3 SiC 2 ; irradiation aux ions lourds ; interactions nucléaires et électroniques ; MEB ; AFM ; DRX sous faible incidence

: Comme pour tout système nucléaire, le concept du combustible pour les réacteurs de IV ème génération consiste en des pastilles de combustible conditionnées dans une matrice qui doit contenir les produits de fission. De par leur excellente réfractarité, les carbures sont pressentis pour constituer cette barrière de confinement. Parmi ceux étudiés, le ternaire Ti 3 SiC 2 se distingue par ses propriétés mécaniques particulières : en effet, sa structure nanolamellaire lui confère une certaine plasticité ainsi qu'une ténacité supérieure aux carbures classiques tels que SiC ou TiC. Cependant, son comportement sous irradiation n'est pas connu. Afin d'appréhender ce comportement, des échantillons ont été irradiés aux ions lourds de différentes énergies puis caractérisés. Le panel d'énergies utilisé a permis de discriminer l'effet des interactions nucléaires -pulvérisation -des interactions électroniques -apparition de monticules et dilatation de la maille cristalline.

Introduction

Dans le cadre du Forum International Génération IV (GIF), de nouveaux systèmes sont étudiés, aussi bien d'un point de vu réacteur que d'un point de vue cycle du combustible. Ils se caractérisent par un niveau de sécurité accru, une meilleure compétitivité économique et une aptitude à recycler tout le combustible afin de valoriser les matières fissiles et minimiser par transmutation la production de déchets à vie longue. Parmi les six systèmes envisagés, le réacteur à caloporteur gaz et à spectre rapide (RNR-Gaz) est étudié en France. Il est conçu pour un fonctionnement nominal sous pression d'hélium et à haute température.

Ces conditions de fonctionnement ont amené à sélectionner les céramiques non oxydes réfractaires comme matériau de structure autour du combustible. Les carbures s'avèrent ainsi être d'excellents candidats ; en effet, ils ont d'excellentes propriétés thermo-mécaniques et sont généralement transparents aux neutrons contrairement aux métaux réfractaires. En revanche, leur comportement sous irradiation est peu connu.

Parmi les carbures étudiés, le ternaire Ti 3 SiC 2 offre des perspectives intéressantes. En effet, en 1972 Nickl et al. [1] ont remarqué que ce matériau, qui fut synthétisé pour la première fois par une équipe viennoise en 1967 [2], est anormalement mou pour un carbure. Au début des années 90, Pampuch, Lis et al. [3][4][5][6][7] ont conduit des travaux sur la synthèse de Ti 3 SiC 2 massifs purs : leurs meilleurs résultats furent des composés purs à 80-90 %, les impuretés étant TiC x et TiSi 2 . Lors de leurs travaux, ils ont montré que ce ternaire est à la fois rigide (module de Young de 325 GPa) et tendre (dureté Vickers de 6 GPa). De plus, comme Goto et al. quelques années auparavant [8], ils ont remarqué que la dureté de ce carbure diminue lorsque la charge appliquée augmente se stabilisant pour une certaine charge fonction du mode d'élaboration et des impuretés présentes dans les échantillons ; cette propriété les a mené à qualifier Ti 3 SiC 2 de « céramique ductile ».

À la fin des années 90, Barsoum et al. se mirent à étudier intensivement ce matériau ainsi que d'autres céramiques aux propriétés similaires [9][10][11][12][13][14]. En 2000 [15], ils regroupèrent tous ces composés dans une même famille de matériaux nanolamellaires, appelée phases MAX, de formule générale M n+1 AX n où n = {1,2,3}, M est un métal de transition, A un élément du groupe A et X est C et/ou N. Les matériaux de cette famille ont la particularité de combiner certaines propriétés des métaux à celles généralement attribuées aux céramiques ; ainsi, ces matériaux sont rigides, conducteurs thermiques et électriques, directement usinables avec des outils pour aciers, relativement tendres et tenaces. Plus spécifiquement, Ti 3 SiC 2 est stable sous atmosphère inerte ou † Coordonnées. Tel. : 04 77 42 66 86 ; fax : 04 77 49 96 94.-e-mail : nappe@emse.fr (JC. Nappé). sous vide jusqu'à 1 700 °C [13,14,16], résiste à des trempes de 1 200 °C à l'ambiante [13,17], a une conductivité thermique de 37 W m -1 K -1 à 25 °C et 32 W m -1 K -1 à 1 500 °C [14], une ténacité de 9 MPa m 1/2 [15,18], une transition mécanique fragile-ductile fonction de la température et de la vitesse de sollicitation [13,[19][20][21], une conductivité électrique de 4,5x10 6 S m -1 [13,15,22,23] et une bonne résistance à la corrosion dans les acides communs et la soude [24][25][26]. En revanche, aucune indication n'est donnée sur son comportement sous irradiation.

