The Stein and Chen-Stein Methods for Functionals of Non-Symmetric Bernoulli Processes

Transcription

1 ALEA, Lat. Am. J. Probab. Math. Stat. 12 (1), (2015) The Stein Chen-Stein Methods for Functionals of Non-Symmetric Bernoulli Processes Nicolas Privault Giovanni Luca Torrisi Division of Mathematical Sciences, Nanyang Technological University, SPMS-MAS-05-43, 21 Nanyang Lin Singapore address: URL: Istituto per le Applicazioni del Calcolo Mauro Picone, CNR, Via dei Taurini 19, Roma, Italy address: URL: Abstract. Based on a new multiplication formula for discrete multiple stochastic integrals with respect to non-symmetric Bernoulli rom wals, we extend the results of Nourdin et al. (2010) on the Gaussian approximation of symmetric Rademacher sequences to the setting of possibly non-identically distributed independent Bernoulli sequences. We also provide Poisson approximation results for these sequences, by following the method of Peccati (2011). Our arguments use covariance identities obtained from the Clar-Ocone representation formula in addition to those usually based on the inverse of the Ornstein-Uhlenbec operator. 1. Introduction Malliavin calculus the Stein method were combined for the first time for Gaussian fields in the seminal paper Nourdin Peccati (2009), whose results have later been extended to other settings, including Poisson processes Peccati (2011); Peccati et al. (2010). In particular, the Stein method has been applied in Nourdin et al. (2010) to Rademacher sequences (X n ) n N of independent identically distributed Bernoulli rom variables with P (X 1 1) P (X 1 1) 1/2, in order to derive bounds on distances between the probability laws of functionals of (X n ) n N the law N(0, 1) of a stard N(0, 1) normal rom variable Z. Those approaches exploit a covariance representation based on the number (or Ornstein-Uhlenbec) operator L its inverse L 1. Received by the editors March 31, 2014; accepted April 6, Mathematics Subject Classification. 60F05, 60G57, 60H07. Key words phrases. Bernoulli processes; Chen-Stein method; Stein method; Malliavin calculus; Clar-Ocone representation. 309

2 310 N. Privault G.L. Torrisi From here onwards, we denote by C 2 b the set of all real-valued bounded functions with bounded derivatives up to the second order. In particular, for h C 2 b, using a chain rule proved in the symmetric case, the bound E[h(F )] E[h(Z)] A 1 min{4 h, h } + h A 2, (1.1) has been derived in Nourdin et al. (2010) (see Theorem 3.1 therein) for centered functionals F of a symmetric Bernoulli rom wal (X n ) n N. Here, (X n ) n N is built as the sequence of canonical projections on Ω : { 1, 1} N [ ] 1 A 1 E DF, DL 1 F l2 (N), A E [ DL 1 F, DF 3 ] l2 (N), where, l 2 (N) is the usual inner product on l 2 (N), D is the symmetric gradient defined as D F (ω) 1 2 (F (ω +) F (ω )), N, where, given ω (ω 0, ω 1,...) Ω, we let ω + (ω 0,..., ω 1, +1, ω +1,...) ω (ω 0,..., ω 1, 1, ω +1,...). The above bound (1.1) can be used to control the Wasserstein distance between N(0, 1) the law of F as in Corollary 3.6 in Nourdin et al. (2010). In addition, the right-h side of (1.1) yields explicit bounds in the case where F is a single discrete stochastic integral (see Corollary 3.3 in Nourdin et al. (2010)) or a multiple discrete stochastic integral (see Section 4 in Nourdin et al. (2010)). In this latter case the derivation of explicit bounds is based on a multiplication formula proved in the symmetric case (see Proposition 2.9 in Nourdin et al. (2010)). In this paper we provide Gaussian Poisson approximations for functionals of not-necessarily symmetric Bernoulli sequences via the Stein Chen-Stein methods, respectively. See Kroowsi et al. (2015) for recent related results on Gaussian approximation, without relying on a multiplication formula for discrete multiple stochastic integrals. The normal Poisson approximations are based on suitable chain rules in Propositions on an extension to the non-symmetric case of the multiplication formula for discrete multiple stochastic integrals (see Proposition 5.1 Section 9 for its proof). In addition to using the Ornstein-Uhlenbec operator L for covariance representations, we also derive error bounds for the normal Poisson approximations using covariance representations based on the Clar-Ocone formula, following the argument implemented in Privault Torrisi (2013). Indeed the operator L is of a more delicate use in applications to functionals whose multiple stochastic integral expansion is not explicitly nown. In contrast with covariance identities based on the number operator, which rely on the divergence-gradient composition, the Clar-Ocone formula only requires the computation of a gradient a conditional expectation. A bound for the Wasserstein distance between a stard Gaussian rom variable a (stardized) function of a finite sequence of independent rom variables has been obtained in Chatterjee (2008), via the construction of an auxiliary rom variable which allows one to approximate the Stein equation. Although the results in our paper are restricted to the Bernoulli case, they may be applied to functionals of an infinite sequence of Bernoulli distributed rom variables. A

3 Chen-Stein method for non-symmetric Bernoulli processes 311 comparison between a bound in our paper that one in Chatterjee (2008) is given at the end of the first example of Section 4. As far as the Gaussian approximation is concerned, using a covariance representation based on the Clar-Ocone formula, in Theorem 3.2 below we find sufficient conditions on centered functionals F of a not necessarily symmetric Bernoulli rom wal so that E[h(F )] E[h(Z)] B 1 min{4 h, h } + h B 2 + h B 3 (1.2) for any h C 2 b, for some positive constants B 1, B 2, B 3 > 0; similarly, using a covariance representation based on the Ornstein-Uhlenbec operator, in Theorem 3.4 below we provide alternate sufficient conditions on centered functionals F of a not necessarily symmetric Bernoulli rom wal so that the bound (1.2) holds for different positive constants C 1, C 2, C 3 > 0, in place of B 1, B 2, B 3 respectively. In Theorem 3.6 below we show that the bound (1.2) can be used to control the Fortet-Mourier distance d FM between F the stard N(0, 1) normal rom variable Z, i.e. we prove d FM (F, Z) 2(B 1 + B 3 )(5 + E[ F ]) + B 2. A similar bound holds, under alternate conditions on F, with the constant B i replaced by C i (i 1, 2, 3). Replacing the Stein method by the Chen-Stein method, we also show that this approach applies to the Poisson approximation in addition to the Gaussian approximation, treat discrete multiple stochastic integrals as examples in both cases. This paper is organized as follows. In Section 2 we recall some elements of stochastic analysis of Bernoulli processes, including chain rules for finite difference operators. In Section 3 we present the two different upper bounds for the quantity E[h(F )] E[h(Z)], h C 2 b, described above the related application to the Fortet-Mourier distance. Section 4 contains explicit first chaos bounds with application to determinantal processes, while Section 5 is concerned with bounds for the nth chaoses. The important case of quadratic functionals (second chaoses) is treated in a separate paragraph. In Section 6 we apply our arguments to the Poisson approximation in Sections 7 8 we investigate the case of single multiple discrete stochastic integrals. Finally, Section 9 deals with the new multiplication formula for discrete multiple stochastic integrals in the non-symmetric case, whose proof is modeled on normal martingales that are solution of a deterministic structure equation. 2. Stochastic analysis of Bernoulli processes In this section we provide some preliminaries. The reader is directed to Privault (2008) references therein for more insight into the stochastic analysis of Bernoulli processes. From now on we assume that the canonical projections X n : Ω { 1, 1}, Ω { 1, 1} N, are considered under the not necessarily symmetric measure P given on cylinder sets by n P ({ε 0,..., ε n } { 1, 1} N ) p (1+ε )/2 q (1 ε )/2, ε { 1, 1}, 0,..., n. 0

