Fourth moment theorems on the Poisson space in any dimension

We extend to any dimension the quantitative fourth moment theorem on the Poisson setting, recently proved by C. D\"obler and G. Peccati (2017). In particular, by adapting the exchangeable pairs couplings construction introduced by I. Nourdin and G. Zheng (2017) to the Poisson framework, we prove our results under the weakest possible assumption of finite fourth moments. This yields a Peccati-Tudor type theorem, as well as an optimal improvement in the univariate case. Finally, a transfer principle"from-Poisson-to-Gaussian"is derived, which is closely related to the universality phenomenon for homogeneous multilinear sums.


Outline
In the recent paper [13], the authors succeeded in proving exact quantitative fourth moment theorems for multiple Wiener-Itô integrals on the Poisson space. Briefly, their method consisted in extending the spectral framework initiated by the remarkable paper [22], and further refined by [1], from the situation of a diffusive Markov generator to the non-diffusive Ornstein-Uhlenbeck generator on the Poisson space. The principal aim of the present article is to extend the results from [13] to the multivariate case of vectors of multiple integrals. In view of the result of Peccati and Tudor [35] on vectors of multiple integrals on a Gaussian space, we are in particular interested in discussing the relationship between coordinatewise convergence and joint convergence to normality. Indeed, one of our main achievements is a complete quantitative version of a Peccati-Tudor type theorem on the Poisson space (see Theorem 1.7 and Corollary 1.8).
Furthermore, still keeping the spectral point of view as in [13], by replacing the rather intrinsic techniques used there with an adaption of a recent construction of exchangeable pairs couplings from [32], we can even remove certain technical conditions, which seem inevitable in order to justify the computations in [13]. In this way, we are able to prove our results under the weakest possible assumption of finite fourth moments. In the univariate case, our strategy provides an optimal improvement of the Wasserstein bound given in Theorem 1.3 of [13] and, a fortiori, of the associated qualitative fourth moment theorem on the Poisson space (see Corollary 1.4 in [13]).
Whereas in the diffusive situation of [22] and [1], the authors were able to identify very general quite weak conditions that ensure the validity of fourth moment theorems, we stress that this is not possible in the general non-diffusive situation. This is because, due to the fact that one only has an approximate chain rule, an additional remainder term in the bound necessarily arises. On the Poisson space, fortunately, it turns out that also this remainder term can be completely controlled in terms of the fourth cumulant (via inequality (2.5) below). However, there are examples of non-diffusive situations where no fourth moment theorem holds. Indeed, the authors of [11] give counterexamples for the setup of infinite Rademacher sequences that show that fourth moment theorems do not always hold in this case but that one also has to take into account the maximal influence of every single member of the random sequence. Thus, we emphasize that we are here dealing with a very peculiar example of a non-diffusive Markov generator where, quite coincidentally, all relevant terms can be reduced to just the fourth cumulant.

Motivation and related works
The so-called fourth moment theorem by Nualart and Peccati [33] states that a normalized sequence of multiple Wiener-Itô integrals of fixed order on a Gaussian space converges in distribution to a standard normal random variable N , if and only if the corresponding sequence of fourth moments converges to 3, i.e. to the fourth moment of N . For future reference, we give a precise statement of this result. Theorem 1.1 ([33]). Let F n = I W q (f n ) be a sequence of multiple Wiener-Itô integral of order q ≥ 2, associated with a Brownian motion (W t , t ∈ R + ) such that f n ∈ L 2 (R q + ) is symmetric for each n ∈ N, and lim n→+∞ E[F 2 n ] = σ 2 > 0. Then, the following statements are equivalent: (1) E[F 4 n ] → 3σ 4 , as n → +∞.
The contraction f n ⊗ r f n is defined as in Section 2. See [27] for any unexplained notions and notation of Gaussian analysis.
Note that such a result significantly simplifies the method of moments for sequences of random variables inside a fixed Wiener chaos. In the years after the appearance of [33], this result has been extended and refined in many respects. While [35] provided a significant multivariate extension (see Theorem 1.10), the paper [26] combined Stein's method of normal approximation and Malliavin calculus in order to yield quantitative bounds for the normal approximation of general smooth functionals on the Wiener space. We refer to the monograph [27] for a comprehensive treatment of the so-called [0, +∞] in the usual way. For B ∈ Z µ , we also define η(B) := η(B) − µ(B) and denote by η = { η(B) : B ∈ Z µ } the compensated Poisson random measure associated with η. Before stating our main results, we need to define some objects from stochastic analysis on the Poisson space. For a detailed discussion see, among others, [19] and [21].
For q ∈ N 0 := {0, 1, 2, . . . } and f ∈ L 2 (µ q ), we denote by I η q (f ) the q-th order multiple Wiener-Itô integral of f with respect to η. Let L be the generator of the Ornstein-Uhlenbeck semigroup with respect to η, then it is well known that −L is diagonalizable on L 2 (P) with discrete spectrum N 0 and that, for q ∈ N 0 , F is an eigenfunction of −L with eigenvalue q, if and only if F = I η q (f ) for some f ∈ L 2 (µ q ). The corresponding eigenspace C q will be called the q-th Poisson Wiener chaos associated with η. In particular, C 0 = R.

