Ivan Nourdin
email: ivan.nourdin@upmc.fr

Giovanni Peccati
email: giovanni.peccati@gmail.com

Gesine Reinert
email: reinert@stats.ox.ac.uk

Invariance Principles for Homogeneous Sums: Universality of Gaussian Wiener Chaos

Keywords: Central Limit Theorems, Chaos, Homogeneous Sums, Lindeberg Principle, Malliavin Calculus, Chi-square Limit Theorems, Stein's Method, Universality, Wiener Chaos. 2000 Mathematics Subject Classification: 60F05, 60F17, 60G15, 60H07

We compute explicit bounds in the normal and chi-square approximations of multilinear homogenous sums (of arbitrary order) of general centered independent random variables with unit variance. Our techniques combine an invariance principle by Mossel, O'Donnell and Oleszkiewicz with a refinement of some recent results by Nourdin and Peccati, about the approximation of laws of random variables belonging to a fixed (Gaussian) Wiener chaos. In particular, we show that chaotic random variables enjoy the following form of universality: (a) the normal and chi-square approximations of any homogenous sum can be completely characterized and assessed by first switching to its Wiener chaos counterpart, and (b) the simple upper bounds and convergence criteria available on the Wiener chaos extend almost verbatim to the class of homogeneous sums. These results partially rely on the notion of "low influences" for functions defined on product spaces, and provide a generalization of central and non-central limit theorems proved by Nourdin, Nualart and Peccati. They also imply a further drastic simplification of the method of moments and cumulants -as applied to the proof of probabilistic limit theoremsand yield substantial generalizations, new proofs and new insights into some classic findings by de Jong and Rotar'. Our tools involve the use of Malliavin calculus, and of both the Stein's method and the Lindeberg invariance principle for probabilistic approximations.

1 Introduction

Overview

The aim of this paper is to study and characterize the normal and chi-square approximations of the laws of multilinear homogeneous sums involving general independent random variables. We shall perform this task by implicitly combining three probabilistic techniques, namely: (i) the Lindeberg invariance principle (in the version developed by Mossel et al. [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF] and Rotar' [START_REF] Rotar | Limit theorems for polylinear forms[END_REF], see also Mossel [START_REF] Mossel | Gaussian bounds for noise correlation of functions[END_REF]), (ii) the Stein's method for the normal and chi-square approximations (see e.g. [START_REF] Chen | Stein's method for normal approximation[END_REF][START_REF] Reinert | Three general approaches to Stein's method[END_REF][START_REF] Ch | A bound for the error in the normal approximation to the distribution of a sum of dependent random variables[END_REF][START_REF] Ch | Approximate computation of expectations[END_REF]), and (iii) the Malliavin calculus of variations on a Gaussian space (see e.g. [START_REF] Malliavin | Stochastic Analysis[END_REF] and [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]). Our analysis will reveal that the Gaussian Wiener chaos (see Section 2 below for precise definitions) enjoys the following properties: (a) the normal and chi-square approximations of any multilinear homogenous sum are completely characterized and assessed by those of its Wiener chaos counterpart, and (b) the strikingly simple upper bounds and convergence criteria available on the Wiener chaos (see [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF][START_REF] Nourdin | Stein's method on Wiener chaos[END_REF][START_REF] Nourdin | Stein's method and exact Berry-Essen bounds for functionals of Gaussian fields[END_REF][START_REF] Nourdin | Multivariate normal approximation using Stein's method and Malliavin calculus[END_REF][START_REF] Nourdin | Second order Poincaré inequalities and CLTs on Wiener space[END_REF][START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF]) extend almost verbatim to the class of homogeneous sums. Our findings partially rely on the notion of "low influences" (see again [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF]) for real-valued functions defined on product spaces. As indicated by the title, we regard the two properties (a) and (b) as an instance of the universality phenomenon, according to which most information about large random systems (such as the "distance to Gaussian" of non-linear functionals of large samples of independent random variables) does not depend on the particular distribution of the components. Other recent examples of the universality phenomenon appear in the already quoted paper [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF], as well as in the Tao-Vu proof of the circular law for random matrices, as detailed in [START_REF] Tao | Random matrices: Universality of ESDs and the circular law[END_REF] (see also the Appendix to [START_REF] Tao | Random matrices: Universality of ESDs and the circular law[END_REF] by Krishnapur). Observe that in Section 7 we will prove analogous results for the multivariate normal approximation of vectors of homogenous sums of possibly different orders.

The approach

In what follows, every random object is defined on a suitable (common) probability space (Ω, F , P ). The symbol E denotes expectation with respect to P . We start by giving a precise definition of the main objects of our study. Definition 1.1 (Multilinear homogeneous sums) Fix some integers N, d 2 and write [N] = {1, ..., N}. Let X = {X i : i 1} be a collection of centered independent random variables, and let f : [N] d → R be a symmetric function vanishing on diagonals (that is, f (i 1 , ..., i d ) = 0 whenever there exist k = j such that i k = i j ). The random variable

Q d (N, f, X) = Q d (X) = 1 i 1 ,...,i d N f (i 1 , ..., i d )X i 1 • • • X i d (1.1) = d! {i 1 ,...,i d }⊂[N ] d f (i 1 , ..., i d )X i 1 • • • X i d (1.2) = d! 1 i 1 <...<i d N f (i 1 , ..., i d )X i 1 • • • X i d (1.3)
is called the multilinear homogeneous sum, of order d, based on f and on the first N elements of X. As in the first equality of (1.1), and when there is no risk of confusion, we will drop the dependence on N and f in order to simplify the notation. Plainly, E[Q d (X)] = 0 and also, if E(X 2 i ) = 1 for every i, then

E[Q d (X) 2 ] = d! f 2 d , (1.4)
where, here and for the rest of the paper, we use the notation

f 2 d = 1 i 1 ,...,i d N f 2 (i 1 , ..., i d ).
In the following, we will systematically use the expression "homogeneous sum" instead of "multilinear homogeneous sum".

Objects such as the RHS of (1.1) are sometimes called "polynomial chaoses", and play a central role in several branches of probability theory and stochastic analysis. For instance:

-In the case d = 2, homogeneous sums are indeed standard quadratic forms with no diagonal terms. Limit theorems for quadratic forms have been the object of an intense study since 30 years. An exhaustive discussion about the best available results on the approximation of the laws of quadratic forms can be found in Götze and Tikhomirov [START_REF] Götze | Asymptotic distribution of quadratic forms and applications[END_REF].

-By setting X i = H q i (G i ), where {G i } is an independent and identically distributed (i.i.d.) centered standard Gaussian sequence and H q is the Hermite polynomial of degree q, one obtains the basic building blocks of the so-called Gaussian Wiener chaos (see Section 2), whose properties are at the core of the Malliavin calculus (see e.g. [START_REF] Janson | Gaussian Hilbert Spaces[END_REF][START_REF] Nualart | The Malliavin calculus and related topics[END_REF]) and of the white noise analysis (see e.g. [START_REF] Obata | White noise calculus and Fock space[END_REF]).

-By setting X i = J q i (P i ), where J q indicates the Charlier polynomial of degree q and {P i } are the values of a centered Poisson measure over disjoint sets, one obtains a class of random elements generating the Poisson Wiener chaos (see e.g. [START_REF] Ogura | Orthogonal functionals of the Poisson process[END_REF][START_REF] Peccati | Stein's method and normal approximation of Poisson functionals[END_REF][START_REF] Privault | Chaotic and variational calculus in discrete and continuous time for the Poisson process[END_REF][START_REF] Rota | Stochastic integrals: a combinatorial approach[END_REF]).

-By setting X to be a Rademacher sequence (that is, the X i 's are i.i.d. and such that P (X 1 = 1) = P (X 1 = -1) = 1/2) then Q d (X) are the constitutive elements of the so-called Walsh chaos, playing a central role e.g. in harmonic analysis (see [START_REF] Blei | Analysis in Integer and Fractional Dimensions[END_REF]), probability on Banach spaces (see [START_REF] Ledoux | Probability on Banach spaces[END_REF]), quantum probability (see [START_REF] Meyer | Quantum Probability for Probabilists[END_REF]Ch. 2]), social choice theory and theoretical computer sciences (see [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF], and the references therein, for an overview of results in these last two areas).

Despite the almost ubiquitous nature of homogeneous sums, results concerning the normal approximation of quantities such as (1.1) in the non-quadratic case (i.e., when d 3) are surprisingly scarce: indeed, to our knowledge, the only general statements in this respect are contained in the paper [START_REF] Jong | A central limit theorem for generalized multilinear forms[END_REF] and the monograph [START_REF] Jong | Central Limit Theorems for generalized multilinear forms[END_REF], both by P. de Jong (as discussed below, and in a different direction, general criteria allowing to assess the proximity of the laws of homogenous sums based on different independent sequences are obtained in [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF][START_REF] Rotar | Limit theorems for multilinear forms and quasipolynomial functions[END_REF][START_REF] Rotar | Limit theorems for polylinear forms[END_REF]).

Remark 1.2 We stress that (1.1) can be viewed as a sum of locally dependent random variables, with index set J = {1, . . . , n} d . Note moreover that the techniques usually adopted in the normal approximation of sums of dependent variables are not appropriate here. For instance, one cannot adopt the approach developed by Chen and Shao in [START_REF] Chen | Normal approximation under local dependence[END_REF]: indeed, in our framework the neighborhoods of dependence A i = {j = (j 1 , . . . , j d ) ∈ J : |i ∩ j| = 0}, i = (i 1 , . . . , i d ) ∈ J , verify the relation ∪ j∈A i A j = J and are consequently "too large".

In this paper, we are interested in controlling objects of the type d H {Q d (X) ; Z}, where: (i) Q d (X) is defined in (1.1), (ii) Z is either a standard Gaussian N (0, 1) or a centered chi-square random variable, and (iii) the distance d H {F ; G}, between the laws of two random variables F, G, is given by

d H {F ; G} = sup E[h(F )] -E[h(G)] : h ∈ H , (1.5)
(E denotes expectation) with H some suitable class of real-valued functions.

Even with some uniform control on the components of X, the problem of directly and generally assessing d H {Q d (X) ; Z} looks very arduous. Indeed, any estimate comparing the laws of Q d (X) and Z capriciously depends on the kernel f , and on the way in which the analytic structure of f interacts with the specific "shape" of the distribution of the random variables X i . One revealing picture of this situation appears if one tries to evaluate the moments of Q d (X) and to compare them with those of Z: see e.g. [START_REF] Peccati | Moments, cumulants and diagram formulae for non-linear functionals of random measure[END_REF][START_REF] Surgailis | CLTs for Polynomials of Linear Sequences: Diagram Formulae with Applications[END_REF] for a discussion of some associated combinatorial structures. In the specific case where Z is Gaussian, one should also observe that Q d (X) is a completely degenerate U-statistic, as E[Q d (X 1 ; x 2 , . . . , x d )] = 0 for all x 2 , . . . , x d , so that the standard results for the normal approximation of U-statistics do not apply.

The main point developed in this paper is that one can successfully overcome these difficulties by implementing the following strategy: first (I) measure the distance

d H {Q d (X) ; Q d (G)},
between the law of Q d (X) and the law of the random variable Q d (G), obtained by replacing X with a centered standard i.i.d. Gaussian sequence G = {G i : i 1}, then (II) assess the distance d H {Q d (G) ; Z}, and finally (III) use the triangle inequality in order to write

d H {Q d (X) ; Z} d H {Q d (X) ; Q d (G)} + d H {Q d (G) ; Z}.
(1.6)

We will see in the subsequent sections that the power of this approach resides in the following two facts.

-The distance evoked at Point (I) can be effectively controlled by means of the techniques developed in [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF], where the authors have produced a general theory, based on a generalization of the so-called Lindeberg invariance principle (see e.g. [START_REF] Trotter | An elementary proof of the central limit theorem[END_REF]), allowing to estimate the distance between multilinear homogeneous sums constructed from different sequences of independent random variables. A full discussion of this point is presented in Section 4 below. In Theorem 4.1 we shall observe that, under the assumptions that E(X 2 i ) = 1 and that the moments E(|X i | 3 ) are uniformly bounded by some constant β > 0 (recall that the X i 's are centered), one can deduce from [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF] that (provided that the elements of H are sufficiently smooth),

d H {Q d (X) ; Q d (G)} C × max 1 i N {i 2 ,...,i d }∈[N ] d-1 f 2 (i, i 2 , ..., i d ), (1.7)
where C is a constant depending on d, β and on the class H . The quantity

Inf i (f ) = {i 2 ,...,i d }∈[N ] d-1 f 2 (i, i 2 , ..., i d ) = 1 (d -1)! 1 i 2 ,...,i d N f 2 (i, i 2 , ..., i d ) (1.8)
is called the influence of the variable i, and roughly quantifies the contribution of X i to the overall configuration of the homogenous sum Q d (X). Influence indices already appear (under a different name) in the papers by Rotar' [START_REF] Rotar | Limit theorems for multilinear forms and quasipolynomial functions[END_REF][START_REF] Rotar | Limit theorems for polylinear forms[END_REF].

-The random variable Q d (G) is an element of the dth Wiener chaos associated with G (see Section 2 for definitions). As such, the distance between Q d (G) and Z (in both the normal and the chi-square cases) can be assessed by means of the results appearing in [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF][START_REF] Nourdin | Stein's method on Wiener chaos[END_REF][START_REF] Nourdin | Stein's method and exact Berry-Essen bounds for functionals of Gaussian fields[END_REF][START_REF] Nourdin | Multivariate normal approximation using Stein's method and Malliavin calculus[END_REF][START_REF] Nourdin | Second order Poincaré inequalities and CLTs on Wiener space[END_REF][START_REF] Nualart | Central limit theorems for multiple stochastic integrals and Malliavin calculus[END_REF][START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF][START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF], which are in turn based on a powerful interaction between standard Gaussian analysis, Stein's method, and the Malliavin calculus on variations. As an example, Theorem 3.1 of Section 3 proves that the above quoted results imply that, if Q d (G) has variance one and Z is standard Gaussian, then

d H {Q d (G) ; Z} C |E[Q d (G) 4 ] -E(Z 4 )| = C |E[Q d (G) 4 ] -3| (1.9)
where C > 0 is some finite constant depending only on H and d. Moreover, if the elements of X have bounded fourth moments, then the RHS of (1.9) can be directly estimated by means of the fourth moment of Q d (X) and some maxima over the influence functions appearing in (1.7) (see Section 5 for a full discussion of this point).