4 312 N. Privault G.L. Torrisi Given ω (ω 0, ω 1,...) Ω ω +, ω defined as above, for any F : Ω R we consider the finite difference operator D F (ω) p q (F (ω +) F (ω )), N, denoting by κ the counting measure on N, we consider the L 2 (Ω N) L 2 (Ω N, P κ)-valued operator D defined for any F : Ω R, by DF (D F ) N. Given n 1 we denote by l 2 (N) n l 2 (N n ) the class of functions on N n that are square integrable with respect to κ n, we denote by l 2 (N) n the subspace of l 2 (N) n formed by functions that are symmetric in n variables. The L 2 domain of D is given by Dom(D) {F L 2 (Ω) : DF L 2 (Ω N)} {F L 2 (Ω) : E[ DF 2 l 2 (N) ] < }. We let (Y n ) n 0 denote the sequence of centered normalized rom variables defined by Y n q n p n + X n 2 p n q n, which satisfies the discrete structure equation Y 2 n 1 + q n p n 2 p n q n Y n. (2.1) Given f 1 l 2 (N) we define the first order discrete stochastic integral of f 1 as J 1 (f 1 ) 0 f 1 ()Y, we let J n (f n ) f n (i 1,..., i n )Y i1... Y in (i 1,...,i n ) n denote the discrete multiple stochastic integral of order n of f n in the subspace l 2 s( n ) of l 2 (N) n composed of symmetric ernels that vanish on diagonals, i.e. on the complement of n {( 1,..., n ) N n : i j, 1 i < j n}, n 1. As a convention we identify l 2 (N 0 ) to R let J 0 (f 0 ) f 0, f 0 R. Hereafter, we shall refer to the set of functionals of the form J n (f) as the n-chaos. The multiple stochastic integrals satisfy the isometry formula E[J n (f n )J m (g m )] 1 {nm} n! f n, g m l 2 s ( n), f n l 2 s( n ), g m l 2 s( m ), cf. e.g. Proposition of Privault (2009). The finite difference operator acts on multiple stochastic integrals as follows: D J n (f n ) nj n 1 (f n (, )1 n (, )) nj n 1 (f n (, )), (2.2) N, f n l 2 s( n ). Due to the chaos representation property any square integrable F may be represented as F n 0 J n(f n ), f n l 2 s( n ), so the L 2 domain of D may be rewritten as Dom(D) F J n (f n ) : n n! f n 2 l 2 (N) < n. n 0 n 1 Next we present a chain rule for the finite difference operator that extends Proposition 2.14 in Nourdin et al. (2010) from the symmetric to the non-symmetric case.

5 Chen-Stein method for non-symmetric Bernoulli processes 313 This chain rule will be used later on for the normal approximation. In the following we write F ± in place of F (ω ±). Proposition 2.1. Let F Dom(D) f : R R be thrice differentiable with bounded third derivative. Assume moreover that f(f ) Dom(D). Then, for any integer 0 there exists a rom variable R F such that where Proof. where D f(f ) f (F )D F D F 2 4 p q (f (F + ) + f (F ))X + R F, a.s. (2.3) R F 5 3! f D F 3, p q a.s. (2.4) By a stard Taylor expansion we have D f(f ) p q (f(f + ) f(f )) p q (f(f + ) f(f )) p q (f(f ) f(f )) p q f (F )(F + F ) + p q f (F )(F + 2 F )2 + R + p q f (F )(F F ) p q f (F )(F 2 F )2 + R f p q (F )D F + f (F )[(F + 2 F )2 (F F )2 ] + R + + R, (2.5) R ± p q f F ± 3! F 3. (2.6) By the mean value theorem we find where f (F ) f (F + ) + f (F ) 2 f (F + ) + f (F ) 2 f + f (F ) f (F + ) + f (F ) f (F ) 2 + R, R ( F + 2 F + F F ). Substituting this into (2.5) we get D f(f ) f p q (F )D F + (f (F + 4 ) + f (F ))[(F + F )2 (F F )2 ] + R + + R + R, (2.7) where R p q f ( F + 4 F + F F )( F + F 2 F F 2 ) p q f ( F + 4 F + F F ) F + F 2. (2.8) Note that F + F (F + F )1 {X 1} + (F + F )1 {X 1} (F + F )1 {X 1} (F + F )1 {X 1}, (2.9)

6 314 N. Privault G.L. Torrisi similarly, F F (F + F )1 {X 1}. (2.10) Therefore we have F ± F D F / p q, combining this with (2.6) (2.8) we find R ± f By (2.9) (2.10) we also have D F 3, R 3!p q f D F 3. (2.11) 2p q (F + F )2 (F + F )2 1 {X 1} (F F )2 (F + F )2 1 {X 1}, therefore (F + F )2 (F F )2 (F + F )2 (1 {X 1} 1 {X 1}) (F + F )2 X D F 2 p q X. The claim follows substituting this expression into (2.7) by using (2.11) to estimate the remainder. Now we present a chain rule for the finite difference operator, which is suitable for integer-valued functionals. This chain rule will be used later on for the Poisson approximation. Given a function f : N R we define the operators f() : f( + 1) f(), 2 f : ( f). Proposition 2.2. Let F Dom(D) be an N-valued rom variable. Then, for any f : N R so that f(f ) Dom(D), we have where R F 2 f 2 D f(f ) f(f )D F + R F, (2.12) ( ( D F D F 1 {X 1} 1) p q p q + D F 1 {X 1} p q ( ) ) D F + 1. p q (2.13) Proof. As shown in the proof of Theorem 3.1 in Peccati (2011), for any f : N R any, a N, f() f(a) f(a)( a) 2 f ( a)( a 1). (2.14) 2 Therefore, taing first F +, a F then F, a F, we deduce D f(f ) p q (f(f + ) f(f )) p q (f(f ) f(f )) p q f(f )(F + F ) + R(1) + p q f(f )(F F ) + R(2), where by (2.14), setting R F : R(1) + R (2), we have R F 2 f ( (F + 2 F )(F + F 1) + (F F )(F F 1) ). The claim follows from (2.9) (2.10).

7 Chen-Stein method for non-symmetric Bernoulli processes 315 Next we give two alternative covariance representation formulas. Let (F n ) n 1 be the filtration defined by F 1 {, Ω}, F n σ{x 0,..., X n }, n 0. By Proposition of Privault (2009), for any F, G Dom(D) with F centered we have the Clar-Ocone covariance representation formula Cov(F, G) E[F G] E E[D F F 1 ]D G. (2.15) 0 The second covariance representation formula involves the inverse of the Ornstein- Uhlenbec operator. The domain Dom(L) of the Ornstein-Uhlenbec operator L : L 2 (Ω) L 2 0(Ω), where L 2 0(Ω) denotes the subspace of L 2 (Ω) composed of centered rom variables, is given by Dom(L) F J n (f n ) : n 2 n! f n 2 l 2 (N) < n n 0 n 1, for any F Dom(L), LF nj n (f n ). n1 The inverse of L, denoted by L 1, is defined on L 2 0(Ω) by L 1 1 F n J n(f n ), n1 with the convention L 1 F L 1 (F E[F ]) in case F is not centered, as in e.g. Peccati (2011). Using this convention, for any F, G Dom(D) we have Cov(F, G) E[G(F E[F ])] E [ DG, DL 1 F l2 (N)], (2.16) cf. Lemma 2.12 of Nourdin et al. (2010) in the symmetric case. For the sae of completeness, we provide an alternative expression for the covariance representation formula (2.16). Let (P t ) t 0 be the semigroup associated to the Ornstein-Uhlenbec operator L (we refer the reader to Section 10 of Privault (2008) for the details). Then P t J n (f n ) e nt J n (f n ), n 1, so for any F n0 J n(f n ) Dom(D) one has e t P t D F dt n e t P t J n 1 (f n (, )) dt 0 n1 n 1 n 0 0 J n 1 (f n (, )) n1 D L 1 F. e t e (n 1)t J n 1 (f n (, )) dt