Main results in the one-dimensional case
Recall that the Wasserstein distance between (the distributions of) two real random variables X and Y in L 1 (P) is defined by where Lip(1) denotes the class of all 1-Lipschitz functions on R.
In the univariate case, our main result reads as follows.    [13], respectively, since they do not require any additional regularity from the involved multiple integrals like e.g. Assumption A in [13]. The main reason for the appearance of such a condition in [13] was that certain intrinsic tools used there, notably the Mecke formula and a pathwise representation of the Ornstein-Uhlenbeck generator L, require L 1 (P ⊗ µ)-integrability conditions. It is the reconciliation of such conditions with the L 2 nature of the objects under consideration which necessitated these assumptions. As will become clear from our proofs in Section 4, such conditions can be completely avoided using an adaptation of exchangeable pairs couplings introduced in [ where F ∈ C q with q ∈ N and C is a numerical constant. See [13] for more details.
In the particular case where η is a Poisson random measure on R + with Lebesgue intensity, we observe the following transfer principle that is of independent interest. EJP 23 (2018), paper 36. Proposition 1.5. Assume (#) and (W t , t ∈ R + ) is a standard Brownian motion. Given p ∈ N, f n ∈ L 2 (R p + ) symmetric for each n ∈ N such that lim n→+∞ p! f n 2 L 2 (R p + ) = 1 , then the following implications holds (N ∼ N (0, 1)) Remark 1.6. This transfer principle "from-Poisson-to-Gaussian" is closely related to the universality of Gaussian Wiener chaos and Poisson Wiener chaos, see Section 1.6. It is also worth pointing out that the transfer principle "from-Gaussian-to-Poisson" does not hold true, due to a counterexample given in [4]; see Proposition 5.4 therein.

Main results in the multivariate case
In this subsection, let us fix integers d ≥ 2 and 1 ≤ q 1 ≤ q 2 ≤ . . . ≤ q d and consider a random vector where F j ∈ C qj , j = 1, . . . , d. We will further assume that F j belongs to L 4 (P) for each j ∈ {1, . . . , d}. Furthermore, we denote by Σ : Note that Σ i,j = 0 whenever q i = q j due to the orthogonality properties of multiple integrals (see Section 2.1), and hence Σ is always a block diagonal matrix. Denote by N = (N 1 , . . . , N d ) T a centred Gaussian random vector with the same covariance matrix Σ.
In order to formulate our bounds, we need to fix some further notation: for a vector x = (x 1 , . . . , x d ) T ∈ R d , we denote by x 2 its Euclidean norm and for a matrix A ∈ R d×d , we denote by A op the operator norm induced by the Euclidean norm, i.e., A op := sup{ Ax 2 : x 2 = 1} .
Recall that for a function h : R d → R, its minimum Lipschitz constant M 1 (h) is given by If h is differentiable, then M 1 (h) = sup x∈R d Dh(x) op . More generally, for k ≥ 1 and a (k − 1)-times differentiable function h : Recall that, for two matrices A, B ∈ R d×d , their Hilbert-Schmidt inner product is  Thus, ·, · H.S. is just the standard inner product on R d×d ∼ = R d 2