Universality

(j) n , f (j) n ) : n 1} be a sequence such that {N (j)
n : n 1} is a sequence of integers going to infinity, and each function f

(j) n : [N (j) n ] d j → R is symmetric and vanishes on diagonals. Define Q d j (N (j) n , f (j)
n , G), n 1, according to (1.1). Assume that, for every j = 1, ...m, the following sequence of variances is bounded:

E[Q d j (N (j) n , f (j) n , G) 2 ], n 1. (1.10)
Let V be a m × m non-negative symmetric matrix, and let N m (0, V ) indicate a centered Gaussian vector with covariance matrix V . Then, as n → ∞, the following two conditions are equivalent.

(1) The vector {Q d j (N

(j) n , f (j)
n , G) : j = 1, ..., m} converges in law to N m (0, V ).

(2) For every sequence X = {X i : i 1} of independent centered random variables, with unit variance and such that sup i E|X i | 3 < ∞, the law of the vector

{Q d j (N (j) n , f (j) n , X) : j = 1, ..., m} converges to the law of N m (0, V ) in the Kol- mogorov distance. Remark 1.4 1. Given random vectors F = (F 1 , ..., F m ) and H = (H 1 , ..., H m ), m 1
, the Kolmogorov distance between the law of F and the law of H is defined as

d Kol (F, H) = sup (z 1 ,...,zm)∈R m |P (F 1 z 1 , ..., F m z m )-P (H 1 z 1 , ..., H m z m )|. (1.11)
Recall that the topology induced by d Kol on the class of all probability measures on R m is strictly stronger than the topology of convergence in distribution.

2. Note that, in the statement of Theorem 1.3, we do not require that the matrix V is positively definite, and we do not introduce any assumption on the asymptotic behavior of influence indices.

3. Due to the matching moments up to second order, one has that

E[Q d i (N (i) n , f (i) n , G)×Q d j (N (j) n , f (j) n , G)] = E[Q d i (N (i) n , f (i) n , X)×Q d j (N (j) n , f (j) n , X)]
for every i, j = 1, ..., m and every sequence X as at Point (2) of Theorem 1.3.

Plainly, the crucial implication in the statement of Theorem 1.3 is (1) ⇒ (2), which basically ensures that any statement concerning the asymptotic normality of (vectors of) general homogeneous sums can be proved by simply focussing on the elements of a Gaussian Wiener chaos. Since Central Limit Theorems (CLTs) on Wiener chaos are by now completely characterized (thanks to the results proved in [START_REF] Nourdin | Stein's method on Wiener chaos[END_REF][START_REF] Nourdin | Multivariate normal approximation using Stein's method and Malliavin calculus[END_REF][START_REF] Nualart | Central limit theorems for multiple stochastic integrals and Malliavin calculus[END_REF][START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF][START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF]), this fact represents a clear methodological breakthrough. As explained later in the paper, and up to the restriction on the third moments appearing at Point (2), we regard Theorem 1.3 as the first exact equivalent -for homogeneous sums -of the usual CLT for linear functionals of i.i.d. sequences. The proof of Theorem 1.3 is completely achieved in Section 7.

Remark 1.5

1. The use of the techniques developed in [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF] makes it unavoidable to require a uniform bound on the third moments of X. However, one advantage of using Lindeberg-type arguments is that we obtain convergence in the Kolmogorov distance, as well as explicit upper bounds on the rates of convergence. We will see below (see Theorem 1.12 for a precise statement) that in the one-dimensional case one can simply require a bound on the moments of order 2 + ǫ, for some ǫ > 0. Moreover, still in the one-dimensional case and when the sequence X is i.i.d., one can alternatively deduce convergence in distribution from a result by Rotar' [40, Proposition 1], for which the existence of moments of order greater than 2 is not required.

2. In this paper we will apply the techniques of [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF] in order to evaluate quantities of the type

|E[Q d (X) m ] -E[Q d (Y) m ]|
, where m 1 is an integer and X and Y are two sequences of independent random variables. From a purely methodological standpoint, we believe that this technique may replace the (sometimes fastidious) use of "diagram formulae" and associated combinatorial devices (see [START_REF] Peccati | Moments, cumulants and diagram formulae for non-linear functionals of random measure[END_REF][START_REF] Surgailis | CLTs for Polynomials of Linear Sequences: Diagram Formulae with Applications[END_REF]). See Section 4 for a full discussion of this point, especially Lemma 4.4.

The role of contractions

The universality principle stated in Theorem 1.3 is based on [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF], as well as on general characterizations of (possibly multidimensional) CLTs on a fixed Wiener chaos. Results of this kind have been first proved in [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF] (for the one-dimensional case) and [START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF] (for the multidimensional case), and make an important use of the notion of "contraction" of a given deterministic kernel (see also [START_REF] Nualart | Central limit theorems for multiple stochastic integrals and Malliavin calculus[END_REF]). When studying homogeneous sums, one is naturally led to deal with contractions defined on discrete sets of the type [N] d , N

1. As already pointed out in [START_REF] Nourdin | Stein's method and stochastic analysis of Rademacher functionals[END_REF], these objects enjoy a number of useful combinatorial properties, related for instance to the notion of fractional dimension or fractional Cartesian product (see Blei [START_REF] Blei | Analysis in Integer and Fractional Dimensions[END_REF]). In this section, we shall briefly explore these issues, in particular by pointing out that discrete contractions are indeed the key element in the proof of Theorem 1.3. More general statements, as well as complete proofs, are given in Section 3.

Definition 1.6 Fix d, N
2. Let f : [N] d → R be a symmetric function vanishing of diagonals. For every r = 0, ..., d, the contraction f ⋆ r f is the function on [N] 2d-2r given by f ⋆ r f (j 1 , ..., j 2d-2r )

(1.12) = 1 a 1 ,...,ar N f (a 1 , ..., a r , j 1 , ..., j d-r )f (a 1 , ..., a r , j d-r+1 , ..., j 2d-2r ).

Observe that f ⋆ r f is not necessarily symmetric and not necessarily vanishes on diagonals. The symmetrization of f ⋆ r f is written f ⋆ r f .

The following result, whose proof will be achieved in Section 7, as a special case of Theorem 7.5, is based on the findings of [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF] and [START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF].

Proposition 1.7 (CLT for chaotic sums) Let the assumptions and notations of Theorem 1.3 prevail, and suppose moreover that, for every i, j = 1, ..., m (as n → ∞)

E[Q d i (N (i) n , f (i) n , G) × Q d j (N (j) n , f (j) n , G)] → V (i, j), (1.13)
where V is a non-negative symmetric matrix. Then, the following three conditions are equivalent, as n → ∞.

(1) The vector {Q d j (N

(j) n , f (j)
n , G) : j = 1, ..., m} converges in law to a centered Gaussian vector with covariance matrix V .

(2) For every j

= 1, ..., m, E[Q d j (N (j) n , f (j) n , G) 4 ] → 3V (i, i) 2 .
(3) For every j = 1, ..., m and every r = 1, ..., d j -1, f

(j) n ⋆ r f (j) n 2d j -2r → 0.
Remark 1.8 Strictly speaking, the results of [START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF] only deal with the case where V is positive definite. The needed general result will be obtained in Section 7 by means of Malliavin calculus.

Let us now briefly sketch the proof of the implication (1) ⇒ (2) in Theorem 1.3. Suppose that the sequence in (1.10) is bounded and that Point (1) in Theorem 1.3 is verified. Then, by uniform integrability (use Proposition 2.7), the convergence (1.13) is satisfied and, according to Proposition 1.7, we have f

(j) n ⋆ d j -1 f (j) n 2 → 0. The crucial remark is now that f (j) n ⋆ d j -1 f (j) n 2 2 = 1 i,k N (j) n    1 i 2 ,...,i d j N (j) n f (j) n (i, i 2 , ..., i d j )f (j) n (k, i 2 , ..., i d j )    2 N (j) n i=1    1 i 2 ,...,i d j N (j) n f (j) n (i, i 2 , ..., i d j ) 2    2 max i=1,...,N (j) n    1 i 2 ,...,i d j N (j) n f (j) n (i, i 2 , ..., i d j ) 2    2 = (d j -1)! max 1 i N (j) n Inf i (f (j) n ) 2 (1.14)
(recall formula (1.8)), from which one immediately obtains that, as n → ∞,

max 1 i N (j) n Inf i (f (j) n ) → 0, for every j = 1, ..., m. (1.15)
The proof of Theorem 1.3 is concluded by using Theorem 7.1, which is a multi-dimensional version of the findings of [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF]. Indeed, this result will imply that, if (1.15) is verified then, for every sequence X as in Point (2) of Theorem 1.3, the distance between the law of {Q d j (N

(j) n , f (j)
n , G) : j = 1, ..., m} and the law of {Q d j (N

(j) n , f (j)
n , X) : j = 1, ..., m} necessarily tends to zero, and therefore the two sequences must converge in distribution to the same limit.

As proved in [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF], contractions play an equally important role in the chi-square approximation of the laws of elements of a fixed chaos of even order. Recall that a random variable Z ν has a centered chi-square distribution with ν 1 degrees of freedom (noted

Z ν ∼ χ 2 (ν)) if Z ν Law = ν i=1 G 2 i -1
, where (G 1 , ..., G ν ) is a vector of i.i.d. centered Gaussian random variables with unit variance. Note that

E(Z 2 ν ) = 2ν, E(Z 3 ν ) = 8ν, and E(Z 4 ν ) = 12ν 2 + 48ν. (1.16)
The following result is proved in [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF].

Theorem 1.9 (Chi-square limit theorem for chaotic sums) Let G = {G i : i 1} be a standard centered i.i.d. Gaussian sequence, and fix an even integer d 2. Let {N n , f n : n 1} be a sequence such that {N n : n 1} is a sequence of integers going to infinity, and each

f n : [N n ] d → R is symmetric and vanishes on diago- nals. Define Q d (N n , f n , G), n
1, according to (1.1), and assume that, as n → ∞,

E[Q d (N n , f n , G) 2 ] → 2ν.
Then, as n → ∞, the following three conditions are equivalent.

(1) Q d (N n , f n , G) Law → Z ν ∼ χ 2 (ν). (2) E[Q d (N n , f n , G) 4 ] -12E[Q d (N n , f n , G) 3 ] → E[Z 4 ν ] -12E[Z 3 ν ] = 12ν 2 -48ν.
(

) f n ⋆ d/2 f n -c d × f n d → 0 and f n ⋆ r f n 2d-2r = 0 for every r = 1, ..., d -1 such that r = d/2 3
, where

c d := 1 (d/2)! d-1 d/2-1 2 = 4 (d/2)! d d/2
2 .

(1.17)

A neat application of Theorem 1.9 is sketched in the next section.

Example: revisiting de Jong criterion

To further clarify the previous discussion (and to motivate the reader), we provide an illustration of how one can use our results in order to refine a remarkable result by de Jong, originally proved in [START_REF] Jong | A central limit theorem for generalized multilinear forms[END_REF].

Theorem 1.10 (See [START_REF] Jong | A central limit theorem for generalized multilinear forms[END_REF]) Let X = {X i : i 1} be a sequence of independent centered random variables such that E(X 2 i ) = 1 and E(X

d (n, X) = Q d (N n , f n , X), n 1, according to (1.1). Assume that E[Q d (n, X) 2 ] = 1 for all n. If, as n → ∞, (i) E[Q d (n, X) 4 ] → 3, and
(ii) max 1 i Nn 1 i 2 ,...,i d Nn f 2 n (i, i 2 , ..., i d ) = (d -1)! max 1 i Nn Inf i (f n ) → 0, then Q d (n, X)
converges in law to a standard Gaussian random variable Z ∼ N (0, 1).

Remark 1.11 As shown in [START_REF] Jong | A central limit theorem for generalized multilinear forms[END_REF], the conclusion of Theorem 1.10 extends to homogeneous sums that are not necessarily multilinear. Also, the fact that E(X 4 i ) < ∞ does not appear in [START_REF] Jong | A central limit theorem for generalized multilinear forms[END_REF] (however, an inspection of the proof of Theorem 1.10 shows that this assumption is indeed necessary).

In the original proof given in [START_REF] Jong | A central limit theorem for generalized multilinear forms[END_REF], Assumption (i) in Theorem 1.10 appears as a convenient (and mysterious) way of reexpressing the asymptotic "lack of interaction" between products of the type X i 1 • • • X i d , whereas Assumption (ii) plays the role of a usual Lindeberg-type assumption. In the present paper, under the slightly stronger assumption that β := sup i E(X 4 i ) < ∞, we will be able to produce bounds neatly indicating the exact roles of both Assumption (i) and Assumption (ii). To see this, define d H according to (1.5), and set H to be the class of thrice differentiable functions whose first three derivatives are bounded by some finite constant B > 0. In Section 5, in the proof of Theorem 5.1 we will show that there exist universal, explicit, finite constants C 1 , C 2 , C 3 > 0, depending uniquely on β, d and B, such that (writing G for an i.i.d. centered standard Gaussian sequence)

d H {Q d (n, X); Q d (n, G)} C 1 × max 1 i Nn Inf i (f n ) + max 1 i Nn Inf i (f n ) (1.18) d H {Q d (n, G); Z} C 2 × |E[Q d (n, G) 4 ] -3| (1.19) |E[Q d (n, X) 4 ] -E[Q d (n, G) 4 ]| C 3 × max 1 i Nn Inf i (f n ) + max 1 i Nn Inf i (f n ) .
(1.20)

In particular, the estimates (1.

: [N n ] d → R is symmetric and vanishes on diagonals. Define Q d (n, G) = Q d (N n , f n , G), n 1, according to (1.1). Assume that E[Q d (n, G) 2 ] → 1 as n → ∞.
Then, the following four properties are equivalent as n → ∞.

(1) The sequence

Q d (n, G) converges in law to Z ∼ N (0, 1). (2) E[Q d (n, G) 4 ] → 3.
(3) For every sequence X = {X i : i 1} of independent centered random variables with unit variance and such that

sup i E|X i | 2+ǫ < ∞ for some ǫ > 0, the sequence Q d (n, X) converges in law to Z ∼ N (0, 1) in the Kolmogorov distance.
(4) For every sequence X = {X i : i 1} of independent and identically distributed centered random variables with unit variance, the sequence Q d (n, X) converges in law to Z ∼ N (0, 1) (not necessarily in the Kolmogorov distance). Remark 1. [START_REF] Ledoux | Probability on Banach spaces[END_REF] 1. Note that at point (4) of the above statement we do not require the existence of moments of order greater than 2. We will see that the equivalence between ( 1) and ( 4) is partly a consequence of Rotar's results [START_REF] Rotar | Limit theorems for polylinear forms[END_REF].

2. Theorem 1.12 is a particular case of Theorem 1.3, and can be seen as refinement of de Jong's Theorem 1.10, in the sense that: (i) since several combinatorial devices are at hand (see e.g. [START_REF] Peccati | Moments, cumulants and diagram formulae for non-linear functionals of random measure[END_REF]), it is in general easier to evaluate moments of multilinear forms of Gaussian sequences than of general sequences, and (ii) when the {X i } are not identically distributed, we only need existence (and uniform boundedness) of the moments of order 2 + ǫ.

