Probability approximation by Clark-Ocone covariance representation

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1 Probability approximation by Clark-Ocone covariance representation Nicolas Privault Giovanni Luca Torrisi October 19, 13 Abstract Based on the Stein method and a general integration by parts framework we derive various bounds on the distance between probability measures. We show that this framework can be implemented on the Poisson space by covariance identities obtained from the Clark-Ocone representation formula and derivation operators. Our approach avoids the use of the inverse of the Ornstein-Uhlenbeck operator as in the existing literature, and also applies to the Wiener space. Keywords: Poisson space, Stein-Chen method, Malliavin calculus, Clark-Ocone formula. Mathematics Subject Classification: 6F5, 6G57, 6H7. 1 Introduction The Stein and Chen-Stein methods have been applied to derive bounds on distances between probability laws on the Wiener and Poisson spaces, cf. [6, [7 and [8. The results of these papers rely on covariance representations based on the number (or Ornstein-Uhlenbeck) operator L on multiple Wiener-Poisson stochastic integrals and its inverse L 1. In particular the bound d W (F, N) E [ 1 DF, DL 1 F (1.1) has been derived for centered functionals of a standard real-valued Brownian motion in [6, Theorem 3.1. Here d W is the Wasserstein distance, N is a random variable distributed according to the standard Gaussian law, D is the classical Malliavin gradient and, is the Division of Mathematical Sciences, Nanyang Technological University, SPMS-MAS-5-43, 1 Nanyang Link Singapore nprivault@ntu.edu.sg Istituto per le Applicazioni del Calcolo Mauro Picone, CNR, Via dei Taurini 19, 185 Roma, Italy. torrisi@iac.rm.cnr.it 1

2 usual inner product on L (R +, B(R + ), l), with l the Lebesgue measure. Although the Ornstein-Uhlenbeck operator L has nice contractivity properties as well as an integral representation, it can be difficult to compute in practice as its eigenspaces are made of multiple stochastic integrals. Thus, although the Ornstein-Uhlenbeck operator applies particularly well to functionals based on multiple stochastic integrals, it is of a more delicate use in applications to functionals whose multiple stochastic integral expansion is not explicitly known. This is due to the fact that the operator L is expressed as the composition of a divergence and a gradient operator, on both the Poisson and Wiener spaces. In this paper we derive bounds on distances between probability laws using covariance representations based on the Clark-Ocone representation formula. In contrast with covariance identities based on the number operator, which relies on the divergence-gradient composition, the Clark-Ocone formula only requires the computation of a gradient and a conditional expectation. In particular, in Corollary 3.4 below we show that (1.1) can be replaced by d W (F, N) E[ 1 D F, E[D F F, (1.) where F is a functional of a normal martingale such that E[F =. Here, D denotes a Malliavin type gradient operator having the chain rule of derivation, and {F t } t is the natural filtration of the normal martingale. In case D is the classical Malliavin gradient on the Wiener space, the bound (1.) offers an alternative to (1.1). For example, if F = I n (f n ) is a multiple stochastic integral with respect to the Brownian motion and the symmetric kernels f n satisfy certain integrability conditions the inequality (1.) gives d W (I n (f n ), N) 1 n! f n L (R n ) + + n n ( ) 4 n 1 (k!) ((n 1) k)! k k= g s n 1,k, gt n 1,k L (R (n 1 k) + ) obtained by the multiplication formula for multiple Wiener integrals, where g t n 1,k( ) = (f n (, t)) k (f n (, t)1 [,t n 1( )), t R +, dsdt (1.3) and the symbol k denotes the canonical symmetrization of the L contraction over k variables, denoted by k. On the other hand, by Proposition 3. of [6 the inequality (1.1)

3 yields d W (I n (f n ), N) 1 n! f n L (R n ) + +n n ( ) 4 n 1 (k!) ((n 1) k)! f n k+1 f n k. L (R (n k 1) + ) k= However, due to its importance, the Wiener case will be the object of a more detailed analysis in a subsequent work. Here our focus will be on the Poisson space, for which (1.) provides an alternative to Theorem 3.1 of [7. Several applications are considered in Section 4. This includes functionals of Poisson jump times (T k ) k 1 of the form f(t k ), for which we obtain the bound cf. d W (f(t k ), N) 1 f (T k ) Tk E[f (T k h + t) h=nt dt L 1 (P ), Proposition 4.1, and a similar result for the gamma approximation, with linear and quadratic functionals of the Poisson jump times as examples. The analogs of (1.3) for Poisson multiple stochastic integrals are treated in Proposition 4.3, and comparisons with the results of [7 are discussed. This paper is organized as follows. In Section we present a general framework for bounds on probability distances based on an abstract integration by parts formula. Next in Section 3 we show that the conditions of this integration by parts setting can be satisfied under the existence of a Clark-Ocone type stochastic representation formula. In Section 4 we apply this general setting to a Clark-Ocone formula stated with a derivation operator on the Poisson space, and consider several examples, including multiple stochastic integrals and other functionals of jump times. In Section 5 we consider the total variation distance between a normalized Poisson compound sum and the standard Gaussian distribution. We close this section by quoting Stein s lemmas for normal and gamma approximations. The following lemma on normal approximation can be traced back to Stein s contribution [1, see also the recent survey [5, and [6. Lemma 1.1 Let h : R [, 1 be a continuous function. The functional equation has a solution f h C 1 b (R) given by f (x) = xf(x) + h(x) E[h(N), x R, f h (x) = e x / x (h(a) E[h(N))e a / da, 3

4 with the bounds f h (x) π and f h(x), x R. The next lemma on the gamma approximation can be found in e.g. Lemma 1.3-(ii) of [6. In the sequel we denote by Γ(ν/) a random variable distributed according to the gamma law with parameters (ν/, 1), ν >. Lemma 1. Let h : R R be a twice differentiable function such that h(x) ce ax, x > ν, for some c > and a < 1/. Then, letting Γ ν := Γ(ν/) ν, the functional equation (x + ν)f (x) = xf(x) + h(x) E[h(Γ ν ), x > ν, (1.4) has a solution f h which is bounded and differentiable on ( ν, ), and such that f h h and f h h. General results.1 Integration by parts The main results of this paper will be derived under the abstract integration by parts (IBP) formula (.1) below. Let T denote a subset of C 1 (R) containing the constant functions. Given F and G two real-valued random variables defined on a probability space (Ω, F, P ) and A F an event with P (A) >, we let Cov A (F, G) := E[F G A E[F AE[G A denote the covariance of F and G given A. The following general Assumption.1 says that the integration by parts formula with weights W 1 and W holds for a random variable F given A on T. Assumption.1 Given F a random variable, we assume that there exist two real-valued random variables W 1 L 1 (P ( A)) and W such that E[W φ (F ) A = Cov A (φ(f ), W 1 ), (.1) for any φ T such that φ(f ), W 1 φ(f ), and W φ (F ) L 1 (P ( A)). 4

