our derivaio provides he higher order depedece of he error rae o, amely O( 5 2 ). This rae ca be reduced o O( 3 2 ) usig a iequaliy of Ball []. The bes order of depedece ow, amely, O( 4 ) is give by Beus[3], usig a differe mehod, which would be difficul o exed o depede cases. As a maer of oaio, he cosas c, wih or wihou subscrips are absolue cosas. The -dimesioal sadard Normal disribuio is deoed by N (, I ) as well as Φ, wih desiy φ... The Geeraor of he ergodic Marov process as a Sei operaor. Suppose Q ad Q are wo probabiliy measures o a measurable space (S, S) ad h is iegrable (wih regards o Q ad Q ). Cosider he problem of esimaig (.) Eh E h hdq hdq. A basic idea of Sei[2] (developed i some examples i [7] ad [8]) is (i) o fid a iverible map L which maps ice fucios o S io he erel or ull space of E, (ii) o fid a perurbaio of L, say L α, which maps ice fucios o S io he erel or ull space of E, (iii) o esimae. usig he ideiy (.2) Eh E h = ELg = E(Lg L α g α ) where g L (h E h), g α L α (h Eh).

5 Bhaacharya ad Holmes/Sei s Mehod for Mulivariae CLT 3 Oe way o fid L is o cosider a ergodic Marov process {X : } o S which has Q as i s ivaria disribuio, ad le L be is geeraor: (.3) T g g Lg = lim, g D L where he limi is i L 2 (S, Q ), ad (T g)(x) = E [g(x ) X = x], or i erms of he rasiios probabiliy p(; x, dy) of he Marov process {X : }, (.4) (T g)(x) = g(y)p(; x, dy) (x S, > ). S Also D L is he se of g for which he limi i (.3) exiss. By he Marov (or, semigroup) propery, T +s = T T s = T s T, so ha (.5) d d T T +s g T g T (T s g g) g = lim = lim = T Lg. s s s s Sice T T s = T s T, T ad L commue so ha (.6) d d T g = LT g. Noe ha ivariace of Q meas ET g(x ) = Eg(X ) = gdq, if he disribuio of X is Q. This implies ha, for every g D L, ELg(X ) =, or [ Lg(x)dQ (x) =, ELg(X ) = E(lim S T g(x ) g(x ) ] ET g(x ) Eg(X ) ) = lim Tha is, L maps D L io he se of mea zero fucios i L 2 (S, Q ). I is ow ha he rage of L is dese i ad if L has a specral gap, he he rage of L is all of. I he laer case L is well defied o (erel of Q ) ad is bouded o i ([4]). Sice T coverges o he ideiy operaor as oe may also use T for small > o smooh he arge fucio h = h hdq. For he case of a diffusio {X : }, L is a differeial operaor ad eve o smooh fucios such as h = B Q (B)(h = B ) are immediaely made smooh by applyig T. Oe may he use he approximaio o h give by (.7) T h = L(L T h) = Lψ, wih ψ = L T h, ad he esimae he error of his approximaio by a smoohig iequaliy, especially if T h may be represeed as a perurbaio by covoluio. For several perspecives ad applicaios of Sei s mehod see [2], [7],[8],[]. (b) The Orsei-Uhlebec Process ad is Gausssia ivaria Disribuio The Orsei-Uhlebec (O-U) process is govered by he Lagevi equaio (see, e.g. [6, pp. 476, 597, 598]) (.8) dx = X d + 2dB