Par conséquent, le but de cette étude est de mieux appréhender le comportement sous irradiation de Ti 3 SiC 2 dont les propriétés mécaniques permettent de l'envisager comme matériaux de conditionnement du futur combustible.

F Fi ig gu ur re e 1 1 : : É Ét ta at t d de e s su ur rf fa ac ce e d de e T Ti i 3 3S Si iC C 2 2 a av va an nt t i ir rr ra ad di ia at ti io on n p pa ar r A AF FM M ( (a a) ) e et t M ME EB B ( (b b) ) e en n é él le ec ct tr ro on ns s r ré ét tr ro od di if ff fu us sé és s

P Pr ré ép pa ar ra at ti io on n d de es s é éc ch ha an nt ti il ll lo on ns s e et t i ir rr ra ad di ia at ti io on ns s Les échantillons ont tout d'abord été préparés par polissage mécanique avec des suspensions diamantées jusqu'à 1 µm. Puis ils ont été irradiés aux ions lourds : ces irradiations ont pour but de simuler les interactions présentes en réacteurs (impacts des produits de fission, des atomes de recul des désintégrations et, dans une moindre mesure, des neutrons) avec l'avantage de ne pas activer les échantillons.

Lorsqu'un ion pénètre dans un matériau, il va ralentir en échangeant son énergie cinétique avec les atomes de la cible. Les interactions sont alors de deux types :

Interactions électroniques : interactions inélastiques avec le nuage électronique des atomes cibles, elles interviennent en début de parcours, si les ions sont suffisamment énergétiques (> 10 MeV).

Interactions nucléaires : interactions élastiques avec les noyaux des atomes cibles, elles interviennent en fin de parcours pour les irradiations à hautes énergies, tout au long du parcours pour les basses énergies. Afin de comprendre l'impact de ces deux interactions sur Ti 3 SiC 2 , deux séries d'irradiations ont été réalisées : une dite « basse énergie » aux ions Au de 4 MeV, réalisée sur l'accélérateur ARAMIS au CSNSM d'Orsay, pour appréhender l'effet des interactions nucléaires, et une dite « haute énergie » aux ions Xe de 90 MeV, réalisée sur l'accélérateur IRRSUD au GANIL à Caen, afin de comprendre l'influence des interactions électroniques. Chaque série d'irradiation a été réalisée à 4 fluences (nombre d'ions projetés sur la cible par unité de surface) différentes : 10 12 , 10 13 , 10 14 et 10 15 ions cm -2 . Les épaisseurs de zone endommagée ont été déterminées par le code de calcul TRIM [START_REF] Ziegler | TRIM -TRansport of Ions in Matter[END_REF]. T Te ec ch hn ni iq qu ue es s d de e c ca ar ra ac ct té ér ri is sa at ti io on n Pour caractériser les modifications de la surface des échantillons, diverses microscopies ont été utilisées : MEB (conventionnel et haute résolution -canon à émission Schottky) et AFM. La structure de la zone irradiée a été caractérisée par DRX sous faible incidence (anticathode de cuivre). Me eV V) ) L'irradiation à basse énergie endommage les 500 premiers nanomètres de l'échantillon par des interactions essentiellement nucléaires. En microscopie, des modifications sont visibles dès 10 15 Au cm -2 . Au MEB (cf. Figure 2), on peut remarquer qu'à certains endroits les joints de grains sont creusés ; il se pourrait que ce phénomène soit dû à une pulvérisation préférentielle de TiSi 2 , localisé essentiellement aux joints de grain, par rapport aux deux autres phases. Par AFM, on peut remarquer une révélation de la microstructure qui peut être expliquée par une pulvérisation préférentielle en fonction de l'orientation cristallographique des grains de Ti 3 SiC 2 ; une orientation des plans compacts de sa structure hexagonale perpendiculairement au faisceau d'ions diminuerait les effets de canalisation des ions dans le réseau et augmenterait la probabilité d'une pulvérisation.

