Third cumulant Stein approximation for Poisson stochastic integrals

Transcription

1 Third cumulan Sein approximaion for Poisson sochasic inegrals Nicolas Privaul Division of Mahemaical Sciences School of Physical and Mahemaical Sciences Nanyang Technological Universiy 2 Nanyang Link Singapore nprivaul@nu.edu.sg February 3, 28 Absrac We derive Edgeworh-ype expansions for Poisson sochasic inegrals, based on cumulan operaors defined by he Malliavin calculus. As a consequence we obain Sein approximaion bounds for sochasic inegrals, which are based on hird cumulans insead of he L 3 norm erm found in he lieraure. The use of he hird cumulan resuls ino a convergence rae faser han he classical Berry-Esseen rae on cerain examples. Key words: Sein approximaion; Malliavin calculus; Poisson sochasic inegral; cumulans; Edgeworh expansions. Mahemaics Subjec Classificaion: 6H7, 62E7, 6H5. Inroducion Edgeworh expansions have been derived on he Wiener space in [9, [2, [4, using a consrucion of cumulan operaors based on he inverse L of he Ornsein- Uhlenbeck operaor [. This approach exend he resuls of [, [3 on Sein approximaion, Berry-Esseen bounds and he fourh momen heorem. Relaed Edgeworh ype expansions have also been derived for he Iô-Skorohod inegral δ(u of a

2 process u on he Wiener space in [8. In his paper we derive Edgeworh ype expansions of he form E [ δ(ug(δ(u E [ u 2 L 2 (R + g (δ(u n + E [ g (k (δ(uγ u k+ + E [ g (n+ (δ(urn u k2 (. for he compensaed Poisson sochasic inegral δ(u u d(n of an adaped process (u R+ wih respec o a sandard Poisson process (N R+, where Γ u k and Rn u are respecively a cumulan ype operaor and a remainder erm defined using he derivaion operaors of he Malliavin calculus on he Poisson space, see Proposiion 3.. From (., in Proposiion 4. and Corollary 4.2 we deduce Sein approximaion bounds of he form d(δ(u, N E [ u 2 and L 2 (R + d(δ(u, N Var[δ(u + [ + E u 3 sds + u, D Var [ [ u 2 H + E u 3 sds + u 2 d + 2E [ R u, u, D u 2 d +2E [ R u, (.2 when he process u is adaped wih respec o he Poisson filraion, where D is a gradien operaor acing on Poisson funcionals, N N (, is a sandard Gaussian random variable and d(f, G : sup E[h(F E[h(G h L is he Wassersein disance beween he laws of wo random variables F and G, where L denoes he class of -Lipschiz funcions on R. In Secion 5 we presen examples of adaped processes (u R+ for which (.2 holds, see Proposiion 5. and Corollary 5.2, in relaion wih classical examples such as he 2

3 normalized sequence ((T k k/ k k, where (T k k is he sequence of jump imes of he sandard Poisson process (N R+. In paricular, when f is a differeniable deerminisic funcion we obain bounds of he form ( d f(d(n, N f 2 L 2 (R + + f 3 (d +2 f L 2 (R + depending on he regulariy of he funcion f, see Corollary 5.3. f ( 2 d, (.3 This alernaive approach, which is based on derivaion operaors, replaces he L 3 (R + norm of f in he classical Sein bound ( d f(d(n, N f 2 L 2 (R + + f 3 ( d, (.4 see Corollary 3.4 of [4, wih he hird cumulan κ f 3 f 3 (d, by removing he inner absolue value in he inegral. The main reason for he appearance of an L 3 norm in (.4 insead of he hird cumulan κ f 3 f 3 (d lies wih he use of finie difference operaors and he replacemen of he chain rule of derivaion wih a Taylor expansion bound, see Theorem 3. of [4 and 4.2 of he recen survey [3. In he presen paper, he use of derivaion operaors allows us insead o use he hird cumulan κ f 3 f 3 (d in (.3. Taking f k of he form f k ( : k g(/k, k, where g C (R is such ha g L 2 (R + and g 3 (d, (.3 shows ha ( d f k (d(n, N 2 g ( 2 d, k, k see (5.4 below, while Corollary 3.4 of [4 only yields he sandard Berry-Esseen rae, see (5.6. This similarly improves on he bound ( ( 2 d f(d(n, N f 2L2(R+ + f ( f(sds 2 d (.5 3