8 316 N. Privault G.L. Torrisi Consequently, the covariance representation (2.16) may be rewritten as [ ] Cov(F, G) E[G(F E[F ])] E e t D GP t D F dt, 0 for any F, G Dom(D), cf. Proposition of Privault (2009). 3. Normal approximation of Bernoulli functionals In this section we present two different upper bounds for the quantity E[h(F )] E[h(Z)], h C 2 b. The first one is obtained by using the covariance representation formula (2.15), while the second one, obtained by using the covariance representation formula (2.16), is a strict extension of the bound given in Theorem 3.1 of Nourdin et al. (2010). Before proceeding further we recall some necessary bacground on the Stein method for the normal approximation refer to Barbour (1990); Götze (1991); Stein (1972, 1986) to Nourdin et al. (2010) for more insight into this technique. Stein s method for normal approximation. Let Z be a stard N(0, 1) normal rom variable consider the so-called Stein s equation associated with h : R R: h(x) E[h(Z)] f (x) xf(x), x R. We refer to part (ii) of Lemma 2.15 in Nourdin et al. (2010) for the following lemma. More precisely, the estimates on the first second derivative are proved in Lemma II.3 of Stein (1986), the estimate of the third derivative is proved in Theorem 1.1 of Daly (2008) the alternative estimate on the first derivative may be found in Barbour (1990) Götze (1991). Lemma 3.1. If h C 2 b, then the Stein equation has a solution f h which is thrice differentiable such that f h 4 h, f h 2 h f h 2 h. We also have f h h. Combining the Stein equation with this lemma, for a generic square integrable centered rom variable F we have 0 E[h(F )] E[h(Z)] E[f h(f ) F f h (F )]. (3.1) Let (F n ) n 1 be a sequence of square integrable centered rom variables. If E[h(F n )] E[h(Z)] 0, h C 2 b then (F n ) n 1 converges to Z in distribution as n tends to infinity, so an upper bound for the right-h side of (3.1) may provide informations about this normal approximation. The results of Sections below are given in terms of bounds for E[h(F )] E[h(Z)], for test functions in C 2 b, they are applied in Section 3.3 to derive bounds for the Fortet-Mourier distance between the laws of two rom variables X Y, which is defined by d FM (X, Y ) sup E[h(X)] E[h(Y )], (3.2) h FM where FM is the class of functions h such that h BL h L + h 1, where L denotes the stard Lipschitz semi-norm. Clearly, any h FM is Lipschitz with Lipschitz constant less than or equal to 1 so it is Lebesgue a.e. differentiable

9 Chen-Stein method for non-symmetric Bernoulli processes 317 h 1. One can also shows that d FM metrizes the convergence in distribution, see e.g. Chapter 11 in Dudley (2002) Clar-Ocone bound. Theorem 3.2. Let F Dom(D) be a centered rom variable assume that B 1 : E 1 E[D F F 1 ]D F, B 2 : 0 B 3 : 5 3 are finite. Then we have p p q E[ E[D F F 1 ] D F 2 ], (3.3) 1 p q E[ E[D F F 1 ] D F 3 ] (3.4) E[h(F )] E[h(Z)] B 1 min{4 h, h } + h B 2 + h B 3 (3.5) for all h C 2 b. Proof. Since the first derivative of f h is bounded we have that f h is Lipschitz. So f h (F ) L 2 (Ω) Consequently we have D f h (F ) p q f h (F + ) f h(f ) f h D F. E[ Df h (F ) 2 l 2 (N) ] f h E[ DF 2 l 2 (N) ] f h (F ) Dom(D). Since F is centered, by the covariance representation (2.15) the chain rule of Proposition 2.1 we have E[F f h (F )] E E[D F F 1 ]D f h (F ) 0 E E[D F F 1 ]D F f h(f ) 0 E E[D F F 1 ] D F 2 4 (f h (F + p q ) + f h (F ))X 0 +E E[D F F 1 ]R F (h). (3.6) 0 Note that the three expectations in (3.6) are finite. The first one since DF L 2 (Ω N) f h is bounded, indeed by Jensen s inequality E E[D F F 1 ]D F f h(f ) 4 h E E[ D F F 1 ] D F h

10 318 N. Privault G.L. Torrisi E E[E[ D F F 1 ] D F F 1 ] 0 4 h E E[E[ D F F 1 ] 2 ] 0 4 h E E[E[ D F 2 F 1 ] 0 4 h E[ D F 2 ] < ; the second relation follows from the boundedness of f h (3.3), while the third one follows from (2.4) (3.4). The rom variables E[D F F 1 ], D F, F ± are independent of X (the first one because it is F 1 -measurable the rom variables (X ) N are independent, the others by their definition). Therefore, the equality (3.6) reduces to 0 E[F f h (F )] E f h(f ) E[D F F 1 ]D F p 4 E[E[D F F 1 ] D F 2 (f h (F + p q ) + f h (F ))] 0 + E E[D F F 1 ]R F (h). 0 Inserting this expression into the right-h side of (3.1) we deduce E[h(F )] E[h(Z)] B 1 min{4 h, h } + h B 2 (3.7) +E E[D F F 1 ] R F (h), 0 (3.8) where to get the term (3.7) we used the inequalities f h min{4 h, h } f h 2 h (see Lemma 3.1). Using (2.4) one may easily see that the term in (3.8) is bounded above by h B 3. The proof is complete. Corollary 3.3. Let F Dom(D) be a centered rom variable assume that B 1 : 1 F 2 L 2 (Ω) + D F, E[D F F 1 ] l2 (N) B 2 : 0 B 3 : 5 3 E[ D F, E[D F F 1 ] l2 (N)] L2 (Ω), 0 1 2p p q D F L2 (Ω) E[ D F 4 ], (3.9) 1 p q E[ D F 4 ] are finite. Then (3.5) holds for all h C 2 b.

11 Chen-Stein method for non-symmetric Bernoulli processes 319 Proof. By the Cauchy-Schwarz the triangular inequalities we have E 1 E[D F F 1 ]D F 1 E[D F F 1 ]D F 0 0 L 2 (Ω) 1 F 2 L 2 (Ω) + D F, E[D F F 1 ] l 2 (N) F 2 L 2 (Ω) L 2 (Ω). By the Clar-Ocone formula (2.15) we have [ ] E[ D F, E[D F F 1 ] l2 (N)] E D F E[D F F 1 ] Therefore 0 F 2 L 2 (Ω). E 1 E[D F F 1 ]D F B 1. 0 By the Cauchy-Schwarz Jensen inequalities we have E[ E[D F F 1 ] D F 2 ] E[ E[D F F 1 ] 2 ] E[ D F 4 ] E[E[ D F 2 F 1 ]] E[ D F 4 ] E[ D F 2 ] E[ D F 4 ], E[ E[D F F 1 ] D F 3 ] E[ E[D F F 1 ] 2 D F 2 ] E[ D F 4 ] E[E[ D F 2 F 1 ] D F 2 ] E[ D F 4 ] E[ D F 4 ] E[ D F 4 ] E[ D F 4 ]. The claim follows from Theorem Semigroup bound. Theorem 3.4. Let F Dom(D) be a centered rom variable let [ ] 1 C 1 : E DF, DL 1 F l 2 (N), C 2 : 0 C 3 : 5 3 be finite. Then for all h C 2 b 0 1 2p p q E[ D L 1 F D F 2 ], (3.10) we have 1 p q E[ D L 1 F D F 3 ] (3.11) E[h(F )] E[h(Z)] C 1 min{4 h, h } + h C 2 + h C 3. (3.12) Proof. Although the proof is similar to that of Theorem 3.2, we give the details since some points need a different justification. As in the proof of Theorem 3.2 one

12 320 N. Privault G.L. Torrisi has f h (F ) Dom(D). Since F is centered, by the covariance representation (2.16) the chain rule of Proposition 2.1 we have E[F f h (F )] E D f h (F )D L 1 F 0 E D F f h(f )D L 1 F 0 +E D F 2 X 4 (f h (F + p q ) + f h (F ))D L 1 F 0 E D L 1 F R F (h). (3.13) 0 Note that the three expectations in (3.13) are finite. The first one since DF L 2 (Ω N) f h is bounded, indeed E D L 1 F D F f h(f ) 4 h E D L 1 F D F h E D L 1 F 2 0 E D F 2 0 1/2 4 h (E DL 1 F 2 l 2 (N) 4 h E[ DF 2 l 2 (N) ] <, where for the latter inequality we used the relation E[ DL 1 F 2 l 2 (N) ] E[ DF 2 l 2 (N) ] 1/2 ) 1/2 ( ) 1/2 E DF 2 l 2 (N) (see Lemma 2.13(3) in Nourdin et al. (2010)); the second one due to the boundedness of f h (3.10); the third one due to (2.4) (3.11). By Lemma 2.13 in Nourdin et al. (2010) we have that the rom variables D L 1 F, D F F ± are independent of X. Therefore, the equality (3.13) reduces to E[F f h (F )] E f h(f )D F D L 1 F p 4 E[ D F 2 (f (F + p q ) + f (F ))D L 1 F ] 0 E R F (h)d L 1 F. 0