6 Bhaacharya ad Holmes/Sei s Mehod for Mulivariae CLT 4 where {B : } is a -dimesioal sadard Browia moio. Is rasiio desiy is (.9) p(; x, y) = [ 2π( 2 ) ] 2 exp{ (y i e x i ) 2 i= 2( 2 ) } x = (x,..., x ), y = (y,..., y ). This is he desiy of a Gaussia (Normal) disribuio wih mea vecor e x ad dispersio marix ( 2 )I where I is he ideiy marix. Oe ca chec (e.g., by direc differeiaio) ha he Kolmogorov bacward equaio holds: (.) p(; x, y) = i= 2 p(; x, y) x 2 i i= x i p(; x, y) x i = p x p = Lp, wih L x where is he Laplacia ad = grad. Iegraig boh sides w.r.. h(y)dy we see ha T h(x) = h(y)p(; x, y)dy saisfies (.) T h(x) = T h(x) x T h(x) = LT h(x), h L 2 (R, Φ). Now o he space L 2 (R, Φ) (where Φ = N(, I ) is he -dimesioal sadard Normal ), L is self adoi ad has a specral gap, wih he eigevalue correspodig o he ivaria disribuio Φ (or he cosa fucio o L 2 (R, Φ)). This may be deduced from he fac ha he Normal desiy p(; x, y) (wih mea vecor e x ad dispersio marix ( 2 )I ) coverges o he sadard Normal desiy φ(y) expoeially fas as, for every iiial sae x. Else, oe ca compue he se of eigevalues of L, amely {,, 2,...} wih eigefucios expressed i erms of Hermie polyomials [6, page 487]. I paricular, L is a bouded operaor o ad is give by (.2) To chec his, oe ha by (.) (.3) h = L h = T s h(x)ds, s T s h(x)ds = h = h LT s h(x)ds = L ( hdφ L 2 (R, Φ). For our purposes h = C : he idicaor fucio of a Borel covex subse C of R. T s h(x)ds ). A smooh approximaio of h is T h for small > (sice T h is ifiiely differeiable). Also, by (.2) (.4) ψ (x) L T h(x) = T s T h(x)ds = T s+ h(x)ds = T s h(x)ds = { h(e s x + 2s z)φ(z)dz}ds R where φ is he -dimesioal sadard Normal desiy. We have expressed T s h(x) E[ h(xs ) X = x] i (.4) as (.5) E[ h(x s ) X = x] = E h(e s x + 2s Z), where Z is a sadard Normal N(, I ). For X s has he same disribuio as e s x + 2s Z. Now oe ha usig (.4), oe may wrie (.6) T h(x) = L(L T h(x)) = (L T h(x)) x (L T h(x)) = ψ (x) x ψ (x).

7 Bhaacharya ad Holmes/Sei s Mehod for Mulivariae CLT 5 For he problem a had (see.) Q = Φ ad Q = Q () is he disribuio of S = (Y + Y Y ) = (X + X X ), (X = Y / ), where Y s are i.i.d. mea-zero wih covariace marix I ad fiie absolue hird mome We wa o esimae (.7) ρ 3 = E Y 3 = E( (Y (i) )2 ) 3 2. i= E h(s ) = Eh(S ) hdφ for h = C, C C he class of all Borel covex ses i R. For his we firs esimae (see (.6)), for small >, (.8) ET h(s ) = E [ ψ (S ) S ψ (S )] This is doe i Secio 3. The ex sep is o esimae, for small >, (.9) ET h(s ) E h(s ) which is carried ou i Secio 4. Combiig he esimaes of (.8) ad (.9), ad wih a suiable choice of >, oe arrives a he desired esimaio of (.7). We will wrie (.2) δ = sup {h= C :C C} hdq () hdφ. 2. Derivaives of ψ L T h Before we egage i he esimaio of (.8) ad (.9), i is useful o compue cerai derivaives of ψ. The, usig (.4), D i ψ (x) = (2.) = = [ [ Le D i =, D ii = 2, D ii x i x i x i = 3, ec.. i x i x i x i R h(y)(2π( 2s )) 2 (y i e s x i ) 2s e s exp{ y e s x 2 ] 2( 2s ) }dy ds h(y)(2π( 2s )) e s 2 (y i e s x i ) exp{ y e s x 2 R 2s 2s 2( 2s ) }dy e s ( ) h(e 2s [R s x + ] 2s z)z i φ(z)dz ds, ] ds z i φ(z) = z i φ(z)) = D i φ(z) usig he chage of variables z = y e s x 2s.