4 obained using derivaion operaors on he Poisson space in Corollary 4.4 of [2, which also yields he sandard Berry-Esseen rae in his case, see (5.5 below. In Secion 2 we recall some background maerial on he Malliavin calculus and cumulan operaors for sochasic inegrals on he Poisson space. In Secion 3 we derive Edgeworh ype expansions, based on a family of cumulan operaors ha are associaed o he process u and specially defined for he Skorohod inegral operaor δ. In Secion 4 we derive Sein ype approximaion bounds for sochasic inegrals, and in Secion 5 we consider adaped and deerminisic examples. 2 Malliavin operaors Le (T k k denoe he sequence of jump imes of a sandard Poisson process (N R+ wih uni inensiy and generaing he filraion (F R+ on a probabiliy space (Ω, F, P, wih T :. The gradien operaor D defined on random funcionals as F S : { F f(t,..., T n : f C b (R n }, D F n k [,Tk ( f x k (T,..., T n, has he derivaion propery, cf. [7, [5, [5. The operaor D defines he Sobolev spaces D p, wih he Sobolev norms p >, where H : L 2 (R +. Covarian derivaive F Dp, : F L p (Ω + DF L p (Ω,H, F S. In addiion o he operaor D, we will also need he following noion of covarian derivaive, see [9 and references herein. In he sequel we le W 2, (R + denoe he Sobolev space of weakly differeniable funcions on R + such ha f 2 W 2, : f( 2 d + 4 f ( 2 d <.

5 Definiion 2. Le he operaor be defined on { n } u U : h i F i : F i S, h i W 2, (R +, i,..., n, n, as i s u : D s u u [, (s, s, R +, where u denoes he ime derivaive of u wih respec o. By closabiliy, he operaor exends o he Sobolev spaces D p, (H of processes, p, by he Sobolev norms u Dp, (H : u L p (Ω,W 2, (R + + Du L p (Ω,H H, u U. We noe ha when a process u D p, (H is (F R+ -adaped we have s u D s u, s >. (2. Definiion 2.2 Given k and u D 2, (H we define he operaor power ( u k in he sense of marix powers wih coninuous indices, as ( u k h s ( k u s k u k u 2 h d d k, s R +, h H. In paricular, for h W 2, (R + and v U we have ( hv s s v h(sd ḣ(s v [,s (d ḣ(s v d, s R +. (2.2 We noe ha when h W 2, (R + is a deerminisic funcion, (2.2 can be ieraed o show ha ( h k f s ( k ḣ(s ( k ḣ(s s ḣ( k k h(s k h( k h( 2 f( d d k k [,s ( k [,k ( k ḣ( k [,2 ( ḣ( 2f( d d k ḣ( 2 2 f( d d k, s R +, k, wih he bound ( k/2 ( h k f L 2 (R + h k L 2 (R 2 f L 2 (R + h ( d 2 f L 2 (R

6 This also yields s ( h k f, h ( k h(sḣ(s 2 ( k h 2 ( k ḣ( k (k +! k k 2 2 ḣ( k ḣ( 2f( d d k ḣ( k ḣ( 2f( d d k h k+ ( f( d, f L 2 (R +, k, and in paricular, for h W 2, (R + we have h p2 Lp (R +, wih h k ( d (k! ( h k 2 h, h (k! h k W 2,, k 2. Noe ha due o (2., when he process u D p, (H is (F R+ -adaped, we have ( u k v s s k 2 ( k u s k u k u 2 v d d k, s R +, and as a consequence, he process (( u k v s s R+ is also (F R+ -adaped. In he sequel we simply denoe,, H, L 2 (R +. Lemma 2.3 Leing n and u D 2, (H, we have Proof. ( u n u, u (n +! n+ u n+2 d + ( k! u n+ k u, D k2 Using he adjoin u of u on H given by ( uv s : wih he dualiy relaion ( s u v d, s R +, v H, v, ( uh ( uv, h, h, v H, we will show by inducion on k n + ha u k d. (2.3 ( u n u k i2 i! u n u u 2 n u n d d n u n i u n+ i D n+ i u i n+2 i d d n i 6

