A Peccati-Tudor type theorem for Rademacher chaoses

Transcription

1 A Peccat-Tudo type theoem fo Rademache chaoses Guangqu Zheng 6 Avenue de la fonte, Mason du Nombe, Unvesté du Luxemboug, Esch-su-Alzette, L4364, Luxemboug Abstact In ths atcle, we pove that n the Rademache settng, a andom vecto wth chaotc components s close n dstbuton to a cented Gaussan vecto, f both the maxmal nfluence of the assocated kenel and the fouth cumulant of each component s small. In patcula, we ecove the unvaate case ecently establshed n Döble and Kokowsk 7). Ou man stategy conssts n a novel adapton of the exchangeable pas couplngs ntated n Noudn and Zheng 7), as well as ts combnaton wth estmates va chaos decomposton. Intoducton. Motvaton Nualat and Peccat s fouth moment theoem states that a nomalsed sequence of fxed-ode multple Wene-Itô ntegals assocated to a Bownan moton conveges n law to the standad Gaussan f and only f the coespondng fouth moment conveges to 3. It was poved n [] usng the Dambs-Dubns- Schwatz andom-tme change technque. Soon afte the appeaance of [], seveal extensons have been made, among whch the pape [3] by Peccat and Tudo povded a sgnfcant multvaate extenson usng the same techque. Roughly speakng, a sequence of chaotc andom vectos on the Wene space conveges n dstbuton to a cented Gaussan vecto wth matched covaance matx f and only f the asymptotc nomalty holds tue fo each component. Note that the necessay condton bols down to the convegence of the fouth moments due to the fouth moment theoem of Nualat and Peccat. In 9, Noudn and Peccat [5] combned the Mallavn calculus and Sten s method of nomal appoxmaton so as to lteally ceate a new feld of eseach, known as the Mallavn-Sten appoach. One of ts many hghlghts s the obtenton of the quanttatve) fouth moment theoem n the total-vaaton dstance. Hee s the bound quoted fom the monogaph [6]: gven a nomalsed q-th Wene-Itô ntegal F assocated to a Bownan moton, one has d TV F, Z) := sup P F A ) P Z A ) q E[F 3 q 4 ] 3 ), A BR) whee Z s a standad Gaussan andom vaable and BR) denotes the Boel σ-algeba on R. As an mmedate consequence, the fouth moment theoem of Nualat and Peccat follows. The success of the Mallavn-Sten appoach stems fom the ntegaton by pats on both sdes, namely, the Sten s lemma wthn the Sten s method and the dualty elaton between Mallavn devatve and Mathematcs Subject Classfcaton. Pmay: 6F5, 6B; Seconday: 47N3. Key wods and phases. Fouth moment theoem; Rademache chaos; Sten s method; exchangeable pas; spectal decomposton; maxmal nfluence. Emal: guangqu.zheng@un.lu

2 G. Zheng Skoohod dvegence on a Gaussan space, see the monogaph [6] fo a compehensve teatment. The only ngedents equed fom the Sten s method ae the Sten s lemma, Sten s equaton and the egulaty popetes of the Sten s soluton, whle exchangeable pas, anothe fundamental tool and notable conestone of Sten s method, had not been touched untl the ecent nvestgaton [] made by Noudn and Zheng. They constucted nfntely many exchangeable pas of Bownan motons and combned them wth E. Meckes abstact esults [, 3] on exchangeable pas to ecove the quanttatve fouth moment theoem on a Gaussan space n any dmenson. Such an elementay stategy was soon adapted by Döble, Vdotto and Zheng n [7] fo the nvestgaton on the Posson space, and they wee able to obtan the quanttatve fouth moment theoem n any dmenson. In fact, the unvaate fouth moment theoem on the Posson space was establshed eale n [6] unde some ntegablty assumptons nvolvng the dffeence opeato, whch ae patally due to the nheent dsceteness of the Posson space. Remakably, the authos of [7] wee able to obtan the exact fouth moment theoem unde the weakest possble assumpton of fnte fouth moment. Ths llustates the powe of the elementay exchangeable pas appoach. In ths wok, unde sutable assumptons, we establsh a Peccat-Tudo type theoem n the Rademache settng usng the elementay exchangeable pas appoach.. Man esult We fst fx a ch pobablty space Ω, F, P ), on whch ou andom objects ae defned. Let E be the assocated expectaton opeato. We wte N := {,,...} and denote by X a sequence of ndependent Rademache andom vaables X k, k N) such that P X k = ) = p k = q k = P X k = ), ). We call t the symmetc case, wheneve p k = / fo each k N; othewse, we call t the geneal case. We wte Y = Y k, k N ) fo the nomalsed veson of X, that s,.) Y k = X k p k + q k p k q k, k N. We wte H = l N), equpped wth usual l -nom and fo p N, H p means the p-th tenso poduct of H and H p ts symmetc subspace. We denote H p := { f H p : f c p = } wth p = {,..., p ) N p : k j fo dffeent k, j }. Clealy, H = H = R and H = H. Let f H d wth d N and Ξ = ξ k, k N) be a genec sequence of ndependent nomalsed andom vaables. We defne the followng homogeneous sum wth ode d, based on the kenel f, by settng,.) Q d f ; Ξ) := f,..., d )ξ ξ d,..., d N and n patcula, Q d f ; Y) s called the dscete) multple ntegal of f. We wte C d = { Q d f ; Y) : f } and call t the d-th Rademache chaos, and as a conventon, we put C = R. In case of no ambguty, H d we wll smple wte Q d f ) fo Q d f ; Y). Let us ntoduce an mpotant noton befoe we state ou man esult: fo a gven kenel f H d, we denote by M f ) the maxmal nfluence of f, namely.3) M f ) := sup k N,..., d N f,..., d, k) fo d and M f ) := sup f k) fo d =. k N Ths noton s adapted fom the boolean analyss see e.g. []), n whch the class of low-nfluence functons s often what s nteestng o necessay n pactce. It s also closely elated to the nvaance