8 Bhaacharya ad Holmes/Sei s Mehod for Mulivariae CLT 6 I he same maer, oe has, usig D i,d ii, ec for derivaives z i, 2 z, ec, i z i e D ii ψ s (x) = ( 2s [R )2 h(e s x + ] 2s z) D ii φ(z)dz ds, D ii i ψ e s (x) = ( 2s [R )3 h(e s x + ] (2.2) 2s z) ( D ii i φ(z))dz ds. The followig esimae is used i he ex secio: sup h( u R R R e s x + e s u + 2s z)φ(x)d ii i φ(z)dxdz (2.3) c e 2s ( 2s ). To prove his, wrie a = es 2s ad chage variables x y = x + az. The (2.4) so ha φ(x) = φ(y az) = φ(y) az φ(y) + a 2 ad he double iegral i (2.3) becomes r,r = z r z r ( v)d rr φ(y vaz)dv, h( e s x + e s u + 2s z) = h( e s y + e s u), (2.5) h( R R e s y + e s u) φ(y) az φ(y) + a 2 r,r = z r z r Noe ha he iegrals of D ii i φ(z) ad z i D ii i φ(z) vaish for i, i, i, i, so ha (2.6) h( R e s y + e s u)(φ(y) az φ(y))d ii i φ(z)dz = ( v)d rr φ(y vaz)dv D ii i φ(z)dzdy The magiude of he las erm o he righ i (2.4) is (2.7) a2 ( v) z r z r (y vaz) r (y vaz) r zr 2 φ(y avz)dv r,r = r= a 2 ( v) z r z r (y vaz) r (y vaz) r + φ(y avz)dv, r,r = sice he sum r,r above is oegaive. Boudig h by, i follows from (2.5)-(2.7) ha he lef side of (2.3) is o more ha a 2 ( v) R (2.8) { =a 2 ( v) r r z r z r 2 R r= from which (2.3) follows. R (y vaz) r (y vaz) r φ(y vaz)dy+ zr D 2 ii i }dv, φ(z) dz r= z 2 r r= z 2 r {(y vaz) 2 r + }φ(y avz)dy D ii i φ(z) dz R dv

9 Bhaacharya ad Holmes/Sei s Mehod for Mulivariae CLT 7 3. Esimaio of T h(s ) By (.6), (3.) T h(s ) = L(L T h)(s ) = Lψ (S ) = ψ (S ) S ψ (S ) Cosider he Taylor expasios ψ (S ) D ii ψ (S ) = i= D ii ψ (S X ) + i= i,i = X (i ) D iii ψ (S X + vx )dv, (3.2) S ψ (S ) = + = X ψ (S ) = = = i,i,i = i= = i= X (i) D i ψ (S X ) + X (i) X (i ) X (i ) X (i) D i ψ (S ) i,i = X (i) X (i ) D ii ψ (S X )+ ( v)d ii i ψ (S X + vx )dv Recallig ha X = Y, EY =, EX (i) X (i ) = (i) EY Y (i ) = δ ii ad X ad S X are idepede, [ ] E ψ (S ) = E D ii ψ (S X ) + E Y (i) (3.3) D iii ψ (S X + vx )dv, i= i,i = (3.4) [ ] ES ψ (S ) = E D ii ψ (S X ) Hece (3.5) ET h(s ) = E i,i = Oe may he wrie i= Y (i) + i,i,i = D iii ψ (S X + vx )dv [ E Y (i) Y (i ) Y (i ) i,i,i = ] ( v)d ii i ψ (S X + vx )dv. Y (i) Y (i ) Y (i ) ( v)d ii i ψ (S X + vx )dv. (3.6) ET h(s ) = E[E( Y )] where is he quaiy wihi square braces i (3.5), i.e., (3.7) E[T h(s ) Y ] = i,i = i,i,i = Y (i ) E [D iii ψ (S X + vx ) Y ] dv Y (i) Y (i ) Y (i ) ( v)e[d ii i ψ (S X + vx ) Y ]dv