5 In particular, we note that if the IBP formula (.1) with weights W 1 and W holds on T for the random variable F given A, then W 1 is centered with respect to P ( A) if and only if we have E[W 1 φ(f ) A = E[W φ (F ) A, φ T, as follows by taking φ = 1 identically. An implementation of this formula on the Poisson space will be provided in Section 4 via the Clark-Ocone representation formula.. Normal approximation Total variation distance The total variation distance between two real-valued random variables Z 1 and Z with laws P Z1 and P Z is defined by d T V (Z 1, Z ) := sup P Z1 (C) P Z (C) = sup P Z1 (C) P Z (C), C B(R) C B b (R) where B(R) and B b (R) stand for the families of Borel and bounded Borel subsets of R, respectively. The following bounds on the total variation distance d T V (F A, N) between the law of F given A and the law of N hold under Assumption.1. Theorem.1 Let A F be such that P (A) > and assume that the IBP formula (.1) holds for F given A on T = C 1 b (R). Then 1. If W = 1 and W 1 is P ( A)-centered we have π d T V (F A, N) E[ W 1 F A. (.). If W 1 = F is P ( A)-centered we have d T V (F A, N) E[ 1 W A. (.3) Proof. 1) Take C B b (R) and let a > be such that C [ a, a. Consider a sequence of continuous functions h n : R [, 1, n 1, such that lim n h n (x) = 1 C (x), µ-a.e. where µ(dx) = (dx + P F A (dx)) [ a,a (restriction to [ a, a of the sum of the Lebesgue measure and the law of F given A), cf. [11 or Corollary 1.1 of [. Lemma 1.1 and the integration by parts formula (.1) show that for any n 1 we have E[h n (F )1 A E[h n (N)P (A) = E[(f h n (F ) F f hn (F ))1 A (.4) 5

6 = E[f hn (F )(W 1 F )1 A π E[ W 1 F 1 A. Dividing first this inequality by P (A) > and then taking the limit as n goes to infinity, the Dominated Convergence Theorem shows that P (F C A) P (N C) π E[ W 1 F A, for any C B b (R). The claim follows taking the supremum over all bounded Borel sets. ) By (.4) and the integration by parts formula, for any n 1, we have E[h n (F )1 A E[h n (N)P (A) = E[f h n (F )(1 W )1 A E[ 1 W 1 A. The claim follows arguing exactly as in case (1) above. Wasserstein distance The Wasserstein distance between the laws of Z 1 and Z is defined by d W (Z 1, Z ) := sup E[h(Z 1 ) E[h(Z ), h Lip(1) where Lip(1) denotes the class of real-valued Lipschitz functions with Lipschitz constant less than or equal to 1. We have the following upper bound for the Wasserstein distance between a centered random variable F and N. Theorem. Assume that the IBP formula (.1) holds for F given A with W 1 = F, on the space T of twice differentiable functions whose first derivative is bounded by 1 and whose second derivative is bounded by. Then we have provided F is P ( A)-centered. d W (F A, N) E[ 1 W A, (.5) Proof. Using the bound (.33) in [7 and the IBP formula (.1), we have d W (F A, N) sup E[φ (F ) F φ(f ) A φ T = sup E[φ (F )(1 W ) A E[ 1 W A. φ T 6

7 .3 Gamma approximation Here we use the distance d H (Z 1, Z ) := sup E[h(Z 1 ) E[h(Z ), (.6) h H where H := {h C b(r) : max{ h, h, h } 1}. The following upper bound for the d H -distance between the centered random variable F given A and a centered gamma random variable holds under the IBP formula (.1) of Assumption.1. Theorem.3 Let F be a P ( A)-centered, a.s. ( ν, )-valued random variable. Given A F such that P (A) >, assume that the IBP formula (.1) holds for F given A on T = C 1 b (R) with W 1 = F. Then we have d H (F A, Γ ν ) E[ (F + ν) W A, (.7) where the random variable Γ ν is defined in Lemma 1.. Proof. Let h H be arbitrarily fixed. Since h is bounded above by 1, there exist c > and a < 1/ such that h(x) ce ax, x > ν (take c > 1 and < a < 1/ so small that 1 < ce aν ). Let f h be solution of (1.4) (its existence is guaranteed by Lemma 1.). By the IBP formula (.1) on C 1 b (R) for the centered random variable F given A with W 1 = F, we have E[h(F )1 A E[h(Γ ν )P (A) = E[((F + ν)f h(f ) F f h (F ))1 A = E [((F + ν)f h(f ) W f h(f ))1 A h E[ (F + ν) W 1 A. The claim follows by dividing the above inequality by P (A) > and then taking the supremum over all functions h H. 3 Integration by parts via the Clark-Ocone formula In this section we consider an implementation of the the IBP formula (.1) of Assumption.1, based on the Clark-Ocone formula for a real-valued normal martingale (M t ) t defined on a 7

8 probability space (Ω, F, P ), generating a right-continuous filtration (F t ) t. In other words, (M t ) t is a square integrable martingale with respect to the natural filtration F t = σ(m s : s t), such that E[ M t M s F s = t s, s < t, and the filtration is rightcontinuous. Let l be the Lebesgue measure on R +. In this section we assume the existence of a gradient operator D : Dom(D) L (Ω, F, P ) L (Ω R +, F B(R + ), P l) with domain Dom(D), defined by DF = (D t F ) t and satisfying the following properties: (i) D satisfies the Clark-Ocone representation formula F = E[F + E[D t F F t dm t, F Dom(D), (3.1) (ii) D satisfies the chain rule of derivation D t φ(f ) = φ (F )D t F, F Dom(D), (3.) for all φ T C 1 (R), cf. e.g. 3.6 of [1 (here T contains the constant functions). This condition will be satisfied in both the Wiener and Poisson settings of Section 4. In addition we will assume that for any F Dom(D) and φ T we have φ(f ) Dom(D). From (3.1) the gradient operator D satisfies the following covariance identity, cf. e.g. Proposition in [1, p. 11. Lemma 3.1 For any F, G Dom(D) we have [ Cov(F, G) = E E[D t F F t D t G dt. (3.3) 3.1 Integration by parts We now implement the IBP formula (.1) for functionals in the domain of D, based on the Clark-Ocone representation formula (3.1). Note that IBP formulas of the form (.1) can also be obtained by the Ornstein-Uhlenbeck semigroup, cf. e.g. Proposition.1 of [3. Proposition 3. If F, G Dom(D) and φ (F )ϕ F,G (F ) L 1 (P ) for any φ T, then the IBP formula (.1) holds on T for F with W 1 = G and W = ϕ F,G (F ), i.e. E[ϕ F,G (F )φ (F ) = Cov(φ(F ), G) 8