13 Chen-Stein method for non-symmetric Bernoulli processes 321 Inserting this expression into the right-h side of (3.1) we deduce E[h(F )] E[h(Z)] C 1 min{4 h, h } + h C 2 (3.14) + E D L 1 F R F (h), 0 (3.15) where to get the term (3.14) we used the inequalities f h min{4 h, h } f h 2 h (see Lemma 3.1). Using (2.4) one may easily see that the term in (3.15) is bounded above by h C 3. The proof is complete. Note that, formally, the upper bound (3.5) may be obtained by (3.12) substituting the term D L 1 F in the definitions of C 1, C 2, C 3, with E[D F F 1 ], vice versa. Corollary 3.5. Let F Dom(D) be a centered rom variable let C 1 : 1 F 2 L 2 (Ω) + D F, D L 1 F l 2 (N) E[ D F, D L 1 F l 2 (N)] L 2 (Ω), C 2 : B 2, where B 2 is defined by (3.9) C 3 defined by (3.11) be finite. Then (3.12) holds for all h C 2 b. Proof. By the Cauchy-Schwarz the triangular inequalities we have [ ] 1 E D F, D L 1 F l 2 (N) 1 D F, D L 1 F l 2 (N) L2 (Ω) 1 F 2 L 2 (Ω) + D F, D L 1 F l 2 (N) F 2 L 2 (Ω) L 2 (Ω). By the covariance representation formula (2.16) we have Therefore Let F Dom(D) be of the form E[ D F, D L 1 F l 2 (N)] F 2 L 2 (Ω). [ ] 1 E D F, D L 1 F l 2 (N) C 1. F n 0 J n (f n ), f n l 2 s( n ). Then D L 1 F n 1 J n 1 (f n (, )) D F n 1 nj n 1 (f n (, )). So, by the isometry formula, we have E[ D L 1 F 2 ] E J n 1 (f n (, )) n 1 n 1 E[ J n 1 (f n (, )) 2 ] 2 n 1(n 1)! f n (, ) 2 l 2 (N) (n 1)

14 322 N. Privault G.L. Torrisi E[ D F 2 ] E nj n 1 (f n (, )) n 1 2 n 1 n 2 E[ J n 1 (f n (, )) 2 ] n 2 (n 1)! f n (, ) 2 l 2 (N). (n 1) n 1 So E[ D L 1 F 2 ] E[ D F 2 ] (3.16) by the Cauchy-Schwarz inequality, we deduce E[ D L 1 F D F 2 ] E[ D L 1 F 2 ] E[ D F 4 ] E[ D F 2 ] E[ D F 4 ]. The claim follows from Theorem Fortet-Mourier distance. In this section we provide bounds in the Fortet- Mourier distance (3.2). Theorem 3.6. Let F Dom(D) be centered. We have: (i) If (3.5) holds for any h C 2 b B 1 + B 3 (5 + E[ F ])/4, then d FM (F, Z) 2(B 1 + B 3 )(5 + E[ F ]) + B 2. (3.17) (ii) If (3.12) holds for any h C 2 b C 1 + C 3 (5 + E[ F ])/4, then d FM (F, Z) 2(C 1 + C 3 )(5 + E[ F ]) + C 2. (3.18) Proof. We only give the details for the proof of (3.17). The inequality (3.18) can be proved similarly. Tae h FM define h t (x) h( ty + 1 tx)φ(y) dy, t [0, 1], R where φ is the density of the stard N(0, 1) normal rom variable Z. As in the proof of Corollary 3.6 in Nourdin et al. (2010), for 0 < t 1/2, one has h t C 2 b the bounds h t 1/ t, (3.19) So E[h(F )] E[h t (F )] ( t 1 + ) E[ F ], E[h(Z)] E[h t (Z)] 3 t. 2 2 E[h(F )] E[h(Z)] (E[h(F )] E[h t (F )]) + (E[h t (F )] E[h t (Z)]) +(E[h t (Z)] E[h(Z)]) E[h(F )] E[h t (F )] + E[h t (F )] E[h t (Z)] t + E[h t (Z)] E[h(Z)] ( 1 + E[ F ] 2 ) + B 1 min{4 h t, h t } + h t B 2

15 Chen-Stein method for non-symmetric Bernoulli processes h t B t 2 ( ) 5 + E[ F ] t + B 1 + B 3 + B 2, (3.20) 2 t where in the latter inequality we used (3.19) that h t 1, for all t. Minimizing in t (0, 1/2] the term in (3.20), we have that the optimal is attained at t 2(B 1 + B 3 )/(5 + E[ F ]) (0, 1/2]. The conclusion follows substituting t in (3.20) then taing the supremum over all the h FM. 4. First chaos bound for the normal approximation In this section we specialize the results of Section 3 to first order discrete stochastic integrals. As we shall see, the bounds (3.5) (3.12) ( the corresponding assumptions) coincide on the first chaos, although they differ on n-chaoses, n 2. Corollary 4.1. Assume that α (α ) 0 is in l 2 (N), 0 Then for the first chaos 1 2p p q α 3 < 0 F J 1 (α) 0 α Y 1 p q α 4 <. (4.1) the bound (3.5) (which in this case coincides with the bound (3.12)) holds with B 1 C 1 1 α 2, B2 C 2 1 2p α 3, p q 0 0 Proof. since B 3 C p q α 4. Since α l 2 (N) we have that F L 2 (Ω). Moreover F is centered, ( q p + 1 D F α p q 2 q ) p 1 p q 2 α, p q we have F Dom(D). The finiteness of the corresponding quantities B 1, B 2 B 3 is guaranteed by α l 2 (N) (4.1). The claim follows from e.g. Theorem 3.2. Example Consider the sequence of functionals (F n ) n 1 defined by Setting F n 1 n 1 Y. n 0 α 1 n, 0,..., n 1, α 0, n,

16 324 N. Privault G.L. Torrisi we have B 1 0 B 2 1 n 1 1 2p n 3/2 B 3 5 p q 3n 2 0 n p q. In the symmetric case p q 1/2 we find B 2 0 the bound is of order 1/n, implying a faster rate than in the classical Berry-Esséen estimate (however here we are using C 2 b test functions; cf. the comment after Corollary 3.3 in Nourdin et al. (2010)). In the non-symmetric case p p q q, p q, the bound is of order n 1/2 as in the classical Berry-Esséen estimate. Indeed we have B 2 B (n) p B 3 B (n) n 3 5 p(1 p) 3n 1 p(1 p) hence the inequality B 1 + B 3 (5 + E[ F n ])/4 of Theorem 3.6 reads [ ( ) ] n n 1 n + 4 3p(1 p) n p E 2 + X, p(1 p) 4 which holds if e.g. n 3p(1 p). Consequently, by (3.17) it follows that for any 4 n 3p(1 p) we have d FM (F n, Z) 2B (n) 3 (5 + E[ F n ]) + B (n) p(1 p) n p(1 p) n E 1 2p 1 + n p(1 p) [ n 1 1 n 0 [ p(1 p) n n 1 3p(1 p) n E 1 n 1 2p 1 + n. p(1 p) 0 ( ( 0 1 2p + X 2 p(1 p) 1 2p + X 2 p(1 p) ) ] ) 2] 1/2 A straightforward computation gives [ ( ) n p + X E n 2] 2 1, p(1 p) hence where 0 d FM (F n, Z) 1 n K 1 (p), n K 1 (p) : p p(1 p) 4 3p(1 p) (4.2) In the general case, if a n : 1 n 1 1 2p n 3/2 0 0 b n : 1 n 1 p q n p q 0, as n, (4.3)

17 Chen-Stein method for non-symmetric Bernoulli processes 325 then F n Z in distribution, the rate depends on the rate of convergence to zero of the sequences (a n ) n 1 (b n ) n 1. For instance, if p ( + 2) α, 0 < α < 1, 0, we have p q (n + 1) α (1 (1/2) α ), 0,..., n 1. Consequently we have n 1 1 n 1 1 2p n 3/2 (1 (1/2) α 1/2 (n + 1)α/2 ) 1 2p p q n 3/ (α 1) (n + 1) α/2, (1 (1/2) α ) 1/2 n 1/2 n n 2 (1 (1/2) α ) 1 n 1 (n + 1) α, p q 0 which yields a bound of order n (1 α)/2. Finally we note that the bound (4.2) in the non-symmetric case p p q q, p q, is consistent with the bound on the Wasserstein distance between F n Z provided by Theorem 2.2 in Chatterjee (2008). Indeed, letting d W denote the Wasserstein distance Y 1 an independent copy of Y 1, a simple computation shows that where since we have d FM (F n, Z) d W (F n, Z) 1 ( 2 n 1 2 n 1 n K 2 (p), E[ Y 1 Y 1 4 ] (E[ Y 1 Y ( E[ Y 1 Y 1 4 ] 4 + E[ Y 1 3 ] K 2 (p) : p 2 p (1 p) 4 p(1 p) E[ Y p ] 2 p(1 p) ) 1 2 ]) 2 + E[ Y 1 3 ] ) E[ Y 1 Y ] p 2 (1 p). We note that when e.g. p is small it holds K 2 (p) > K 1 (p). Application to determinantal processes. Let E be a locally compact Hausdorff space with countable basis B(E) the Borel σ-field. We fix a Radon measure λ on (E, B(E)). The configuration space Γ E is the family of non-negative N-valued Radon measures on E. We equip Γ E with the topology which is generated by the functions Γ E ξ ξ(a) N, A B(E), where ξ(a) denotes the number of points of ξ in A. The existence uniqueness of a determinantal process with Hermitian ernel K is due to Macchi (1975) Soshniov (2000) can be summarized as follows (we refer the reader to Blan et al. (1994) for notions of functional analysis).