Résultats et discussion

F Fi ig gu ur re e 2 2 : : É Ét ta at t d de e s su ur rf fa ac ce e a ap pr rè ès s u un ne e i ir rr ra ad di ia at ti io on n à à 1 10 0 1 15 5 A Au u c cm m - -2 2 p pa ar r M ME EB B ( (a a) ) e et t A AF FM M ( (b b) ) e et t ( (c c) )

La DRX sous une incidence de 1° (cf. Figure 3), qui correspond à une pénétration des rayons X de 230 nm dans Ti 3 SiC 2 , révèle une diminution de l'intensité des raies des trois phases en fonction de la fluence dès 10 14 Au cm -2 , ainsi qu'une élévation de la ligne de base. Ces modifications des diffractogrammes sont certainement dues à une amorphisation partielle du matériau, phénomène souvent rencontré lors de l'irradiation d'autres matériaux tels que SiC [START_REF] Audren | [END_REF] ou certains oxydes [START_REF] Dalmasso | [END_REF].

F Fi ig gu ur re e 3 3 : : D Di if ff fr ra ac ct to og gr ra am mm me es s p po ou ur r l l' 'i ir rr ra ad di ia at ti io on n à à l l' 'o or r

I Ir rr ra ad di ia at ti io on ns s h ha au ut te e é én ne er rg gi ie e ( (X Xe e d de e 9 90 0 M Me eV V) ) L'irradiation à haute énergie endommage les 5 premiers microns de l'échantillon par des interactions électroniques. Au MEB, aucune évolution de l'état de surface n'a pu être observée. Par contre, l'AFM a permis de mettre en évidence deux phénomènes à partir de 10 15 Xe cm -2 (cf. Figure 4) : L'apparition de monticules à la surface de Ti 3 SiC 2 , d'une taille de 50 à 100 nm pour une dizaine de nanomètres de hauteur.

Ces monticules sont également visibles en MEB haute résolution lorsqu'on réalise des images dans les porosités créées par arrachage de grains lors du polissage. L'origine de ce phénomène n'est pas encore comprise. Cependant, certains auteurs ont remarqué des reliefs similaires pour des conditions d'irradiation analogues sur des monocristaux de silicium [START_REF] Srivastava | [END_REF], de InP [31] et d'un supraconducteur [32]. Mais dans chaque cas, les doses sont moins élevées (de 2,6x10 10 cm -2 pour le silicium à 5x10 12 cm -2 pour InP) et les matériaux non polycristallins. Dans leurs articles, Srivastava [START_REF] Srivastava | [END_REF] et Dudey [31] expliquent les reliefs par un transfert d'énergie de l'ion à la cible, élevé au point de fondre localement la matière et de la rendre amorphe par un brusque refroidissement. Ces zones étant fortement contraintes, elles impliqueraient des transferts de matière engendrant la formation de monticules en surface.

F Fi ig gu ur re e 5 5 : : D Di if ff fr ra ac ct to og gr ra am mm me es s p po ou ur r l l' 'i ir rr ra ad di ia at ti io on n a au u x xé én no on n

La DRX sous une incidence de 3° (cf. Figure 5), qui correspond à une pénétration des rayons X de 720 nm dans Ti 3 SiC 2 , révèle également un phénomène d'amorphisation mais uniquement pour Ti 3 SiC 2 : ainsi, contrairement à SiC [START_REF] Audren | [END_REF], il semblerait que le désordre créé par l'irradiation aux ions lourds dans le carbure ternaire soit davantage fonction de la fluence que de l'énergie de l'ion utilisé. De plus, on observe une modification de la maille de Ti 3 SiC 2 ; en effet, la raie initialement située à 40,9°, qui correspond à la diffraction des plans (00l) de Ti 3 SiC 2 , se décale vers les 2θ plus faibles en fonction de la fluence. Une estimation des paramètres de maille du ternaire a permis de mettre en évidence une dilatation de la maille suivant l'axe c en fonction de la fluence, ainsi qu'une légère contraction suivant a, impliquant une augmentation du volume de la maille.

Conclusion

Cette première étude sur les dommages sous irradiations de Ti 3 SiC 2 a permis de mettre en exergue un impact différent des interactions nucléaires et électroniques sur ce matériau aux propriétés remarquables. Ainsi, si le phénomène d'amorphisation ne semble qu'une question de fluence, les interactions nucléaires provoquent un endommagement par pulvérisation des joints de grains (certainement dû à la présence de TiSi 2 ) et pulvérisation préférentielle en fonction de l'orientation cristalline des grains. Quant aux interactions électroniques, elles induisent la formation de monticules dont la formation reste encore à comprendre, ainsi qu'une dilatation de la maille de Ti 3 SiC 2 suivant l'axe c.