9 where ϕ F,G is the function [ ϕ F,G (z) := E D t F E[D t G F t dt F = z, z R. (3.4) Proof. By Lemma 3.1 and the properties of the gradient operator, for any φ T and F, G Dom(D), we have [ Cov(φ(F ), G) = E E[D t G F t D t φ(f ) dt [ = E φ (F ) D t F E[D t G F t dt [ [ = E E φ (F ) D t F E[D t G F t dt F = E[φ (F )ϕ F,G (F ). (3.5) 3. Normal and gamma approximation We now apply Theorems.1 and. using the Clark-Ocone formula (3.3). For any F Dom(D), we define [ ϕ F (z) := ϕ F,F (z) = E D t F E[D t F F t dt F = z, z R (3.6) and note that by Jensen s inequality ϕ F (F ) L 1 (P ) DF L (P l) = E [ D t F dt <, F Dom(D). The next proposition follows as a simple consequence of Theorems.1,.,.3 and Proposition 3. and uses the definition (.6) of the distance d H. Proposition 3.3 For any F Dom(D) such that E[F =, we have d T V (F, N) E[ 1 ϕ F (F ) and d W (F, N) E[ 1 ϕ F (F ), where ϕ F is defined in (3.6). If moreover F is a.s. ( ν, )-valued then we have d H (F, Γ ν ) E[ (F + ν) ϕ F (F ). 9

10 Letting, denote the usual inner product on L (R + ), from Proposition 3.3 we also have the following corollary: Corollary 3.4 For any F Dom(D) such that E[F =, we have and d T V (F, N) 1 D F, E[D F F L (P ) 1 F L (P ) + D F, E[D F F E[ D F, E[D F F L (P ) d W (F, N) 1 D F, E[D F F L (P ) 1 F L (P ) + D F, E[D F F E[ D F, E[D F F L (P ). If moreover F is a.s. ( ν, )-valued then we have d H (F, Γ ν ) (F + ν) D F, E[D F F L (P ) Proof. (F + ν) F L (P ) L (P ) + D F, E[D F F E[ D F, E[D F F L (P ). The first inequality follows by Proposition 3.3 and the Cauchy-Schwarz inequality. The second inequality follows by the triangle inequality noticing that by the Itô isometry and the Clark-Ocone formula we have [ E[ D F, E[D F F = E D t F E[D t F F t dt [ = E E[D t F F t dt [ ( ) = E E[D t F F t dm t = F L (P ). The counterpart of this statement for the Wasserstein and d H distances is proved similarly. In this work our main focus will be on the Poisson space, and in Section 4, we shall compare the upper bound on the Wasserstein distance with the bound obtained in [7 on the Poisson space. 1

11 4 Analysis on the Poisson space In this section we apply the results of Section 3 to functionals of a standard Poisson process (N t ) t with jump times (T k ) k 1 defined on an underlying probability space (Ω, F, P ). We let S = {F = f(t 1,..., T d ) : d 1, f C 1 p(r d +)}, where C 1 p(r d +) denotes the space of continuously differentiable functions such that f and its partial derivatives have polynomial growth, i.e. for any i {, 1,..., d} there exist α (i) j, j = 1,..., d, such that sup (x 1,...,x d ) R d + x α(i) x α(i) d d i f(x 1,..., x d ) <, where f := f. Given F = f(t 1,..., T d ) S, we consider the gradient on the Poisson space defined as D t F = d 1 [,Tk (t) k f(t 1,..., T d ), t (4.1) k=1 (see e.g. Definition 7..1 in [1 p. 56). We recall that the gradient operator D : S L (Ω, F, P ) L (Ω R +, F B(R + ), P l) is closable (see [1 p. 59). We shall continue to denote by D its minimal closed extension, whose domain Dom(D) coincides with the completion of S with respect to the norm F 1, = F L (P ) + DF L (P l). By Proposition 7..8 in [1 p. 6 the operator D satisfies the Clark-Ocone representation formula, i.e. for any F Dom(D) we have F = E[F + E[D t F F t (dn t dt), (4.) where (F t ) t is the filtration generated by (N t ) t. We note that the gradient D satisfies the chain rule on the set T of real-valued functions which have polynomial growth and are continuously differentiable with bounded derivative, i.e. for any g T and F Dom(D) we have g(f ) Dom(D) and Dg(F ) = g (F )DF, cf. Lemma 6.1 in the Appendix. Before turning to some concrete examples of Poisson functionals we note that identifying the process E[D t F F t in (4.) amounts to finding the predictable representation of the random 11

12 variable F. For example if F = X T stochastic differential equation is the terminal value of the solution (X t ) t [,T to the X t = X + t σ(s, X s )(dn s ds), (4.3) where σ : [, T Ω R is a measurable function, then we immediately have E[D t F F t = σ(t, X t ) and Corollary 3.4 shows that e.g. T d T V (X T, N) 1 σ(t, X t )D t X T dt L (P ) 1 X T T L (P ) + (σ(t, X t )D t X T E[σ(t, X t )D t X T ) dt L (P ) provided the terminal value X T belongs to Dom(D) and E[X =. In particular, the domain condition can be achieved under a usual Lipschitz condition on σ(, x), x R, and a usual sub-linear growth condition on σ(t, ), t [, T. We refer the reader to e.g. Proposition 3. of [4 for an explicit solution of (4.3) which is suitable for D-differentiation when σ(t, x) vanishes at t = T, for any x R. 4.1 Approximation of Poisson jump times functionals Proposition 4.1 Let f C 1 p(r + ) be such that E[f(T k ) =, k 1. Then d T V (f(t k ), N) 1 f (T k ) Tk E[f (T k h + t) h=nt dt L 1 (P ), (4.4) and Tk d W (f(t k ), N) 1 f (T k ) E[f (T k h + t) h=nt dt (4.5) L 1 (P ). If moreover f(t k ) > ν a.s. we have Tk d H (f(t k ), Γ ν ) (f(t k ) + ν) f (T k ) E[f (T k h + t) h=nt dt (4.6) L 1 (P ). Proof. We have f(t k ) Dom(D) and D t f(t k ) = f k (T k)d t T k = f k (T k)1 [,Tk (t), t. By the formula in [1 p. 61 we have E[D t f(t k ) F t = t f (x)p k 1 Nt (x t) dx, where p k (t) = P (N t = k). So [ ϕ f(tk )(f(t k )) = E D t f(t k )E[D t f(t k ) F t dt f(t k ) 1