18 326 N. Privault G.L. Torrisi Theorem 4.2. Let K be a self-adjoint integral operator on L 2 (E, λ) with ernel K. Suppose that the spectrum of K is contained in [0, 1] that K is locally of traceclass, i.e. for any relatively compact Λ E, K Λ P Λ KP Λ is of trace-class (here P Λ f f 1 Λ is the orthogonal projection.) Then there exists a unique probability measure µ K on Γ E with n-th correlation measure λ n (dx 1,..., dx n ) det(k(x i, x j )) 1 i,j n λ(dx 1 )... λ(dx n ), where det(k(x i, x j )) 1 i,j n is the determinant of the n n matrix with ij-entry K(x i, x j ). The probability measure µ K is called determinantal process with ernel K. Given a relatively compact set Λ E, we focus on the rom variable ξ(λ) recall the following basic result (see e.g. Proposition 2.2 in Shirai (2006)). Theorem 4.3. Let K be as in the statement of Theorem 4.2 denote by κ [0, 1], 0, the eigenvalues of K Λ. Under µ K the rom variable ξ(λ) has the same distribution of 0 Z, where Z 0, Z 1,... are independent rom variables such that Z obeys the Bernoulli distribution with mean κ, i.e. Z n X n {0, 1}, n N where the X s tae values on { 1, 1} are independent with P (X n 1) κ n. The central limit theorem for the number of points on a relatively compact set of a determinantal process may be obtained in different manners, see Costin Lebowitz (1995), Shirai (2006) Soshniov (2002). In the following we provide an alternate derivation which gives the rate of the normal approximation. Corollary 4.4. Let K be as in the statement of Theorem 4.2 (Λ n ) n 0 E be an increasing sequence of relatively compact sets such that where κ (n) Var µk (ξ(λ n )) 0 κ (n) (1 κ(n) ), as n [0, 1], 0, are the eigenvalues of K Λn, n 0. Setting for any h C 2 b, we have where B (n) 2 F n ξ(λ n) E µk [ξ(λ n )], VarµK (ξ(λ n )) E µk [h(f n )] E[h(Z)] h B (n) 2 + h B (n) 3, n 0 0 κ(n) (1 κ(n) ) 1 2κ(n) 1 ( ) 3/2 ( ) 1/2 0 κ(n) (1 κ(n) ) 0 κ(n) (1 κ(n) ) B (n) κ(n) So we have a bound of order [Var µk (ξ(λ n ))] 1/2. (1 κ(n) ).

19 Chen-Stein method for non-symmetric Bernoulli processes 327 Proof. For n 0, let (Z (n) ) 0 be a sequence of independent {0, 1}-valued rom variables with Z (n) Be(κ (n) (n) ) (Y ) 0 defined by By Theorem 4.3 we have ξ(λ n ) d 0 Z (n) Y (n) Z(n) κ (n) 0 κ (n) κ (n). (1 κ(n) ) (1 κ(n) (n) )Y + 0 κ (n), where d denotes the equality in distribution. Then where F n ξ(λ n) E µk [ξ(λ n )] VarµK (ξ(λ n )) α (n) d 0 κ (n) (1 κ(n) ). 0 κ(n) (1 κ(n) ) α (n) Y (n), We are going to apply Corollary 4.1. Clearly, for any n 0, the sequence (α (n) is in l 2 (N). Moreover, for any n 0, 0 κ (n) α (n) 3 (1 κ(n) ) α (n) 4 0 κ (n) 1 VarµK (ξ(λ n )) < (1 κ(n) ) 1 Var µk (ξ(λ n )) <. ) 0 So condition (4.1) is satisfied. Moreover, a straightforward computation gives B 1 B (n) 1 0, B 2 B (n) 2 B 3 B (n) 3, the proof is completed. Example Let E C λ the stard complex Gaussian measure on C, i.e. λ(dz) 1 π e z 2 dz, where dz is the Lebesgue measure on C. The Ginibre process µ exp is the determinantal process with exponential ernel K(z, w) e zw, where z is the complex conjugate of z C. Let b(o, n) be the complex ball centered at the origin with radius n. By Theorem 1.3 in Shirai (2006) we have 4n 2 Var µexp (ξ(b(o, n))) n π 0 (1 x/(4n 2 )) 1/2 x 1/2 e x dx n, π as n. So for the Ginibre process Corollary 4.4 provides a bound of order n 1/2.

20 328 N. Privault G.L. Torrisi 5. nth chaos bounds for the normal approximation In this section we give explicit upper bounds for the constants B i C i, i 1, 2, 3, involved in (3.5) (3.12), when F J n (f n ), f n l 2 s( n ). Our approach is based on the multiplication formula (5.3) below, which extends formula (2.11) in Nourdin et al. (2010) (see the discussion after Proposition 5.1). Given f n l 2 s( n ) g m l 2 s( m ), the contraction f n l g m, 0 l, is defined to be the function of n + m l variables where f n l g m (a l+1,..., a n, b +1,..., b m ) : ϕ(a l+1 ) ϕ(a )f n l g m (a l+1,..., a n, b +1,..., b m ), (cf. the structure equation (2.1)) a 1,...,a l N ϕ(n) q n p n 2 p n q n, n N (5.1) f n l g m (a l+1,..., a n, b +1,..., b m ) : f n (a 1,..., a n )g m (a 1,..., a, b +1,..., b m ) is the contraction considered in Nourdin et al. (2010) for the symmetric case, see p therein. By convention, we define ϕ(a l+1 ) ϕ(a ) 1 if l (even when ϕ 0). Denote by fn l g m, 0 l, the symmetrization of f n l g m. Then, we shall consider the contraction f n l g m (a l+1,..., a n, b +1,..., b m ) : 1 n+m l (a l+1,..., a n, b +1,..., b m ) fn l g m(a l+1,..., a n, b +1,..., b m ). (5.2) Note that in the symmetric case p n q n 1/2 we have f n g m f n g m. However, f n l g m 0 if l < so f n l g m f n l g m for l <. The following multiplication formula holds. Proposition 5.1. We have the chaos expansion provided the functions h n,m,s : J n (f n )J m (g m ) s 2i 2(s n m) 2(n m) s0 ( n i! i J n+m s (h n,m,s ), (5.3) )( m i )( i s i ) f n s i i belong to l 2 s( n+m s ), 0 s 2(n m). Here the symbol s 2i 2(s n m) means that the sum is taen over all the integers i in the interval [s/2, s n m]. Since it is not obvious that formula (5.3) extends the product formula (2.11) in Nourdin et al. (2010), it is worthwhile to explain this point in detail. In the symmetric case p n q n 1/2, for any f n l 2 s( n ), g m l 2 s( m ), we have g m