  (a) École Nationale Supérieure des Mines, Centre SPIN, Saint-Étienne (b) Commissariat à l'Énergie Atomique de Cadarache, St Paul lez Durance (c) École Nationale Supérieure des Mines, Centre CIS, Saint-Étienne (d) Université de Nice -Sophia Antipolis, Laboratoire LPES-CRESA, Nice R Ré és su um mé é :

  Matériau et techniques expérimentales M Ma at té ér ri ia au u Le matériau étudié est un composé commercial polycristallin de formule Ti 3 SiC 2 fourni par la société 3-one-2 basée au New Jersey, aux États-Unis. Une estimation de la proportion en phases secondaires a été faite par Diffraction des Rayons X (DRX) : les échantillons sont constitués d'environ 80 % de Ti 3 SiC 2 , 15 % de TiC x et 5 % de TiSi 2 . La Figure 1 montre l'état de surface des échantillons après polissage : par AFM, on peut remarquer la faible dureté de Ti 3 SiC 2 par rapport à TiC x et TiSi 2 .

  I Ir rr ra ad di ia at ti io on ns s b ba as ss se e é én ne er rg gi ie e ( (A Au u d de e 4 4 M

F

  Fi ig gu ur re e 4 4 : : É Ét ta at t d de e s su ur rf fa ac ce e a ap pr rè ès s u un ne e i ir rr ra ad di ia at ti io on n ha au ut te e r ré és so ol lu ut ti io on n ( (c c) )Une révélation de la microstructure par le biais d'un comportement différent des phases présentes dans le matériau.

Remerciements

Les auteurs souhaitent remercier Isabelle Monnet (GANIL) ainsi que Lionel Thomé (CSNSM) pour leur grande aide lors de l'irradiation des échantillons.


Ciprian A Tudor
email: ciprian.tudor@univ-lille.fr

Jérémy Zurcher
email: jeremy.zurcher@univ-lille.fr

Multidimensional Stein's method for Gamma approximation

Keywords: 2010 AMS Classification Numbers: 60F05, 60G15, 60H05, 60H07 Stein's method, Stein's equation, Gamma approximation, Malliavin calculus, multiple stochastic integrals, asymptotic independence

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Introduction

The Stein's method represents a popular probabilistic collection of techniques that allows to evaluate the distances between the probability distributions of random variables. Given two random variables F, G, the Stein's method allows to obtain sharp estimates for the quantities of the form

sup h∈H |E[h(F )] -E[h(G)]| (1)
where H constitutes a large enough class of functions. Of particular importance is the case when one of the two random variables follows the Gaussian distribution but the cases of other target distributions are also of interest. We refer to the monographs and surveys [START_REF] Chen | Normal approximation by Stein's method[END_REF], [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 | Approximate computation of expectations[END_REF] for a detailed description of the techniques of Stein's method and for its applications.

Our work concerns a variant of the Stein's method that allows to measure the distance between two random vectors with the same marginals but with different correlations between their components. This variant has been initiated in [START_REF] Pimentel | Integration by parts and the KPZ two-point function[END_REF], where the author, by combining the Stein's heuristics with the tools of Malliavin calculus, obtained bounds for the Wasserstein distance between the probability distribution of a random vector (X, Y ) where X ∼ N (0, 1) and Y is an arbitrary random variable and P Z ⊗ P Y , i.e. the law of the vector (Z, Y ) where Z ∼ N (0, 1) and Z is independent by Y . We denote by P X the probability distribution of a random vector X. In some sense, this approach allows to evaluate how far are the components X and Y from being independent. The method has been extended in [START_REF] Tudor | Multidimensional Stein method and quantittative asymptotic independence[END_REF], by giving asymptotic results and by focusing on the case of random vectors with components in Wiener chaos.