13 Finally, by Proposition 3.3 we deduce [ Tk = E f (T k ) E[D t f(t k ) F t dt f(t k ) [ Tk = E f (T k ) f (x)p k 1 Nt (x t) dxdt f(t k ) t [ Tk = E f (T k ) f (x + t)p k 1 Nt (x) dxdt f(t k ) [ Tk = E f (T k ) E[f (T k h + t) h=nt dt f(t k ). (4.7) d T V (f(t k ), N) E[ 1 ϕ f(tk )(f(t k )) [ E [ Tk = E 1 f (T k ) E[f (T k h + t) h=nt dt f(t k ) 1 f (T k ) Tk E[f (T k h + t) h=nt dt L 1 (P ). The inequalities concerning d W and d H can be proved similarly. Example - Linear Poisson jump times functionals Proposition 4.1 can be applied to linear functionals of Poisson jump times. Consider first the normal approximation. Take e.g. f(x) = (x k)/ k, i.e. f(t k ) = (T k k)/ k, k 1, and note that T k /k is gamma distributed with mean 1 and variance 1/k. All hypotheses of Proposition 4.1 are satisfied and we have Tk 1 f (T k ) E[f (T k h + t) h=nt dt = T k L 1 (P ) k 1 L Var(T k /k) = 1, 1 (P ) k where the latter inequality follows by the Cauchy-Schwarz inequality. So (4.4) and (4.5) recovers the classical Berry-Esséen bound. For the gamma approximation we take e.g. ν := k and f(x) = (x k), k 1. In such a case Γ ν has the same law of f(t k ) and we check that d H (f(t k ), Γ ν ) =. Indeed, (f(t k ) + ν) f (T k ) Tk E[f L (T k h + t) h=nt dt = 4T k 4T k =. L 1 (P ) 1 (P ) Example - Quadratic Poisson jump times functionals Proposition 4.1 can also be applied to quadratic functionals of Poisson jump times. Consider first the normal approximation, take e.g. f(x) = αx β, with α = 1 k 3/ and β = k + 1 k, k 1. 13

14 Recall that if X is gamma distributed with parameters a and b, then E[X k = (a+k 1)(a+ k ) (a + 1)a/b k, k 1. One easily sees that all the assumptions of Proposition 4.1 are satisfied, and we find 1 f (T k ) Tk E[f (T k h + t) h=nt dt 1 k + 4 k + 1 k + 6 k k L 1 (P ) ( 4 3k k + 3k 3 + ) 4 k + 1 k + 6 k +, 3 k (4.8) cf. the Appendix. Note that this upper bound is asymptotically equivalent to ( )/ k as k, and so we recover the Berry-Esséen bound. 4. Approximation of multiple Poisson stochastic integrals We present some applications of Corollary 3.4 to Poisson functionals. For n 1, we denote by I n (f n ) = n! t n t f n (t 1,..., t n ) (dn t1 dt 1 ) (dn tn dt n ) the multiple Poisson stochastic integral of the symmetric function f n L (R n +) with I n (f n ) = I n ( f n ) when f n is not symmetric, where f n denotes the symmetrization of f n in n variables (see e.g. Section 6. in [1). As a convention we identify L (R +) with R, and let I (f ) = f, f L (R +). Moreover, we shall adopt the usual convention j k=i = if i > j. Let the space Sn 1, of weakly differentiable functions be defined as the completion of the symmetric functions f n C 1 c([, ) n ) under the norm f n 1, = f n L (R n + ) + = f n L (R n + ) + t 1 f n (s 1,..., s n ) ds 1 dtds ds n 1 f n[t (s 1,..., s n ) ds 1 dtds ds n (4.9) where i f n[t (s 1,..., s n ) = i f n (s 1,..., s n )1 [t, ) (s i ). The next lemma is proved in the Appendix, cf. Proposition 8 of [9 or Proposition 7.7. page 79 of [1. Lemma 4. For any function f n S 1, n Dom(D) with symmetric in its n variables we have I n (f n ) D t I n (f n ) = ni n 1 (f n (, t)) ni n ( 1 f n[t ), t R +, (4.1) 14

15 and DI n (f n ) L (P l) = n (n 1)! +n n! t f n (t 1,..., t n ) dt 1 dt n 1 f n (t 1,..., t n ) dt 1 dtdt dt n. We recall the multiplication formula for multiple Poisson stochastic integrals, cf. e.g. Proposition of [1. For symmetric functions f n L (R n +) and g m L (R m +), we define f n l k g m, l k, to be the function (x l+1,..., x n, y k+1,..., y m ) f n (x 1,..., x n )g m (x 1,..., x k, y k+1,..., y m ) dx 1 dx l R l + of n+m k l variables. We denote by f n l k g m the symmetrization in n+m k l variables of f n l k g m, l k. Note that if k = l then k := k k is the classical L contraction over k variables and k := k k is the canonical symmetrization of k. We have if the functions h n,m,k = I n (f n )I m (g m ) = k i (k n m) (n m) k= ( n i! i )( m i I n+m k (h n,m,k ) )( i k i ) f n k i i belong to L (R n+m k + ), k (n m). In particular, letting 1 {(t1,...,t n)<t} denote the function 1 [,t n(t 1,..., t n ), for any symmetric function f n Sn 1,, we have if the functions g (1,t) n 1,n 1,k = I n 1 (f n (, t))i n 1 (f n (, t)1 { <t} ) = k i (k (n 1)) belong to L (R n k + ), k n, and if the functions g (,t) n,n 1,k = n k= g m I n k (g (1,t) n 1,n 1,k ) (4.11) ( ) i! n 1 ( ) i f n (, t) k i i f n (, t)1 { <t} (4.1) i k i I n ( 1 f n[t )I n 1 (f n (, t)1 { <t} ) = k i (k (n 1)) n k= ( )( )( ) n n 1 i i! 1 f n[t k i i i i k i I n 1 k (g (,t) n,n 1,k ) (4.13) f n (, t)1 { <t} belong to L (R n 1 k + ), k n. Part () of the next proposition proposes an alternative to the Gamma bound of Theorem.6 of [8. 15