21 Chen-Stein method for non-symmetric Bernoulli processes 329 f n s i i g m 0 if s < 2i, 0 s 2(n m). Therefore, for any fixed 0 s 2(n m) we have h n,m,s 0 if s/2 is not an integer ( )( ) n m h n,m,s (s/2)! f n s/2 s/2 s/2 s/2 g m, if s/2 is an integer. Note that if s/2 is an integer, we have f n s/2 s/2 g m l 2 s ( n+m s) f n s/2 s/2 g m l 2 s ( n+m s) where we used the straightforward relation f n s/2 s/2 g m l2 ( n+m s ), f l2 (N) n f l 2 (N) n, (5.4) being f the symmetrization of f. Therefore, by Lemma 2.4(1) in Nourdin et al. (2010) we have f n s/2 s/2 g m l 2 s( n+m s ), so h n,m,s l 2 s( n+m s ), for any 0 s 2(n m). By (5.3) we have J n (f n )J m (g m ) 2(n m) s0 J n+m s (h n,m,s ) 2(n m) ( )( ) n m (s/2)! J n+m s (f n s/2 s/2 s/2 s/2 g m) s0 n m ( )( ) n m r! J n+m 2r (f n r r g m ), r r r0 which is exactly formula (2.11) in Nourdin et al. (2010). We conclude this part with the following lemma. Lemma 5.2. For any f n l 2 s( n ), g m l 2 s( m ), we have f n (, q) l g m (, q) f n l+1 +1 g m. Proof. Note that q 0 f n (, q) l g m (, q)(a l+1,..., a n 1, b +1,..., b m 1 ) ϕ(a l+1 )... ϕ(a ) f n (a 1,..., a n 1, q)g m (a 1,..., a, b +1,..., b m 1, q), a 1,...,a l N so summing up over q N we deduce f n (, q) l g m (, q)(a l+1,..., a n 1, b +1,..., b m 1 ) q 0 ϕ(a l+1 )... ϕ(a ) a 1,...,a l,q N f n (a 1,..., a n 1, q)g m (a 1,..., a, b +1,..., b m 1, q) ϕ(a l+1 )... ϕ(a )f n l+1 +1 g m(a l+1,..., a n 1, b +1,..., b m 1 ) f n l+1 +1 g m(a l+1,..., a n 1, b +1,..., b m 1 ).

22 330 N. Privault G.L. Torrisi 5.1. Clar-Ocone bound. By e.g. Lemma 4.6 in Privault (2008), for the nth-chaos J n (f n ), n 2, f n l 2 s( n ), we have Therefore E[J n (f n ) F ] J n (f n 1 [0,] n), N. E[D J n (f n ) F 1 ] ne[j n 1 (f n (, )) F 1 ] nj n 1 (f n] ), (5.5) where f n] ( ) : f n (, )1 [0, 1] n 1( ). (5.6) So by the isometric properties of discrete multiple stochastic integrals we have that the constants B i of Corollary 3.3 are equal, respectively, to B 1 : 1 n! f n 2 l 2( n) + n2 J s n 1 (f n (, )), J n 1 (f n] ( )) l 2 (N) B 2 : n 3 (n 1)! 0 B 3 : 5n4 3 E[ J n 1 (f n (, )), J n 1 (f n] ( )) l 2 (N)] L 2 (Ω), (5.7) 1 2p p q f n (, ) l 2 s ( n 1 ) E[ Jn 1 (f n (, )) 4 ], (5.8) 0 1 p q E[ J n 1 (f n (, )) 4 ]. (5.9) In the proof of the next theorem we show that these constants can be bounded above by computable quantities. Theorem 5.3. Let n 2 be fixed let f n l 2 s( n ). Assume that for any N the functions h () n 1,n 1,s : ( ) 2 ( ) n 1 i i! f n (, ) s i i f n] ( ) (5.10) i s i h () n 1,n 1,s : s 2i 2(s (n 1)) s 2i 2(s (n 1)) belong to l 2 s( 2n 2 s ), 0 s 2n 2, that B 1 : 1 n! f n 2 l 2 s ( n) ( 2n 3 + n 2 (2n 2 s)! s0 ( ) 2 ( n 1 i1 i 2 s i 1 ( ) 2 ( ) n 1 i i! f n (, ) s i i f n (, ) (5.11) i s i )( i2 s {2i 1, 2i 2 } 2(s (n 1)) s i 2 ) 0 0 f n (, ) s i2 i 2 f n] ( ) l2 ( 2n 2 s )) 1/2, ( n 1 i 1!i 2! i 1 ) 2 f n (, ) s i1 i 1 f n] ( ) l2 ( 2n 2 s ) B 2 : n 3 (n 1)! 0 s {2i 1, 2i 2 } 2(s (n 1)) ( 2n 2 1 2p f n (, ) l 2 p q s ( n 1) (2n 2 s)! s0 ( ) 2 ( ) 2 ( n 1 n 1 i1 i 1!i 2! i 1 i 2 s i 1 )( i2 s i 2 )

23 Chen-Stein method for non-symmetric Bernoulli processes 331 f n (, ) s i 1 i 1 f n (, ) l2 ( 2n 2 s ) f n (, ) s i 2 i 2 f n (, ) l2 ( 2n 2 s )) 1/2, (5.12) B 3 : 5n4 3 (2n 2 s)! 2n 2 s0 s {2i 1, 2i 2} 2(s (n 1)) 0 ( ) 2 ( ) 2 ( n 1 n 1 i1 i 1!i 2! i 1 i 2 s i 1 1 p q f n (, ) s i 1 i 1 f n (, ) l2 ( 2n 2 s ) f n (, ) s i 2 i 2 f n (, ) l2 ( 2n 2 s ) are finite. Then for all h C 2 b we have )( i2 s i 2 E[h(J n (f n ))] E[h(Z)] B 1 min{4 h, h } + h B 2 + h B 3. ) (5.13) Proof. The claim follows from Corollary 3.3 if we show that the constants B i defined by (5.7), (5.8) (5.9) are bounded above by the constants B i defined in the statement, respectively. Step 1: Proof of B 1 B 1. By the hypotheses on the functions h () n 1,n 1,s the multiplication formula (5.3), we deduce J n 1 (f n (, ))J n 1 (f n] ) 2n 2 s0 s 2i 2(s (n 1)) ( n 1 i! i ) 2 ( i s i ) J 2n 2 s (f n (, ) s i i f n] ( )) (n 1)!f n (, ) n 1 n 1 f n]( ) 2n 3 ( ) 2 ( ) n 1 i + i! J 2n 2 s (f n (, ) s i i f n] ( )). i s i s0 s 2i 2(s (n 1)) Since the constant B 1 in the statement is finite, we have f n (, ) s i i f n] ( ) l2 ( 2n 2 s ) <, 0 0 s 2n 3, s 2i 2(s (n 1)). By (5.4) this in turn implies f n (, ) s i i f n] ( ) l 2 s ( 2n 2 s ) <, 0 0 s 2n 3, s 2i 2(s (n 1)), so f n (, ) s i i f n] ( ) l 2 s( 2n 2 s ), 0 (5.14) 0 s 2n 3, s 2i 2(s (n 1)) (it is worthwhile to note that one can not use Lemma 5.2 to express the infinite sum 0 f n(, ) s i i f n] ( ) since the function

24 332 N. Privault G.L. Torrisi of n variables f n] ( ) is not symmetric). Therefore, summing up over 0 in the equality (5.14), we get J n 1 (f n (, )), J n 1 (f n] ( )) l2 (N) (n 1)! 0 2n 3 + f n (, ) n 1 n 1 f n]( ) s0 s 2i 2(s (n 1)) ( n 1 i! i ) 2 ( i s i ) J 2n 2 s 0 f n (, ) s i i f n] ( ). Taing the mean noticing that discrete multiple stochastic integrals are centered, we have E[ J n 1 (f n (, )), J n 1 (f n] ( )) l2 (N)] (n 1)! 0 f n (, ) n 1 n 1 f n]( ), so J n 1 (f n (, )), J n 1 (f n] ( )) l2 (N) E[ J n 1 (f n (, )), J n 1 (f n] ( )) l2 (N) 2n 3 ( ) 2 ( ) n 1 i i! J 2n 2 s f n (, ) s i i f n] ( ). i s i 0 s0 s 2i 2(s (n 1)) By means of the orthogonality isometric properties of discrete multiple stochastic integrals, we have 2n 3 ( ) 2 ( ) 2 E n 1 i i! J 2n 2 s f n (, ) s i i f n] ( ) i s i 0 s0 s 2i 2(s (n 1)) 2n 3 s0 E 0,2n 3 + s 1 s 2 s 2i 2(s (n 1)) J 2n 2 s E [( ( ) 2 ( ) n 1 i i! i s i 0 f n (, ) s i i s 1 2i 2(s 1 (n 1)) J 2n 2 s1 ( 2 f n] ( ) ( ) 2 ( ) n 1 i i! i s 1 i ) 0 f n (, ) s 1 i i s 2 2i 2(s 2 (n 1)) J 2n 2 s2 f n] ( ) ( ) 2 ( ) n 1 i i! i s 2 i )] 0 f n (, ) s 2 i i f n] ( )