3 Peccat-Tudo theoem fo Rademache chaoses 3 pncple establshed n [4] and the unvesalty phenomenon of Gaussan Wene chaos [8]. See also Secton 4 fo moe detals. In ths wok, we ae manly concened wth andom vaables n a Rademache chaos and andom vectos wth components n Rademache chaoses. Moe pecsely, we establsh the followng esult. Theoem.. Fx nteges d and q... q d, and consde the sequence of andom vectos F n) = F n),..., Fn) )T := Q q f,n ),..., Q qd f d,n ) ) T d wth kenels f j,n n H q j fo each n N, j {,..., d}. Assume that the covaance matx Σ n of F n) conveges n Hlbet-Schmdt nom to a nonnegatve defnte symmetc matx Σ = Σ, j,, j d ), as n +. Suppose that the followng condton holds: lm n + d M f j,n ) =. j= If fo each j {,..., d}, E [ F n)) 4 ] j conveges to 3Σ j, j, as n +, then F n) conveges n dstbuton to Z N, Σ), as n +. The above theoem s analogous to the Peccat-Tudo theoem on a Gaussan space [3], so we call t a Peccat-Tudo type theoem, whch explans ou ttle. One of the man tools we need fo the poof s the followng ngedent fom Sten s method of exchangeable pas. As one wll see easly, we can obtan a quanttatve veson of Theoem., whch wll be an analogue to [7, Theoem.7] and left fo nteested eades. Recall fst that two andom vaables W and W, defned on a common pobablty space, ae sad to fom an exchangeable pa, f W, W ) has the same dstbuton as W, W). Poposton. Poposton 3.5 n [7]). Fo each t >, let F, F t ) be an exchangeable pa of cented d-dmensonal andom vectos defned on a common pobablty space. Let G be a σ-algeba that contans σ{f}. Assume that Λ R d d s an nvetble detemnstc matx and Σ s a symmetc, non-negatve defnte detemnstc matx such that a) lm t t E[ F t F G ] = ΛF n L Ω), b) lm t E[ F t F)F t F) T G ] = ΛΣ + S n L Ω, H.S. ) fo some matx S = S F), and wth H.S. the Hlbet-Schmdt nom, t c) fo each {,..., d}, thee exsts some eal numbe ρ F) such that lm t E[ F,t F ) 4] = ρ F), whee F,t esp. F ) stands fo the -th coodnate of F t esp. F). Then, fo g C 3 R d ) such that gf), gz) L P), we have, wth Z N, Σ), E[gF)] E[gZ)] Λ op d M g) d E S 4, j + dm3 g) Λ d d op Λ, Σ, + E[S, ] ρ F), 8 whee M k g) := sup x R d, j= D k gx) op wth op the opeato nom. = t =

4 4 G. Zheng The est of ths pape s ogansed as follows: Secton.3 s devoted to a bef ovevew of elated esults and we sketch ou stategy of povng Theoem. n Secton.4; n Secton, we povde pelmnay knowledge on Rademache chaos and a cucal exchangeable pas couplng. The poof of ou man esult wll be gven n Secton 3 and some dscusson about unvesalty aound Rademache chaos wll be pesented n Secton 4..3 A bef ovevew of lteatue Soon afte the appeaance of [5], Noudn, Peccat and Renet combned Sten s method and a dscete veson of Mallavn calculus to study the Gaussan appoxmaton of Rademache functonals n the symmetc case. Ths analyss s known as the dscete Mallavn-Sten appoach. It has been genealsed by the authos of [9, ] not only n the multvaate settng but also n the geneal case whee functonals nvolvng non-symmetc, non-homogeneous Rademache andom vaables wee nvestgated. Recently, Döble and Kokowsk [5] gave the followng fouth-moment-nfluence bound and ponted out that t s optmal n the sense that thee ae examples, n whch the fouth moment condton alone would not guaantee the asymptotc nomalty. Theoem. Theoem. n [5]). Fx p N and f H p satsfyng p! f =. Let Z be a standad H p Gaussan and F = Q p f ; Y) L 4 P), then we have the followng bound n Wassesten dstance: ) d W F, Z := sup E [ hf) hz) ] E[F C 4 ] 3 + C M f ), h whee C, C ae two numecal constants. Ths esult echoes the emakable de Jong s cental lmt theoem [4]. Besdes the afoementoned efeences, Kokowsk [8] deved a multplcaton fomula that genealses the one n [7], and applyng as well the Chen-Sten s method, he studed the Posson appoxmaton of Rademache functonals. Independently, Pvault and Tos [6] also deved a multplcaton fomula and moeove, they obtaned a genealsaton of the appoxmate chan ule fom [7], and appled them to study Gaussan and Posson appoxmaton of Rademache functonals n the geneal case. Concenng the nomal appoxmaton n [7] o [6], the authos wee only able to obtan the bounds n some smooth-veson dstance, due to egulaty nvolvng n the chan ules and Sten s soluton. In a follow-up wok, Zheng [8] obtaned a neate chan ule that eques mnmal egulaty see [8, Remak.3]), fom whch he obtaned the bound n Wassesten dstance as well as an almost sue cental lmt theoem fo Rademache chaos. It s wothy pontng out that wthout usng any chan ule, the authos of [9, ] used caefully a epesentaton of the dscete Mallavn gadent and the fundamental theoem of calculus to deduce the Bey-Esseen bound fo nomal appoxmaton. Usng smla deas, Döble and Kokowsk [5] also povded the Bey-Esseen bound fo the fouth-moment-nfluence theoem, whch s of the same ode as the above Wassesten bound..4 Stategy of povng Theoem. Sten s method of exchangeable pas was fst systematcally pesented n Chales Sten s 986 monogaph [7], whch was subsequently developed and amfed by many authos. Concenng ou wok, we menton n patcula E. Meckes dssetaton [], n whch she developed an nfntesmal veson of ths method to obtan total-vaaton bound n nomal appoxmaton. Ths nfntesmal veson of Sten s method of exchangeable pas was late genealsed n [3, 3] fo the multvaate nomal appoxmaton. As announced, Poposton. s one of ou man tools, and t can be seen as a genealsaton of [3]. To use t, we need to constuct a sutable famly of andom vectos F t, t such that F t, F)