. The corresponding
Hilbert-Schmidt norm will be denoted by · H.S. . With this notion at hand, following [7] and [24], for k = 2 we finally define where Hess h is the Hessian matrix corresponding to h. Note that for a symmetric matrix The next statement is our main result in the multivariate setting.
Theorem 1.7. Under the above assumptions and notation, we have the following bounds: (ii) If in addition Σ is positive definite, then for every g ∈ C 2 (R d ), we have The qualitative statement in the multivariate situation reads as follows.
EJP 23 (2018), paper 36. Corollary 1.8. Fix d ∈ N and q 1 , . . . , q d ∈ N and suppose that, for each n ∈ N, belongs to the q k -th Poisson Wiener chaos. Moreover, assume that C = C(i, j) 1≤i,j≤d is a fixed nonnegative definite matrix and that N = (N 1 , . . . , N d ) T is a centred Gaussian vector with covariance matrix C. Assume that the following two conditions hold true: Then, as n → ∞, the random vector F (n) converges in distribution to N . Remark 1.9. (a) Comparing the bounds in Theorem 1.7 with the one provided in Theorem 1.2, one observes that in the multivariate case the order of dependence on the fourth cumulants of the respective coordinates is 1/4 instead of 1/2. This phenomenon, which technically results from an application of the Cauchy-Schwarz inequality in order to disentangle certain joint moments of the coordinate variables, is nothing peculiar of the Poisson framework but also arises in the Gaussian situation [25] and in the recent multivariate de Jong type CLT for vectors of degenerate non-symmetric U -statistics [12]. Moreover, this phenomenon only arises in the case when there are components belonging to different chaoses (see Remark 4.3).
(b) We stress that it is remarkable that, as in the Gaussian case [35], the bounds and conditions in Theorem 1.7 and Corollary 1.8 can be expressed just in terms of the individual fourth cumulants of the components of the random vectors. Indeed, both in the general situation of diffusive Markov generators (see [5,Theorem 1.2]) and for the multivariate CLT for vectors of degenerate non-symmetric U -statistics (see [12,Theorem 1.7]), one additionally needs to assume the convergence of mixed fourth moments of those entries, which are of the same chaos and Hoeffding order, respectively.
(c) Corollary 1.8 is a full Poisson space analogue of the Peccati-Tudor theorem [35] for vectors of multiple integrals on a Gaussian space, which boils down the question about joint convergence to conditions guaranteeing coordinatewise convergence (via Corollary 1.3).
For the convenience of later reference, we state below the theorem of Peccati-Tudor on a Gaussian space, whose proof can be found in [27]. Theorem 1.10 ( [35]). Let (W t , t ∈ R + ) be a real standard Brownian motion, and we fix integers d ≥ 2 and 1 ≤ q 1 ≤ . . . ≤ q d . Let C = C(i, j) 1≤i,j≤d be a d × d symmetric nonnegative definite matrix and for any n ≥ 1, i ∈ {1, . . . , d}, let f n,i ∈ L 2 (R qi + , dx) be symmetric. Assume that the d-dimensional random vectors . . , d} . Then, as n → +∞, the following assertions are equivalent: (1) The vector F (n) converges in distribution to a d-dimensional Gaussian vector N (0, C); converges in distribution to a real Gaussian random variable N 0, C(i, i) ;

Universality of homogeneous sums
The transfer principle in Proposition 1.5 is closely related to the universality phenomenon for homogeneous multilinear sums in independent Poisson random variables. We refer to the papers [29], [37], [28] and [2] for the universality results on homogeneous multilinear sums and it is worth pointing out that the reference [28] also provides fourth moment theorems for homogeneous sums in general random variables satisfying some moment conditions. Before we can state the result, we need to introduce some notation.
Notation. Suppose that d ≥ 2, N ∈ N, and that f ∈ 2 (N d ) is a function, which is symmetric in its arguments and vanishes on diagonals, i.e. for any i 1 , . . . , i d ∈ N, of real random variables, we define the multilinear homogeneous sum of order d, based on the kernel f and on the first N elements of X by Now let us consider an independent sequence P = (P i , i ∈ N) of normalised Poisson random variables, which can be realised via our Poisson random measure η on R + . More precisely, let (t i , i ∈ N) be a strictly increasing sequence of positive numbers. Set We are now in the position to state the universality result. Theorem 1.11. Let the above notation prevail. Fix integers d ≥ 2 and q d ≥ . . . ≥ q 1 ≥ 2. For each j ∈ {1, . . . , d}, let (N n,j , n ≥ 1) be a sequence of natural numbers diverging to infinity, and let f n,j : {1, . . . , N n,j } qj → R be symmetric and vanishing on diagonals such (A 1 ) Let G be a sequence of i.i.d. standard Gaussian random variables, then, as n → +∞, are all equivalent and any of them is equivalent to the following assertion: Remark 1.12. The authors of [37] established a fourth moment theorem for sequences of homogeneous sums in independent Poisson random variables whose variance is bounded away from zero, namely, inf{t i+1 − t i : i ∈ N} > 0 in our language. In particular, in order to get the implication "(A 0 ) ⇒ (A 1 )", they relied heavily on the assumption that inf{t i+1 − t i : i ∈ N} > 0, which is inevitable due to their use of the product formula. As a consequence, our Theorem 1.11 is an improvement of the results in [37].
Plan of the paper. In Section 2, we review some necessary definitions and facts about multiple integrals and Malliavin operators on the Poisson space. Section 3 is devoted to the essential construction of a suitable family of exchangeable pairs for the concrete purpose of establishing fourth moment bounds on the Poisson space. In order to make use of it, we also state two new abstract plug-in results for such families of exchangeable pairs. In Section 4 we give the proofs of our main results, whereas Section 5 presents the proofs of Proposition 1.5, Theorem 1.11 as well as certain technical auxiliary results.
2 Some stochastic analysis on the Poisson space