16 Proposition 4.3 1) For any symmetric function f n S 1, n such that g (1,t) n 1,n 1,k L (R n k + ), k n and g (,t) n,n 1,k L (R n 1 k + ), k n, we have d T V (I n (f n ), N) 1 n! f n L (R n ) + ( n 3 +n (n k)! g (1,t) n 1,n 1,k g(,t) n,n 1,k+1, g(1,s) n 1,n 1,k g(,s) n,n 1,k+1 L (R n k (, ) + ) dsdt and + k= (n 1)! g (,t) n,n 1,, g (,s) n,n 1, L (R n 1 + ) dsdt ) 1/ d W (I n (f n ), N) 1 n! f n L (R n ) + ( n 3 +n (n k)! g (1,t) n 1,n 1,k g(,t) n,n 1,k+1, g(1,s) n 1,n 1,k g(,s) n,n 1,k+1 L (R n k (, ) + ) dsdt + k= (n 1)! g (,t) n,n 1,, g (,s) n,n 1, L (R n 1 + ) dsdt ) 1/. ) If moreover I n (f n ) is a.s. ( ν, )-valued then we have d H (I n (f n ), Γ ν ) ( n 3 +n (n k)! + Proof. k= 4ν + n! f n 4 L (R n + ) + 4n!(1 ν) f n L (R n + ) g (1,t) n 1,n 1,k g(,t) n,n 1,k+1, g(1,s) n 1,n 1,k g(,s) n,n 1,k+1 L (R n k (, ) + ) dsdt ) 1/. (n 1)! g (,t) n,n 1,, g (,s) n,n 1, L (R n 1 + ) dsdt By Lemma 4. we have I n (f n ) Dom(D) and D t I n (f n ) = ni n 1 (f n (, t)) ni n ( 1 f n[t ), t R +. So by Lemma.7. p. 88 of [1 and the definition of 1 f n[t, we have E[D t I n (f n ) F t = ni n 1 (f n (, t)1 { <t} ) ni n ( 1 f n[t 1 [,t n) = ni n 1 (f n (, t)1 { <t} ), (4.14) 16

17 since f n (, t)1 { <t} =, t R +. Combining this with the multiplication formulas (4.11) and (4.13) for multiple Poisson stochastic integrals, we deduce D I n (f n ), E[D I n (f n ) F = n I n 1 (f n (, t))i n 1 (f n (, t)1 { <t} ) dt n I n ( 1 f n[t )I n 1 (f n (, t)1 { <t} ) dt n = n n = n k= k= n n = n k= n = n I n k (g (1,t) n 1,n 1,k ) dt n I n k (g (1,t) n 1,n 1,k ) dt n I n 1 (g (,t) n,n 1,) dt I n k (g (1,t) n 1,n 1,k ) dt n I n 1 (g (,t) n,n 1,) dt I (g (1,t) n 1,n 1,n ) dt + n n I n 1 (g (,t) = n (n 1)! n 3 +n k= = n (n 1)! n 3 +n k= n,n 1,) dt n 3 k= n k= n k=1 n 3 s= f n (, t) n 1 n 1 f n (, t)1 { <t} dt I n k (g (1,t) n 1,n 1,k g(,t) I n 1 k (g (,t) n,n 1,k ) dt I n 1 k (g (,t) n,n 1,k ) dt I n s (g (,t) n,n 1,s+1) dt I n k (g (1,t) n 1,n 1,k g(,t) n,n 1,k+1 ) dt n,n 1,k+1 ) dt n I n 1 (g (,t) n,n 1,) dt [,t n 1 f n (t 1,... t n 1, t)f n (t 1,... t n 1, t) dt 1 dt n 1 dt I n k (g (1,t) n 1,n 1,k g(,t) n,n 1,k+1 ) dt n I n 1 (g (,t) n,n 1,) dt. Using the first equality above and the isometry formula for multiple Poisson stochastic integrals (see Proposition.7.1 p. 87 in [1) we have Hence E[ D I n (f n ), E[D I n (f n ) F = n (n 1)! f n (t 1,... t n 1, t)f n (t 1,... t n 1, t) dt 1 dt n 1 dt. [,t n 1 DI n (f n ), E[DI n (f n ) F E[ DI n (f n ), E[DI n (f n ) F 17

18 n 3 = n k= I n k (g (1,t) n 1,n 1,k g(,t) n,n 1,k+1 ) dt n I n 1 (g (,t) n,n 1,) dt. We conclude by Corollary 3.4, noticing that I n (f n ) is a centered random variable (see [1 pp ), I n (f n ) L (P ) = n! f n L (R n + ), for any ν > and n 3 k= (I n (f n ) + ν) n! f n L (R n + ) L (P ) = n! f n 4 L (R n + ) + 4n!(1 ν) f n L (R n + ) + 4ν, = E +E I n k (g (1,t) n 1,n 1,k g(,t) ( n 3 k= [ ( n 3 = E = = = +E n 3 k= +E n 3 k= [ E k= ( n 3 k= [ ( E n,n 1,k+1 ) dt I n k (g (1,t) n 1,n 1,k g(,t) ) I n 1 (g n,n 1,) (,t) dt [ ( [ ( ) n,n 1,k+1 ) dt I n 1 (g (,t) n,n 1,) dt L (P ) I n k (g (1,t) n 1,n 1,k g(,t) n,n 1,k+1 )I n 1(g n,n 1,) (,s) ds dt I n k (g (1,t) n 1,n 1,k g(,t) ) I n 1 (g n,n 1,) (,t) dt [ E [ +E ) n,n 1,k+1 ) dt ) I n k (g (1,t) n 1,n 1,k g(,t) n,n 1,k+1 ) dt ) I n 1 (g n,n 1,) (,t) dt n 3 (n k)! k= +(n 1)! I n k (g (1,t) n 1,n 1,k g(,t) n,n 1,k+1 )I n k(g (1,s) n 1,n 1,k g(,s) n,n 1,k+1 ) dsdt I n 1 (g (,t) n,n 1,)I n 1 (g (,s) n,n 1,) dsdt g (1,t) n 1,n 1,k g(,t) n,n 1,k+1, g(1,s) n 1,n 1,k g(,s) n,n 1,k+1 L (R n k + ) dsdt g (,t) n,n 1,, g (,s) n,n 1, L (R n 1 + ) dsdt. 18

19 Single Poisson stochastic integrals In the particular case n = 1, the space S 1, 1 is the completion of C 1 c([, )) under the norm f 1, = f L (R + ) + f[ L (R + ) L (R + ) : = f(t) dt + f (s) dsdt = f(t) dt + s f (s) ds where f [t (s) := f (s)1 [t, ) (s) and we have I 1 (f) Dom(D) with and D t I 1 (f) = f(t) I 1 (1 [t, ) f ), t R +, DI 1 (f) L (P l) = f(t) dt + t f (s) dsdt. The following result is a simple consequence of Proposition 4.3 for n = 1. Corollary 4.4 For any f S 1, 1, we have and ( t d T V (I 1 (f), N) 1 f L (R + ) + f (t) f(z) dz) dt ( t d W (I 1 (f), N) 1 f L (R + ) + f (t) f(z) dz) dt. If moreover I 1 (f) is a.s. ( ν, )-valued then we have t (4.15) ( t d H (I 1 (f), Γ ν ) 4ν + f 4L (R+) + 4(1 ν) f L (R+) + f (t) f(z) dz) dt. Note that Corollary 3.4 of [7 states that for any f L (R + ), which shows that d W (I 1 (f), N) 1 f L (R + ) + f 3 L 3 (R + ), I 1 (f k ) N in law provided f k L (R + ) 1 and f k L 3 (R + ) as k goes to infinity. Next we consider a couple of examples for comparison with Corollary