7 + u k k! n+ k u n k u n+ k d d n+ k. (2.4 This relaion holds for k. Nex, assuming ha he ideniy (2.4 holds for some k {,..., n}, and using he relaion we have n k u n+ k D n k u n+ k [,n+ k ( n k u n+ k, n k, n+ k R +, ( u n u + k! k i! i2 + k! k! k i2 k i! i2 i! + (k +! k+ i2 (k +! i! + (k +! k+ i2 u n i u n+ i D n+ i u i n+2 i d d n+2 i u k n+ k u n k u n+ k d d n+ k u n i u n+ i D n+ i u i n+2 i d d n+2 i u k n+ k u n k u n k D n k u n+ k d d n+ k n k u n+ k u k n+ k u n k u n k d d n+ k u n i u n+ i D n+ i u i n+2 i d d n+2 i u n k u n k D n k u k+ n+ k d d n+ k u n k u n k (u k+ n k dd d n k u n i u n+ i D n+ i u i n+2 i d d n+2 i i! ( u n+ i D u k+ n k u n k u n k d d n k u i sds + (k +! ( u n k u k+, which shows by inducion ha (2.4 holds for k,..., n. In paricular, for k n we have ( u n n+ u (n +! un+ + i! ( u n+ i D u i sds, R +, i2 which yields (2.3 by inegraion wih respec o R + and dualiy. 7

8 Cumulan operaors We recall he definiion of he cumulan operaors inroduced in 6 of [7 on he Poisson space. Given k 2 and u D 2, (H an (F R+ -adaped process we define he cumulan operaors Γ u k : D 2, L 2 (Ω, k 2, by Γ u kf : F ( u k 2 u, u + ( u k u, DF, k 2. As a consequence of Lemma 2.3 we have Γ u k (k! u k d + and Γ u k u k W 2, (R +, a.s., k 2. k i2 ( i! u k i u, D u i d, When h W 2, (R + is a deerminisic funcion we find Γ h k (k! h k (d (k! κ k(h, k 2, where κ k (h h k (sds is he cumulan of order k 2 of he compensaed Poisson sochasic inegral h(d(n. Poisson-Skorohod inegral In addiion, D has a closable adjoin operaor δ wih domain Dom(δ, ha saisfies he dualiy relaion [ E u D F d E[F δ(u, F D 2,, u Dom(δ, (2.5 and coincides wih he compensaed Poisson sochasic inegral on square-inegrable processes (u R+ adaped o he filraion (F R+ generaed by (N R+, i.e., we have δ(u u d(n. 8

9 In addiion, he operaors, δ and D saisfy he commuaion relaion D δ(u u + δ( u, (2.6 for u D 2, (H an (F R+ -adaped process, cf. Lemma 4.5 in [7 and Relaions (2.6, (2.9 and Lemma 2.4 in [9. Recall ha when (u R+ inegral of u. is an (F R+ -adaped process, δ(u coincides wih he Iô 3 Edgeworh ype expansions Classical Edgeworh expansions are used in paricular as asympoic expansions around he Gaussian cumulaive disribuion funcion Φ(x for he cumulaive disribuion funcion P (F x of a cenered random variable F wih E[F 2, as Φ(x + c φ(xh (x + + c m φ(xh m (x +, where φ(x, x R, is he sandard Gaussian densiy, H k (x is he Hermie polynomial of degree k, and c k is a coefficien depending on he sequence of cumulans (κ n n of a random variable F, cf. Chaper 5 of [8 and A.4 of [2. Edgeworh ype expansions of he form n κ l+ E[F g(f E[g (l (F + E[g (n+ (F Γ n+ F, n, l! l have been obained by he Malliavin calculus in [9, [2, [4, wrien here for F a cenered random variable, where Γ n+ is a cumulan ype operaor on he Wiener space such ha n!e[γ n F coincides wih he cumulan κ n+ of order n +, n N, cf. [, exending resuls of [. An infinie Edgeworh ype expansion for he compensaed Poisson sochasic inegral of a deerminisic funcion f p2 Lp (R + can be wrien as E [ δ(fg(δ(f [ E f(s ( g(δ(f + f(s g(δ(f ds (3. 9