Basic operators and notation
For a positive integer p, we denote by L 2 (µ p ) the Hilbert space of all square-integrable and real-valued functions on Z p , and we denote by L 2 s (µ p ) the subspace of L 2 (µ p ) whose elements are µ p -a.e. symmetric. Moreover, we indicate by · 2 and ·, · 2 respectively the usual norm and scalar product on L 2 (µ p ) for any value of p. We also set L 2 (µ 0 ) := R. For f ∈ L 2 (µ p ), we define I η p (f ) to be the multiple Wiener-Itô integral of f with respect to the compensated Poisson random measure η. If p = 0, then, by convention, I η 0 (c) := c for each c ∈ R.
The multiple Wiener-Itô integrals satisfy the following properties: where S p is the symmetric group acting on {1, . . . , p}. Note c = c for any c ∈ R.
See Section 3 of [19] for the proofs of the above well known results.
For p ∈ N 0 , the Hilbert space C p := {I η p (f ), f ∈ L 2 (µ p )}, is called the p-th Poisson Wiener chaos associated with η. The well-known Wiener-Itô chaotic decomposition states that every F ∈ L 2 (P) := L 2 (Ω, σ{η}, P) admits a unique representation Let F ∈ L 2 (P) and p ∈ N 0 , then we define by J p (F ) the orthogonal projection of F on C p . Note that, if F has the chaotic decomposition as in (2.1), For F ∈ L 2 (P) with the chaotic decomposition as in (2.1), we define This gives us the Ornstein-Uhlenbeck semigroup (P t , t ∈ R + ). The domain dom L of the Ornstein-Uhlenbeck generator L is the set of those F ∈ L 2 (P) with the chaotic decomposition (2.1) verifing ∞ p=1 p 2 p! f p 2 2 < +∞, and for F ∈ dom L, one has pI η p (f p ) . We conclude from (2.2) that LF is always centred, N 0 is the spectrum of −L and F ∈ dom L is an eigenfunction of −L with corresponding eigenvalue p if and only if F = I η p (f p ) for some f p ∈ L 2 s (µ p ), i.e. C p = Ker(L + pI). Moreover, it is easy to see that L is symmetric in the sense that E[GLF ] = E[F LG] for all F, G ∈ dom L. Finally, for F, G ∈ dom L with F G ∈ dom L, we define the carré du champ operator Γ associated with L by and it is easy to verify that E[Γ(F, . It follows from Lemma 2.1 below that Γ(F, G) is always well-defined whenever F, G ∈ L 4 (P) and both have a finite chaotic decomposition.
In the book [3], the authors develop Dirichlet form method for the Poisson point process, and starting from the Dirichlet form associated with the Ornstein-Uhlenbeck structure, they obtain an expression of carré du champs operator that is close to the one derived in [13]. As readers will see, we only need the spectral decomposition rather than the intrinsic tools in [3,13]. This highlights the elementary feature of our method.
Observe that f ⊗ r g ∈ L 2 (µ p+q−2r ) is in general not symmetric and that f ⊗ 0 g = f ⊗ g is simply the tensor product of f and g.  [13]). Let p, q ∈ N and F = I η p (f ), G = I η q (g) be in L 4 (P) with f, g symmetric, then F G has a finite chaotic decomposition of the form where h r ∈ L 2 s (µ r ) for each r. In particular, h p+q = f ⊗g.