20 Example - Single Poisson stochastic integrals with specific kernels 1. Take g k (t) = (/k) 1/ e t/k t, k 1. We shall show later on that g k S 1, 1. We have g k L (R + ) = 1 and ( ( t ) 1/ g k(t) g k (z) dz) dt = ( k = ( 1/ (e t/k e 3t/k + e )dt) 4t/k = k hence by Corollary 4.4 we get while Corollary 3.4 of [7 yields and 8/3 > 1. d W (I 1 (g k ), N) d W (I 1 (g k ), N) ) 1/ e t/k 1 e t/k dt 1 3k, g k (t) 3 dt = k 1 3k, To check that g k S 1, 1 it suffices to verify that g S 1, 1, where g(t) = e t, t. Let {χ k } k 1 be a sequence of functions in C 1 c([, )) with χ k (t) 1, for any t, χ k (t) = 1, for any t [, k, and sup k 1,t χ k (t) <. Then one may easily see that {χ k g} k 1 is a sequence in C 1 c([, )) converging to g in the norm 1,.. Take g k (t) = (k + 1 )1/ 1 [, 1 (k+ 1 )(t) cos(πt), t, k 1. Note that g k is continuous and piecewise differentiable (with a piecewise continuous derivative) and so g k is weakly differentiable. We shall show later on that g k S 1, 1. We have and g k L (R + ) = 1 (k+ 1 ) (k + 1 cos(πt) dt = 1, )1/ ( ( t g k(t) g k (z) dz) dt = ) 1/ ( 4π 1 (k+ 1 ) ( t (k + 1 sin(πt) g k (z) dz) dt )1/ ) 1/

21 ( = 8π 1 (k+ 1 ) ( t k + 1 sin(πt) ( 4 1 (k+ 1 ) = sin(πt) 4 dt k + 1 = 4 k + 1 ( (k+ 1 ) ( = 3 1 (k + 1 ) k = k + 1 hence by Corollary 4.4 we get k + 1 = 3 k + 1 sin(πt) dt 1 (k+ 1 ) d W (I 1 (f k ), N) whereas by Corollary 3.4 of [7 we have Note that 16/(3π) < 3. d W (I 1 (f k ), N) cos(πz) dz) dt ) 1/ ) 1/ 3 k + 1, cos(πt) dt g k (t) 3 dt = π. k + 1 ) 1/ ) 1/ To check that g k S 1, 1 it suffices to verify that g S 1, 1, where g(t) = 1 [,π/ (t) cos t, t. Let ρ be a smooth probability density on [, ) with support in [, 1, ρ k (t) = kρ(kt) and G k (t) := g ρ k (t) = t ρ k (t s)g(s) ds, t. Then one may easily see that {G k } k 1 is a sequence in C 1 c([, )) converging to g in the norm 1,. Double Poisson stochastic integrals For the case of double Poisson stochastic integrals we have the following corollary. Corollary 4.5 For any symmetric function f S 1,, we have d T V (I (f), N) 1 f L (R + ) ( +8! f(x, t)f(x, s) dx (, ) 1 t s f(y, t)f(y, s) dydsdt

22 and + (, ) +3! (, ) t s f(x, t)f(x, s) dxdsdt t s 1 f(x, y) dxdy t s f(z, t)f(z, s) dzdsdt ) 1/ d W (I (f), N) 1 f L (R + ) ( +4! f(x, t)f(x, s) dx + (, ) +3! (, ) t s (, ) f(x, t)f(x, s) dxdsdt t s 1 f(x, y) dxdy t s t s If moreover I (f) is a.s. ( ν, )-valued then we have Proof. f(y, t)f(y, s) dydsdt f(z, t)f(z, s) dzdsdt) 1/. d H (I (f), Γ ν ) ( f 4 L (R + ) + 8(1 ν) f L (R + ) + 4ν ) 1/ ( t s +4! f(x, t)f(x, s) dx f(y, t)f(y, s) dydsdt + (, ) +3! (, ) t s (, ) f(x, t)f(x, s) dxdsdt t s 1 f(x, y) dxdy t s f(z, t)f(z, s) dzdsdt) 1/. It follows after a direct computation taking n = in Proposition 4.3 and noticing that, for any f S 1,, g (1,t) 1,1,(x, y) = f(x, t)f(y, t)1 (,t) (y), and g (1,t) 1,1,1(x) = f(x, t) 1 (,t) (x), g (,t),1,(x, y, z) = 1 f(x, y)f(z, t)1 [t, ) (x)1 (,t) (z), g (1,t) 1,1, = t g (,t),1,1 g (,t),1,. f(x, t) dx 5 Normal approximation of the compound Poisson distribution In this section we present an application of formula (.) to the compound Poisson distribution. Let (Z k ) k 1 be a sequence of real-valued i.i.d. random variables independent of a

23 Poisson distributed random variable N n with parameter n 1. We assume that Z 1 has moments of any order and that its distribution has a continuously differentiable density p Z1 (z) with respect to the Lebesgue measure, such that lim z ± z p p Z1 (z) = for all p 1. We also assume that d dz log p Z 1 (z) = p Z 1 (z)/p Z1 (z) has at most polynomial growth. Consider the sequence F n := Nn k=1 Z k ne[z 1, n 1. ne[z 1 It is well-known that F n N, in law, as n. In the following we are going to upper bound the total variation distance between F n and N. The following lemma applies the IBP formula of [1 to each F n N n = m, m, n 1. Lemma 5.1 Let m, n 1 be fixed integers. Under the foregoing assumptions on the law of the jump amplitude Z 1, we have that the IBP formula (.1) holds on C 1 b (R) for F n N n = m with P ( N n = m)-centered ne[z W 1 = W n (m) := 1 m and W = 1. m k=1 q Z1 (Z k ) where q Z1 (z) := p Z 1 (z) p Z1 (z) 1 {pz1 (z)>} Proof. See Theorem 3.1 and Section 4 in [1. We have the following bound for the total variation distance. Proposition 5. Under the foregoing assumptions on the law of the jump amplitude Z 1, we have where Proof. d T V (F n, N) e n + For any n 1, we have d T V (F n, N) = sup C B(R) = sup C B(R) R n = n N n P (F n C) P (N C) π (1 R n)1 {Nn 1} L (P ), n 1 E[Z1 N n k=1 q Z 1 (Z k ) Nn k=1 Z k ne[z 1. [P (F n C N n = m)p (N n = m) P (N C)P (N n = m) m m d T V (F n N n = m, N)P (N n = m) e n + π E[ W n (m) F n N n = mp (N n = m) (5.1) m 1 3