10 The firs erm o he righ side i (3.7) equals (3.8) i,i = = ( Y (i ) i,i = Y (i ) Bhaacharya ad Holmes/Sei s Mehod for Mulivariae CLT 8 e s ( 2s )3 ( { e s ( 2s )3 oig ha he disribuio of S X = Therefore, (3.8) is equal o { [ E h(e s (S X ) + e s vx + ]} ) 2s z) ( D iii φ(z))dz Y dv ds R R [ s h(e R x + e s vx + ] } ) 2s z)dq ( ) (x) D iii φ(z)dz dv ds, ( Y2+Y3+ +Y ) is ha of V, where V has disribuio Q ( ). (3.9) i,i = Y (i ) e s { ( 2s )3 R [ R s h(e x + e s vx + ] 2s z)(d(q ( ) (x) Φ(x)) + dφ(x)) } D iii φ(z)dz dvds Sice he class of fucios h = C, where C rages over all Borel covex subses of R, is ivaria uder raslaio, ad bc is covex if C is covex (bc = {bx : x C}, b > ), s h(e R x + e s vx + 2s (3.) z)(d(q ( ) (x) Φ(x))) δ. Similarly, he secod erm o he righ i (3.7) equals (3.) i,i,i = Y (i) Y (i ) Y (i ) ( d(q ( ) (x) Φ(x)) + dφ(x) e s { ( 2s )3 } )].D ii i φ(z)dz dv R [ R s h(e x + e s vx + 2s z) Agai, he ier iegral i (3.) wih regard o Q ( ) Φ is esimaed by (3.). Therefore, usig (2.3) for he remaiig iegraio wih regard o Φ i (3.8), (3.). ET h(s ) i,i = ( E Y (i ) ( e s [ 2s )3 ] δ D iii φ(z) dz + c e R 2s ( 2s ) ) ds (3.2) + i,i,i = E Y (i) Y (i ) Y (i ) ( e s [ ( ( 2s )3 δ ) ( v)dv ] ) D ii i φ(z) dz+ c e 2s ( 2s ) ds. R Nex, he firs wo erms o he righ i (3.2) may be esimaed by usig D ii i φ(z) dz = E (Z (i) )2 E Z (i ) i i, i = iori, R E (Z (i) )3 Z (i) 6 i = i = i, ) D ii i φ(z) dz = E Z(i) Z(i Z (i ) (3.3) if i, i, i are all disic. R

11 Bhaacharya ad Holmes/Sei s Mehod for Mulivariae CLT 9 Fially, oe ha (3.4) e s 2s 2s (s > ), so ha (3.5) e s 2s ds = c <, Hece, usig (3.2)-(3.5), ogeher wih he esimaes oe has E i,i,i = Y (i) Y (i ) Y (i ) 3 2 ρ3, e s ( 2s )3 ds (2) 2. E i,i = Y (i ) 2 ρ3, (3.6) ET h(s ) c 3/2 ρ 3 ( δ ) + c 2 5/2 ρ 3 /2 4. The smoohig iequaliy ad he Esimaio of δ Le H = { C, C C}, where C is he class of all Borel covex subses of R. As before, h = h hdφ. We also wrie G b as he disribuio of bw, if W has disribuio G(b > ). Recall ha (see.5) T h(x) = E h(e x+ 2 Z), where Z has he sadard Normal disribuio Φ = N(, I ), which we ae o be idepede of S. The ET h(s ) = E h(e S + 2 Z) = h(e x + 2 z)dq () (x)φ(z)dz R R (4.) = hd((q() ) e Φ R e 2) = hd((q() ) e Φ e ) Φ R e 2 The iroducio of he exra erm Φ e Φ e 2 = Φ does o affec he iegraio i he las sep sice R hdφ =. Sice he las iegraio is wih respec o he differece bewee wo probabiliy measures, is value is uchaged if we replace h by h. Hece (4.2) ET h(s ) = hd[q () ) e Φ e ] Φ. R e 2 Also he class C is ivaria uder muliplicaio C bc where b > is give. Therefore, δ = sup E h(s ) = sup hd(q () Φ) = sup hd [ (4.3) (Q () ) e Φ e ]. h H h H h H Thus (4.2) is a perurbaio (or, smoohig) of he iegral i (4.3) by covoluio wih Φ e 2. If ɛ > is a cosa such ha (4.4) Φ e 2 ({ z < ɛ}) = 7 8, he he smoohig iequaliy below applies, wih µ = (Q () ) e, ν = Φ e, K = Φ e 2, f = h = C, α = 7/8, ad ɛ as i (4.4).