Useful estimates via spectral decomposition
To conclude the section, we state several lemmas that are useful for our proofs.

Lemma 2.2.
Let F ∈ L 4 (P) ∩ C p and G ∈ L 4 (P) ∩ C q for p, q ∈ N. Then, (2.5) In particular, for F = G, we obtain Note that the authors of [13] provided a proof of (2.5) under the Assumption A therein, while we only require the assumption of finite fourth moment. Although (2.6) is the content of Lemma 3.1 in [13], we will provide another proof, in which we deduce a nice relation between contractions of kernels and the fourth cumulant. Such a relation is crucial for us to obtain the transfer principle "from-Poisson-to-Gaussian". The proof of Lemma 2.2 as well as that of the next lemma will be presented in Section 5.

Lemma 2.3.
Under the same assumptions of Lemma 2.2, we have that

Stein's method of exchangeable pairs
The exchangeable pairs approach within Stein's method was first used in the paper [9] which, however, attributes the method to Charles Stein himself. Later, this technique was presented in a systematic way in Stein's monograph [43]. We recall that a pair (X, X ) of random elements on a common probability space is said to be exchangeable, if (X, X ) has the same distribution as (X , X). In the book [43], it is highlighted that a given real random variable W is close in distribution to a standard normal variable N , whenever one can construct an exchangeable pair (W, W ) such that W is close to W in some sense and that the linear regression property In recent years, the method of exchangeable pairs has been generalised for other distributions and multi-dimensional settings in many papers like [39,40,42,38,7,8,6,10,15], to name a few.
Moreover, the articles [23,7,24,14] develop versions of the exchangeable pairs method suitable for situations, in which one can construct a continuous family (W, W t ) t>0 of exchangeable pairs. By their continuity assumptions, these papers succeed in reducing the order of smoothness of test functions and hence in obtaining bounds in more sophisticated probabilistic distances. For instance, the bounds from [23] are expressed in terms of the total variation distance. It is this framework of exchangeable pairs that is most closely related to the variant of the method developed in the present paper. In contrast to the quoted papers, however, our abstract results on exchangeable pairs do not make such strong continuity assumptions and hence, allow us to deal with the inherent discreteness of the Poisson space, which, in general, does not even allow for convergence in the total variation distance, see Section 3.2 for details. EJP 23 (2018), paper 36.

Exchangeable pairs constructed via continuous thinning
Recall that, in our general framework, η is a Poisson random measure on some σ-finite measure space Z, Z , µ . As a consequence, we can assume that η is a proper Poisson point process, that is, almost surely η = κ n=1 δ Xn , (3.1) where X n , n ≥ 1 are random variables with values in Z and κ is a N 0 ∪ {+∞}-valued random variable. Indeed, according to Corollary 3.7 in [21], any Poisson random measure η on some σ-finite measure space is equal in distribution to some proper Poisson point process. As in this work, we are only concerned with distributional properties, we will always assume that η is of the form (3.1).
Let N σ be the collection of σ-finite measures ν : Z → N 0 ∪ {+∞} and N σ (Z) be the σ-algebra generated by the maps ν ∈ N σ −→ ν(B), B ∈ Z . We consider the Poisson point process η as a random element in N σ , N σ (Z) . Moreover, for any F ∈ L 0 (Ω, σ{η}, P), one can find a (P-a.s. unique) representative f of F such that F = f(η), see [19] for more details. Now let Q be a standard exponential measure on R + with density exp(−y) dy, and let (Y n ) n∈N be a sequence of i.i.d. random variables with distribution Q, independent of (κ, X n ). Then the marked point process ξ, given by is a Poisson point process with control µ ⊗ Q. For each t ∈ R + , we define η e −t (A) := ξ A × [t, +∞) , which is called the e −t -thinning of η: it is obtained by removing the atoms (X n ) in η independently of each other with probability 1 − e −t . Moreover, η e −t and η − η e −t are two independent Poisson point processes with control measure e −t µ, (1 − e −t )µ respectively. One can refer to Chapter 5 in [21] for more details.
For any fixed t ≥ 0, let η 1−e −t be a Poisson point process on Z with control (1 − e −t )µ such that it is independent of (η, η e −t ). The Mehler formula gives a useful representation of the Ornstein-Uhlenbeck semigroup (P t ): for F ∈ L 2 (Ω, σ{η}, P), where f is a representative of F , see [19] for more details. We remark that the Mehler formula on the Poisson space has already been effectively used in [20] in order to obtain a pathwise representation for the pseudo-inverse of the Ornstein-Uhlenbeck generator L on the Poisson space, which has led to second-order Poincaré inequalities.
We record an important observation in the following lemma. Proof. To prove this lemma, it suffices to notice that η = η e −t + η − η e −t and that η − η e −t , η 1−e −t have the same law, and are both independent of η e −t .
Let f : N σ → R be N σ (Z)-measurable; then, for any Borel subsets A 1 , A 2 of R, one has that This implies the exchangeability of (η, η t ). The following result is a consequence of Lemma 3.1 and Mehler formula: it is a key ingredient for us to obtain exact fourth moment theorems in any dimension. Indeed, it fits extremely well with the abstract results for exchangeable pairs that are presented in Section 3.2.
Proof. The exchangeability of F, F t is an immediate consequence of Lemma 3.1. Relation (a) is a direct consequence of the Mehler formula: and such a quantity converges almost surely, and in L 4 (P) to LF = −qF , as t ↓ 0.
converges almost surely and in L 2 (P) to p+q k=0 −k J k (F G) = L(F G), as t ↓ 0. Hence almost surely and in L 2 (P), we infer that so (c) follows immediately from (a),(b) and the fact that F ∈ L 4 (P).