Our purpose is to develop a similar theory for the case of the centered Gamma distribution with parameter ν > 0, denoted F (ν) in the sequel. The basic observation is that if X ∼ F (ν) and X is independent from an arbitrary n-dimensional random vector Y, then

E 2(X + ν) ∂f ∂x (X, Y) -Xf (X, Y) = 0 (2)
for a large class of differentiable functions f : R × R n → R. Also, if [START_REF] Arras | Stein characterizations for linear combinations of gamma random variables[END_REF] holds true for a large class of functions f , then X ∼ F (ν) and X is independent of Y. Then, we introduce a multidimensional counterpart of the standard Sten's equation for the Gamma law, i.e.

where Z ν ∼ F (ν) and h : R × R n → R belongs to suitable class of functions. We analyze in details the existence, the uniqueness and the regularity of the solution to (3) and of its partial derivatives.

In particular, we prove that there is a unique bounded solution to [START_REF] Arras | A stroll along the gamma[END_REF] and the infinity norm of this solution and of its first order partial derivatives are controlled by the infinity norm of h and of its first and second order partial derivatives. By combining the Stein's equation with the Malliavin's integration by parts, we obtain bounds for

d 2 P (X,Y) , P Zν ⊗ P Y ,
whered 2 is the second Wasserstein distance (defined later in our work), Z ν ∼ F (ν), and X, Y are arbitrary random vectors in R and R n respectively, sufficiently regular in the sense of Malliavin calculus. We also analyze the case of random sequences in Wiener chaos where those general results translate into easy-to-apply criteria for Gamma approximation. We consider a sequence (X k , k 1) and a sequence of n-dimensional random vectors (Y k , k 1) such that for each k 1, X k and the components of Y k belongs to a Wiener chaos and we assume that, in distribution, X k converges to Z ν ∼ F (ν) and Y k converges to an arbitrary random vector Y as k → ∞. Under some rather natural conditions (and easy to check in particular case), we prove that the vector ((X k , Y k ), k 1) converges in law, as k → ∞, to F (ν) ⊗ P Y . We also evaluate the corresponding rate of convergence under the so-called second Wasserstein distance. The method is illustrated by some examples. Our results extend the methods and findings in the literature related to the Gamma approximation, see e.g. [START_REF] Arras | A bound on the 2-Wasserstein distance between linear combinations of independent random variables[END_REF], [START_REF] Arras | Stein characterizations for linear combinations of gamma random variables[END_REF], [START_REF] Arras | A stroll along the gamma[END_REF], [START_REF] Azmoodeh | Optimal Gamma approximation on Wiener space[END_REF], [START_REF] Azmoodeh | Convergence towards linear combinations of chisquared random variables: a Malliavin-based approach[END_REF], [START_REF] Döbler | The Gamma Stein equation and noncentral de Jong theorems[END_REF], [START_REF] Döbler | New developments in Stein's method with applications[END_REF], [START_REF] Kusuoka | Stein's method for invariant measures of diffusions via Malliavin calculus[END_REF], [START_REF] Kusuoka | Characterization of the convergence in total variation and extension of the fourth moment theorem to invariant measures of diffusions[END_REF], [START_REF] Nourdin | Stein's method on Wiener chaos[END_REF], [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF]. We organized our paper as follows. In Section 2 we give a characterization for the probability distribution of a random vector whose first marginal is a centered Gamma law and the rest of the vector is independent by the first component. This naturally leads to a multidimensional Stein's equation corresponding to the law of a such multidimensional random vectors. We then analyze in details the solution to the Stein's equation on the whole real line and we prove suitable bounds for this solution and for its partial derivatives. In Section 3 we combine our findings on the solution to the Stein's equation with the techniques of Malliavin calculus, in the spirit of [START_REF] Nourdin | Stein's method on Wiener chaos[END_REF], in order to obtain bounds for the second Wasserstein distance between an arbitrary random vector and a random vector with the first component following the F (ν) law and the rest of the vector independent by this first component. Section 4 is devoted to the study of sequences of random variables in Wiener chaos. If (X n , n ≥ 1) converges in distribution to F (ν) and (Y n , n ≥ 1) converges in distribution to an arbitrary law U on R n , we give criteria to check the joint convergence of the vector ((X n , Y n ), n ≥ 1) to F (ν) ⊗ U . We illustrate our result by two concrete examples in Section 5. Finally, the Appendix contains the basics of Malliavin calculus and the proof of as technical result.