12 Bhaacharya ad Holmes/Sei s Mehod for Mulivariae CLT Smoohig Iequaliy Le µ, ν, K be probabiliy measures o R, K({x : x < ɛ}) = α > 2.The for every bouded measurable f oe has (4.5) fd(µ ν) (2α [ ) γ (f : ɛ) + ωf (2ɛ : ν) ] R where, h ɛ f ɛ + (x) = sup{f(y) : y x < ɛ}, fɛ (x) = if{f(y) : y x < ɛ}, { ad γ(f : ɛ) = max f + ɛ d(µ ν), f ɛ d(µ ν) }, R R γ (f : ɛ) = sup γ(f y : ɛ), f y (x) f(x + y), y R ω f (x : ɛ) = sup{ f(y) f(x) : y x < ɛ}, ω f (ɛ : v) = ω f (x : ɛ)dν(x), ω (f : ɛ) = sup y R ω fy (ɛ : ν). For a proof of he iequaliy (4.5) see Bhaacharya ad Rao [5], Lemma.4. Wih h = C oe ges h + ɛ = C ɛ = C ɛ, where C ɛ = {x : dis(x, C) < ɛ},c ɛ = {x : ope ball of radius ɛ ad ceer x is coaied i C} are boh covex, so ha { γ(h : ɛ) max C ɛd [ (Q () ) e Φ e ] Φ e 2, Sice C is ivaria uder raslaio oe he obais C ɛd [ } (Q () ) e Φ e ] Φ sup ET e h(s ). 2 h H (4.6) γ (h : ɛ) sup ET h(s ). h H Also, leig Z be sadard Normal N(, ), ω h(2ɛ : Φ e ) = P (e Z ( C) 2ɛ ) (4.7) = P (Z e ( C) 2ɛ ) c 3 2ɛe. where c 3 > is a cosa (see Bhaacharya ad Rao [5], Theorem 3.). From (4.4) oe ges P ( ) 2 Z < ɛ = 7 ( ) 8, P ɛ Z < = so ha ɛ/ 2 = a, where a saisfies P ( Z < a ) = 7 8. I is simple o chec ha a = O( ), as, ad (4.8) a c 4, ɛ = a 2 c 4 2 c 4 2 Usig his esimae of ɛ i (4.9), oe obais (4.9) ω h(2ɛ : Φ e ) c 5 e

13 Bhaacharya ad Holmes/Sei s Mehod for Mulivariae CLT The smoohig iequaliy ow yields (use (4.3),(4.6), (4.9) i (4.5)) δ 4 [ sup ET h(s ) +c 5 ] (4.) e 3 h H Now use (3.4) i (4.) o ge (4.) δ (c 6 3/2 ρ 3 ) δ + c 7 5/2 ρ 3 /2 + c 8 e. By comparig he firs ad hird erms o he righ, a opimal order of is obaied as { } δ ρ 3 = mi,. I follows ha (4.2) δ (c 9 5/4 ρ /2 3 ) δ/2 4 + c 7 3/2 ρ 3 /2. Cosider ow he iducio hypohesis : The iequaliy (4.3) δ c5/2 ρ 3 holds for some ad a absolue cosa c specified below. Noe ha (4.3) clearly holds for c 2 5 ρ 2 3 Sice c 2 5 ρ 2 3 > 8, suppose he (4.3) holds for some = 8. The by (4.2) we ca ae 3 : uder he iducio hypohesis, ad (4.2), δ + c 5 9 c ρ3 (( ( + )) 4 c 5 c 2 ρ 3 ( + ) 2 + c 7 3/2 ρ 3 ( + ) 2 + c 7 5/2 ρ ( + ) 2 (c = c 9 +, , for 2). (4.4) Now, choose c o be he greaer of ad he posiive soluio of c = c c + c7, o chec ha (4.3) holds for = +. Hece (4.3) holds for all. We have proved he followig resul. Theorem There exiss a absolue cosa c > such ha (4.5) δ c 5 2 ρ 3 5. The No-Ideically Disribued Case For he geeral case cosidered i [9], X s ( ) are idepede wih zero meas ad = CovX = I. Assume (5.) β 3 E X 3 <