10 k k k! f k+ (sdse [ g (k (δ(f k! κ k+(fe [ g (k (δ(f, g C b (R, using a sandard inegraion by pars for finie difference operaors on he Poisson space. In his secion we esablish an Edgeworh ype expansion of any finie order wih an explici remainder erm for he compensaed Poisson sochasic inegral δ(u of an (F R+ -adaped process u. Before proceeding o he saemen of general expansions in Proposiion 3., we derive an expansion of order one for a deerminisic inegrand f W 2, (R +. In he sequel we le Cb n(r + denoe he space of Cb n funcions bounded ogeher wih heir derivaives on R +, n. By he dualiy relaion (2.5 beween D and δ, he chain rule of derivaion for D and he commuaion relaion (2.6 we ge, for g Cb 2(R, E [ δ(fg(δ(f E [ g (δ(ff, Dδ(f E [ g (δ(ff, f + E [ g (δ(ff, δ( f E [ g (δ(ff, f + E [ f, D(g (δ(ff E [ g (δ(ff, f + E [ g (δ(f( ff, Dδ(f E [ g (δ(ff, f + 2 f 3 (de [ g (δ(f + E [ g (δ(f( ff, δ( f κ 2 (fe [ g (δ(f + 2 κ 3(fE [ g (δ(f + E [ g (δ(f( ff, δ( f, since by Lemma 2.3 we have ( ff, f f(f ( f(sdsd 2 f 3 (d 2 κf 3. In he nex proposiion we derive general Edgeworh ype expansions for adaped inegrands processes (u R+.

11 Proposiion 3. Le n and assume ha u D n+, (H is an (F R+ -adaped process. Then for all g C n+ b (R and bounded G D 2, we have E [ Gδ(ug(δ(u E [ g(δ(uu, DG n + E [ g (k (δ(uγ u k+g + E [ Proof. Gg (n+ (δ(u ( k u n+2 n+ s (n +! ds + k2 k! ( u n+ k u, D u k d + ( u n u, δ( u. By he dualiy relaion (2.5 beween D and δ, he chain rule of derivaion for D and he commuaion relaion (2.6, we ge E [ Gg(δ(u( u k u, Dδ(u E [ Gg (δ(u( u k+ u, Dδ(u E [ Gg(δ(u( u k u, u + E[Gg(δ(u( u k u, δ( u E [ Gg (δ(u( u k+ u, Dδ(u E [ Gg(δ(u( u k u, u + E [ u, D(Gg(δ(u( u k u E [ Gg (δ(u( u k+ u, Dδ(u E [ Gg(δ(u( u k u, u + E [ g(δ(u( u k+ u, DG + E [ Gg(δ(u u, D(( u k u E [ g(δ(uγ u k+2g, since u, D(( u k u as u is an (F R+ -adaped process, see Lemma 4.4 of [6. Therefore, we have E [ Gδ(ug(δ(u E [ Gg (δ(uu, Dδ(u + E [ g(δ(uu, DG E [ g(δ(uu, DG + E [ Gg (n+ (δ(u( u n u, Dδ(u n + (E [ Gg (k+ (δ(u( u k u, Dδ(u E [ Gg (k+2 (δ(u( u k+ u, Dδ(u k E [ g(δ(uu, DG + E [ g(δ(uu, DG + n E [ g (k (δ(uγ u k+g + E [ Gg (n+ (δ(u( u n u, Dδ(u k n E [ g (k (δ(uγ u k+g k + E [ Gg (n+ (δ(u( u n u, u + E [ Gg (n+ (δ(u( u n u, δ( u, and we conclude by Lemma 2.3. When G, Proposiion 3. shows ha E [ δ(ug(δ(u n+ [ k! E u k+ s dsg (k (δ(u k

12 n+ + k2 k i2 [ i! E ( u k i u, D u i d g (k (δ(u +E [ g (n+ (δ(u( u n u, δ( u, n. On he oher hand, when f W 2, (R + is a deerminisic funcion and g C b find E [ δ(fg(δ(f n+ k n+ k k! (R we f k+ (sdse [ g (k (δ(f + E [ g (n+ (δ(f( f n f, δ( f k! κ k+(fe [ g (k (δ(f + E [ g (n+ (δ(f( f n f, δ( f, n, showing, as n ends o +, ha E [ δ(fg(δ(f f k+ (s dse [ g (k (δ(f [ E f(s ( g(δ(f + f(s g(δ(f ds, k! k which recovers (3. and he sandard inegraion by pars ideniy for finie difference operaors on he Poisson space. 4 Sein approximaion We le N N (, denoe a sandard Gaussian random variable. In comparison wih he resuls of [2, our bounds apply o a differen sochasic inegral represenaion. In he case n, Proposiion 3. reads E [ g (δ(uu, u δ(ug(δ(u E [ g (δ(uu, δ( u, for u D 2, (H and g Cb (R. Applying his relaion o he soluion g x of he Sein equaion (,x (z Φ(z g x(z zg x (z, z R, which saisfies g x 2π/4 and g x, cf. Lemma 2.2-(v of [6, yields he expansion P (δ(u x Φ(x E [ ( u, u g x(δ(u E [ u, δ( u g x(δ(u, x R, 2