In Section 7, we will generalize the content of this section to multivariate Gaussian approximations. By using Proposition 1.9 and [40, Proposition 1], one can also obtain the following universal chi-square limit result.

Theorem 1.14 We let the notation of Theorem 1.12 prevail, except that we now assume that d 2 is an even integer and

E[Q d (n, G) 2 ] → 2ν
, where ν 1 is an integer. Then, the following conditions are equivalent as n → ∞.

(1) The sequence

Q d (n, G) converges in law to Z ν ∼ χ 2 (ν). (2) E[Q d (n, G) 4 ] -12E[Q d (n, G) 3 ] → E(Z 4 ν ) -12E(Z 3 ν ) = 12ν 2 -48ν.
(3) For every sequence X = {X i : i 1} of independent centered random variables with unit variance and such that

sup i E|X i | 2+ǫ < ∞ for some ǫ > 0, the sequence Q d (n, X) converges in law to Z ν .
(4) For every sequence X = {X i : i 1} of independent and identically distributed centered random variables with unit variance, the sequence Q d (n, X) converges in law to Z ν .

Two counterexamples 1.6.1 There is no universality for sums of order one

One striking feature of Theorem 1.3 and Theorem 1.12 is that they do not have any equivalent for sums of order d = 1. To see this, consider an array of real numbers {f n (i) :

1 i n} such that n i=1 f 2 n (i) = 1. Let G = {G i : i 1} and X = {X i : i
1} be, respectively, a centered i.i.d. Gaussian sequence with unit variance, and a sequence of independent random variables with zero mean and unit variance. Then,

Q 1 (n, G) := n i=1 f n (i)G i ∼ N (0, 1) for every n, but it is in general not true that Q 1 (n, X) := n i=1 f n (i)X i converges
in law to a Gaussian random variable. An example of this situation is obtained by taking X 1 to be non-Gaussian, f n (1) = 1 and f n (j) = 0 for j > 1. As it is well-known, to ensure that Q 1 (n, X) has a Gaussian limit one customarily adds the Lindeberg-type requirement that max 1 i n |f n (i)| → 0 (see e.g. [START_REF] Kallenberg | Foundations of modern probability[END_REF]Th. 4.12]). A closer inspection indicates that the fact that no Lindeberg conditions are required in Theorem 1.3 and Theorem 1.12 is due to the implication (1) ⇒ (3) in Proposition 1.7, as well as to the inequality (1.14).

Walsh chaos is not universal

We shall now prove that one cannot replace the Gaussian sequence G with a Rademacher one in the statements of Theorem 1.3 and Theorem 1.12. Let X = {X i : i 1} be an i.i.d. Rademacher sequence, and fix d 2. For every N d, consider the homogeneous sum

Q d (N, X) = X 1 X 2 • • • X d-1 N i=d X i √ N -d + 1 (1.21)
It is easily seen that each Q d (N, X) can be written in the form (1.1), for some symmetric

f = f N vanishing on diagonals and such that d! f N 2 d = 1. Since X 1 X 2 • • • X d-1 is a random sign independent of {X i : i d}, a simple application of the Central Limit Theorem yields that, as N → ∞, Q d (N, X)
Law → N (0, 1). On the other hand, a direct computation reveals that, for every N d, max 2 . This shows that homogenous sums associated with Rademacher sequences are not universal. For instance, by replacing X in (1.21) with a i.i.d. standard Gaussian sequence G = {G i : i 1}, one sees that, for every N 2,

1 i N Inf i (f N ) = (1/d!)
Q d (N, G) = G 1 G 2 • • • G d-1 N i=d G i √ N -d + 1 Law = G 1 • • • G d . (1.22)
Since (for d 2) the random variable G 1 • • • G d is not Gaussian, this yields that the sequence Q d (N, G), N 1, does not verify a Central Limit Theorem.

Three caveats on generality

In order to enhance the readability of the forthcoming material, we decided not to state some of our findings in full generality. In particular:

(i) It will be clear later on that the results of this paper easily extend to the case of infinite homogeneous sums (obtained by putting N = +∞ in (1.1)). This requires, however, a somewhat heavier notation, as well as some distracting digressions about convergence.

(ii) Our findings do not hinge at all on the fact that N is an ordered set: it follows that our results exactly apply to homogeneous sums of random variables indexed by a general finite set.

(iii) Our results on chi-square approximations could be slightly modified in order to accommodate approximations by arbitrary centered Gamma laws. To do this, one can use the results about Gamma approximations proved in [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF][START_REF] Nourdin | Stein's method on Wiener chaos[END_REF].

Wiener chaos

In this section, we briefly introduce the notion of (Gaussian) Wiener chaos, and point out some of its crucial properties. The reader is referred to [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]Ch. 1] or [START_REF] Janson | Gaussian Hilbert Spaces[END_REF]Ch. 2] for any unexplained definition or result.

Definitions

Let G = {G i : i 1} be a sequence of i.i.d. centered Gaussian random variables with unit variance.

Definition 2.1 (Hermite polynomials and Wiener chaos)

1. The sequence of Hermite polynomials {H q : q 0} is defined as follows: H 0 = 1, and, for q 1,

H q (x) = (-1) q e x 2 2 d q dx q e -x 2 2 , x ∈ R.
Recall that the class {(q!) -1/2 H q : q 0} is an orthonormal basis of

L 2 (R, (2π) -1/2 e -x 2 /2 dx).
2. A multi-index q = {q i : i 1} is a sequence of nonnegative integers such that q i = 0 only for a finite number of indices i. We also write Λ to indicate the class of all multi-indices, and use the notation |q| = i 1 q i , for every q ∈ Λ.

3. For every d 0, the dth Wiener chaos associated with G is defined as follows: C 0 = R, and, for d 1, C d is the L 2 (P )-closed vector space generated by random variables of the type

Φ(q) = ∞ i=1 H q i (G i ), q ∈ Λ and |q| = d. (2.23) Example 2.2 (i) The first Wiener chaos C 1 is the Gaussian space generated by G, that is, F ∈ C 1 if and only if F = ∞ i=1 λ i G i , for some sequence {λ i : i 1} ∈ ℓ 2 .
(ii) Fix d, N 2 and let f : [N] d → R be symmetric and vanishing on diagonals. Then, the d-homogeneous sum

Q d (G) = d! {i 1 ,...,i d }⊂[N ] d f (i 1 , ..., i d )G i 1 • • • G i d = 1 i 1 ,...,i d N f (i 1 , ..., i d )G i 1 • • • G i d (2.24) is an element of C d .
It is easily seen that two random variables belonging to Wiener chaoses of different orders are orthogonal in L 2 (P ). Moreover, since linear combinations of polynomials are dense in L 2 (P, σ(G)), one has that L 2 (P, σ(G)) = d 0 C d , that is, any square integrable functional of G can be written as an infinite sum, converging in L 2 and such that the dth summand is an element of C d . This representation result is known as the Wiener-Itô chaotic decomposition of L 2 (P, σ(G)).

It is often useful to encode the properties of random variables in the spaces C d by using increasing tensor powers of Hilbert spaces (see e.g. [11, Appendix E] for a collection of useful facts about tensor products). To do this, introduce a (arbitrary) real separable Hilbert space H and, for d 2, denote by H ⊗d (resp. H ⊙d ) the dth tensor power (resp. symmetric tensor power) of H; write moreover H ⊗0 = H ⊙0 = R and H ⊗1 = H ⊙1 = H. Let {e j : j 1} be an orthonormal basis of H. With every multi-index q ∈ Λ, we associate the tensor e(q) ∈ H ⊗|q| given by e(q) = e

⊗q i 1 i 1 ⊗ • • • ⊗ e ⊗q i k i k ,
where {q i 1 , ..., q i k } are the non-zero elements of q. We also denote by ẽ(q) ∈ H ⊙|q| the canonical symmetrization of e(q). It is well-known that, for every d 2, the collection {ẽ(q) : q ∈ Λ, |q| = d} defines a complete orthogonal system in H ⊙d . For every d 1 and every h ∈ H ⊙d with the form h = q∈Λ, |q|=d c q ẽ(q), we define

I d (h) = q∈Λ, |q|=d c q Φ(q), (2.25)
where Φ(q) is given in (2.23

E[I d (h)I d (h ′ )] = d! h, h ′ H ⊗d .
As proved e.g. in [27, Section 1. (ii) The random variable Q d (G) defined in (2.24) is such that

1.2], if H = L 2 (A, A , µ),
Q d (G) = I d (h), where h = d! {i 1 ,...,i d }⊂[N ] d f (i 1 , ..., i d )e i 1 ⊗ • • • ⊗ e i d ∈ H ⊙d . (2.26)

Multiplication formulae and hypercontractivity

The notion of "contraction" is the key to prove the general bounds stated in the forthcoming Section 3.

Definition 2.5 (Contractions) Let {e i : i 1} be a complete orthonormal system in H. Given f ∈ H ⊙p and g ∈ H ⊙q , for every r = 0, . . . , p ∧ q, the rth contraction of f and g is the element of H ⊗(p+q-2r) defined as

f ⊗ r g = ∞ i 1 ,...,ir=1
f, e i 1 ⊗ . . . ⊗ e ir H ⊗r ⊗ g, e i 1 ⊗ . . . ⊗ e ir H ⊗r .

(2.27)

Plainly, f ⊗ 0 g = f ⊗ g equals the tensor product of f and g while, for p = q, f ⊗ p g = f, g H ⊗p . Note that, in general (and except for trivial cases), the contraction f ⊗ r g is not a symmetric element of H ⊗(p+q-2r) . The canonical symmetrization of f ⊗ r g is written f ⊗ r g.

Contractions appear in multiplication formulae

I p (f )I q (g) = p∧q r=0 r! p r q r I p+q-2r (f ⊗ r g).
(2.28)

Note that the previous statement implies that multiple integrals admit finite moments of every order. The next result (whose proof is given e.g. in [START_REF] Janson | Gaussian Hilbert Spaces[END_REF]Th. 5.10]) establishes a more precise property, namely that random variables living in a finite sum of Wiener chaoses are hypercontractive. Proposition 2.7 (Hypercontractivity) Let d 1 be a finite integer and assume that

F ∈ d k=0 C k . Fix reals 2 p q < ∞. Then, E[|F | q ] 1 q (q -1) d 2 E[|F | p ] 1 p . (2.29)
We will use hypercontractivity in order to deduce some of the bounds stated in the next section.

Normal and chi-square approximation on Wiener chaos

Starting from this section, and for the rest of the paper, we adopt the following notation for distances between laws of real-valued random variables.

-The symbol d T V (F, G) indicates the total variation distance between the law of F and G, obtained from (1.5) by taking H equal to the class of all indicators of the Borel subsets of R.

-The symbol d W (F, G) denotes the Wasserstein distance, obtained from (1.5) by choosing H as the class of all Lipschitz functions with Lipschitz constant less or equal to 1.

-The symbol d BW (F, G) stands for the bounded Wasserstein distance (or Fortet-Mourier distance), deduced from (1.5) by choosing H as the class of all Lipschitz functions that are bounded by 1, and with Lipschitz constant less than or equal to 1.

While [START_REF] Gibbs | On choosing and bounding probability metrics[END_REF] for an overview.

d Kol (F, G) d T V (F, G) and d BW (F, G) d W (F, G), in general d T V (F, G) and d W (F, G) are not comparable, see
In what follows, we consider as given an i.i.d. centered standard Gaussian sequence G = {G i : i 1}, and we shall adopt the Wiener chaos notation introduced in Section 2.

Bounds on the normal approximation

In the recent series of papers [START_REF] Nourdin | Stein's method on Wiener chaos[END_REF][START_REF] Nourdin | Stein's method and exact Berry-Essen bounds for functionals of Gaussian fields[END_REF][START_REF] Nourdin | Second order Poincaré inequalities and CLTs on Wiener space[END_REF], it has been shown that one can effectively combine Malliavin calculus with Stein's method, in order to evaluate the distance between the law of an element of a fixed Wiener chaos, say F , and a standard Gaussian distribution. In this section, we state several refinements of these results, by showing in particular that all the relevant bounds can be expressed in terms of the fourth moment of F . The proofs involve the use of Malliavin calculus and are deferred to Section 8. [START_REF] Blei | Analysis in Integer and Fractional Dimensions[END_REF], and write

Theorem 3.1 (Fourth moment bounds) Fix d 2. Let F = I d (h), h ∈ H ⊙d , be an element of the dth Gaussian Wiener chaos C d such that E(F 2 ) = 1, let Z ∼ N (0,
T 1 (F ) := d 2 d-1 r=1 (r -1)! 2 d -1 r -1 4 (2d -2r)! h ⊗ r h 2 H ⊗2(d-r) T 2 (F ) := d -1 3d |E(F 4 ) -3|.
We have

T 1 (F ) T 2 (F ). (3.30)
Moreover, the following bounds hold:

1.

d T V (F, Z) 2T 1 (F ). (3.31) 2. d BW (F, Z) d W (F, Z) T 1 (F ). (3.32) 3. Let ϕ : R → R be a thrice differentiable function such that ϕ ′′′ ∞ < ∞. Write K * = K * |ϕ ′ (0)|, |ϕ ′′ (0)|, ϕ ′′′ ∞ (3.33) = max 2 |ϕ ′′ (0)| + ϕ ′′′ ∞ 3 2 √ 2 √ π ; 2 |ϕ ′ (0)| + 1 3 ϕ ′′′ ∞ C * = C * d, |ϕ ′ (0)|, |ϕ ′′ (0)|, ϕ ′′′ ∞ = 3 + (2 √ 2) d × K * . (3.34)
Then, one has that

|E[ϕ(F )] -E[ϕ(Z)]| C * × T 1 (F ).
Proof. See Section 8.3.

Remark 3.2 If E(F ) = 0 and F has a finite fourth moment, then the quantity κ 4 (F ) = E(F 4 ) -3E(F 2 ) 2 is known as the fourth cumulant of F . See e.g. [START_REF] Surgailis | CLTs for Polynomials of Linear Sequences: Diagram Formulae with Applications[END_REF] for a general introduction to cumulants. One can also prove (see e.g. [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF]) that, if F is a non-zero element of the dth Wiener chaos of a given Gaussian sequence (d 2), then κ 4 (F ) > 0.

Central limit theorems

Now fix d 2, and consider a sequence of random variables of the type

F n = I d (h n ), n 1, such that, as n → ∞, E(F 2 n ) = d! h n 2 H ⊗d → 1.
In [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF] it is proved that the following double implication holds: as n → ∞, (1) F n converges in law to Z ∼ N (0, 1).

h n ⊗ r h n H ⊗2(d-r) → 0, ∀r = 1, ..., d -1 ⇔ h n ⊗ r h n H ⊗2(d-r) → 0, ∀r = 1, ..., d -1. ( 3
(2) E(F 4 n ) → E(Z 4 ) = 3.