5 Peccat-Tudo theoem fo Rademache chaoses 5 s exchangeable fo each t and satsfes seveal asymptotc egesson condtons. In fact, we wll fst constuct a famly of Rademache sequences X t such that X t, X ) s an exchangeable pa of {±} N -valued andom vaables fo each t. Moe pecsely, let X be an ndependent copy of X and Θ = θ k, k N) be a sequence of..d. standad exponental andom vaables such that X, X and Θ ae ndependent. Fo each t [, + ), we defne X t k := X k θk t) + X k θ k <t). It has been ponted out n [] that X t has the same dstbuton as X, see also Remak 3.4 n [7] fo the symmetc case. Howeve, both of these two atcles dd not explctly state the exchangeablty of X t and X, whch wll be poved n Lemma.. Assumng ths and wtng F = fx) fo some epesentatve f : {±} N R d, we can set F t = fx t ). It s easy to see that the exchangeablty can be passed to F, F t ) now. If F = Q p f ; Y),..., Q pd f d ; Y) ), then we can wte F t = Q p f ; Y t ),..., Q pd f d ; Y t ) ) wth Y t the nomalsed veson of X t n the sense of.). Moeove, ths exchangeable pas couplng fts well wth the Mehle s fomula, whch gves a nce epesentaton of the dscete Onsten-Uhlenbeck semgoup P t, t ) : gven F L Ω, σ{x}, P ), we can fst wte F = fx) fo some f : {±} N R, then the Mehle fomula [, Poposton 3.]) states that.4) P t F = E [ f X t) σ{x} ]. Fo ξ C p, as we wll see n Secton, P t ξ = e pt ξ, then the asymptotc lnea egesson a) n Poposton. follows easly, and wth slghtly moe effot, the hghe ode egessons can also be obtaned, see Poposton.. Anothe mpotant ngedent n ou poof s Ledoux spectal pont-of-vew fo fouth moment theoem [], whch was late efned e.g. n [, ]. Such a spectal vewpont helps one get d of some computatonal deadlock that s usually caused by the complcated multplcaton fomula. In patcula, ou poof s motvated by some aguments n []. As a bypoduct of ou stategy, we wll povde a shot poof of Theoem. n the begnnng of Secton 3. Some estmate fom ths poof wll also be helpful fo ou multvaate case. Acknowledgement. Pat of ths wok was done dung a vst at Natonal Unvesty of Sngapoe. I thank vey much Pofesso Lous H. Y. Chen at NUS fo hs vey geneous suppot and knd hosptalty. The gattude also goes to Pofesso Govann Peccat fo shang hs altenatve poof of Lemma.4 n [6], whch motved ou poof of Lemma.. Pelmnaes Denote by σ{x} the σ-algeba geneated by the sequence X, and note that σ{x} = σ{y}. The Wene- Itô-Wash chaos decomposton assets that any andom vaable F L Ω, σ{x}, P ) admts a unque epesentaton.) F = E[F] + Q p f p ) p wth f p H p fo each p N, whee the above sees conveges n L P). We denote by J k ) the pojecton onto the k-th Rademache chaos C k : fo F gven n.), J p F) = Q p f p ) fo each p N, and J F) = E[F]. It s not dffcult to check that fo f H p and g H q, t holds that E [ Q p f )Q q g) ] = {p=q} p! f, g H p.

6 6 G. Zheng Ths s known as the othogonalty popety of the multple ntegals. One can efe to N. Pvault s suvey [5] fo moe detals and elevant dscete Mallavn calculus. The authos of [7] establshed a multplcaton fomula fo dscete multple ntegal n the symmetc case: gven f H p and g H q, one has.) p q p Q p f )Q q g) =! = ) q ) Q p+q f g p+q ), whee the -contacton f g of f and g s defned by f g) ),..., p, j,..., j q := f ) ),..., p, k,..., k g j,..., j q, k,..., k k,...,k N and f g s the canoncal symmetsaton of f g,.e. fo any h H p, h s gven by h,..., p ) = h ) σ),..., σp), p! σ S p wth S p the pemutaton goup ove {,..., p}. We follow the conventon that c = c fo each c R. Note t s easy to deduce fom the Cauchy-Schwaz nequalty that h H p h H p fo each h H p, then applyng the above othogonalty popety and mathematcal nducton gves us a weak fom of the hypecontactvty popety n the symmetc case, namely, E [ F ] < + fo any F C p, p, N. Howeve, n the geneal case, one can not even guaantee the exstence of fnte fouth moment of a genec multple ntegal. Such a phenomenon, due to the asymmety, s also evealed n the coespondng multplcaton fomulae, see Poposton. n [8] and Poposton 5. n [6]. As aleady ponted out n [5], gven F C p L 4 P), one can not dectly deduce fom these multplcaton fomulae that F admts a fnte chaotc decomposton. Adaptng the nducton aguments fom the poof of [6, Lemma.4], Döble and Kokowsk gave the followng postve esult. Lemma. Lemma.3 n [5]). Let F = Q p f ) L 4 P) and G = Q q g) L 4 P) fo some f H p and g H q. Then FG L P) admts a fnte chaos decomposton of the fom FG = E[FG] + p+q In patcula, f Q h) belongs to L 4 P) fo some h H, then k= J k FG) + Q p+q f g p+q ). Q h) = h H + Q w) + Q h h ) wth wk) = hk) q k p k ) pk q k, k N. As ths lemma s cucal fo ou wok and fo the sake of completeness, we povde n Secton 3.3 anothe and dect poof suggested by Govann Peccat.). Onsten-Uhlenbeck Stuctue and caé du champs opeato Denote by doml) the set of those F n.) vefyng p E [ Q p f p ) ] = p= p p! f p H < +. p p=