14 Bhaacharya ad Holmes/Sei s Mehod for Mulivariae CLT 2 Le { X : } be a idepede copy of {X : }. The, wriig S = = X, as before, E D ii ψ (S ) = E i= (5.2) = i,i = = E = i,i = D ii ψ (S ) X (i) X (i ) D ii ψ (S X ) X (i) X (i ) + = i,i,i = X (i) X (i ) X (i ) D ii i ψ (S X + vx )dv, ad (5.3) E [S ψ (S )] = E X ψ (S ) = = E X ψ (S X )+ = i,i = Subsracig (5.3) from (5.2) ad oig ha X (i) X (i ) D ii ψ (S X )+ i,i,i = X (i) X (i ) X (i ) ( v)d ii i ψ (S X + vx )dv EX ψ (S X ) =, oe obais (5.4) ET h(s ) = E = i,i,i = = i,i,i = X (i) X (i ) X (i ) X (i) X (i ) X (i ) D ii i ψ (S X + vx )dv ] ( v)d ii i ψ (S X + vx )dv The esimaio of he codiioal expecaio of he iegrals i (5.4), give X, proceeds as i Secio 3 (wih X i place of X ). The oly sigifica chage is i he ormalizaio i he argume of h (see (3.8) - (3.)) where, wriig N as he posiive square roo of he iverse of Cov(S X ), (5.5) E [ h(e s (S X ) + e s vx + ] 2s z X = E [ h(e s N (N (S X )) + e s vx + ] 2s z X = h(e s N x + e s vx + 2s z)dq ( ), (x) = R where Q () deoes he disribuio of S h(e s N R (x + N e s 2s z) + e s vx )dq ( ), (x), = X, ad Q ( ), ha of N (S X ), which has mea zero, covariace I. As i Secio 3, he las iegraio is divided io wo pars: d(q ( ), Φ)(x) + dφ(x). Sice he class of Borel covex ses is ivaria uder o-sigular affie liear rasformaios, he iegral wih regards o Q ( ), Φ is bouded by δ. For he iegral wih regards o Φ, we chage variables x y = x + A z, where A = e s 2s N. The esimaio of he iegral ow proceeds as i (2.3) (2.8), wih scalar a replaced by he

15 Bhaacharya ad Holmes/Sei s Mehod for Mulivariae CLT 3 marix A. The effec of his is simply o chage he sum a 2 r,r z rz r D rr φ(y vaz) i (2.4) o (A z) r (A z) r D rr φ(y va z) r,r = Arguig as i (2.3) (2.8) oe arrives a he upper boud for (5.5) give by usig c A 2 = c e 2s ( 2s ) N 2 c e 2s ( 2s )( β ), (5.6) N 2 = (I CovX ) 2 2 = I CovX, I CovX = sup u (I CovX )u = sup ( E(u.X ) 2 ) u = u = E X 2 (E X 3 ) 2 3 β ad assumig (5.7) β 3 < Proceedig as i Secio 4 oe arrives a he boud: (5.8) δ c 5 2 β3. If oe aes he absolue cosa c >, he he β 3 may be assumed o be smaller or equal o c 5 2, ad ( β ) ( ) = c. The iducio argume is similar. c 2 3 Remar: If oe defies (5.9) γ 3 E( X (i) ) 3, = i= he = i,i,i = E X (i) X (i ) X (i ) = γ 3, Sice γ 3 ow replaces 3 2 β 3 i he compuaios, i follows ha (5.) δ cγ 3 Sice, γ β 3, (5.) provides a beer boud ha (5.8) or (4.3).

16 Bhaacharya ad Holmes/Sei s Mehod for Mulivariae CLT 4 Refereces [] Ball, K. (993) The reverse isoperimeric problem of Gaussia measure, Discree Compu. Geom.,, pp [2] Barbour, A. D. (988) Sei s Mehod ad Poisso Process Covergece, Joural of Applied Probabiliy, Vol. 25, pp [3] Beus, V. (23) O he depedece of he Berry-Essee boud o dimesio. J. Sa. Pla. If., 3, pp [4] Bhaacharya, R. N.(982) O he fucioal ceral limi heorem ad he law of he ieraed logarihm for Marov processes, Zei. Wahr. Ver. Geb. 6, [5] Bhaacharya, R. N. ad Raga Rao, R. (976) Normal Approximaio ad Asympoic Expasios, Wiley, New Yor. [6] Bhaacharya, R.N. ad Waymire, E.C. (29) Sochasic Processes wih Applicaios. SIAM, Philadelphia. [7] Diacois, P. ad Holmes, S. (24) Sei s Mehod: Exposiory Lecures ad Applicaios, IMS Lecure Noes, 46, pp [8] Holmes, S. (24) Sei s mehod for birh ad deah chais. i Sei s Mehod: Exposiory Lecures ad Applicaios, IMS Lecure Noes, 46, pp [9] Göze, F. (99) O he rae of covergece i he mulivariae CLT. A. Probab. 9, [] Raic, M (24) O he rae of covergece of he mulivariae CLT, Persoal Commuicaio. [] Rio, Y. ad Roar, V. (997). O couplig cosrucios ad raes i he cl for depede summads wih applicaios o he ai-voer model ad weighed U-saisics. A. Applied Probabiliy 7, 8 5. [2] C. Sei (986). Approximae Compuaio of Expecaios. IMS, Hayward, Califoria.