(

) For every r = 1, ..., d -1, h n ⊗ r h n H ⊗2(d-r) → 0. 3
Proof. Since sup n E(F 2 n ) < ∞, one deduces from Proposition 2.7 that, for every M > 2, sup n E|F n | M < ∞. By uniform integrability it follows that, if (1) is in order, then necessarily E(F 4 n ) → E(Z 4 ) = 3. The rest of the proof is a consequence of the bounds (3.30)-(3.31) and of the fact that the topology induced by the total variation distance (on the class of all probability measures on R) is stronger than the topology of convergence in distribution.

The following (elementary) result is one of the staples of the present paper. It relates the Hilbert space contractions of formula (2.27) with the discrete contraction operators defined in (1.12). We state it in a form which is also useful for the chi-square approximation of the forthcoming Section 3.3.

h ⊗ r h H ⊗(2d-2r) = f ⋆ r f 2d-2r , (3.36)
where we have used the notation (1.12). Also, if d is even, then for every

α 1 , α 2 ∈ R α 1 (h ⊗ d/2 h) + α 2 h H ⊗d = α 1 (f ⋆ d/2 f ) + α 2 f d . (3.37)
Proof. Fix r = 1, ..., d -1. Using (2.26) and the fact that {e j : j 1} is an orthonormal basis of H, one infers that

h ⊗ r h = 1 i 1 ,...,i d N 1 j 1 ,...,j d N f (i 1 , ..., i d )f (j 1 , ..., j d )[e i 1 ⊗ • • • ⊗ e i d ] ⊗ r [e j 1 ⊗ • • • ⊗ e j d ] = 1 a 1 ,...,ar N 1 k 1 ,...,k 2d-2r N f (a 1 , ...a r , k 1 , ..., k d-r ) × ×f (a 1 , ..., a r , k d-r+1 , ..., k 2d-2r )e k 1 ⊗• • • ⊗e k 2d-2r = 1 k 1 ,...,k 2d-2r 1 f ⋆ r f (k 1 , ..., k 2d-2r )e k 1 ⊗ • • • ⊗ e k 2d-2r . (3.38)
Since the set {e k 1 ⊗ • • • ⊗ e k 2d-2r : k 1 , ..., k 2d-2r 1, mutually distinct} is an orthonormal basis of H ⊗(2d-2r) , one deduces immediately (3.36). The proof of (3.37) is analogous.

Bounds on the chi-square approximation

As demonstrated in [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF][START_REF] Nourdin | Stein's method on Wiener chaos[END_REF], the combination of Malliavin calculus and Stein's method also allows to estimate the distance between the law of an element F of a fixed Wiener chaos and a (centered) chi-square distribution χ 2 (ν) with ν degrees of freedom. Analogously to the previous section for Gaussian approximations, we now state a number of refinements of the results proved in [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF][START_REF] Nourdin | Stein's method on Wiener chaos[END_REF]. In particular, we will show that all the relevant bounds can be expressed in terms of a specific linear combination of the third and fourth moments of F . Once again, the proofs are deferred to Section 8.

T 3 (F ) := 4d! h - d! 2 4(d/2)! 3 h ⊗ d/2 h 2 H ⊗d +d 2 r=1,...,d-1 r =d/2 (r -1)! 2 d -1 r -1 4 (2d -2r)! h ⊗ r h 2 H ⊗2(d-r) 1 2 T 4 (F ) := d -1 3d |E(F 4 ) -12E(F 3 ) -12ν 2 + 48ν|.
Then,

T 3 (F ) T 4 (F ) (3.39)
and Proof. See Section 8.4.

d BW (F, Z ν ) max 2π ν , 1 ν + 2 ν 2 T 3 (F ). ( 3

Chi-square limit theorems

Now fix an even integer d 2, and consider a sequence of random variables of the type

F n = I d (h n ), n 1, such that, as n → ∞, E(F 2 n ) = d! h n 2 H ⊗d → 2ν.
In [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF] it is proved that the following double implication holds: as n → ∞,

h n ⊗ r h n H ⊗2(d-r) → 0, ∀r = 1, ..., d -1, r = d/2 ⇐⇒ h n ⊗ r h n H ⊗2(d-r) → 0, ∀r = 1, ..., d -1, r = d/2.
(3.41) Theorem 3.6, combined with (3.41), allows therefore to recover the following characterization of chi-square limit theorems on Wiener chaos. Note that this is a special case of a 'non-central limit theorem'; one usually calls 'non-central limit theorem' any result involving convergence in law to a non-Gaussian distribution.

Theorem 3.8 (See [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF]) Fix an even integer d 2, and let

F n = I d (h n ), n 1
be a sequence in the dth Wiener chaos of G. Assume that lim n→∞ E(F 2 n ) = 2ν. Then, the following conditions are equivalent, as n → ∞.

(1) (1.17)) and, for every r = 1, ..., d-1

F n converges in law to a Z ν ∼ χ 2 (ν). (2) E(F 4 n ) -12E(F 3 n ) → E(Z 4 ν ) -12E(Z 3 ν ) = 12ν 2 -48ν. (3) h n ⊗ d/2 h n -c d ×h n H ⊗d → 0 (for c d as given in
such that r = d/2, h n ⊗ r h n H ⊗2(d-r) → 0.
Proof.

Since sup n E(F 2 n ) < ∞, one deduces from Proposition 2.7 that, for every M > 2, sup n E|F n | M < ∞. By uniform integrability it follows that, if (1) holds, then necessarily

E(F 4 n ) -12E(F 3 n ) → E(Z 4 ν ) -12E(Z 3 ν ) = 12ν 2 -48ν.
The rest of the proof is a consequence of the bound (3.39) and of the fact that the topology induced by the bounded Wasserstein distance (on the class of all probability measures on R) is equivalent to the topology of convergence in distribution.

Remark 3.9 By using (3.37) in the case α 1 = 1 and α 2 = -c d , one sees that Theorem 3.8 yields an immediate proof of Proposition 1.9.

Low influences and proximity of homogeneous sums

We now turn to some remarkable invariance principles by Rotar' [START_REF] Rotar | Limit theorems for polylinear forms[END_REF] and Mossel, O' Donnell and Oleszkiewicz [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF]. As already discussed, the results proved in [START_REF] Rotar | Limit theorems for polylinear forms[END_REF] yield sufficient conditions in order to have that the laws of homogeneous sums (or, more generally, polynomial forms) that are built from two different sequences of independent random variables are asymptotically close, whereas in [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF] one can find explicit upper bounds on the distance between these laws. Since in this paper we adopt the perspective of deducing general convergence results from limit theorems on a Gaussian space, we will state the results of [START_REF] Rotar | Limit theorems for polylinear forms[END_REF] and [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF] in a slightly less general form, namely by assuming that one of the sequences is i.i.d. Gaussian. See also Davydov and Rotar' [START_REF] Davidov | On asymptotic proximity of distributions[END_REF], and the references therein, for some general characterizations of the asymptotic proximity of probability distributions.

Theorem 4.1 (See [20]) Let X = {X i , i
1} be a collection of centered independent random variables with unit variance, and let G = {G i , i 1} be a collection of standard centered i.i.d. Gaussian random variables. Fix d 1, and let {N n , f n : n 1} be a sequence such that {N n : n 1} is a sequence of integers going to infinity, and each

f n : [N n ] d → R is symmetric and vanishes on diagonals. Define Q d (N n , f n , X) and Q d (N n , f n , G) according to (1.1). 1. If sup i 1 E[|X i | 2+ǫ ] < ∞ for some ǫ > 0 and if max 1 i Nn Inf i (f n ) → 0 as n → ∞
(with Inf i (f n ) given by (1.8)) then we have

sup z∈R P Q d (N n , f n , X) z -P Q d (N n , f n , G) z → 0 as n → ∞. (4.42)
2. If the random variables X i are identically distributed and if

max 1 i Nn Inf i (f n ) → 0 as n → ∞, then E ψ(Q d (N n , f n , X))] -E ψ(Q d (N n , f n , G))] → 0 as n → ∞, (4.43)
for every continuous and bounded function ψ : R → R.

If

β := sup i 1 E[|X i | 3 ] < ∞ then, for all thrice differentiable ϕ : R → R such that ϕ ′′′ ∞ < ∞ and for every fixed n, E ϕ(Q d (N n , f n , X)) -E ϕ(Q d (N n , f n , G)) (4.44) ϕ ′′′ ∞ (30β) d d! max 1 i Nn Inf i (f n ).
Proof.

Point 1 is Theorem 2.2 in [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF]. Point 2 is Proposition 1 in [START_REF] Rotar | Limit theorems for polylinear forms[END_REF]. Point 3 is Theorem 3.18 (under Hypothesis H2) in [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF]. Note that our polynomials Q d relate to polynomials d!Q in [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF], hence the extra factor of d! in 4.44.

Remark 4.2 We stress that Point 3 in Theorem 4.1 is a non-asymptotic result. In particular, the estimate (4.44) holds for every n independently of the asymptotic behavior of the quantities max 1 i Nn Inf i (f n ).

In the sequel, we will also need the following technical lemma, which is a combination of several results stated and proved in [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF].

Lemma 4.3 Let X = {X i , i 1} be a collection of centered independent random variables with unit variance. Assume moreover that

γ := sup i 1 E[|X i | q ] < ∞ for some q > 2.
Fix N, d 1, and let f : [N] d → R be a symmetric function (here, observe that we do not require that f vanishes on diagonals). Define

Q d (X) = Q d (N, f, X) by (1.1). Then E Q d (X) q γ d 2 q -1 qd E Q d (X) 2 q/2 .
Proof. Combine Propositions 3.11, 3.16 and 3.12 in [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF].

As already evoked in the Introduction, one of the key elements in the proof of Theorem 4.1 given in [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF] is the use of an elegant probabilistic technique, which is in turn inspired by the well-known Lindeberg's proof of the Central Limit Theorem (see [START_REF] Lindeberg | Eine neue Herleitung des exponential-Gesetzes in der Warscheinlichkeitsrechnung[END_REF]). Very lucid accounts of Lindeberg's ideas can be found in Trotter [START_REF] Trotter | An elementary proof of the central limit theorem[END_REF] and Feller [START_REF] Feller | An Introduction to Probability Theory and its Applications[END_REF]Section VIII]. As an illustration of the Lindeberg-type approach developed in [START_REF] Mossel | Noise stability of functions with low influences: variance and optimality[END_REF], we will now state and prove a useful lemma, concerning moments of homogeneous sums. As suggested before, we believe that the employed technique may replace "diagram formulae" and similar combinatorial devices, that are customarily used in order to estimate moments of nonlinear functionals of independent sequences (see e.g. [START_REF] Peccati | Moments, cumulants and diagram formulae for non-linear functionals of random measure[END_REF][START_REF] Surgailis | CLTs for Polynomials of Linear Sequences: Diagram Formulae with Applications[END_REF], as well as the estimates on moments of homogeneous sums given in de Jong's paper [START_REF] Jong | A central limit theorem for generalized multilinear forms[END_REF]). Lemma 4.4 Let X = {X i : i 1} and Y = {Y i : i 1} be two collections of centered independent random variables with unit variance. Fix some integers N, d 1, and let f :

[N] d → R be a symmetric function vanishing on diagonals. Define Q d (X) = Q d (N, f, X) and Q d (Y) = Q d (N, f, Y) according to (1.1). 1. Suppose k 2 is such that: (a) X i and Y i belong to L k (Ω) for all i 1; (b) E(X l i ) = E(Y l i ) for all i 1 and l ∈ {2, . . . , k}. Then Q d (X) and Q d (Y) belong to L k (Ω), and E(Q d (X) l ) = E(Q d (Y) l
) for all l ∈ {2, . . . , k}.

Suppose m > k 2 are such that: (a) α

:= max sup i 1 E|X i | m , sup i 1 E|Y i | m < ∞; (b) E(X l i ) = E(Y l i
) for all i 1 and l ∈ {2, . . . , k}. Assume moreover (for simplicity) that:

(c) E[Q d (X) 2 ] 1/2
M for some finite constant M 1. Then Q d (X) and Q d (Y) belong to L m (Ω) and, for all l ∈ {k + 1, . . . , m},

E(Q d (X) l ) -E(Q d (Y) l ) c d,l,m,α × M l-k+1 × max 1 i N max Inf i (f ) k-1 2 ; Inf i (f ) l 2 -1 , where c d,l,m,α = 2 l+1 (d-1)! α dl m 2 √ l -1 (2d-1)l d! l-1 .
Proof. Point 1 could be verified by a direct (elementary) computation. However, we will obtain the same conclusion as the by-product of a more sophisticated construction (based on Lindeberg-type arguments) which will also lead to the proof of Point 2. We shall assume, without loss of generality, that the two sequences X and Y are stochastically independent. For i = 0, . . . , N, let Z (i) denote the sequence (Y 1 , . . . , Y i , X i+1 , . . . , X N ). Fix a particular i ∈ {1, . . . , N}, and write

U i = 1 i 1 ,...,i d N ∀k: i k =i f (i 1 , . . . , i d )Z (i) i 1 . . . Z (i) i d V i = 1 i 1 ,...,i d N ∃k: i k =i f (i 1 , . . . , i d )Z (i) i 1 . . . Z (i) i . . . Z (i) i d ,
where

Z (i)
i means that this particular term is dropped (observe that this notation bears no ambiguity: indeed, since f vanishes on diagonals, each string i 1 , ..., i d contributing to the definition of V i contains the symbol i exactly once). Note that U i and V i are independent of the variables X i and Y i , and that

Q d (Z (i-1) ) = U i + X i V i and Q d (Z (i) ) = U i + Y i V i .
By using the independence of X i and Y i from U i and V i (as well as the fact that E(X l i ) = E(Y l i ) for all i and all 1 l k) we infer from the binomial formula that, for l ∈ {2, . . . , k},

E (U i + X i V i ) l = l j=0 l j E(U l-j i V j i )E(X j i ) (4.45) = l j=0 l j E(U l-j i V j i )E(Y j i ) = E (U i + Y i V i ) l . That is, E Q d (Z (i-1) ) l = E Q d (Z (i)
) l for all i ∈ {1, . . . , N} and l ∈ {2, . . . , k}. The desired conclusion of Point 1 follows by observing that

Q d (Z (0) ) = Q d (X) and Q d (Z (N ) ) = Q d (Y).
To prove Point 2, let l ∈ {k + 1, . . . , m}. Using (4.45) and then Hölder's inequality, we can write

E Q d (Z (i-1) ) l -E Q d (Z (i) ) l = l j=k+1 l j E(U l-j i V j i ) E(X j i ) -E(Y j i ) l j=k+1 l j (E|U i | l ) 1-j/l (E|V i | l ) j/l E|X i | j + E|Y i | j .
But, by Lemma 4.3 and since E(

U 2 i ) E(Q d (X) 2 ) M 2 , we have E|U i | l α dl/m 2 √ l -1 ld E(U 2 i ) l/2 α dl/m 2 √ l -1 ld M l . Similarly, since E(V 2 i ) = d! 2 Inf i (f ) (see (1. 8
)), we have

E|V i | l α (d-1)l/m 2 √ l -1 l(d-1) E(V 2 i ) l/2 α (d-1)l/m 2 √ l -1 l(d-1) d! l Inf i (f ) l/2 .
Hence, since

E|Y i | j + E|X i | j 2α j/m , E Q d (Z (i-1) ) l -E Q d (Z (i) ) l 2 l j=k+1 l j α dl/m 2 √ l -1 ld M l 1-j l α (d-1)/m 2 √ l -1 d-1 d! Inf i (f ) j α j/m 2 l+1 α dl m 2 √ l -1 l(2d-1) d! l M l-k-1 × max Inf i (f ) k+1 2 ; Inf i (f ) l 2 .
Finally, summing for i over 1, . . . , N and using that N i=1 Inf i (f ) =

f 2 d (d-1)! M 2 d!(d-1)! yields E Q d (X) l -E Q d (Y) l 2 l+1 α dl m 2 √ l -1 l(2d-1) d! l M l-k-1 × max 1 i N max Inf i (f ) k-1 2 ; Inf i (f ) l 2 -1 N i=1 Inf i (f ) c d,l,m,α × M l-k+1 × max 1 i N max Inf i (f ) k-1 2 ; Inf i (f ) l 2 -1 .
5 Normal approximation of homogeneous sums

Bounds

The following statement provides an explicit upper bound on the normal approximation of homogenous sums, when the test function has a bounded third derivative.