25 Chen-Stein method for non-symmetric Bernoulli processes 333 2n 3 s0 E s 2i 2(s (n 1)) J 2n 2 s ( ) 2 ( ) n 1 i i! i s i 0 f n (, ) s i i 2 f n] ( ) 2n 3 s0 s {2i 1, 2i 2 } 2(s (n 1)) (2n 2 s)! 2n 3 s0 ( i2 s i 2 [ ( ) 2 ( n 1 i1 E i 1! i 1 s i 1 J 2n 2 s J 2n 2 s ) i 2! 0 f n (, ) s i 1 i 1 0 f n (, ) s i 2 i 2 ( ) 2 ( n 1 i2 i 2 f n] ( ) f n] ( ) s i 2 ] ( ) 2 ( ) 2 ( ) n 1 n 1 i1 i 1!i 2! i 1 i 2 s i 1 s {2i 1, 2i 2 } 2(s (n 1)) ) f n (, ) s i1 i 1 f n] ( ), f n (, ) s i2 i 2 f n] ( ) l 2 s ( 2n 2 s ). 0 0 (5.15) ) By the above relations (5.7), we deduce ( 2n 3 B 1 1 n! f n 2 l 2 s ( n) + n2 (2n 2 s)! 0 f n (, ) s i1 i 1 s0 f n] ( ), 0 ( ) 2 ( n 1 n 1 Now, note that by the Cauchy-Schwarz inequality i 1 s {2i 1, 2i 2 } 2(s (n 1)) ) 2 ( )( i1 i2 i 2 s i 1 i 1!i 2! s i 2 f n (, ) s i2 i 2 f n] ( ) l 2 s ( 2n 2 s )) 1/2. ) (5.16) f, g l2 (N) n f l 2 (N) n g l 2 (N) n, for any f, g l2 (N) n. (5.17) By this relation, (5.4) (5.16) we easily get B 1 B 1. Step 2: Proof of B i B i, i 2, 3. By the hypotheses on the functions the multiplication formula (5.3), we deduce J n 1 (f n (, )) 2 2n 2 s0 s 2i 2(s (n 1)) ( n 1 i! i ) 2 ( i s i ) J 2n 2 s (f n (, ) s i i h () n 1,n 1,s f n (, )).

26 334 N. Privault G.L. Torrisi By a similar computation as for (5.15), we have 2n 2 ( ) 2 ( ) E[ J n 1 (f n (, )) 4 ] E n 1 i i! i s i s0 s 2i 2(s (n 1)) ( J 2n 2 s fn (, ) s i i f n (, ) )) 2 (2n 2 s)! 2n 2 s0 ( ) 2 ( n 1 i1 i 2 s i 1 f n (, ) s i1 i 1 so by (5.8) (5.9) we deduce B 2 n 3 (n 1)! 0 B 3 5n4 3 s {2i 1, 2i 2 } 2(s (n 1)) s {2i 1, 2i 2 } 2(s (n 1)) )( ) i2 s i 2 f n (, ), f n (, ) s i2 i 2 1 2p f n (, ) l 2 p q s ( n 1) ( 2n 2 ( ) 2 n 1 i 1!i 2 i 1 f n (, ) l 2 s ( 2n 2 s), (2n 2 s)! s0 ( ) 2 ( ) 2 ( n 1 n 1 i1 i 1!i 2! i 1 i 2 s i 1 )( i2 s i 2 f n (, ) s i 1 i 1 f n (, ), f n (, ) s i 2 i 2 f n (, ) l 2 s ( 2n 2 s) (2n 2 s)! 2n 2 s0 s {2i 1, 2i 2 } 2(s (n 1)) 0 ( ) 2 ( ) 2 ( n 1 n 1 i1 i 1!i 2! i 1 i 2 s i 1 )( i2 s i 2 ) (5.18) ) 1/2 1 f n (, ) s i 1 i p q 1 f n (, ), f n (, ) s i 2 i 2 f n (, ) l 2 s ( 2n 2 s). The claim follows from the above equalities relations (5.17) (5.4). (5.19) ) (5.20) 5.2. Semigroup bound. For the nth-chaos J n (f n ), n 2, f n l 2 s( n ), we have D L 1 J n (f n ) n 1 D J n (f n ) J n 1 (f n (, )) (5.21) the constants C i of Corollary 3.5 are equal, respectively, to C 1 : 1 n! f n 2 l 2 s ( n) + n J n 1(f n (, )), J n 1 (f n (, )) l 2 (N)

27 Chen-Stein method for non-symmetric Bernoulli processes 335 E[ J n 1 (f n (, )), J n 1 (f n (, )) l2 (N)] L2 (Ω), (5.22) C 2 : B 2, where B 2 is defined by (5.8) C 3 : B 3 n, where B 3 is defined by (5.9). In the next theorem we show that these constants can be bounded above by computable quantities. Theorem 5.4. Let n 2 be fixed let f n l 2 s( n ). Assume that for any N () the functions h n 1,n 1,s defined by (5.11) belong to l2 s( 2n 2 s ), 0 s 2n 2, that C 1 : 1 n! f n 2 l 2 s ( n) + n ( 2n 3 (2n 2 s)! s0 s {2i 1, 2i 2 } 2(s (n 1)) ( ) 2 ( ) 2 ( ) n 1 n 1 i1 i 1!i 2! i 1 i 2 s i 1 ( ) 1/2 i2 f n s i1+1 i s i 1 +1 f n l2 ( 2n 2 s ) f n s i2+1 i 2 +1 f n l2 ( 2n 2 s )), 2 C 2 : B 2, where B 2 is defined by (5.12), C 3 : B 3 /n where B 3 is defined by (5.13), are finite. Then for all h C 2 b we have E[h(J n (f n ))] E[h(Z)] C 1 min{4 h, h } + h C 2 + h C 3. Proof. The claim follows from Corollary 3.5 if we show that the constant C 1 defined by (5.22) is bounded above by the constant C 1 defined in the statement (for the bounds C i C i, i 2, 3, see Step 2 of the proof of Theorem 5.3). Along a similar computation as in the Step 1 of the proof of Theorem 5.3, we have J n 1 (f n (, )), J n 1 (f n (, )) l2 (N) (n 1)! 0 2n 3 f n (, ) n 1 f n(, ) + s0 s 2i 2(s (n 1)) J 2n 2 s ( ) 2 ( ) n 1 i i! i s i f n (, ) s i i f n (, ) 0 (n 1)!f n n n f n 2n 3 + s0 s 2i 2(s (n 1)) ( n 1 i! i ) 2 ( i s i ) J 2n 2 s ( fn s i+1 i+1 f n ), (5.23) where the latter equality follows from Lemma 5.2. By a similar computation as for (5.15), we have J n 1 (f n (, )) 2 l 2 (N) E[ J n 1(f n (, )) 2 l 2 (N) ] 2 L 2 (Ω) 2n 3 ( ) 2 ( ) 2 E n 1 i i! J 2n 2 s (f n s i+1 i+1 f n ) i s i s0 s 2i 2(s (n 1))

28 336 N. Privault G.L. Torrisi (2n 2 s)! 2n 3 s0 ( i1 s i 1 )( i2 s {2i 1, 2i 2 } 2(s (n 1)) s i 2 By this relation (5.22) we deduce C 1 1 n! f n 2 l 2 s ( n) + n ( 2n 3 (2n 2 s)! s0 ( ) 2 ( n 1 n 1 i 1!i 2! i 1 ) f n s i1+1 i 1 +1 f n, f n s i2+1 i 2 +1 f n l 2 s ( 2n 2 s ). (5.24) s {2i 1, 2i 2} 2(s (n 1)) i 2 ) 2 ( ) 2 ( ) 2 ( ) n 1 n 1 i1 i 1!i 2! i 1 i 2 s i 1 ( ) 1/2 i2 f n s i 1+1 i s i 1 +1 f n, f n s i 2+1 i 2 +1 f n l 2 s ( 2n 2 s )). 2 By this equality (5.17) (5.4) we finally have C 1 C 1. Connection with Theorem 4.1 in Nourdin et al. (2010). In this subsection we refine a little the bound given by Theorem 5.4 in order to strictly extend the bound provided by Theorem 4.1 in Nourdin et al. (2010). For the nth chaos J n (f n ), n 2, f n l 2 s( n ), we have that the constants C i of Theorem 3.4 are equal, respectively, to ] C 1 : E [ 1 n J n 1 (f n (, )) 2 l 2 (N), (5.25) C 2 : n p p q E[ J n 1 (f n (, )) 3 ], (5.26) C 3 : B 3 n, where B 3 is defined by (5.9) In the next theorem we show that these constants can be bounded above by computable quantities. Theorem 5.5. Let n 2 be fixed let f n l 2 s( n ). Assume that for any N () the functions h n 1,n 1,s defined by (5.11) belong to l2 s( 2n 2 s ), 0 s 2n 2, that ( C 1 : 1 n! f n 2 l 2 s ( n) 2 2n 3 + n 2 (2n 2 s)! s0 s {2i 1, 2i 2 } 2(s (n 1)) ( ) 2 ( ) 2 ( ) n 1 n 1 i1 i 1!i 2! i 1 i 2 s i 1 ( ) 1/2 i2 f n s i1+1 i s i 1 +1 f n l 2 ( 2n 2 s) f n s i2+1 i 2 +1 f n l 2 ( 2n 2 s)), 2 C 2 : B 2 /n, where B 2 is defined by (5.12) C 3 : B 3 /n, where B 3 is defined by (5.13) are finite. Then for all h C 2 b we have E[h(J n (f n ))] E[h(Z)] C 1 min{4 h, h } + h C 2 + h C 3.