13 around he Gaussian cumulaive disribuion funcion Φ(x, wih u D 2, (H. On he oher hand, given h : R R an absoluely coninuous funcion wih bounded derivaive, he funcional equaion h(z E[h(N g (z zg(z, z R, (4. has a soluion g h Cb (R which is wice differeniable and saisfies he bounds g h h and g h 2 h, x R, cf. Lemma.2-(v of [ and references herein. Proposiion 4. Le u D 2, (H be adaped wih respec o he Poisson filraion (F R+. We have d(δ(u, N E [ u, u [ + E u 3 sds + u, D u 2 d +2E [ ( uu, δ( u. (4.2 Proof. For n and G, Proposiion 3. shows ha E[δ(ug(δ(u E[g (δ(uu, u + ( [g 2 E (δ(u u 3 sds + +E [ g (δ(u( uu, δ( u, u, D u 2 d hence for any absoluely coninuous funcion h : R R wih bounded derivaive, denoing by g h he soluion o (4. we have hence E[h(δ(u E[h(N E[δ(ug h (δ(u g h(δ(u E[g h(δ(u(u, u + ( [g 2 E (δ(u u 3 sds + +2E [ g h(δ(u( uu, δ( u, E[δ(uh(δ(u E[h(N h E [ u, u [ + h E u 3 sds + u, D u, D +2 h E [ ( uu, δ( u, u 2 d u 2 d which yields (4.2. 3

14 As a consequence of Proposiion 4. and he Iô isomery we have he following corollary. Corollary 4.2 Le u D 2, (H be adaped wih respec o he Poisson filraion (F R+. We have d(δ(u, N Var[δ(u + Var [ [ u 2 H + E u 3 sds + u, D u 2 d +2E [ ( uu, δ( u. (4.3 Proof. hence By he Iô isomery we have [ ( Var[δ(u E 2 u d(n E[u, u, E [ u, u E [ E[u, u + E [ u, u E[u, u Var[δ(u + E[(u, u E[u, u 2 Var[δ(u + Var [ u H 2. In paricular, when Var[δ(u, (4.3 shows ha d(δ(u, N Var [ [ u 2 H +E u 3 sds + u, D 5 Examples Adaped inegrands u 2 d +2E [ ( uu, δ( u. Alhough he presen approach does no apply direcly o he classical normalized sequence F k : T k k k k [,Tk (d(n, k, due o he lack of ime differeniabiliy of he adaped inegrand [,Tk (, examples of his form can be reaed via smoohed processes u k as in (5. below, see also Corollary

15 Proposiion 5. Le u k ( : g( [,Tk ( + g(t k (Tk, (f( T k, R +, (5. k N, where g is a Lipschiz funcion on R + and f W 2, (R + saisfies f(. Then for every k N we have d(δ(u k, N (5.2 [ Tk [ Tk E g 2 (d f 2 L 2 (R + g2 (T k + E (g 2 (s g 2 (T k g(sds + E [ g 3 (T k [ Tk f 3 (d + 2 f 2 L 2 (R + E g(t k g (T k g(d [ Tk + 2E g ( g(sds g (sd(n s sd [ Tk + 2 f L 2 (R + E g (T k g ( g(sdsd [ Tk + 2 f L 2 (R + E g(t k g (T k g(sds f ( 2 d + 2 f L 2 (R + E [ g 3 (T k f ( 2 d. Proof. and δ(u k We have u k (d(n Var[δ(u k E On he oher hand, we have [ g(d(n + g(t k g 2 (d + E[g 2 (T k T k f( T k d(n, f 2 (d. s u k ( D s u k ( ( g ( [,Tk ( + g(t k f ( T k (Tk, ( [, (s ( g (T k (Tk, (f( T k g(t k (Tk, (f ( T k [,Tk (s ( g ( [,Tk ( + g(t k f ( T k (Tk, ( [, (s ( g ( [,Tk ( + g (T k (Tk, (f( T k [, (s [,Tk (s g(t k {Tk <s<}f ( T k, and ( u k u k ( g(s ( g ( [,Tk ( + g (T k (Tk, (f( T k [, (s [,Tk (sds 5