7 Peccat-Tudo theoem fo Rademache chaoses 7 Fo such a F doml), we defne LF = p pq p f p ). In patcula, f F C p, LF = pf. In othe wods, L has pue spectum N {} and each egenvalue p {} N coesponds to the egenspace C p. And we call L the Onsten-Uhlenbeck opeato, equpped wth ts doman doml). Fo F, G doml) such that FG doml), we defne the caé du champs opeato ΓF, G) by settng ΓF, G) := LFG) FLG GLF ). In patcula, fo F, G as n Lemma., one has FG doml) and.3) ΓF, G) = [ ] p+q p + q) + L J k FG) = p + q p+q E[FG] + and as a consequence of the othogonalty popety, one deduces that k= k= p + q k J k FG),.4) Va ΓF, G) ) = p+q k= p + q k) 4 Va J k FG) ) p+q max{p, q } Va J k FG) ), k= whch s all we need about the caé du champs. Fo each t [, + ) and F as n.), we defne P t F := E[F] + e pt Q p f p ). p= P t, t ) s called the Onsten-Uhlenbeck semgoup, whch can be epesented altenatvely by the Mehle fomula.4). To vefy.4), one can fst consde F = Q p f p ) n a Rademache chaos wth f p H p havng fnte suppot and then use the standad appoxmaton agument. Note that fo F doml), t s not dffcult to check t P t F F) conveges n L P) to LF, as t.. Exchangeable pas of Rademache sequences Lemma.. Let X t and X be gven as befoe, then X, X t) has the same dstbuton as X t, X ). In patcula, fo any f j H p j wth p j N, j =,..., d, Qp f ; Y),..., Q pd f d ; Y) ) and Q p f ; Y t ),..., Q pd f d ; Y t ) ) fom an exchangeable pa, whee Y t stands fo the nomalsed veson of X t n the sense of.). Poof. Note fst that X t s a sequence of ndependent Rademache andom vaables fo each t [, + ). Fo each k N, t s easy to check that P X t k =, X k = ) = P X t k =, X k = ) = e t )p k q k. Ths gves us the exchangeablty of X k, Xk t ) fo each k N. Let a = a, N), b = b, N) {±} N, then usng the ndependence wthn those two sequences X, X t, we obtan P X = a, X t = b ) = P X k = a k, Xk t = b ) k = P X k = b k, Xk t = a ) k by exchangeablty of X k, Xk t k N k N = P X = b, X t = a ).

8 8 G. Zheng Ths poves the exchangeablty of X, X t. The est follows fom a standad appoxmaton agument: t s clea that afte tuncaton, wth [N] := {,..., N}) Qp f [N] p ; Y),..., Q pd f d [N] p d ; Y) ) and Q p f [N] p ; Y t ),..., Q pd f d [N] p d ; Y t ) ) fom an exchangeable pa; lettng N + and keepng n mnd that the exchangeablty s peseved n lmt, we get the desed esult. The followng esult bngs moe connectons between ou exchangeable pas and Onsten-Uhlenbeck opeato. Poposton.. Let F = Q p f ; Y) L 4 P) fo some f H p and defne F t = Q p f ; Y t ). Then, F, F t ) s an exchangeable pa fo each t R +. Moeove, a) lm t t E[ F t F σ{x} ] = LF = pf n L 4 P). b) If G = Q q g; Y) L 4 P) and G t = Q q g; Y t ) fo some g H q, then we have lm t t E[ F t F)G t G) σ{x} ] = ΓF, G), wth the convegence n L P). c) lm t t E[ F t F) 4] = 4p E[F 4 ] + E [ F ΓF, F) ]. Poof. By the Mehle fomula.4), we have t E[ F t F σ{x} ] = P tf) F t = e pt F, t conveges n L 4 P) to pf = LF, as t. As a consequence of Lemma., FG has a fnte chaos expanson of the fom FG = E[FG] + p+q k= Q k hk ; Y ) fo some h k H k. Theefoe, F tg t = E[FG] + p+q k= Q k hk ; Y t), mplyng t E[ F t G t FG σ{x} ] p+q = t E[ Q k hk ; Y t) Q k hk ; Y ) σ{x} ] k= conveges n L P) to p+q k= k J kfg) = LFG), as t. Hence, we nfe that n L P) and as t, t E[ F t F)G t G) σ{x} ] = t E[ F t G t FG σ{x} ] F E[G t G σ{x}] G E[F t F σ{x}] t t LFG) FLG GLF = ΓF, G). Snce the pa F, F t ) s exchangeable, we can wte E [ F t F) 4] = E [ Ft 4 + F 4 4Ft 3 F 4F 3 F t + 6Ft F ] = E[F 4 ] 8E [ F 3 ] [ F t + 6E F Ft ] by exchangeablty of F, Ft ) ) = 4E [ F 3 F t F) ] + 6E [ F F t F) ] afte eaangement) = 4E [ F 3 E[F t F σ{x}] ] + 6E [ F E[F t F) σ{x}] ]. so c) follows mmedately fom a),b) and the fact that F L 4 P).