Theorem 5.1 Let X = {X i , i

1} be a collection of centered independent random variables with unit variance. Assume moreover that sup i E(X 4 i ) < ∞ and let α := max{3 ; sup i E(X 4 i )}. Fix N, d 1, and let f : [N] d → R be symmetric and vanishing on diagonals. Define Q d (X) = Q d (N, f, X) according to (1.1)

and assume that E[Q d (X) 2 ] = 1. Let ϕ : R → R be a thrice differentiable function such that ϕ ′′′ ∞ B.
Then, for Z ∼ N (0, 1), we have

E ϕ(Q d (X)) -E ϕ(Z) B(30β) d d! max 1 i N Inf i (f ) (5.46) +C * d -1 3d E Q d (X) 4 -3 + 4 √ 2 × 144 d-1 2 α d 2 √ dd! max 1 i N Inf i (f ) 1/4
, with C * defined as in Theorem 3.1.

Proof. Let G = (G i ) i 1 be a standard centered i.i.d. Gaussian sequence. We have

E ϕ(Q d (X)) -E ϕ(Z) δ 1 + δ 2 (5.47) with δ 1 = E ϕ(Q d (X)) -E ϕ(Q d (G)) and δ 2 = E ϕ(Q d (G)) -E ϕ(Z) . By The- orem 4.1 point 3, we have δ 1 B(30β) d d! max 1 i N Inf i (f ). By Theorem 3.1 and since E Q d (X) 2 = E Q d (G) 2 = 1, we have δ 2 C * d -1 3d E Q d (G) 4 -3 .
By Lemma 4.4 point 2 (with M = 1, k = 2 and l = m = 4) and since Inf i (f ) 1 for all i, we have

E Q d (X) 4 -E Q d (G) 4 32 × 144 2d-1 α d dd! 2 max 1 i N Inf i (f ) (5.48) so that δ 2 C * d -1 3d E Q d (X) 4 -3 + 4 √ 2 × 144 d-1 2 α d 2 √ dd! max 1 i N Inf i (f ) 1/4
.

Finally, the desired conclusion follows by putting the bounds for δ 1 and δ 2 in (5.47).

Remark 5.2 As a corollary of Theorem 5.1, we immediately recover de Jong's Theorem 1.10, under the additional hypothesis that sup i E(X 4 i ) < ∞.

Converse statements: universality of Wiener chaos

Here, we prove a slightly stronger version of Theorem 1.12 stated in Section 1.5. More precisely, we have the following result, where an additional condition on contractions (see assumption (3) in Theorem 5.3 just below and Definition 1.6) have been added with respect to Theorem 1.12, making the criterion more easily applicable in practice.

: [N n ] d → R is symmetric and vanishes on diagonals. Define Q d (n, G) = Q d (N n , f n , G), n 1, according to (1.

1), and assume that

E[Q d (n, G) 2 ] → 1 as n → ∞.
Then, the following are equivalent as n → ∞.

(1) The sequence

Q d (n, G) converges in law to Z ∼ N (0, 1). (2) E[Q d (n, G) 4 ] → 3. (3) For all r = 1, . . . , d -1, f n ⋆ r f n 2d-2r → 0.
(4) For every sequence X = {X i : i 1} of independent centered random variables with unit variance and such that sup i E|X i | 2+ǫ < ∞ for some ǫ > 0, the sequence Q d (n, X) converges in law to Z ∼ N (0, 1), in the Kolmogorov distance.

(5) For every sequence X = {X i : i 1} of independent and identically distributed centered random variables with unit variance, the sequence Q d (n, X) converges in law to Z ∼ N (0, 1) (not necessarily in the Kolmogorov distance).

Proof. The equivalences (1) ⇔ (2) ⇔ (3) are a mere reformulation of Theorem 3.3, deduced by taking into account the equality (3.36). On the other hand, it is trivial that each one of conditions ( 4) and ( 5) implies [START_REF] Blei | Analysis in Integer and Fractional Dimensions[END_REF]. So, it remains to prove the implication (1), ( 2), (3) ⇒ (4), [START_REF] Jong | Central Limit Theorems for generalized multilinear forms[END_REF]. Fix z ∈ R. We have

P Q d (n, X) z -P Z z δ (a) n (z) + δ (b) n (z) with δ (a) n (z) = P Q d (n, X) z -P Q d (n, G) z δ (b) n (z) = P Q d (n, G) z -P Z z .
By assumption ( 2

Bounds on the Wasserstein distance

The following statement shows that our techniques allow to directly control the Wasserstein distance between the law of a homogenous sum and the law of a standard Gaussian random variable.

Proposition 5.4 As in Theorem 5.1, let X = {X i , i
1} be a collection of centered independent random variables with unit variance. Assume moreover that

sup i E(X 4 i ) < ∞ and note α := max{3 ; sup i E(X 4 i )}. Fix N, d 1, and let f : [N] d → R be symmetric and vanishing on diagonals. Define Q d (X) = Q d (N, f, X) according to (1.1) and assume that E[Q d (X) 2 ] = 1. Put B 1 = 2(30β) d d! max 1 i N Inf i (f )
and

B 2 = 3 × 3 + (2 √ 2) d d -1 3d × × E Q d (X) 4 -3 + 4 √ 2 × 144 d-1 2 α d 2 √ dd! max 1 i N Inf i (f ) 1 4
.

Then, for Z ∼ N (0, 1), we have

d W (Q d (X), Z) 4(B 1 + B 2 ) 1 3 , provided B 1 + B 2 3 4 √ 2 .
Proof. Let h ∈ Lip(1) be a Lipschitz function with Lipschitz constant 1. By Rademacher's theorem, h is Lebesgue-almost everywhere differentiable. So, if we denote by h ′ its derivative, then h ′ ∞ 1. For t > 0, define

h t (x) = ∞ -∞ h( √ ty + √ 1 -tx)φ(y)dy,
where φ denotes the standard normal density. The triangle inequality gives

E[h(Q d (X))] -E[h(Z)] E[h t (Q d (X))] -E[h t (Z)] + E[h(Q d (X))] -E[h t (Q d (X))] + E[h(Z)] -E[h t (Z)] .
Now, let us differentiate and integrate by parts, using that φ ′ (x) = -xφ(x), to get

h ′′ t (x) = 1 -t √ t ∞ -∞ yh ′ ( √ ty + √ 1 -tx)φ (y) dy.
Hence, for 0 < t < 1, we may bound 2/t (as it can be shown by using the same arguments as in the proof of (5.49) above). Hence, as 2 1 t and √ 2 t -1 2 , we have

h ′′ t ∞ 1 -t √ t h ′ ∞ ∞ -∞ |y|φ(y)dy 1 √ t . (5.49) For 0 < t 1 2 (so that √ t √ 1 -t), we have |E[h(Q d (X))] -E[h t (Q d (X))]| E ∞ -∞ h( √ ty + √ 1 -tQ d (X)) -h( √ 1 -tQ d (X)) φ(y)dy +E h( √ 1 -tQ d (X)) -h(Q d (X)) h ′ ∞ √ t ∞ -∞ |y|φ(y)dy + h ′ ∞ t 2 √ 1 -t E |Q d (X)| 3 2 √ t. Similarly, |E[h(Z)] -E[h t (Z)]|
C * = 3 + (2 √ 2) d × K * 3 + (2 √ 2) d × max 2t -1 2 + 4 √ 2 3 √ π t -1 ; 2 + 2 3 t -1 3 + (2 √ 2) d × 3 t . Due to h ′′′ t ∞
2/t, Theorem 5.1 gives the bound

E h t (Q d (X)) -E h t (Z) 3 √ t + (B 1 + B 2 ) 1 t . Minimising 3 √ t + (B 1 + B 2 ) 1 t in t gives that t = 2 3 (B 1 + B 2 ) 2 3
. Plugging in the values and bounding the constant part ends the proof.

6 Chi-square approximation of homogeneous sums

Bounds

The next result provides upper bounds on the chi-square approximation of homogeneous sums.

Theorem 6.1 Let X = {X i , i
1} be a collection of centered independent random variables with unit variance. Assume moreover that sup i E(X 4 i ) < ∞ and note α := max{3; sup i E(X 4 i )}. Fix an even integer d 2 and, for N

1, let f : [N] d → R be symmetric and vanishing on diagonals. Define Q d (X) = Q d (N, f, X) according to (1.1) and assume that E[Q d (X) 2 ] = 2ν for some integer ν 1. Let ϕ : R → R be a thrice differentiable function such that ϕ ∞ 1, ϕ ′ ∞ 1 and ϕ ′′′ ∞ B. Then, for Z ν ∼ χ 2 (ν), we have E ϕ(Q d (X)) -E ϕ(Z ν ) B(30β) d d! max 1 i N Inf i (f ) + max 2π ν , 1 ν + 2 ν 2 × d -1 3d E Q d (X) 4 -12E Q d (X) 3 -12ν 2 + 48ν +4 √ dd! √ 2 × 144 d-1 2 α d 2 + √ ν(2 √ 2) 3(2d-1) 2 α 3d 2 max 1 i N Inf i (f ) 1/4
.

Proof. We proceed as in Theorem 5.

1. Let G = (G i ) i 1 denote a standard centered i.i.d. Gaussian sequence. We have E ϕ(Q d (X)) -E ϕ(Z ν ) δ 1 + δ 2 with δ 1 = E ϕ(Q d (X)) -E ϕ(Q d (G)) δ 2 = E ϕ(Q d (G)) -E ϕ(Z ν ) .
By Theorem 4.1 (Point 3), we have

δ 1 B(30β) d d! max 1 i N Inf i (f ). By Theorem 3.6,
we have, with

C # = max 2π ν , 1 ν + 2 ν 2 : δ 2 d BW (Q d (G), Z ν ) C # d -1 3d E Q d (G) 4 -12E Q d (G) 3 -12ν 2 + 48ν .
Additionally to (5.48), we have, by Lemma 4.4 (Point 2, with M = √ 2ν, k = 2, l = 3 and m = 4):

E Q d (X) 3 -E Q d (G) 3 16ν(2 √ 2) 3(2d-1) α 3d 4 dd! max 1 i N Inf i (f ).
Hence

δ 2 C # d -1 3d E Q d (X) 4 -12E Q d (X) 3 -12ν 2 + 48ν +4 √ dd! √ 2 × 144 d-1 2 α d 2 + √ ν (2 √ 2) 3(2d-1) 2 α 3d 2 max 1 i N Inf i (f ) 1/4
.

The desired conclusion follows.

As an immediate corollary of Theorem 6.1, we deduce the following new criterion for the asymptotic non-normality of homogenous sums -compare with de Jong's Theorem 1.10. Corollary 6.2 Let X = {X i : i 1} be a sequence of independent centered random variables with unit variance such that sup i E(X 4 i ) < ∞. Fix an even integer d 2, and let {N n , f n : n 1} be a sequence such that {N n : n 1} is a sequence of integers going to infinity, and each

f n : [N n ] d → R is symmetric and vanishes on diagonals. Define Q d (n, X) = Q d (N n , f n , X) according to (1.1). If, as n → ∞, (i) E(Q d (n, X) 2 ) → 2ν, (ii) E[Q d (n, X) 4 ] -12E[Q d (N n , f n , X) 3 ] → 12ν 2 -48ν, and (iii) max 1 i Nn Inf i (f n ) → 0, then Q d (n, X) converges in law to Z ν ∼ χ 2 (ν).

A converse statement

The following statement contains a universal chi-square limit theorem result: it is a general version of Theorem 1.14.

Theorem 6.3 Let G = {G i : i
1} be a standard centered i.i.d. Gaussian sequence, and fix an even integer d 2. Let {N n , f n : n 1} be a sequence such that {N n } is a sequence of integers going to infinity, and each

f n : [N n ] d → R is symmetric and vanishes on diagonals. Define Q d (n, G) = Q d (N n , f n , G), n 1, according to (1.1), and assume that E[Q d (n, G) 2 ] → 2ν as n → ∞.
Then, the following are equivalent as n → ∞.

(1) The sequence (1.17)) and, for every r = 1, ..., d -1 such that r = d/2, f n ⋆ r f n 2d-2r → 0.

Q d (n, G) converges in law to Z ν ∼ χ 2 (ν). (2) E[Q d (n, G) 4 ] -12E[Q d (n, G) 3 ] → 12ν 2 -48ν. (3) f n ⋆ d/2 f n -c d × f n d → 0 (for c d defined in
(4) For every sequence X = {X i : i 1} of independent centered random variables with unit variance and such that sup i E|X i | 2+ǫ < ∞ for some ǫ > 0, the sequence Q d (n, X) converges in law to Z ν ∼ χ 2 (ν).

(5) For every sequence X = {X i : i 1} of independent and identically distributed centered random variables with unit variance, the sequence Q d (n, X) converges in law to Z ν .