29 Chen-Stein method for non-symmetric Bernoulli processes 337 Proof. The claim follows from Theorem 3.4 if we show that the constants C i, i 1, 2, defined by (5.25) (5.26) are bounded above by the constants C i, i 1, 2, defined in the statement, respectively (for the bound C 3 C 3 see Step 2 of the proof of Theorem 5.3). Step 1: Proof of C 1 C 1. By the Cauchy-Schwarz inequality, (5.23) (5.24) we have C 1 E [ 1 n J n 1 (f n (, )) 2 l 2 (N) 2] 1/2 1 n! f n 2 l 2 s ( n) 2 + n 2 E ( 1 n! f n 2 l 2 s ( n) 2 2n 3 + n 2 (2n 2 s)! s0 2n 3 s0 s 2i 2(s (n 1)) J 2n 2 s (f n s i+1 i+1 f n ) s {2i 1, 2i 2 } 2(s (n 1)) ( n 1 i! i ) 2 ) 2 ( i s i 1/2 ( ) 2 ( n 1 n 1 i 1!i 2! ( )( ) 1/2 i1 i2 f n s i 1+1 i s i 1 s i 1 +1 f n, f n s i 2+1 i 2 +1 f n l 2 s ( 2n 2 s)). 2 The claim follows from Relations (5.17) (5.4). Step 2: Proof of C 2 C 2. By the Cauchy-Schwarz inequality we have E[ J n 1 (f n (, )) 3 ] (E[ J n 1 (f n (, )) 2 ]) 1/2 (E[ J n 1 (f n (, )) 4 ]) 1/2. By the isometry for discrete multiple stochastic integrals we have J n 1 (f n (, )) L2 (Ω) (n 1)! f n (, ) l 2 s ( n 1 ). By the above relations (5.18) we have C 2 B 2 /n, where B 2 is defined by (5.19). The claim follows from (5.17) (5.4). Since it is not obvious that the above theorem extends Theorem 4.1 in Nourdin et al. (2010), it is worthwhile to explain this point in detail. Tae f n l 2 s( n ), () n 2, let h n 1,n 1,s be defined by (5.11). In the symmetric case p q 1/2, by the same arguments as those one after the statement of Proposition 5.1 we have () that, for any fixed 0 s 2(n 1) N, h n 1,n 1,s 0 if s/2 is not an integer ( ) 2 h () n 1 n 1,n 1,s (s/2)! f n (, ) s/2 s/2 s/2 f n(, ) otherwise. If s/2 is an integer we also have f n (, ) s/2 s/2 f n(, ) l 2 s ( 2n 2 s) f n (, ) s/2 s/2 f n(, ) l 2 ( 2n 2 s) i 1 i 2 ) 2 f n (, ) s/2 s/2 f n(, ) l2 ( 2n 2 s ) f n (, ) 2 l 2 s ( n 1) <, )

30 338 N. Privault G.L. Torrisi where the latter relation follows from Lemma 2.4(1) in Nourdin et al. (2010). So h () n 1,n 1,s l2 s( 2n 2 s ). In the symmetric case, by the definition of the contraction, for 0 s 2n 3 s 2i 2(s (n 1)), we have f n s i+1 i+1 f n l2 ( 2n 2 s ) 0, if s < 2i f n s i+1 i+1 f n l2 ( 2n 2 s ) f n i+1 i+1 f n l2 ( 2n 2 2i ) f n 2 l 2 s ( n), if s 2i where the latter relation follows from Lemma 2.4(1) in Nourdin et al. (2010). Consequently, the constant C 1 in the statement of Theorem 5.5 is finite reduces to C 1 2(n 2) ( 1 n! f n 2l2s( n) 2 + n 2 s0 1{s/2 N}(2n 2 s)! f n s/2+1 s/2+1 f n 2 l 2 ( 2n 2 s)) 1/2, ( [ n 1 ( ) ] 2 2 n 1 1 n! f n 2 l 2 s ( n) 2 + n 2 (2n 2s)! (s 1)! s 1 s1 f n s s f n 2 l 2 ( 2n 2s)) 1/2. ( s ) ( ) 2 4 n 1 2! s/2 As far as the constant C 2 in the statement of Theorem 5.5 is concerned, in the symmetric case one clearly has C 2 0. Finally, consider the constant C 3 in the statement of Theorem 5.5. The following bound holds: C n 2 3 n3 (2n 2 s)! s0 s {2i 1, 2i 2} 2(s (n 1)) ( ) 2 ( ) 2 ( n 1 n 1 i1 i 1!i 2! i 1 i 2 s i 1 0 f n (, ) s i 1 i 1 )( i2 f n (, ) l2 (N) 2n 2 s s i 2 ) 20n3 3 2n 2 s0 f n (, ) s i 2 i 2 f n (, ) l2 (N) 2n 2 s ( s ) ( ) 2 4 n 1 1{s/2 N}(2n 2 s)! 2! s/2 0 f n (, ) s/2 s/2 f n(, ) 2 l 2 (N) 2n 2 s

31 Chen-Stein method for non-symmetric Bernoulli processes n3 3 20n3 3 [ n ( ) ] 2 2 n 1 (2n 2s)! (s 1)! s 1 s1 0 [ n ( n 1 (2n 2s)! (s 1)! s 1 s1 f n (, ) s 1 s 1 f n(, ) 2 l 2 (N) 2n 2s ) 2 ] 2 f n s 1 s f n 2 l 2 (N) 2n 2s+1, where the latter equality follows from Lemma 2.4(2) (relation (2.4)) in Nourdin et al. (2010) the constant C 3 is finite again by Lemma 2.4(1) in Nourdin et al. (2010). We recovered the bound provided by Theorem 4.1 in Nourdin et al. (2010) Convergence to the normal distribution. The next theorems follow by Theorems , respectively. Theorem 5.6. Let n 2 be fixed let F m J n (f m ), m 1, be a sequence of discrete multiple stochastic integrals such that f m l 2 s( n ), for any N the functions h () () n 1,n 1,s h n 1,n 1,s defined by (5.10) (5.11) with f m in place of f n belong to l 2 s( 2n 2 s ), 0 s 2n 2, 0 n! f m 2 l s( n) 1, as m (5.27) f m (, ) s i i f m] ( ) l2 ( 2n 2 s ) 0, 0 as m, for any 0 s 2n 3 s 2i 2(s (n 1)) (5.28) 1 2p f m (, ) l 2 p q s ( n 1) f m (, ) s i 1 i 1 f m (, ) l 2 ( 2n 2 s) f m (, ) s i 2 i 2 f m (, ) l 2 ( 2n 2 s) 0, 0 as m, for any 0 s 2n 2 s {2i 1, 2i 2 } 2(s (n 1)) (5.29) 1 f m (, ) s i 1 i p q 1 f m (, ) l2 ( 2n 2 s ) f m (, ) s i 2 i 2 f m (, ) l2 ( 2n 2 s ) Then 0, as m, for any 0 s 2n 2 s {2i 1, 2i 2 } 2(s (n 1)). (5.30) F m Law N(0, 1). Theorem 5.7. Let n 2 be fixed let F m J n (f m ), m 1, be a sequence of discrete multiple stochastic integrals such that f m l 2 s( n ), for any N the () functions h n 1,n 1,s defined by (5.11) with f m in place of f n belong to l 2 s( 2n 2 s ), 0 s 2n 2, f m s i+1 i+1 f m l2 ( 2n 2 s ) 0, as m, for any 0 s 2n 3 s 2i 2(s (n 1)) (5.31)