9 Peccat-Tudo theoem fo Rademache chaoses 9 3 Poofs We begn wth the followng lemma, whose poof s postponed to Secton 3.3. Lemma 3.. Gven F = Q p f ) wth f H p and G = Q q g) wth g H q, we assume that F, G L4 P). Then we have the followng estmates: 3.) p+q k= Va J k FG) ) E [ F G ] E[FG] VaF)VaG) + p + q)! f g c p+q H p+q, and n patcula, 3.) max p k= Va J k F ) ) p, p! = ) p f f H p E[ F 4] 3E[F ] + p)! f f c p, H p wth 3.3) p q f g c p p+q! H p+q = ) q ) mn { f H p Mg), g H q M f ) }. As a conventon, we put =.) = Befoe we pove ou multvaate lmt theoem, we wll gve a shot poof of the unvaate case n Wassesten dstance, usng ou exchangeable pas couplng. 3. Altenatve poof of Theoem. We need the followng esult, whch s the unvaate analogue of Poposton.. Poposton 3.. Let F and a famly of eal andom vaables F t ) t be defned on a common pobablty law space Ω, F, P) such that F t = F fo evey t. Assume that F L 4 Ω, G, P) fo some σ-algeba G F and that n L P), a) lm t t E[ F t F G ] = λ F fo some λ >, b) lm t t E[ Y t Y) G ] = λ + S )VaF) fo some andom vaable S ; c) and lm t t E[ F t F) 4] = ρf)vaf) fo some ρf). Then, wth Z N, VaF) ), we have d W F, Z) VaF) λ π E[ S ] + λ + E[S ])VaF) ρf). 3λ Fo the poof, one can efe to [7, Poposton 3.3]. One may also want to efe to Theoem 3.5 of [7] fo a dffeent couplng bound.

10 G. Zheng Now gven F = Q p f ; Y ) L 4 P) wth E [ F ] = ), we can get by usng.4) and 3.) that 3.4) Va p ΓF, F) ) p k= Va J k F ) ) E[F 4 ] 3E[F ] + p)! f f c p H p E[F 4 ] 3E[F ] + γ p E[F ]M f ) wth γ p := p)! p! p ) p!. = Also usng the chaos expanson of F and ΓF, F) as well as the othogonalty popety, we have 3E [ F ΓF, F) ] pe[f 4 ] = 3E [ F ΓF, F) p ) ] p E[F 4 ] 3 ) p p = 3E J k F p k ) J k F ) p E[F 4 ] 3 ) p 3p Va J k F ) ) p E[F 4 ] 3 ). k= It follows fom 3.4) that k= k= 3.5) 3E [ F ΓF, F) ] pe[f 4 ] p E[F 4 ] 3 ) + 3pγ p M f ). Now defne F t = Q p f ; Y t ) fo each t [, + ), then by Poposton., F t, F) s an exchangeable pa satsfyng the condtons n Poposton 3. wth G = σ{x}, λ = p, S = ΓF, F) p and ρf) = 4p E[F 4 ] + E [ F ΓF, F) ]. Theefoe, d W F, N) p π E[ ΓF, F) p ] + π Va p ΓF, F) ) + p 3p p 3p /π E[F 4 ] 3 + γ p M f ) + 3 4p E[F 4 ] + E [ F ΓF, F) ] 4p E[F 4 ] + E [ F ΓF, F) ] snce E[ΓF, F)] = p) E[F 4 ] 3 ) + 3γ p M f ) 4) /π + E[F 3 4 ] 3 + 6) /π + γp M f ) 3 Ths poves Theoem. wth C = /π and C = /π + 6) p)! 3 p! p ) p!. Remak 3.. ) Fo F n the fst Rademache chaos, one can dectly pove Theoem. wthout usng the exchangeable pas. Indeed, f F = Q h) L 4 P) fo some h H wth h H = and Z N, ), then by [8, Theoem 3.], = d W F, Z) k= p k q k hk) 4. By Lemma., F ) = + Q w) + Q h h wth wk) = hk) q k p k ) pk q k, k N. Ths mples E [ F 4] = + k= hk) 4 q k p k ) p k q k + h h H h h c H