The multidimensional Stein's equation and its solution

Let Z ν be a random variable following the centered Gamma distribution F (ν) with parameter ν > 0, whose probability density function is

p ν (x) := (x + ν) ν 2 -1 e -x+ν 2 2 ν 2 Γ ν 2 1 (-ν,+∞) (x). ( 4
)
When ν is an integer, then Z ν coincides in law with ν i=1 (N 2 i -1), where N 1 , ..., N ν are standard Gaussian independent random variables. We recall that in the standard Stein's method for Gamma approximation, the development of the method is based by the fact that Z ν ∼ F (ν) if and only if

E 2(Z ν + ν) + f ′ (Z ν ) = E [Z ν f (Z ν )] (5)
for a large class of differentiable functions f : R → R, where x + = max(x, 0). Let us state and prove an analogous result in the multidimensional context. We recall that if X ∼ F (ν), then its characteristic function ψ is the unique solution to the differential equation

∀λ ∈ R, (1 -2iλ)ψ ′ (λ) + 2λνψ(λ) = 0, (6)
with ψ(0) = 1.

Lemma 1. Let Y = (Y 1 , ..., Y n ) be a n-dimensional real random vector. 1. Assume Z ν ∼ F (ν) and Z ν and Y are independent. Then E 2(Z ν + ν) ∂f ∂x (Z ν , Y) = E [Z ν f (Z ν , Y)] ,
for any function

f : R × R n → R such that E (Z ν + ν) ∂f ∂x (Z ν , Y) < ∞ and E [|Z ν f (Z ν , Y)|] < ∞. 2. Conversely, assume that Z is an integrable random variable such that E 2(Z + ν) + ∂f ∂x (Z, Y) = E [Zf (Z, Y)] , ( 7
)
for any function

f : R × R n → R such that E (Z + ν) ∂f ∂x (Z, Y) < ∞ and E [|Zf (Z, Y)|] < ∞. Then Z ∼ F (ν) and Z is independent of Y.
Proof: Assume that Z ν ∼ F (ν) and Z ν and Y are independent. Then, by the Stein's characterization of the centered Gamma law (5), we have for every

y ∈ R n , E 2(Z ν + ν) ∂f ∂x (Z ν , y) = E [Z ν f (Z ν , y)] . So, ˆ∞ -ν 2(x + ν) ∂f ∂x (x, y)dP Zν (x) = ˆ∞ -ν xf (x, y)dP Zν (x),
for every y ∈ R n . We integrate the above relation with respect to P Y on R n and we find for the left-hand side, by using the independence of Z ν and Y,

ˆRn ˆ∞ -ν 2(x + ν) ∂f ∂x (x, y)dP Zν (x) dP Y (y) = ˆ(-ν,∞)×R n 2(x + ν) ∂f ∂x (x, y)d(P Zν ⊗ P Y )(x, y) = E 2(Z ν + ν) ∂f ∂x (Z ν , Y) .
Similarly for the right-hand side

ˆRn ˆ∞ -ν xf (x, y)dP Zν (x) dP Y (y) = ˆ(-ν,∞)×R n xf (x, y)d(P Zν ⊗ P Y )(x, y) = E [Z ν f (Z ν , Y)] .
Let us now prove the second point of the lemma. By taking a function f with support contained in (-∞, -ν), we found that P(Z ν ≤ -ν) = 0. Let ϕ be the characteristic function of the random vector (Z, Y), i.e.

ϕ(λ 1 , λ 2 ) := E e i(λ 1 Z+λ 2 •Y) , for λ 1 ∈ R, λ 2 ∈ R n
, where x • y denotes the Euclidean scalar product of x, y ∈ R n . We take the derivative of ϕ with respect to λ 1 and we apply (7) (for the real and imaginary parts of ϕ). By denoting g(x, y) = e i(λ 1 x+λ 2 •y) , then since Z is integrable, g satisfies the conditions to have [START_REF] Chen | Stein's method for normal approximation[END_REF] and we get

∂ϕ ∂λ 1 (λ 1 , λ 2 ) = iE Ze i(λ 1 Z+λ 2 •Y) = iE[Zg(X, Y)] = iE 2(Z + ν) ∂g ∂x (Z, Y) = -2λ 1 E (Z + ν)e i(λ 1 Z+λ 2 •Y) = 2iλ 1 ∂ϕ ∂λ 1 (λ 1 , λ 2 ) -2νλ 1 ϕ(λ 1 , λ 2 ).
Consequently, for every λ 1 ∈ R, λ 2 ∈ R n , we have

(1 -2iλ 1 ) ∂ϕ ∂λ 1 (λ 1 , λ 2 ) + 2λ 1 νϕ(λ 1 , λ 2 ) = 0.
By noticing that for every

λ 2 ∈ R n ϕ(0, λ 2 ) = E e iλ 2 •Y = ϕ Y (λ 2 ),
where ϕ Y stands for the characteristic function of the random vector Y, we obtain from (6), for every

λ 1 ∈ R, λ 2 ∈ R n , ϕ(λ 1 , λ 2 ) = ψ(λ 1 )ϕ Y (λ 2 ).
This means that Z ∼ F (ν) and Z is independent of Y.