24 π = e n + F n 1 {Nn 1} [ π = e n + E F n 1 W n (Nn) 1{Nn 1} F n ( ) π e n + F n L (P ) 1 W n (Nn) F n ( ) π = e n + 1 W n (Nn) 1 {Nn 1} F n L (P ), E[ W (Nn) n 1 {Nn 1} L (P ) where the inequality (5.1) follows by (.). Example - Normal approximation of Poisson compound sums with Gaussian addends Suppose that Z 1 is a standard Gaussian random variable. Then all the hypotheses of Proposition 5. are satisfied and R n = n/n n. So, for any n 1, π ) d T V (F n, N) e n + (1 nnn We have ) (1 nnn 1 {Nn 1} L (P ) = e n/ m 1 1 {Nn 1} ( 1 n ) n m m m! (5.) L (P ). = 1 e n ne n S 1 (n) + n e n S (n), (5.3) where and S 1 (n) := m 1 S (n) := m 1 n m m m! n m m m!. Note that these two series converge (by e.g. the ratio test), but their sum do not have a closed form. After some manipulations, one can realize that S 1 (n) and S (n) have the following series expansion at n = : S 1 (n) = e n (n 1 + n + O(n 3 )) + log n 1 iπ γ + O(n 7 ) and S (n) = e n (n +3n 3 +O(n 4 )) 1 log (n 1 )+(γ+iπ) log n 1 +5π /1 iγπ γ /+O(n 6 ) 4

25 where γ denotes the Euler-Mascheroni constant. So n(1 e n ne n S 1 (n) + n e n S (n)) = 1 + ω(n), where ω(n) is a suitable function converging to zero as n. Therefore, combining (5.) and (5.3) we get a Berry-Esseen type upper bound for F n, which is asymptotically equivalent to π, as n. n 6 Appendix Lemma 6.1 Let g : R R be with polynomial growth and continuously differentiable with bounded derivative and F Dom(D), we have g(f ) Dom(D) and Dg(F ) = g (F )DF. Proof. For any F S we clearly have g(f ) S and the claim is an immediate consequence of the action of D t on S and the chain rule for the derivative. Now, take F Dom(D). Then there exists a sequence (F (n) ) n 1 S such that F (n) F in L (P ) and DF (n) DF in L (P l). Since the claim holds for functionals in S, for g : R R with polynomial growth and continuously differentiable with bounded derivative, we have Dg(F (n) ) = g (F (n) )DF (n), n 1. By the convergence in L (P ) we have that there exists a subsequence {n } of {n} such that F (n ) F a.s.. The claim follows if we check that Dg(F (n ) ) g (F )DF in L (P l). By the boundedness of g, for a positive constant C > we have Dg(F (n ) ) g (F )DF L (P l) = g (F (n ) )DF (n ) g (F )DF L (P l) g (F (n ) )DF (n ) g (F (n ) )DF L (P l) + g (F (n ) )DF g (F )DF L (P l) ( [ 1/ C DF (n ) DF L (P l) + E g (F (n ) ) g (F ) D t F dt). This latter quantity tend to zero as n. Indeed, the first term goes to zero since DF n DF in L (P l); the second term goes to zero by the Dominated Convergence Theorem since for some constant C > we have g (F (n ) ) g (F ) D t F C D t F, P l-a.e, DF L (P l) and g (F (n ) ) g (F ) a.s. by the continuity of g. Proof of Lemma 4.. Let f n C 1 c([, ) n ) be a symmetric function and define the sequence F (m) = n ( ) n ( 1) n k k k= 1 l 1 =l k m f n (T l1,..., T lk, s k+1,..., s n ) ds k+1 ds n 5

26 = I n (f n 1 n) [,Tm n ( ) n + ( 1) n k k k= 1 l 1 =l k m R n k + \[,Tmn k f n (T l1,..., T lk, s k+1,..., s n ) ds k+1 ds n, (6.1) m 1. We have (F (m) ) m 1 S and F (m) converges to F = I n (f n ) in L (P ). Indeed, by the isometry formula for multiple Poisson stochastic integrals (see e.g. Proposition 6..4 in [1), we have [ (In E[ F I n (f n 1 [,Tm n) = E (f n (1 1 n))) [,Tm (6.) [ = n!e f n (t 1,..., t n ) (1 1 n(t [,Tm 1,..., t n )) dt 1 dt n and this latter term tends to as m goes to infinity by the Dominated Convergence Theorem. Moreover, each of the remaining terms R n k + \[,Tmn k f n (T l1,..., T lk, s k+1,..., s n ) ds k+1 ds n in (6.1) is bounded and tends to a.s. as m goes to infinity, since f n has compact support. Next, by (4.1) we have n ( ) n D t F (m) = ( 1) n k k = k= k 1 [,Tli (t) i=1 k= + k= = k= n ( ) n ( 1) n k k 1 l 1 =l k m 1 l 1 =l k m i f n (T l1,..., T lk, s k+1,..., s n ) ds k+1 ds n n i f n[t (T l1,..., T lk, s k+1,..., s n ) ds k+1 ds n i=1 n ( ) n ( 1) n k k n i=k+1 n ( ) n ( 1) n k k 1 l 1 =l k m i f n[t (T l1,..., T lk, s k+1,..., s n ) ds k+1 ds n 1 l 1 =l k m n i f n[t (T l1,..., T lk, s k+1,..., s n ) ds k+1 ds n i=1 6