Proof. The proof follows exactly the same lines of reasoning as in Theorem 5.

Multivariate Extensions

Bounds

We recall here the standard multi-index notation. A multi-index is a vector α ∈ {0, 1, . . .} m . We write

|α| = m j=1 α j , α! = m j=1 α j !, ∂ j = ∂ ∂x j , ∂ α = ∂ α 1 1 . . . ∂ α d d , x α = m j=1 x α j j .
Note that by convention, 0 0 = 1. Also note that |x α | = y α , where y j = |x j | for all j.

Finally, for ϕ : R m → R regular enough and k 1, we put

ϕ (k) ∞ = max |α|=k 1 α! sup z∈R m |∂ α ϕ(z)|.
First we derive a multivariate version of (4.44); see also [START_REF] Mossel | Gaussian bounds for noise correlation of functions[END_REF]Theorem 4.1] for a similar statement.

Theorem 7.1 Let X = {X i , i
1} be a collection of centered independent random variables with unit variance and such that β

:= sup i 1 E[|X i | 3 ] < ∞. Let G = {G i : i 1} be a standard centered i.i.d. Gaussian sequence. Fix integers m 1 and d m . . . d 1
1 as well as N 1 , . . . , N m 1. For every j = 1, ..., m, let

f j : [N j ] d j → R be a symmetric function vanishing on diagonals. Define Q j (G) = Q d j (N j , f j , G) and Q j (X) = Q d j (N j , f j , X) according to (1.

1), and assume that

E[Q j (G) 2 ] = E[Q j (X) 2 ] = 1 for all j = 1, . . . , m. Assume that there exists a C > 0 is such that max j N j i=1 max 1 j m Inf i (f j ) C. Then, for all thrice differentiable ϕ : R m → R with ϕ ′′′ ∞ < ∞, we have E ϕ(Q 1 (X), . . . , Q m (X)) -E ϕ(Q 1 (G), . . . , Q m (G)) C ϕ ′′′ ∞ β + 8 π m j=1 (16 √ 2β) d j -1 3 d j ! 3 max 1 j m max 1 i max j N j Inf i (f j ).
In the one-dimensional case (i.e. m = 1),

max j N j i=1 max 1 j m Inf i (f j ) = [d!(d -1)!] -1 , so we can choose C = [d!(d -1)!] -1 .
In this case, when β is large the bound from Theorem 7.1 essentially differs from (4.44) by a constant times a factor d.

Proof. Abbreviate Q(X) = Q 1 (X), . . . , Q m (X) ,
and define analogously Q(G). We proceed as for Lemma 4.4, with similar notation. For i = 0, . . . , max j N j , let Z (i) denote the sequence G 1 , . . . , G i , X i+1 , . . . , X max j N j . Using the triangle inequality,

E ϕ(Q(X)) -E ϕ(Q(G)) max j N j i=1 E ϕ(Q(Z (i-1) )) -E ϕ(Q(Z (i) )) .
Fix a particular i ∈ {1, . . . , max j N j }, and write, for j = 1, . . . , m, U

(j) i = 1 i 1 ,...,i d N j ∀k: i k =i f j (i 1 , . . . , i d )Z (i) i 1 . . . Z (i) i d V (j) i = 1 i 1 ,...,i d N j ∃k: i k =i f j (i 1 , . . . , i d )Z (i) i 1 . . . Z (i) i . . . Z (i) i d ,
where

Z (i) i
means that this particular term is dropped. We write

U i = (U (1)
i , . . . , U

) and

V i = (V (1) i , . . . , V (m) i
). Note that U i and V i are independent of the variables X i and G i , and that Q(

Z (i-1) ) = U i + X i V i and Q(Z (i) ) = U i + G i V i
. By Taylor's theorem, using the independence of X i from U i and V i as well as E(X i ) = 0 and E(X 2 i ) = 1, we have

E ϕ(U i + X i V i ) - |α| 2 1 α! E ∂ α ϕ(U i )V α i E(X |α| i ) |α|=3 1 α! sup z∈R m |∂ α ϕ(z)| E(|X i | 3 )E(|V α i |) ϕ ′′′ ∞ E(|X i | 3 ) |α|=3 E(|V α i |).
Similarly,

E ϕ(U i + G i V i ) - |α| 2 1 α! E ∂ α ϕ(U i )V α i E(G |α| i ) 8 π |α|=3 1 α! sup z∈R m |∂ α ϕ(z)| E(|V α i |) ϕ ′′′ ∞ 8 π |α|=3 E(|V α i |).
Due to the independence and the matching moments up to second order, we obtain that

E ϕ(Q(n, Z (i-1) )) -E ϕ(Q(n, Z (i) )) = E ϕ(U i + X i V i ) -E ϕ(U i + G i V i ) β + 8 π ϕ ′′′ ∞ |α|=3 E(|V α i |).
Abbreviate τ i = max 1 j m Inf i (f j ). Next we use that, for j = 1, . . . , m, by Lemma 4.3 (with q = 3), we have

E[|V (j) i | 3 ] (16 √ 2β) d j -1 E[(V (j) i ) 2 ] 3 2 = (16 √ 2β) d j -1 d j ! 3 τ 3/2 i . Thus |α|=3 E |(V i ) α | = m j,k,l=1 E(|V (j) i V (k) i V (l) i |) m j,k,l=1 E(|V (j) i | 3 ) 1 3 E(|V (k) i | 3 ) 1 3 E(|V (l) i | 3 ) 1 3 = m j=1 E(|V (j) i | 3 ) 1 3 3 m j=1 (16 √ 2β) d j -1 3 d j ! 3 τ 3/2 i .
Collecting the bounds, summing over i, and using that max j N j i=1 τ i C gives the desired result.

The next theorem associates bounds to the normal approximation of the vector (Q 1 (X), . . . , Q m (X)). Theorem 7.2 Let X = {X i : i 1} be a collection of centered independent random variables with unit variance. Assume moreover that β :

= sup i E |X i | 3 < ∞. Fix integers m 1, d m . . . d 1 2
and N 1 , . . . , N m 1. For every j = 1, ..., m, let f j : [N j ] d j → R be a symmetric function vanishing on diagonals. Define Q j (X) = Q d j (N j , f j , X) according to (1.1), and assume that E[Q j (X) 2 ] = 1 for all j = 1, . . . , m. Let V be the m × m symmetric matrix given by

V (i, j) = E Q i (X)Q j (X) . Let C be as in Theorem 7.1. Let ϕ : R 3 → R be a thrice differentiable function such that ϕ ′′ ∞ < ∞ and ϕ ′′′ ∞ < ∞. Then, for Z V = (Z 1 V , . . . , Z m V ) ∼ N m (0, V ) (centered Gaussian vector with covariance matrix V ), we have E ϕ(Q 1 (X), . . . , Q m (X)) -E ϕ(Z V ) ϕ ′′ ∞ m i=1 ∆ ii + 2 1 i<j m ∆ ij +C ϕ ′′′ ∞ β + 8 π m j=1 (16 √ 2β) d j -1 3 d j ! 3 max 1 j m max 1 i N j Inf i (f j ),
for ∆ ij given by

d j √ 2 d i -1 r=1 (r -1)! d i -1 r -1 d j -1 r -1 (d i + d j -2r)! f i ⋆ d i -r f i H ⊗(2r) + f j ⋆ d j -r f j H ⊗(2r) +1 {d i <d j } d j ! d j d i f j ⋆ d j -d i f j H ⊗2d i . (7.50)
Proof. The proof is divided into several steps.

Step 1: Reduction of the problem. Let G = (G i ) i 1 be a standard centered i.i.d. Gaussian sequence. We have

E ϕ(Q 1 (X), . . . , Q m (X)) -E ϕ(Z V ) δ 1 + δ 2 (7.51)
with

δ 1 = E ϕ(Q 1 (X), . . . , Q m (X)) -E ϕ(Q 1 (G), . . . , Q m (G)) δ 2 = E ϕ(Q 1 (G), . . . , Q m (G)) -E ϕ(Z V ) .
Step 2: Bounding δ 1 . By Theorem 7.1, we have

δ 1 C ϕ ′′′ ∞ β + 8 π m j=1 (16 √ 2β) d j -1 3 d j ! 3 max 1 j m max 1 i N j Inf i (f j ).
Step 3: Bounding δ 2 . We will not use the result proved in [START_REF] Nourdin | Multivariate normal approximation using Stein's method and Malliavin calculus[END_REF], since here we do not assume that the matrix V is positive definite. Instead, we will rather use an interpolation technique à la Talagrand [START_REF] Talagrand | Spin Glasses: A Challenge for Mathematicians[END_REF]. Without loss of generality, we assume in this step that Z V is independent of G. Also, for any j = 1, . . . , m, observe that Q j (G) = I d j (h j ) where

h j = d j ! {i 1 ,...,i d j }⊂[N j ] d j f j (i 1 , ..., i d j ) e i 1 ⊗ . . . ⊗ e i d j ∈ H ⊙d ,
see also (2.26). For t ∈ [0, 1], set

Ψ(t) = E ϕ √ 1 -t(I d 1 (h 1 ), . . . , I dm (h m )) + √ tZ V , so that δ 2 = Ψ(1) -Ψ(0) sup t∈(0,1)
Ψ ′ (t) .

We easily see that Ψ is differentiable on (0, 1), and that Ψ ′ (t) equals

m i=1 E ∂ϕ ∂x i √ 1 -t(I d 1 (h 1 ), . . . , I dm (h m )) + √ tZ V 1 2 √ t Z i V - 1 2 √ 1 -t I d i (h i ) .
By integrating by parts, we can write

E ∂ϕ ∂x i √ 1 -t(I d 1 (h 1 ), . . . , I dm (h m )) + √ tZ V Z i V = E E ∂ϕ ∂x i √ 1 -tz + √ tZ V Z i V |z=(I d 1 (h 1 ),...,I dm (hm)) = √ t m j=1 V (i, j) E E ∂ 2 ϕ ∂x i ∂x j √ 1 -tz + √ tZ V |z=(I d 1 (h 1 ),...,I dm (hm)) = √ t m j=1 V (i, j) E ∂ 2 ϕ ∂x i ∂x j √ 1 -t(I d 1 (h 1 ), . . . , I dm (h m )) + √ tZ V .
By using e.g. (8.61) in order to perform the integration by parts, we can also write

E ∂ϕ ∂x i √ 1 -t(I d 1 (h 1 ), . . . , I dm (h m )) + √ tZ V I d i (h i ) = E E ∂ϕ ∂x i √ 1 -t(I d 1 (h 1 ), . . . , I dm (h m )) + √ tz I d i (h i ) |z=Z V = √ 1 -t d i m j=1 E E ∂ 2 ϕ ∂x i ∂x j √ 1 -t(I d 1 (h 1 ), . . . , I dm (h m )) + √ tz D[I d i (h i )], D[I d j (h j )] H |z=Z V = √ 1 -t d i m j=1 E ∂ 2 ϕ ∂x i ∂x j √ 1 -t(I d 1 (h 1 ), . . . , I dm (h m )) + √ tZ V D[I d i (h i )], D[I d j (h j )] H . Hence Ψ ′ (t) equals 1 2 m i,j=1 E ∂ 2 ϕ ∂x i ∂x j √ 1 -t(I d 1 (h 1 ), . . . , I dm (h m )) + √ tZ V V (i, j) - 1 d i D[I d i (h i )], D[I d j (h j )] H ,
so that we get:

δ 2 ϕ ′′ ∞ m i,j=1 E V (i, j) - 1 d i D[I d i (h i )], D[I d j (h j )] H ϕ ′′ ∞ m i,j=1 E V (i, j) - 1 d i D[I d i (h i )], D[I d j (h j )] H 2 = ϕ ′′ ∞ m i,j=1 1
d i Var D[I d i (h i )], D[I d j (h j )] H .
Step 4: Bounding Var D[I d i (h i )], D[I d j (h j )] H . Assume, for instance, that i j. We have

D[I d i (h i )], D[I d j (h j )] H = d i d j R I d i -1 h i (•, a) I d j -1 h j (•, a) da = d i d j R d i -1 r=0 r! d i -1 r d j -1 r I d i +d j -2-2r h i (•, a) ⊗ r h j (•, a) da by (2.28) = d i d j R d i -1 r=0 r! d i -1 r d j -1 r I d i +d j -2-2r h i (•, a) ⊗ r h j (•, a) da = d i d j d i -1 r=0 r! d i -1 r d j -1 r I d i +d j -2-2r R h i (•, a) ⊗ r h j (•, a)da = d i d j d i -1 r=0 r! d i -1 r d j -1 r I d i +d j -2-2r h i ⊗ r+1 h j = d i d j d i r=1 (r -1)! d i -1 r -1 d j -1 r -1 I d i +d j -2r h i ⊗ r h j = d i d j d i r=1 (r -1)! d i -1 r -1 d j -1 r -1 I d i +d j -2r h i ⊗ r h j . Hence, if d i < d j , then Var D[I d i (h i )], D[I d j (h j )] H equals d 2 i d 2 j d i r=1 (r -1)! 2 d i -1 r -1 2 d j -1 r -1 2 (d i + d j -2r)! h i ⊗ r h j 2 H ⊗(d i +d j -2r)
while, if d i = d j , it equals

d 4 i d i -1 r=1 (r -1)! 2 d i -1 r -1 4 (2d i -2r)! h i ⊗ r h j 2 H ⊗(2d i -2r) .
Now, let us stress the two following estimates. If r < d i d j then

h i ⊗ r h j 2 H ⊗(d i +d j -2r) h i ⊗ r h j 2 H ⊗(d i +d j -2r) = h i ⊗ d i -r h i , h j ⊗ d j -r h j H ⊗2r h i ⊗ d i -r h i H ⊗2r h j ⊗ d j -r h j H ⊗2r 1 2 h i ⊗ d i -r h i 2 H ⊗2r + h j ⊗ d j -r h j 2 H ⊗2r . If r = d i < d j , then h i ⊗ d i h j 2 H ⊗(d j -d i ) h i ⊗ d i h j 2 H ⊗(d j -d i ) h i 2
H ⊗d i h j ⊗ d j -d i h j H ⊗2d i .
By putting all these estimates in the previous expression for Var D[

I d i (h i )], D[I d j (h j )] H ,
we get, using also (3.36), that

1 d i Var D[I d i (h i )], D[I d j (h j )] H ∆ ij ,
for ∆ ij defined by (7.50).

This completes the proof of the theorem.

We now translate the bound in Theorem 7.2 into a bound for indicators of convex sets.