Abstract results for exchangeable pairs
As indicated in the introductory part of this section, the following two Propositions should be seen as complements to [23,Theorem 1.4] and [24,Theorem 4] as well as [7, Theorem 2.4], respectively. The main difference with respect to these results, as mentioned above, is that we do not assume any continuity from the respective families of exchangeable pairs, which precisely means that we allow for non-zero limits in the respective conditions (c) below. EJP 23 (2018), paper 36. Proposition 3.3. Let Y and a family of random variables (Y t ) t≥0 be defined on a common probability space (Ω, F, P) such that Y t law = Y for every t ≥ 0. Assume that Y ∈ L 4 (Ω, G , P) for some σ-algebra G ⊂ F and that, in L 1 (P), The following result is a multivariate extension of Proposition 3.3. The proofs will be postponed to Section 5.5 and 5.6. Proposition 3.5. For each t > 0, let (X, X t ) be an exchangeable pair of centred d-dimensional random vectors defined on a common probability space (Ω, F, P). Let G be a σ-algebra that contains σ{X}. Assume that Λ ∈ R d×d is an invertible deterministic matrix and Σ is a symmetric, non-negative definite deterministic matrix such that where X i,t (resp. X i ) stands for the i-th coordinate of X t (resp. X).
Then, with N ∼ N (0, Σ), we have the following bounds: where the constants Θ 1 (g) and Θ 2 (g) are given by (2) If, in addition, Σ is positive definite, then for g ∈ C 2 (R d ) such that g(X), g(N ) ∈ L 1 (P), one has E g(X) − g(N ) where the constants K 1 (g) and K 2 (g) are given by (3.4)

Proof of Theorem 1.2
Without loss of generality, we assume F = I η q (f ) for some f ∈ L 2 s (µ q ), and we define F t = I η t q (f ) for t ∈ R + . Then, by Proposition 3.2, (F, F t ) is an exchangeable pair and the assumptions (a), (b), (c) in Proposition 3.3 are satistified with More precisely, Therefore, one has (using that E Γ(F, The desired result follows immediately from Lemma 2.2.

Proof of Theorem 1.7
Assume that . . ≤ q d and f j ∈ L 2 s (µ qj ) for each j, and for each t ∈ R + , define Then, by Lemma 3.1, (F t , F ) is an exchangeable pair and by Proposition 3.2, we deduce It is easy to see that for each j ∈ {1, . . . , d}, from which we deduce that as t ↓ 0 and in L 2 (P), we have . . , d}, and observe in particular that S i,j has zero mean. Thus, where the last equality is a consequence of the fact that (see e.g. (4.2) in [31]) In particular, if q 1 = . . . = q d , one has, Proof of Lemma 4.1. Let us first consider the particular case where q 1 = . . . = q d . One obtains from Lemma 2.3 that In the general case where q 1 ≤ . . . ≤ q d , Lemma 2.3 implies One can rewrite