11 Peccat-Tudo theoem fo Rademache chaoses = 3 + k= hk) 4 q k p k ) p k q k hk) 4 = 3 + k= k= hk) 4 q k + p k p k q k 4 Notcng p k + q k / fo each k N, we have hk) 4 4 hk) 4 + E [ F 4] 3 4Mh) + E [ F 4] 3. p k q k k= k= hk) 4. Hence, d W F, Z) E[F 4 ] 3 + Mh). Moeove, usng the so-called second-ode Poncaé nequalty n [, Theoem 4.], we can have the Bey-Esseen bound d Kol F, Z ) := sup z R P F z ) P Z z ) k= k= hk) p k q 4 E[F 4 ] Mh). k ) Contnung the dscusson n pevous pont and assumng p k = p = q = q k fo each k, we have E [ F 4] 3 = p + q 4 4pq 3.6) hk) 4. pq If p, ) \ { ± }, then we have the exact fouth moment bounds: 3 d W F, Z) pq hk) 4 E[F 4 ] 3 p + q 4pq k= see also Coollay.4 n [5]. 3. Poof of Theoem. k= ) / and d KolF, Z) E[F 4 ) / ] 3 p + q, 4pq Wthout losng any genealty, we assume that Σ n = Σ and each component of F n) belongs to L 4 P). Recall that F n) = F n),..., Fn) d )T := Q q f,n ; Y ),..., Q qd fd,n ; Y ) ) T and we defne F n) t = F n),t,..., Fn) d,t )T wth F n),t := Q q f,n ; Y t) so that by Lemma. and Poposton., ) F, F t := F n), F n) ) t fom an exchangeable pa satsfyng the condtons n Poposton. wth G = σ{x}, Λ = dagq,..., q d ) and S = Γ F n), F n) ) ) j q j Σ, j ρ F n) ) = 4q E [ F n) ) 4] + E [ F n) ) ΓF n), F n) ) ]., j d, Indeed, the condton c) n Poposton. follows fom the elaton c) n Poposton., and fo each, j {,..., d}, we have and It follows that lm t t E[ F n),t F n) ) F n) lm t t E[ F n),t F n) σ{x} ] = q F n) n L 4 P), j,t F n) ) ] j σ{x} = q j Σ, j + [ Γ F n), F n) ) ] j q j Σ, j t E[ F n) t F n) σ{x} ] + ΛF n) d = R d = t E[ F n),t F n) σ{x} ] ) + q F n) n L P).

12 G. Zheng conveges to zeo n L P), as t ; and = t E[ F n) d, j= t t E[ F n),t F n)) F n) t F n) ) F n) F n))t σ{x} ] ΛΣ S H.S. j,t F n) ) ] j σ{x} Γ F n), F n) ) ) j conveges to zeo n L P), as t. Hence we can apply Poposton. and consequently, t suffces to show E [ d ] S H.S. + ρ F n) ) d =, j= Va Γ F n), F n) )) j / d + ρ F n) ), = as n +. In vew of 3.4) and 3.5), t educes to pove lm n + Va Γ F n), F n) )) j = fo < j. We splt ths pat nto two steps. Step. Suppose F, G ae two eal andom vaables gven as n Lemma. wth p q, then we have E [ F G ] = E[FG] + and by.4) and Lemma 3., we get q Va ΓF, G) ) p+q k= p+q k= Va J k FG) ) Cov F, G ) E[FG] + q)! Thus, we can futhe educe ou poblem to show Va J k FG) ) + p + q)! f g p+q H p+q p p! = ) q ) mn { f H p Mg), g H q M f ) }. 3.7) lm n + Cov F n) ), F n) j ) ) E[F n) F n) j ] ) = fo any < j d, whch wll be caed out n the next step. Step. Let F, G be gven as n pevous step, we have E [ F G ] q = E F E[G ] + J k G ) + J q G ) k= = VaF)VaG) + E J k G ) + p=q)e [ J q F )J q G ) ]. F q k= If p < q, then E[FG] = and Cov F, G ) E [ q F 4] Va J k G ) ) k= E [ F 4] E [ G 4] 3E[G ] + γ p E[G ]Mg),

13 Peccat-Tudo theoem fo Rademache chaoses 3 whee the second nequalty follows fom 3.4) and the constant γ p s gven theen. If p = q, then E [ J q F )J q G ) ] = q)! f f, g g = q)! q f f, g g q)! f f, g g H q H q c q H q q q = q! f, g H + q! ) f q g, g f H q q)! f f, g g c, q H q = whee the last equalty follows fom Lemma. n [9]. Consequently, Cov F, G ) E[FG] s equal to q q q E F J k G ) + q! ) 3.8) f g, g f H q q)! f f, g g c. q H q k= = q The fst tem n 3.8) can be ewtten as E J k F )J k G ), whch can be bounded by q k= Va J k F ) ) q k= k= Va J k F ) ) E [ F 4] 3E[F ] + γ q E[F ]M f ) E [ G 4] 3E[G ] + γ q E[G ]Mg) ; and the second tem n 3.8) can be bounded by q ) q q! f g q q = q! 3.9) H f q f, g ) q q g H 3.) 3.) = q = q q! ) f q f H g q g H = q q = q! ) f f H q g g H q = q ) q q! f f q H q = = E [ F 4] 3E[F ] + γ q M f )E[F ] q! q ) g g H q E [ G 4] 3E[G ] + γ q Mg)E[G ], whee 3.9) follows fom the easy fact that f g = f H q q f, g q g H, and we used Cauchy- Schwaz nequalty n 3.), whle 3.) can be deduced fom Lemma 3. and 3.4); fnally, the thd tem n 3.8) can be bounded by f q)! H g g H q c q q f q)!γ H q q E[G ]Mg). To conclude ths case, we obtan Cov F, G ) E[FG] E [ F 4 ] 3E[F ] + γ q M f )E[F ] ) E [ G 4] 3E[G ] + γ q Mg)E[G ] ) + f H q q)!γ q E[G ]Mg). Combnng the above two cases, we get mmedately the elaton 3.7), and hence we fnsh the poof of Theoem..