Let us introduce the multidimensional Stein's equation for the centered Gamma law F (ν)

2(x + ν) ∂f ∂x (x, y) -xf (x, y) = h(x, y) -E [h(Z ν , y)] , (8)
with

Z ν ∼ F (ν) and h : R × R n → R such that E [|h(F (ν), y)|] < ∞ for any y ∈ R n .
In order to solve [START_REF] Döbler | New developments in Stein's method with applications[END_REF], let us recall some facts concerning the one-dimensional Stein's equation (on the whole real line) for the Gamma law

2(x + ν)f ′ (x) -xf (x) = h(x) -E[h(F (ν))], (9)
with h : R → R satisfying E [|h(F (ν))|] < ∞. Recall that the density of this law is given by ( 4) and let us also introduce the function

q ν (x) := (-(x + ν)) ν 2 -1 e -x+ν 2 2 ν 2 Γ ν 2 1 (-∞,-ν) (x). ( 10
)
Both functions p ν and q ν are positive. Note that, contrary to p ν , the function q ν does not define a probability measure. We consider F ν the cumulative probability function of p ν . We denote by Fν the following function defined on (-∞, -ν]:

Fν (x) = ˆ-ν x q ν (u) du. ( 11
)
Then Fν is a positive decreasing map on (-∞, -ν].

Lemma 2. Let

h : R → R be measurable such that E [|h(Z ν )|] < ∞.
Then the Stein's equation ( 9) admits a unique bounded solution given by

f h (x) = ˆx -ν p ν (u) 2(x + ν)p ν (x) (h(u) -E [h(Z ν )]) du 1 (-ν,+∞) (x) - ˆ-ν x q ν (u) 2(x + ν)q ν (x) (h(u) -E[h(Z ν )]) du 1 (-∞,-ν) (x) (12)
+ h(-ν) -E [h(Z ν )] ν 1 {-ν} (x).
Proof: See Section 2.2 in [START_REF] Döbler | The Gamma Stein equation and noncentral de Jong theorems[END_REF].

Remark 1. We can show that this unique solution is C 1 on R, by using the l'Hôpital's rule several times, when h is C 1 with h ′ absolutely continuous. Indeed, for instance, for x going to -ν from above :

lim x↓-ν f ′ h (x) = lim x↓-ν 2(x + ν)p ν (x) h(x) + x ´x -ν h(u)p ν (u) du 4(x + ν) 2 p ν (x) ( Ĥ) = lim x↓-ν 2(x + ν)p ν (x)h ′ (x) -xp ν (x) h(x) + ´x -ν h(u)p ν (u) du + x h(x)p ν (x) 2(2 -x)(x + ν)p ν (x) = 1 2 + ν lim x↓-ν 2(x + ν)p ν (x)h ′ (x) + ´x -ν h(u)p ν (u) du 2(x + ν)p ν (x) ( Ĥ) = 1 2 + ν lim x↓-ν 2h ′′ (x)(x + ν)p ν (x) -xp ν (x)h ′ (x) + h(x)p ν (x) -xp ν (x) = h ′ (-ν) 2 + ν + h(-ν) ν(2 + ν) , where h(x) = h(x) -E[h(Z ν )], " ( Ĥ)
= " means that we used the l'Hôpital's rule and we also applied the fundamental property

2(x + ν)p ν (x) = - ˆx -ν up ν (u) du. ( 13
)
This works analogously when x goes to -ν from below, via a similar relation for q ν : 2(x + ν)q ν (x) =

ˆ-ν

x uq ν (u) du for every x < -ν.