Remark 4.2.
Note that in the same way, we can provide another proof of the quantitative Peccati-Tudor Theorem in the Gaussian setting. In particular, keeping the indicator functions in (4.4), we can obtain the bound in a similar form as in [25,Theorem 1.5].
End of the proof of Theorem 1.7. First we obtain from (4.2)-(4.3) and Lemma 4.1 that If g ∈ C 3 (R d ) and g(F ), g(N ) are integrable, then by Proposition 3.5, we deduce where the last inequality follows from (4.5) and (4.1). It is easy to check that Assertion (i) of Theorem 1.7 follows immediately. Assertion (ii) can be proved in the same way, by using moreover the relations: Wiener chaos, then we can obtain better bounds, namely: (ii) If, in addition, Σ is positive definite, then for every g ∈ C 2 (R d ), we have EJP 23 (2018), paper 36.

Proofs of technical and auxiliary results
In this section, we first provide the proofs of Lemma 2.2, Lemma 2.3. The following result from [31] will be helpful. Lemma 5.1 (Lemma 2.2 of [31]). Given p, q ∈ N, f ∈ L 2 s (µ p ) and g ∈ L 2 s (µ q ), then and in the case of p = q, one has Here we follow the convention that 0 r=1 = 0.

Proof of Lemma 2.2
Without loss of generality, we assume F = I η p (f ) and G = I η q (g) for some f ∈ L 2 s (µ p ) and g ∈ L 2 s (µ q ). It follows from Lemma 2.1 and the definition of Γ that J p+q (F G) = I η p+q (f ⊗g) and Var J k (F G) .
Similarly, as F G ∈ L 2 (P), It follows from Lemma 5.1 that Var J k (F G) In particular, Lemma 5.1, applied to p = q and f = g, gives us This proves (2.6) and It is also clear from (5.2) that So by orthogonality, we have 3. Let F, N be given as in Theorem 1.7, then from (4.3) it follows that E F 4 2 ≥ E N 4 2 . Moreover, if one of the components F j in F has nonzero variance, it follows from the above two points and again (4.3) that E F 4 2 > E N 4 2 .

Proof of Lemma 2.3
Assume F = I η p (f ) and G = I η q (g) are in L 4 (P) for some f ∈ L 2 s (µ p ), g ∈ L 2 s (µ q ). Then it follows from Lemma 2.1 that J 2p (F 2 ) = I η 2p (f ⊗f ) and J 2q (G 2 ) = I η 2q (g ⊗g). Moreover, one has If p < q, then E F 2 J 2q (G 2 ) = 0, so that where the above inequality follows from Cauchy-Schwarz inequality and isometry property. The desired result (2.7) follows from (5.4).
Now we consider the case where p = q, From now on, we identify f with f in case of no confusion. Observe that the sequence G of i.i.d standard Gaussian random variables can be realised via the Brownian motion (W t , t ∈ R + ). That is, for each i ∈ N, we put G i = I W 1 (g i ). As a consequence, the homogeneous sum Q d (f, N, G) can be rewritten as I W d ( f ), with f given in (5.6). With these notions at hand and in view of our transfer principle, if (A 0 ) holds, then, for each j ∈ {1, . . . , d} and every r ∈ {1, . . . , q j − 1}, f n,j ⊗ r f n,j 2 → 0, as n → +∞. Then, (A 1 ) is an immediate consequence of Theorem 1.10.

Proof of Proposition 3.3
Without loss of any generality, we may and will assume that Var(Y ) = 1 and N ∼ N (0, 1). Let f : R → R be 1-Lipschitz function and consider g(x) = e x 2 /2 x −∞ f (y) − E[f (N )] e −y 2 /2 dy , x ∈ R , which satisfies the Stein's equation as well as g ∞ ≤ 2/π, g ∞ ≤ 2, see e.g. Section 2.3 in [44]. In what follows, we fix such a pair (f, g) of functions. Let G : R → R be a differentiable function such that G = g. Then due to Y t law = Y and Y ∈ L 4 (P), one has By assumption (c) and as t ↓ 0, Therefore as t ↓ 0, assumptions (a) and (b) imply that The above equation shall be understood as "the limit lim t↓0 t −1 E[R g ] exists and is equal to −λE g (Y ) − Y g(Y ) − 1 2 E g (Y )S , bounded by 1