27 ( ) n ( 1) n k k n 1 k= n ( ) n = ( 1) n k k k= n 1 ( ) n ( 1) n k (n k) k k= n 1 l 1 =l k m i=k+1 f n (T l1,..., T lk, s k+1,..., s i 1, t, s i+1,... s n ) ds k+1 ds i 1 ds i+1 ds n 1 1 l 1 =l k m 1 l 1 =l k m n i f n[t (T l1,..., T lk, s k+1,..., s n ) ds k+1 ds n f n (T l1,..., T lk, t, z 1,... z n k 1 ) dz 1 dz n k 1 n ( ) n n = ( 1) n k i f n[t (T l1,..., T lk, s k+1,..., s n ) ds k+1 ds n k k= 1 l 1 =l k m i=1 n 1 ( ) n 1 n ( 1) n k k k= 1 l 1 =l k m i=1 f n (T l1,..., T lk, t, z 1,..., z n k 1 ) dz 1 dz n k 1 = ni n 1 (f n (, t)1 n 1) ni [,Tm n( 1 f n[t 1 n) [,Tm n ( ) n ( 1) n k k k= R n k + \[,Tmn k 1 l 1 =l k m n i f n[t (T l1,..., T lk, s k+1,..., s n ) ds k+1 ds n i=1 n 1 ( n 1 +n ( 1) n 1 k k k= ) 1 l 1 =l k m R n 1 k + \[,T m n 1 k f n (T l1,..., T lk, t, z 1,..., z n 1 k ) dz 1 dz n k 1, t R +. To conclude we note that, as in (6.), ni n 1 (f n (, t)1 n 1) ni [,Tm n( 1 f n[t 1 [,Tm n) converges in L (P l) to D t F := ni n 1 (f n (, t)) ni n ( 1 f n[t ), t R +, (6.5) as m goes to infinity by the isometry formula for multiple Poisson stochastic integrals, and the two terms in (6.3) and (6.4) converge to since f n C 1 c([, ) n ). (6.3) (6.4) In order to complete the proof of the first part of the lemma by closability, given f n Sn 1, we choose a sequence (f n (m) ) m N C 1 c([, ) n ) converging to f n for the norm (4.9) and we 7

28 define the sequence of functionals (F (m) ) m 1 in Dom(D) by F (m) := I n (f (m) n ), m 1. Then we note that by the isometry formula for multiple Poisson stochastic integrals and the convergence of f (m) n to f n in L (R + ), we have I n (f (m) n ) I n (f n ) in L (P ) as m. Moreover, D t F defined by (4.1) and (6.5) satisfies [ E D t F D t F (m) dt [ n E I n ( 1 f n[t ) I n ( 1 f (m) n[t ) dt [ +n E I n 1 (f n (, t)) I n 1 (f n (m) (, t)) dt = n n! +n (n 1)! t 1 f n (s 1,..., s n ) 1 f (m) n (s 1,..., s n ) ds 1 dtds ds n f n (s 1,..., s n ) f (m) n (s 1,..., s n ) ds 1 ds n. So DF (m) converges to DF in L (P l) by the convergence of f (m) n to f n with respect to the norm 1,. Finally, using again the isometry formula for multiple Poisson stochastic integrals, we have [ [ ( E D t F dt = n E In 1 (f n (, t)) I n ( 1 f n[t ) ) dt [ [ = n E (I n 1 (f n (, t))) ( dt + n E In ( 1 f n[t ) ) dt [ n E I n 1 (f n (, t))i n ( 1 f n[t ) dt = n (n 1)! +n n! Proof of (4.8). We have Tk f (T k ) t f n (t 1,..., t n ) dt 1 dt n 1 f n (t 1,..., t n ) dt 1 dtdt dt n. Tk E[f (T k h + t) h=nt dt = 4α T k (t + E[T k h h=nt ) dt Tk = 4α T k (t + k N t ) dt 8

29 Therefore Tk Tk = α Tk 3 + 4kα Tk 4α T k N t dt k = α Tk 3 + 4kα Tk 4α T k (h 1)(T h T h 1 ). 1 f (T k ) E[f (T k h + t) h=nt dt L 1 (P ) [ E[ 1 4kα Tk + α k E T k (h 1)(T h T h 1 ) Tk h=1 E[ 1 4kα Tk + α T k L (P ) [ 1 T = E k + 1 k + 1 k k k k h=1 h=1 k (h 1)(T h T h 1 ) Tk h=1 L (P ) (h 1)(T h T h 1 ) T k L. k (P ) We shall provide an upper bound for both these addends. We have [ 1 T [ E k Tk (k + 1)k = E + 1 k k k Now, consider the other term. We have k k h=1 (h 1)(T h T h 1 ) T k k 1 k + 1 k Var(T k ) = 1 k + 4 k + 1 k + 6 k 3. = 1 k ((k + 1)k T k ) + k (h 1)(T k h T h 1 1) + k (h 1) k h=1 h=1 (k + 1)k k, hence k k (h 1)(T h T h 1 ) T k L k (P ) h=1 + T k (k + 1)k L + k (P ) k h=1 (h 1)(T h T h 1 1) k L (P ) k h=1 (h 1) (k + 1)k k h=1 (h 1)(T h T h 1 1) L + k (P ) k T k (k + 1)k L + k (P ) k. 9

30 Note that k h=1 (h 1)(T h T h 1 1) L = k (P ) ( ) k Var h=1 (h 1)(T h T h 1 ) = k h 1 k = h=1 k 4 3k k + 3k 3, and T k (k + 1)k L = 1 4 Var (T k (P ) k k ) = k + 1 k + 6 k. 3 Collecting all these inequalities leads to (4.8). References [1 V. Bally, M.-P. Bavouzet-Morel, and M. Messaoud. Integration by parts formula for locally smooth laws and applications to sensitivity computations. Ann. Appl. Probab., 17(1):33 66, 7. [ F. Hirsch and G. Lacombe. Elements of functional analysis, volume 19 of Graduate Texts in Mathematics. Springer-Verlag, New York, [3 C. Houdré and N. Privault. Concentration and deviation inequalities in infinite dimensions via covariance representations. Bernoulli, 8(6):697 7,. [4 J. León, J.L. Solé, and J. Vives. A pathwise approach to backward and forward stochastic differential equations on the Poisson space. Stochastic Anal. Appl., 19(5):81 839, 1. [5 I. Nourdin. Lectures on Gaussian approximations with Malliavin calculus. Preprint, 1. [6 I. Nourdin and G. Peccati. Stein s method on Wiener chaos. Probab. Theory Related Fields, 145(1- ):75 118, 9. [7 G. Peccati, J. L. Solé, M. S. Taqqu, and F. Utzet. Stein s method and normal approximation of Poisson functionals. Ann. Probab., 38(): , 1. [8 G. Peccati and C. Thäle. Gamma limits and U-statistics on the Poisson space. Preprint arxiv: , 13. [9 N. Privault. A calculus on Fock space and its probabilistic interpretations. Bull. Sci. Math., 13():97 114, [1 N. Privault. Stochastic Analysis in Discrete and Continuous Settings, volume 198 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 39 pp., 9. [11 W. Rudin. Real and Complex Analysis. Mc graw-hill, [1 C. Stein. Estimation of the mean of a multivariate normal distribution. Ann. Stat., 9(6): ,