14 4 G. Zheng 3.3 Poofs of techncal lemmas Poof of Lemma. Let us fst ntoduce some notaton: f F = f X), we wte F k = f X,..., X k, +, X k+,... ) and F k = f X,..., X k,, X k+,... ), we defne the dscete gadent D k F = p k q k F k F k), n patcula, D k Y k =. We can defne the teated gadents D m) k,...,k m = D k D m ) k,...,k m wth D ) ) k = D k. Fo example, D k Q d f ) = dq d f k, ) and D ) k,l Q ) d f ) = dd )Q d f k, l, ) fo d and f H d, see [] fo moe detals. Poof. It s clea that FG L P) has the chaotc expanson FG = E[FG] + Q m h m ), whee fo each m N, the kenel h m H m s gven by h m k,..., k m ) := m! E[ D m) k,...,k m FG) ], due to the Stoock s fomula Poposton. n []). So t suffces to show that 3.) m D p+q) k,...,k p+q FG) = p + q)! f g)k,..., k p+q ) p+q k,..., k p+q ) and D s) k,...,k s FG) = fo any s > p + q. Note that the second pat follows mmedately fom the fst one. Recall the poduct fomula see e.g. [,.4)]) fo the dscete gadent D k : fo F, G L P), D k FG) = D k F)G + FD k G) X k pk q k D k F)D k G) =: D L k FG) + DR k FG) + DM k FG), that s, we decompose D k nto thee opeatons Dk L, DR k and DM k. Theefoe, we can wte fo k <... < k p+q, D p+q) k,...,k p+q FG) = D A p+q k p+q FG) = D A p+q k p+q FG), A,...,A p+q {L,M,R} D A k A,...,A p+q {L,R} whee the last equalty follows fom the fact that fo k l, D l X k F) = X k D l F. Moeove, D A k D A p+q k p+q FG) = unless L appeas exactly p tmes and R appeas exactly q tmes n the wods A,..., A p+q, so that one can futhe ewte D p+q) σ S p+q : σ)<...<σp) σp+)<...<σp+q) k,...,k p+q FG) as p) D k σ),...,k σp) F ) D q) k σp+),...,k σp+q) G ) = D A k σ S p+q f k σ),..., k σp) ) g kσp+),..., k σp+q) ), whee the last equalty follows fom the symmety of f and g, and t gves us D p+q) k,...,k p+q FG) = p + q)! f g)k,..., k p+q ). Ths poves 3.), whle the patcula case follows fom agan the Stoock s fomula. Moe pecsely, one can fst deduce fom the pevous dscusson that Q h) = h H + Q w) + ) [ Q h h fo some w H gven by wk) := E Dk Q h) )]. By the defnton of dscete gadent, one has D k Q h) ) = p k q k h j)y j + hk) p k + q k p j k k q k h j)y j + hk) p k + q k p j k k q k = hk) q k p k + hk) h j)y j, pk q k whch concludes ou poof of Lemma.. j k

15 Peccat-Tudo theoem fo Rademache chaoses 5 Poof of Lemma 3.: It follows fom Lemma. that FG = E[FG] + theefoe by othogonalty popety, one has E [ F G ] = E[FG] + = E[FG] + p+q k= p+q k= Recall fom [9, Lemma.] that 3.3) p + q)! f g H p+q = p!q! Va J k FG) ) + p + q)! f g p+q H p+q p+q k= J k FG) + Q p+q f g p+q ), Va J k FG) ) + p + q)! f g H p+q p + q)! f g c p+q H p+q. p q = p ) q ) f g H p+q p!q! f H p g H q + p=q) p! f, g H p, thus 3.) follows by notcng that E[FG] = p=q) p! f, g H p and VaF)VaG) = p!q! f H p g H q. Usng 3.3) agan, we have 3.4) p+q k= Va J k F ) ) = E [ F 4] p 3E[F ] p! = ) p f f + p)! H f f p c p, H p whch mples 3.). It emans to pove 3.3) and we ll use the same aguments as n the poof of [5, Lemma 3.3]: f g c p+q H f g p+q c p+q = f H p+q,..., p ) g j,..., j q ) 3.5) p q ) ) p q =! =,..., p, j,..., j q ) c p+q,..., p ) p j,..., j q ) q cad{,..., p } { j,..., j q })= f,..., p ) g j,..., j q ), whee cada) means the cadnalty of the set A, and the combnatoal constant! p q ) ) s the numbe of ways one can buld pas of dentcal ndces out of,..., p ) p and j,..., j q ) q. Theefoe, t s enough to notce that fo each {,..., p q}, the nne sum n 3.5) s bounded by f,..., p, k,..., k ) g j,..., j q, k,..., k ),..., p,k,...,k ) p j,..., j q,k,...,k ) q,..., p,k) p j,..., j q,k) q The poof of Lemma 3. s complete. f,..., p, k) g j,..., j q, k) mn { f H p Mg), g H q M f ) }. 4 Unvesalty of Homogeneous sums Fx d and a dvegent sequence N n, n ) of natual numbes. Consde the kenels f n : {,..., N n } d R symmetc and vanshng on dagonals and d! f n =, then accodng to.), H d Q d f n ; Ξ) = f n,..., d )ξ ξ d.,..., d N n