Corollary 7.3 Let the notation and assumptions from Theorem 7.2 prevail. We consider the class H(R m ) of indicator functions of measurable convex sets in R m . Let

B 1 = 1 2 m i=1 ∆ ii + 1 i<j m ∆ ij and B 2 = C β + 8 π m j=1 (16 √ 2β) d j -1 3 d j ! 3 max 1 j m max 1 i N j Inf i (f j ).
1. Assume that the covariance matrix V is the m × m identity matrix I m . Then

sup h∈H(R m ) E h Q 1 (X), . . . , Q m (X) -E[h(Z V )] 8(B 1 + B 2 ) 1 4 m 3 8 .
2. Assume that V is of rank k m, and let Λ = diag(λ 1 , . . . , λ k ) be the diagonal matrix with the non-zero eigenvalues of V on the diagonal. Let B be a m×k column orthonormal matrix (that is, B T B = I k and BB T = I m ), such that V = BΛB T , and let

b = max i,j Λ -1 2 B T i,j . (7.52) Then sup h∈H(R m ) E h Q 1 (X), . . . , Q m (X) -E[h(Z V )] 8(b 2 B 1 + b 3 B 2 ) 1 4 m 3 8 .
Remark 7.4 1. We note that sup

z∈R m |P Q 1 (X), . . . , Q m (X) z -P [Z V z]| sup h∈H(R m ) E h Q 1 (X), . . . , Q m (X) -E[h(Z V )] .
Thus Corollary 7.3 immediately gives a bound for Kolmogorov distance.

2. By using the bound for δ 2 derived in the proof of Theorem 7.2 above, and following the same line of reasoning as in the proof of Corollary 7.3, we have, by keeping the notation of Theorem 7.2, that if ∆ ij → 0 for all i, j = 1, . . . , m and max 1 j m max 1 i N j Inf i (f j ) → 0, then

Q d 1 (N 1 , f 1 , G), . . . , Q dm (N m , f m , G) → N m (0, V ), as N 1 , . . . , N j → ∞,
in the Kolmogorov distance.

Proof. First assume that V is the identity matrix. We partially follow [START_REF] Rinott | A multivariate CLT for local dependence with n -1/2 log n rate and applications to multivariate graph related statistics[END_REF], and let Φ denote the standard normal distribution in R m , and φ the corresponding density function.

For h ∈ H(R m ), define the following smoothing:

h t (x) = R m h √ ty + √ 1 -tx Φ(dy), 0 < t < 1.
The key result, found for example in [START_REF] Götze | On the rate of convergence in the multivariate CLT[END_REF]Lemma 2.11], is that, for any probability measure Q on R m , for any W ∼ Q and Z ∼ Φ, and for any 0 < t < 1, we have:

sup h∈H(R m ) |E[h(W )] -E[h(Z)]| 4 3 sup h∈H(R m ) |E[h t (W )] -E[h t (Z)]| + 2 √ m √ t . (7.53)
Similarly as in [15, page 24], put

u(x, t, z) = (2πt) -m 2 exp - m i=1 (z i - √ 1 -tx i ) 2 2t ,
and observe that u(x, t, z) is the density function of the Gaussian vector Y ∼ N (0, tI m ) taken in z -√ 1 -tx. Substituting z = √ ty + √ 1 -tx, we get, for 0 < t < 1,

h t (x) = R m h(z)u(x, t, z)dz.
By dominated convergence, we may differentiate under the integral and obtain, for the second derivatives,

∂ 2 h t ∂x 2 i (x) = - 1 -t t R m h(z)u(x, t, z)dz + 1 -t t 2 R m h(z)(z i - √ 1 -tx i ) 2 u(x, t, z)dz
and, for i = j,

∂ 2 h t ∂x i ∂x j (x) = 1 -t t 2 R m h(z)(z i - √ 1 -tx i )(z j - √ 1 -tx j )u(x, t, z)dz.
For the third partial derivatives,

∂ 3 h t ∂x 3 i (x) = -3 (1 -t) 3 2 t 2 R m h(z)(z i - √ 1 -tx i )u(x, t, z)dz + (1 -t) 3 2 t 3 R m h(z)(z i - √ 1 -tx i ) 3 u(x, t, z)dz
and, for i = j,

∂ 3 h t ∂x 2 i ∂x j (x) = - (1 -t) 3 2 t 2 R m h(z)(z j - √ 1 -tx j )u(x, t, z)dz + (1 -t) 3 2 t 3 R m h(z)(z i - √ 1 -tx i ) 2 (z j - √ 1 -tx j )u(x, t, z)dz,
and, for i, j, k all distinct,

∂ 3 h t ∂x i ∂x j ∂x k (x) = (1 -t) 3 2 t 3 R m h(z)(z i - √ 1 -tx i )(z j - √ 1 -tx j )(z k - √ 1 -tx k )u(x, t, z)dz.
Because 0 h(z) 1 for all z ∈ R m , we may bound

∂ 2 h t ∂x 2 i (x) 1 -t t + 1 -t t 2 E[Y 2 i ] = 2(1 -t) t .
Similarly, for i = j,

∂ 2 h t ∂x i ∂x j (x) 1 -t t 2 E[|Y i |]E[|Y j |] = 2(1 -t) πt .
Thus, we have h ′′ t ∞ 1/t 1/t 3/2 . Bounding the third derivatives in a similar fashion yields, for all i, j, k not necessarily distinct,

∂ 3 h t ∂x i ∂x j ∂x k (x) (1 -t) 3 2 t 3 max 3 E[|Y i |] t + E[|Y i | 3 ]; E[|Y j |] t + E[Y 2 i ]E[|Y j |]; E[|Y i |]E[|Y j |]E[|Y k |] ,
and so h ′′′ t ∞ 1/t

(R m ) E h Q 1 (X), . . . , Q m (X) -E[h(Z V )] 4 3 sup h∈H(R m ) E h t Q 1 (X), . . . , Q m (X) -E[h t (Z V )] + 2 √ m √ t 8 3 √ m √ t + 4 3 (B 1 + B 2 )t -3 2 .
This function is minimised for t

= 3(B 1 +B 2 ) 2 √ m , yielding that sup h∈H(R m ) E h Q 1 (X), . . . , Q m (X) -E[h(Z V )] 8(B 1 + B 2 ) 1 4 m 3 8 ,
as required.

For Point 2, write W = (Q 1 (X), . . . , Q m (X)) for simplicity. For h ∈ H(R m ), we have

E[h(W )] -E[h(Z V )] = E[h(BΛ 1 2 × Λ -1 2 B T W )] -E[h(BΛ 1 2 × Λ -1 2 B T Z V )]. Put g(x) = h(BΛ 1 2
x). Then, g ∈ H(R k ) and, thanks to the inequality (7.53), we can write

sup h∈H(R m ) |E[h(W )] -E[h(Z V )]| sup g∈H(R k ) |E[g(Λ -1 2 B T W )] -E[g(Λ -1 2 B T Z V )]| 4 3 sup g∈H(R k ) |E[g t (Λ -1 2 B T W )] -E[g t (Λ -1 2 B T Z V )]| + 2 √ k √ t .
We may bound the partial derivatives of f t (x) = g t (Λ -1 2 B T x) using the chain rule and (7.52), to give that

f ′′ t ∞ b 2 t -3 2 and f ′′′ t ∞ b 3 t -3 2 .
Using Theorem 7.2 and minimising the bound in t as before gives the assertion; the only changes are that B 1 gets multiplied by b 2 and B 2 gets multiplied by b 3 .

More universality

Here, we prove a slightly stronger version of Theorem 1.3 stated in Section 1.3. Precisely, we add the two conditions ( 2) and ( 3), making the criterion contained in Theorem 1.3 more effective for potential applications.

n ) : n 1} be a sequence such that {N (j) n : n 1} is a sequence of integers going to infinity, and each function f

(j) n : [N (j) n ] d j → R is symmetric and vanishes on diagonals. Define Q j (n, G) = Q d j (N (j) n , f (j) n , G), n
1, according to (1.1). Assume that, for every j = 1, ...m, the sequence E[Q j (n, G) 2 ], n 1, of variances is bounded. Let V be a m × m non-negative symmetric matrix, and let N m (0, V ) be a centered Gaussian vector with covariance matrix V . Then, as n → ∞, the following two conditions are equivalent.

(1) The vector {Q j (n, G) : j = 1, ..., m} converges in law to N m (0, V ).

(2) For all i, j = 1, . . . , m, we have

E Q i (n, G)Q j (n, G) → V (i, j) and E Q i (n, G) 4 → 3V (i, i) 2 as n → ∞.
(3) For all i, j = 1, . . . , m, we have

E Q i (n, G)Q j (n, G) → V (i, j
) and, for all 1 i m and 1 r d i -1, we have f

(i) n ⋆ r f (i) n 2d i -2r → 0.
(4) For every sequence X = {X i : i 1} of independent centered random variables, with unit variance and such that sup i E|X i | 3 < ∞, the vector {Q j (n, X) : j = 1, ..., m} converges in law to N m (0, V ) for the Kolmogorov distance.

For the proof of Theorem 7.5, we need the following result, which consists in a collection of some of the findings contained in the papers by Peccati and Tudor [START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF]. Strictly speaking, the original statements contained in [START_REF] Peccati | Gaussian limits for vector-valued multiple stochastic integrals[END_REF] only deal with positive definite covariance matrices: however, the extension to a non-negative matrix can be easily achieved by using the same arguments as in the "Step 3" of the proof of Theorem 7.2.

F (n) = (F (n) 1 , . . . , F (n) m ) := (I d 1 (h (n) 1 ), . . . , I dm (h (n) m )) n 1, is such that lim n→∞ E[F (n) i F (n) j ] = V (i, j), 1 i, j m. ( 7

.54)

Then, as n → ∞, the following four assertions are equivalent:

chaos by means of increasing (tensor) powers of a fixed real separable Hilbert space H. We recall that the first Wiener chaos of G (which coincides with the Gaussian space generated by G) is the L 2 -closed Hilbert space composed of random variables of the type I 1 (h), where h ∈ H. We shall denote by L 2 (G) the space of all R-valued random elements F that are measurable with respect to σ{G} and verify E[F 2 ] < ∞. Also, L 2 (Ω; H) denotes the space of all H-valued random elements u, that are measurable with respect to σ{G} and verify the relation E u 2 H < ∞. We stress (see again [START_REF] Nualart | The Malliavin calculus and related topics[END_REF]Ch. 1] or [START_REF] Janson | Gaussian Hilbert Spaces[END_REF]) that any random variable F belonging to L 2 (G) admits the following chaotic expansion:

F = E[F ] + ∞ d=1 I d (h d ), (8.55)
where the series converges in L 2 (G) and the symmetric kernels h d ∈ H ⊙d , d 1, are uniquely determined by F . For convenience we also put

I 0 (f 0 ) = E[F ],
so that we can write

F = ∞ d=0 I d (h d ).
In the particular case where H = L 2 (A, A , µ), with (A, A ) a measurable space and µ a σ-finite and non-atomic measure, H ⊙d = L 2 s (A d , A ⊗d , µ ⊗d ) is the space of symmetric and square integrable functions on A d .

Let S be the set of all smooth cylindrical random variables of the form

F = g I 1 (φ 1 ), . . . , I 1 (φ n )
where n 1, g : R n → R is a smooth function with compact support and φ i ∈ H. The Malliavin derivative of F (with respect to G) is the element of L 2 (Ω; H) defined as

DF = n i=1 ∂g ∂x i I 1 (φ 1 ), . . . , I 1 (φ n ) φ i .
By iteration, one can define the mth derivative D m F (which is an element of L 2 (Ω, H ⊗m )) for every m 2. For m 1, D m,2 denotes the closure of S with respect to the norm • m,2 , defined by the relation

F 2 m,2 = E[F 2 ] + m i=1 E D i F 2 H ⊗i .
We have DI 1 (h) = h for every h ∈ H. The Malliavin derivative D satisfies the following chain rule: if g : R n → R is continuously differentiable and has bounded partial derivatives, and if (F 1 , ..., F n ) is a vector of elements of D

| c u E[F 2 ] for any F ∈ S ,
for some constant c u depending only on u. If u ∈ Domδ, then the random variable δ(u) is defined by the duality relationship (customarily called "integration by parts formula"):

E(F δ(u)) = E DF, u H , (8.57)
which holds for every

F ∈ D 1,2 . If H = L 2 (A, A , µ) (with µ non-atomic), then the derivative of a random variable F = E(F ) + d 1 I d (h d ) can be identified with the element of L 2 (A × Ω) given by D a F = ∞ d=1 dI d-1 h d (•, a) , a ∈ A. ( 8

Relations following from Stein's method

Originally introduced in [41, 42], Stein's method can be described as a collection of probabilistic techniques, allowing to compute explicit bounds on the distance between the laws of random variables by means of differential operators. The reader is referred to [START_REF] Chen | Normal approximation under local dependence[END_REF] and [START_REF] Reinert | Three general approaches to Stein's method[END_REF], and the references therein, for an introduction to these techniques. The following statement contains four bounds which can be obtained by means of a combination of Malliavin calculus and Stein's method. Points 1, 2 and 4 have been proved in [START_REF] Nourdin | Stein's method on Wiener chaos[END_REF], whereas the content of Point 3 is new. Our proof of such a bound gives an explicit example of the interaction between Stein's method and Malliavin calculus. We also introduce the following notation: for every centered F ∈ D 1,2 , we write where C * is given in (3.34).

If E(F 2 ) = 2ν then the following bound is in order: where K * is given in (3.33). Now use (8.61) with g = f ϕ and G = F , as well as a standard aproximation argument to take into account that f ′ ϕ is not necessarily bounded, in order to write

|E[ϕ(F )] -E[ϕ(Z)]| = |E[f ′ ϕ (F ) -F f ϕ (F )]| = E f ′ ϕ (F ) 1 - 1 d DF 2 H K * E 1 + |F | + |F | 2 + |F | 3 1 - 1 d DF 2

H

.

By applying the Cauchy-Schwarz inequality four times, and by exploiting Proposition 2.7, one infers

K * E 1 + |F | + |F | 2 + |F | 3 1 - 1 d DF 2 H C * E 1 - 1 d DF 2 H 2 = C * T 0 (F ),
which is the desired conclusion.

Lemma 8.2

The function f ϕ defined in (8.66) verifies relation (8.67).

Proof. We denote by Z a Gaussian random variable with zero mean and unit variance.

Observe that

E |Z| 3 = 2 √ 2 √ π and E |Z| = √ 2 √ π . ( 8

.68)

We want to bound the quantity f ′ ϕ (x) , where: (i) ϕ is such that ϕ (x) = ϕ (0) + ϕ ′ (0) x + ϕ ′′ (0) x 2 /2 + R (x), with |R (x) | ϕ ′′′ ∞ |x| 3 /6, and (ii) f ϕ is given by (8.66), and consequently We now evaluate the four terms Y i separately (observe that each of them is positive): It is now immediate to deduce that T 3 (F ) T 4 (F ).