16 Given ha δ( su k his yields g 2 (T k f ( T k [,Tk (g ( ( u k u k, δ( u k ( g(t k + g 3 (T k g ( + g (T k + (Tk, (g 2 (T k f ( T k ( s + {Tk <s}g(t k f(s T k {Tk <s<}ds g(sds (Tk, (g (T k f( T k T k f(s T k ds. g(sds g (d(n + g (T k f( T k d(n [,Tk (s T k g (sd(n s s + g (T k T k T k g(t k g (T k + g 3 (T k s f ( T k d(n, T k f(s T k d(n s s g ( f (s T k d(n s sg (T k f( T k f (s T k d(n s sf ( T k g (sd( ˆN s s f(sd( ˆN s s f ( g(sds f( g(sdsd g ( f (sd( ˆN s s g(sdsd f (sd( ˆN s sd f(sdsd, g(sdsd T k f(s T k dsd g(sdsd where denoes equaliy in disribuion and ( ˆN R+ is a sandard Poisson process independen of (T k k. We also have ( D s u 2 k(d D s g 2 (d + g 2 (T k ( D s g 2 (d + g 2 (T k 6 T k f 2 ( T k d f 2 (d

17 consequenly we have u k, D u 2 k(d and u 3 k(sds + u k, D g 2 (T k D s T k + 2g(T k g (T k ( g 2 (T k + 2g(T k g (T k ( g 2 (T k + 2g(T k g (T k u 2 k(d (g 2 (s g 2 (T k g(sds + g 3 (T k Hence by Corollary 4.2 we find f 3 (d 2g(T k g (T k f 2 (dd s T k f 2 (d [,Tk (s, f 2 (d g(d, f 2 (d g(d. d(δ(u k, N Var[δ(u k + Var [ [ u k 2 H + E u 3 k(sds + u k, D u 2 k(d +2E [ ( u k u k, δ( u k [ Tk E g 2 (d g 2 (T k f 2 (d [ Tk +E (g 2 (s g 2 (T k g(sds +E [ g 3 (T k [ Tk f 3 (d + 2E g(t k g (T k g(d f 2 (d [ Tk +2E g (sd(n s sg ( g(sdsd [ Tk [ +2E g (T k g ( g(sdsd E f(sd(n s s [ Tk [ +2E g(t k g (T k g(sds E f( f (sd(n s sd +2E [ g 3 (T k [ E f ( f (sd(n s s f(sdsd. Finally, we noe ha [ E f( f (sd(n s sd E[ f, δ( f 7

18 and [ E f ( f L 2 (R + E[ δ( f 2 L 2 (R + [ f L 2 (R + E δ( f 2 d f L 2 (R + f L 2 (R + f(s 2 dsd f L 2 (R + s f (s 2 ds, f (sd(n s s f(sdsd E [ ( ff, δ( f [ ( ff L 2 (R + E δ( f 2 d f L 2 (R + f 2 L 2 (R 2 + f L 2 (R + The above bound can also be obained as E [ ( ff, δ( f E [ δ(( f 2 f s f (s 2 ds. f (s 2 [,s (dsd E [ δ(( f 2 f 2 ( f 2 f L 2 (R + f L 2 (R + f 2 L 2 (R 2 + f L 2 (R + s f (s 2 ds. When u k is an adaped process of he form u k ( : g k ( [,Tk (, R +, where g k is Lipschiz, and e.g. f( : e n, Proposiion 5.2 shows, by leing n end o infiniy, ha ( d g k (d(n, N E [ 8 [ Tk gk(d 2 + E (gk(s 2 gk(t 2 k g k (sds