16 6 G. Zheng The followng cental lmt theoem due to de Jong [4] gave suffcent condtons fo asymptotc nomalty of Q d f n ; Ξ). Theoem 4.. Unde the above settng, let Ξ = ξ, ) be a sequence of ndependent cented andom vaables wth unt vaance and fnte fouth moments. If E [ Q d f n ; Ξ) 4] 3 and the maxmal nfluence M f n ) as n +, then Q d f n ; Ξ) conveges n law to a standad Gaussan. The above esult exhbts the unvesalty phenomenon as well as the mpotance of the noton maxmal nfluence. Anothe stkng esult wth smla natue s the nvaance pncple establshed n [4], n whch the authos wee able to contol dstbutonal dstance between homogeneous sums ove dffeent sequences of ndependent andom vaables n tems of maxmal nfluence, see e.g. Theoem. theen. Let us estct ouselves to the Gaussan settng fo a whle: when G s a sequence of..d. standad Gaussans, Q d f n ; G) belongs to the d-th Gaussan Wene chaos, and the fouth moment theoem [] mples that f Q d f n ; G) conveges n law to a standad Gaussan o equvalently E [ Q d f n ; G) 4] 3), then f n d f n H. Whle M f n ) f n d f n H due to [7, Lemma.4], so that M f n ). Ths hnts the unvesalty of the Gaussan Wene chaos, see [8] fo moe detals. The followng esult s slghtly) adapted fom Theoem 7.5 n [8]. Theoem 4.. Fx nteges d and q d... q. Fo each j {,..., d}, let N j,n, n ) be a sequence of natual numbes dvegng to nfnty, and let f j,n : {,..., N j,n } q j R be symmetc and vanshng on dagonals.e. f j,n H q j wth suppot contaned n {,..., N j,n } q j ) such that lm q n + k =q l )q k! f k,n,..., qk ) f l,n,..., qk ) = Σ k,l,,..., qk N k,n whee Σ = Σ, j,, j d) s a symmetc nonnegatve defnte d by d matx. Then the followng statements ae equvalent: A ) Gven a sequence G of..d. standad Gaussans, Q q f,n ; G),..., Q qd f d,n ; G) )T conveges n dstbuton to N, Σ), as n +. A ) Fo evey sequence Ξ = ξ, N ) of ndependent cented andom vaables wth unt vaance and sup N E [ ξ 3] < +, the sequence of d-dmensonal andom vectos Q q f,n ; Ξ),..., Q qd f d,n ; Ξ) ) T conveges n dstbuton to N, Σ), as n +. Smla unvesalty esult fo Posson chaos was fst establshed n [4] and efned ecently n [7]. It was ponted out n [4] and [8] that homogeneous sums nsde the Rademache chaos ae not unvesal wth espect to nomal appoxmaton and a counteexample s avalable e.g. n [4, Poposton.7]: A Counteexample: Let Y be a sequence of..d. andom vaables wth PY = ) = PY = ) = / that s, n the symmetc settng). Fx q and fo each N q, we set f N,..., q ) = q! N q +, f {,..., q } = {,,..., q, s} fo q s N;, othewse. Then n the symmetc case, Q q f N ; Y) = Y Y Y q N =q Y N q +

17 Peccat-Tudo theoem fo Rademache chaoses 7 conveges n law to the standad Gaussan, whle f G s a sequence of..d. standad Gaussans, then fo evey N, Q q f N ; G) law = G G G q fals to be Gaussan. It s easy to check that the maxmal nfluence M f N ) of the kenel f N s equal to /qq!) fo evey N, whch s consstent wth de Jong s theoem. In the end of ths secton, we povde a patally) unvesal esult fo Rademache chaos that complements [7, 8, 4]. Poposton 4.. Let the assumptons n Theoem 4. peval. Then, the followng statement s equvalent to A ) and A ) n Theoem 4.: A 3 ) n the symmetc case, as n +, Q q f,n ; Y),..., Q qd f d,n ; Y) )T conveges n dstbuton to N, Σ), and M f j,n ) fo each j {,..., d}. Poof. Suppose A ) holds tue, then Q q f,n ; Y),..., Q qd f d,n ; Y) )T conveges n dstbuton to N, Σ) by A ) A ) ; and by the fouth moment theoem on a Gaussan space [], A ) mples that f j,n q j f j,n H, as n +. Recall fom [7, Lemma.4] that M f ) f d f H fo each f H d, theefoe M f j,n) fo each j {,..., d}. Ths poves the mplcaton A ) A 3 ). It emans to show A 3 ) A ). Now we assume that A 3 ) s tue, then by a weak fom of the hypecontactvty popety see Secton ), we have lm n + E [ Q q j f n, j ; Y) 4] = 3Σ j, j fo each j =,..., d. It follows fom Lemma 3. that f j,n f H j,n q j fo each =,..., q j, and any j =,..., d. Hence, A ) follows mmedately fom the Peccat-Tudo theoem [3]. Ths concludes ou poof. Refeences [] E. Azmoodeh, S. Campese and G. Poly. Fouth moment theoems fo Makov dffuson geneatos. J. Funct. Anal. 66 4), [] S. Campese, I. Noudn, G. Peccat and G. Poly. Multvaate Gaussan appoxmatons on Makov chaoses. Electon. Commun. Pobab. Volume 6), no. 48, -9. [3] S. Chattejee and E. Meckes. Multvaate nomal appoxmaton usng exchangeable pas. ALEA 4, 57-83, 8. [4] P. de Jong. A cental lmt theoem fo genealzed multlnea foms. J. Multvaate Anal [5] C. Döble and K. Kokowsk. On the fouth moment condton fo Rademache chaos. axv pepnt, 7) [6] C. Döble and G. Peccat. The fouth moment theoem on the Posson space. Ann. Pobab. to appea 7+) [7] C. Döble, A. Vdotto and G. Zheng. Fouth moment theoems on the Posson space n any dmenson. axv pepnt, 7) [8] K. Kokowsk. Posson appoxmaton of Rademache functonals by the Chen-Sten method and Mallavn calculus. Commun. Stoch. Anal. 7), no., 95- [9] K. Kokowsk, A. Rechenbachs and Ch. Thäle. Bey-Esseen bounds and multvaate lmt theoems fo functonals of Rademache sequences. Ann. Inst. Hen Poncaé Pobab. Stat. 5 6), no.,

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