Y 1 (x) = C ′ xe

  with µ σ-finite and non-atomic, then the operators I d are indeed (multiple) Wiener-Itô integrals.

Example 2 . 4

 24 (i) By definition, G i = I 1 (e i ), for every i 1.

. 35 )

 35 Theorem 3.1, combined with(3.35), allows therefore to recover the following characterization of CLTs on Wiener chaos. It has been first proved (by completely different methods) in[START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF].Theorem 3.3 (See[START_REF] Nualart | Central limit theorems for multiple stochastic integrals and Malliavin calculus[END_REF][START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF]) Fix d 2, and let F n = I d (h n ), n 1 be a sequence in the dth Wiener chaos of G. Assume that lim n→∞ E(F 2 n ) = 1. Then, the following conditions are equivalent, as n → ∞.

Lemma 3 . 4

 34 Fix d 2, and suppose that h ∈ H ⊙d is given by (2.26), with f : [N] d → R symmetric and vanishing on diagonals. Then, for r = 1, ..., d -1,

Remark 3 . 5

 35 Theorem 3.3 and formula (3.36) yield immediately a proof of Proposition 1.7 in the case m = 1.

Theorem 3 . 6 (

 36 Third and fourth moment bounds) Fix an even integer d 2 as well as an integer ν 1. Let F = I d (h) be an element of the dth Gaussian chaos C d such that E(F 2 ) = 2ν, let Z ν ∼ χ 2 (ν), and write

. 40 ) 3 . 7

 4037 Remark Observe that d! 2 4(d/2)! 3 = 1 c d , where c d is given in (1.17).

  ) and (3.30)-(3.31), we have sup z∈R δ (b) n (z) → 0. By combining assumption (3) (for r = d -1) with (1.14)-(1.15), we get that max 1 i Nn Inf i (f n ) → 0 as n → ∞. Hence, Theorem 4.1 (Point 1) implies that sup z∈R δ (a) n (z) → 0, and the proof of the implication (1), (2), (3) ⇒ (4) is complete. To prove that (1) ⇒ (5), one uses the same line of reasoning, the only difference being that we need to use Point 2 of Theorem 4.1 instead of Point 1.

  3. Details are left to the reader. Let us just mention that the only differences consist in the use Theorem 3.8 instead of Theorem 3.3, and the use of (3.39)-(3.40) instead of (3.30)-(3.31).

Theorem 7 . 5

 75 Let G = {G i : i 1} be a standard centered i.i.d. Gaussian sequence, and fix integers m 1 and d 1 , ..., d m 2. For every j = 1, ..., m, let {(N

Theorem 7 . 6 . d 1 1 .

 761 Fix integers m 1 and d m . . Let V = {V (i, j) : i, j = 1, ..., m} be a m × m non-negative symmetric matrix. For any n 1 and i = 1, . . . , m, let h (n) i belong to the d th i Gaussian chaos C d i . Assume that

  .58) Next we define the operator L, called the infinitesimal generator of the Ornstein-Uhlenbeck semigroup. It acts on square-integrable random variables as follows: a random variableF = E(F ) + d 1 I d (h d ) is in the domain of L, noted DomL, if and only if d 1 dI d (h d ) is convergent in L 2 ,and in this case LF =d 1 dI d (h d ). The operator L verifies the following crucial properties: (i) DomL = D 2,2 , and (ii) a random variable F is in DomL if and only if F ∈ DomδD (i.e. F ∈ D 1,2 and DF ∈ Domδ), and in this case one has that δ(DF ) = -LF . We also define the operator L -1 , which is the pseudo-inverse of L, as follows: for every centered random variableF = d 1 I d (h d ) of L 2 (G), we set L -1 F = d 1 -1 d I d (h d ).Note that L -1 is an operator with values in D 2,2 . Now consider a centered F ∈ L 2 (G). By using the relations F = L(L -1 F ) = δ(-DL -1 F ), one deduces from (8.57) and (8.56) that, for every G ∈ D 1,2 and for every continuously differentiable g : R → R with a bounded derivative, we haveE[g(G)F ] = E[g ′ (G) DG, -DL -1 F H ].(8.59)In particular, with g(x) = x, we getE[GF ] = E[ DG, -DL -1 F H ]. (8.60) Moreover, if F = I d (h), with h ∈ H ⊙d , then L -1 F = -1 d F ,and therefore (8.59) and (8.60) become E[g(G)F ] = 1 d E[g(G) DG, DF H ] and E[GF ] = 1 d E[ DG, DF H ]. (8.61) Let h ∈ H ⊙d with d 2, and let s 0 be an integer. The following identity is obtained by taking F = I d (h) and G = F s+1 in formula (8.61), and then by applying (8.56): E I d (h) s+2 = s + 1 d E I d (h) s DI d (h) 2 H . (8.62)

T 0 Proposition 8 . 1

 081 (F ) = Var( DF, -DL -1 F H ),(8.63) where Var indicates variance (observe that, due to(8.60), E( DF, -DL -1 F H ) = E(F 2 )). Note that, if F = I d (f ), then T 0 (F ) = Var1 Consider F = I d (h) with d 1 and h ∈ H ⊙d , and let Z and Z ν have respectively a N (0, 1) and a χ 2 (ν) distribution (ν 1). If E(F 2 ) = 1 then the following three bounds hold:1. d T V (F, Z) 2T 0 (F ).

2 .

 2 d W (F, Z) T 0 (F ).

3 .

 3 For every thrice differentiable functionϕ : R → R such that ϕ ′′′ < ∞, |E[ϕ(F )] -E[ϕ(Z)]| C * × T 0 (F ),

4 ., 1 ν + 2 ν 2 E

 422 d BW (F, Z ν ) max 2π ν [(2ν + 2F -DF, -DL -1 F H ) 2 ].Proof. Point 1 and 2 are proved in[START_REF] Nourdin | Stein's method on Wiener chaos[END_REF] Theorem 3.1], while Point 4 is proved in[START_REF] Nourdin | Stein's method on Wiener chaos[END_REF] Theorem 3.11]. To prove Point 3, fix ϕ as in the statement, and consider the Stein equationf ′ (x) -xf (x) = ϕ(x) -E[ϕ(Z)], x ∈ R. (8.65)It is easily seen that a solution of (8.65) is given byf ϕ (x) = e x 2 /2 x -∞ (ϕ(y) -E[ϕ(Z)])e -y 2 /2 dy. (8.66) According to the forthcoming Lemma 8.2, the following bound is in order: |f ′ ϕ (x)| K * (1 + |x| + |x| 2 + |x| 3 ), (8.67)

C

  f ′ ϕ (x) = ϕ (x) -Eϕ (Z) + xf ϕ (x) = A (x) + B (x) ,(8.69)with A (x) := ϕ (x) -Eϕ (Z) and B (x) := xf ϕ (x). It will become clear later on that our bounds on f ′ ϕ (x) do not depend on the sign of x, so that in what follows we will uniquely focus on the case x > 0. Due to the assumptions on ϕ, we have thatA (x) = ϕ ′ (0) x + 1 2 ϕ ′′ (0) x 2 + R (x) + C := ax + bx 2 + R (x) + C, where -C = ϕ ′′ (0) 2 + ER (Z) ,(note that the term ϕ (0) simplifies) and also, by using (8.68), and (recall that x > 0)|A (x)| |ϕ ′ (0)| x + 1 2 |ϕ ′′ (0)| x 2 + 1 6 ϕ ′′′ ∞ x 3 + C ′ = |a| x + |b| x 2 + γx 3 + C ′ Onthe other hand, since EA (Z) = 0 by construction, ′ + |a| y + |b| y 2 + γy 3 e -y 2 2 dy := Y 1 (x) + Y 2 (x) + Y 3 (x) + Y 4 (x) .

|b| y 2 e -y 2 2 dy e x 2 2 +∞ x |b| y 3 e -y 2 2 dy = |b| x 2 + 2 = 2 |b| + |b| x 2 ;Y 4 (x) = xe x 2 2 +∞ x γy 3 e -y 2 2 dy = γx x 2 + 2 =

 2222242322 γx 3 + 2γx. By combining the above bounds with (8.69), one infers thatf ′ ϕ (x) 2C ′ + 2 |b| + x (2 |a| + 2γ) + x 2 2 |b| + x 3 2γ max {2C ′ + 2 |b| ; 2 |a| + 2γ; 2 |b| ; 2γ} × 1 + x + x 2 + x 3 = max {2C ′ + 2 |b| ; 2 |a| + 2γ} × 1 + x + x 2 + x 3 ,which yields the desired conclusion.

8. 3 2 I 1 4( 2 I 2 ( 4 )

 321224 Proof of Theorem 3.1 Let F = I d (h), h ∈ H ⊙d . In view of Proposition 8.1, it is sufficient to show thatT 0 (F ) = T 1 (F ) T 2 (2d-2r h ⊗ r h ,(8.70)which, by taking the orthogonality of multiple integrals of different orders into account, yields 1)! 2 d -1 r -2d -2r)! h ⊗ r h 2 H ⊗2(d-r) ,(8.71)and consequently T 0 (F ) = T 1 (F ). From the multiplication formula (2.28)) we get 2d-2r (h ⊗ r h).(8.72)To conclude the proof, we use (8.62) with s = 2, combined with (8.70) and (8.72), as well as the assumption that E(F 2 ) = 1, to get thatE[F 4 ] -2d -2r)! h ⊗ r h 2 H ⊗2(d-r) . -3],thus yielding the desired inequality T 1 (F ) T 2 (F ).

8. 4 2 = 2 (r - 1 )! 2 d -1 r - 1 4( 2 H 2 H 2 . 2 ( 2 H

 4221122222 Proof of Theorem 3.6 Let F = I d (h), h ∈ H ⊙d . In view of Proposition 8.1 and since L -1 F = -1 d F , it is sufficient to T 3 (F ) T 4 (F ). By taking into account the orthogonality of multiple integrals of different orders, relation (2d -2r)! h ⊗ r h ⊗(2d-2r) , and consequently T 3 (F ) = E 2ν + 2F -1 d DF On the other hand, by combining (8.62) (for s = 1 and s = 2) with (8.72), we get, still by taking into account the orthogonality of multiple integrals of different orders:E[F 4 ] -12E[F 3 ] = 12ν 2 -48ν + 24d! h -d! 2 4(d/2)! 3 h ⊗ d/2 h 2d -2r)! h ⊗ r h ⊗(2d-2r) .

  Bounds such as(1.6),(1.7) and (1.9) only partially account for the term "universality" appearing in the title of the present paper. Our techniques allow indeed to prove the following statement, involving vectors of homogeneous sums of possibly different orders, see Theorem 7.5.

Theorem 1.3 (Universality of Wiener chaos) Let G = {G i : i 1} be a standard centered i.i.d. Gaussian sequence, and fix integers m 1 and d 1 , ..., d m 2. For every j = 1, ..., m, let {(N

  4 i ) < ∞ for every i. Fix d 2, and let {N n , f n : n 1} be a sequence such that {N n : n 1} is a sequence of integers going to infinity, and each f n : [N n ] d → R is symmetric and vanishes on diagonals. Define Q

  [START_REF] Meyer | Quantum Probability for Probabilists[END_REF] and(1.20) show that Assumption (ii) in Theorem 1.10 ensures that both the laws and the fourth moments of Q d (n, X) and Q

d (n, G) are asymptotically close: this fact, combined with Assumption (i) implies that the LHS of (1.19) converges to zero. This gives an alternate proof of Theorem 1.10 in the case of uniformly bounded fourth moments. Also, by combining the universality principle stated in Theorem 1.3 with (1.19) (or, alternatively, with Proposition 1.7 in the case m = 1) and with [40, Proposition 1], one obtains the following "universal version" of de Jong's criterion. Theorem 1.12 Let G = {X i : i 1} be a centered i.i.d. Gaussian sequence with unit variance. Fix d 2, and let {N n , f n : n 1} be a sequence such that {N n : n 1} is a sequence of integers going to infinity, and each f n

  Theorem 5.3 Let G = {G i : i 1} be a standard centered i.i.d. Gaussian sequence, and fix an even integer d 2. Let {N n , f n : n 1} be a sequence such that {N n } is a sequence of integers going to infinity, and each f n

  3/2 . With (7.53) and Theorem 7.2, this gives that sup h∈H

  , ..., F n ) is a vector of multiple integrals (of possibly different orders) and g is a polynomial in n variables. We denote by δ the adjoint of the operator D, also called the divergence operator. A random element u ∈ L 2 (Ω; H) belongs to the domain of δ, noted Domδ, if and only if it satisfies |E DF, u H

	Dg(F 1 , . . . , F n ) =	n i=1	∂g ∂x i	(F 1 , . . . , F n )DF i .	(8.56)
	A careful application of the multiplication formula (2.28) shows that (8.56) continues to
	hold when (F 1

1,2

, then g(F 1 , . . . , F n ) ∈ D 1,2 and

Acknowledgement. Part of this paper has been written while the three authors were visiting the Institute for Mathematical Sciences of the National University of Singapore, in the occasion of the program "Progress in Stein's method" (January 5 -February 6 2009). We heartily thank Andrew Barbour, Louis Chen and Kwok Pui Choi for their kind hospitality and generous support.

(i) For every 1 i m, F (n) i converges in distribution to a centered Gaussian random variable with variance V (i, i).

(ii) For every 1 i m, E (F

(iii) For every 1 i m and every 1

Proof of Theorem 7.5. The equivalences (1) ⇔ (2) ⇔ (3) only consist in a reformulation of the previous Theorem 7. [START_REF] Jong | A central limit theorem for generalized multilinear forms[END_REF], by taking into account the equality (3.36) and the fact that (since we suppose that the sequence E[Q j (n, G) 2 ] of variances is bounded, so that an hypercontractivity argument can be applied), if Point ( 1) is verified, then (7.54) holds. On the other hand, it is completely obvious that ( 4) implies ( 1), since G is a particular case of such an X. So, it remains to prove the implication (1), ( 2), ( 3)

By assumption (3), we have that ∆ ij → 0 for all i, j = 1, . . . , m (with ∆ ij defined by (7.50)). Hence, Remark 7.4 (point 2) implies that δ

n → 0. By assumption (3) (for r = d j -1) and (1.14)-(1.15), we get that max

n ) → 0 as n → ∞ for all j = 1, . . . , m. Hence, Corollary 7.3 implies that δ (a) n → 0, which completes the proof.

8 Some proofs based on Malliavin calculus and Stein's method

The language of Malliavin calculus

Let G = {G i : i 1} be an i.i.d. sequence of Gaussian random variables with zero mean and unit variance. In what follows, concerning the Wiener Gaussian chaos, we will systematically use the definitions and notation that have been introduced in Section 2. In particular, we shall encode the structure of random variables belonging to some Wiener