19 [ Tk +2E g k( g k(sd(n s s As a consequence, similarly o Corollary 4.2 we have he following resul. g k (sdsd. Corollary 5.2 Le g k be a Lipschiz funcion on R + for k. We have ( [ [ Tk [ Tk d g k (d(n, N E E gk(d 2 + Var gk 2(d [ Tk +E (gk(s 2 gk(t 2 k g k (sds [ Tk +2E g k( g k(sd(n s s g k (sdsd. For example, when g k ( / k is consan, Corollary 5.2 recovers he sandard Berry-Esseen rae ( Tk k d, N k Deerminisic inegrands Var[Tk k k, k. Taking k and g ( in Proposiion 5., we also find he following corollary. Corollary 5.3 Le f W 2, (R + be a deerminisic funcion. We have ( d f(d(n, N f 2 (d + f 3 (d +2 f L 2 (R + Considering for example f k ( of he form f k ( : k g(/k, k, f ( 2 d. where g C (R is such ha g L 2 (R + f k L 2 (R +, we have ( d f k (d(n, N fk 3 (sds + 2 g 3 (d + 2 k k f k( 2 d g ( 2 d, (5.3 9

20 k. When g has consan sign, e.g. g(x : 2be b wih b > and f k ( 2b/ke b/k, (5.3 does no improve on he sandard Berry-Esseen convergence rae ( d g k (d(n, N 2 ( 2b b 3 k, and d g k (d(n, N 3k, respecively obained from (.4 and (.5, see 4.2 of [2. choosing g such ha we find he bound ( d g 3 (d, f k (d(n, N 2 k On he oher hand, g ( 2 d, k, (5.4 while Corollary 4.4 of [2 and Corollary 3.4 of [4 only yield he Berry-Esseen raes ( d f k (d(n, N f k 2 L 2 (R + + f k L 2 (R + f k ( 2 d by (.5, and ( d f k (d(n, N by (.4. For anoher example, choosing f k ( : C k g ( 2 d, k, (5.5 k f k 2 L 2 (R + + fk 3 (s ds g( 3 d, (5.6 k m a i e bi/k, k, wih f k L 2 (R +, for b i >, a i R, i,..., m, such ha i,j,l m (5.4 yields he bound ( d f k (d(n, N i a i a j a l b i + b j + b l, C 2 k 2 i,j m a i a j b i b j (b i + b j 2, k.

21 In paricular, when f k ( : k /2 (e /k ae b/k /C wih a, b > and C 2 : 2 2a + b + a2 2b b(b + 4ab + a2 ( + b 2b(b + for any b > we can choose a > saisfying he equaion which yields he bound ( d f 3 k (d C 3 k f k (d(n, N where c(a, b depends only on a, b >. References ( 3 3a 2 + b + 3a2 + 2b a3 3b >,, c(a, b, k, k [ A.D. Barbour. Asympoic expansions based on smooh funcions in he cenral limi heorem. Probab. Theory Rela. Fields, 72(2:289 33, 986. [2 H. Biermé, A. Bonami, I. Nourdin, and G. Peccai. Opimal Berry-Esseen raes on he Wiener space: he barrier of hird and fourh cumulans. ALEA La. Am. J. Probab. Mah. Sa., 9(2:473 5, 22. [3 S. Bourguin and G. Peccai. The Malliavin-Sein mehod on he Poisson space. In Sochasic analysis for Poisson poin processes, volume 7 of Bocconi & Springer Series, pages Springer-Verlag, 26. [4 S. Campese. Opimal convergence raes and one-erm Edgeworh expansions for mulidimensional funcionals of Gaussian fields. ALEA La. Am. J. Probab. Mah. Sa., (2:88 99, 23. [5 E. Carlen and É. Pardoux. Differenial calculus and inegraion by pars on Poisson space. In S. Albeverio, Ph. Blanchard, and D. Tesard, ediors, Sochasics, Algebra and Analysis in Classical and Quanum Dynamics (Marseille, 988, volume 59 of Mah. Appl., pages Kluwer Acad. Publ., Dordrech, 99. [6 L.H.Y. Chen, L. Goldsein, and Q.-M. Shao. Normal approximaion by Sein s mehod. Probabiliy and is Applicaions (New York. Springer, Heidelberg, 2. [7 R.J. Ellio and A.H. Tsoi. Inegraion by pars for Poisson processes. J. Mulivariae Anal., 44(2:79 9, 993. [8 P. McCullagh. Tensor mehods in saisics. Monographs on Saisics and Applied Probabiliy. Chapman & Hall, London, 987. [9 I. Nourdin and G. Peccai. Sein s mehod and exac Berry-Esseen asympoics for funcionals of Gaussian fields. Ann. Probab., 37(6: , 29. [ I. Nourdin and G. Peccai. Sein s mehod on Wiener chaos. Probab. Theory Relaed Fields, 45(-2:75 8, 29. 2

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