In a first step, we deduce the existence of a unique bounded solution of the multidimensional Stein's equation [START_REF] Döbler | New developments in Stein's method with applications[END_REF]. 8) admits an unique bounded solution which can be expressed as

Proposition 1. Let h : R × R n → R be a test function such that E [|h(F (ν), y)|] < ∞ for any y ∈ R n . Then (
f h (x, y) = ˆx -ν p ν (u) 2(x + ν)p ν (x) (h(u, y) -E [h(Z ν , y)]) du 1 (-ν,+∞) (x) - ˆ-ν x q ν (u) 2(x + ν)q ν (x) (h(u, y) -E [h(Z ν , y)]) du 1 (-∞,-ν) (x), (14)
+ h(-ν, y) -E [h(Z ν , y)] ν 1 {-ν} (x).
Proof: Let us prove first that ( 14) satisfies [START_REF] Döbler | New developments in Stein's method with applications[END_REF]. For x > -ν, we have by differentiating [START_REF] Nourdin | Normal Approximations with Malliavin Calculus From Stein's Method to Universality[END_REF] with respect to x:

∂f h ∂x (x, y) = h(x, y) -E[h(Z ν , y)] 2(x + ν) + x 4(x + ν) 2 p ν (x) ˆx -ν p ν (u)(h(x, y) -E[h(Z ν , y)]) du,
and so we conclude that we have ( 8) in (-ν, +∞). For x < -ν :

∂f h ∂x (x, y) = h(x, y) -E[h(Z ν , y)] 2(x + ν) - x 4(x + ν) 2 q ν (x) ˆ-ν x q ν (u)(h(x, y) -E[h(Z ν , y)]) du,
giving the same conclusion on (-∞, -ν). Finally, we can check by l'Hôpital's rule, in the same way as in [START_REF] Döbler | New developments in Stein's method with applications[END_REF], that when x goes to -ν, the solution ( 14) goes to h(-ν,y)-E[h(Zν ,y)] ν , either the taking the lft or the right limit. Hence, x -→ f h (x, y) is continuous on R. Moreover, we can prove, when h is C 1 with ∂h ∂x absolutely continuous, by the same argument as in Remark 1, that is also the case of

∂f h ∂x . So f is C 1 with respect to x.
This concludes the existence part.

Let us prove the unicity. Assume f h , g h are two bounded solutions to [START_REF] Döbler | New developments in Stein's method with applications[END_REF]. Then for every

(x, y) ∈ R × R n , 2(x + ν) ∂(f h -g h ) ∂x (x, y) -x(f h -g h )(x, y) = 0.
By solving the above equations, we get

(f h -g h )(x, y) =      c 1 (y) e x 2 (x+ν) ν 2 if (x, y) ∈ (-ν, ∞) × R n c 2 (y) e x 2 (-(x+ν)) ν 2 if (x, y) ∈ (-∞, -ν) × R n , so f h (x, y) =      g h (x, y) + c 1 (y) e x 2 (x+ν) ν 2 if (x, y) ∈ (-ν, ∞) × R n g h (x, y) + c 2 (y) e x 2 (-(x+ν)) ν 2 if (x, y) ∈ (-∞, -ν) × R n .
Consequently, f h is bounded if and only if c 1 (y) = c 2 (y) = 0 for every y ∈ R n . The next step is to find suitable estimates for the solution ( 14) and for its partial derivatives. We follow the methodology proposed in [START_REF] Döbler | The Gamma Stein equation and noncentral de Jong theorems[END_REF] and [START_REF] Döbler | New developments in Stein's method with applications[END_REF] in the one-dimensional case. Let us start by an useful result that gives an alternative expression for the solution to [START_REF] Döbler | New developments in Stein's method with applications[END_REF]. The proof of the below lemma can be found in the Appendix.

Lemma 3.

1. Let f h be given by [START_REF] Nourdin | Normal Approximations with Malliavin Calculus From Stein's Method to Universality[END_REF]. Then for x > -ν and y ∈ R n ,

f h (x, y) (15) = 1 -F ν (x) ´x -ν up ν (u) du ˆx -ν ∂h ∂x (w, y)F ν (w) dw + F ν (x) ´x -ν up ν (u) du ˆ+∞ x ∂h ∂x (w, y)(1 -F ν (w)) dw,
and for every x < -ν and y ∈ R n , we have:

f h (x, y) = 1 ´-ν x uq ν (u) du ˆ-ν x ∂h ∂x (w, y) Fν (x) -Fν (w) dw + Fν (x) ˆ+∞ -ν ∂h ∂x (w, y)(1 -F ν (w)) dw . ( 16
)
2. If F ν denotes the cumulative distribution function of the centered Gamma law F (ν), then for any x ∈ (-ν, ∞) :

ˆx -ν F ν (s) ds = xF ν (x) - ˆx -ν sp ν (s) ds.
and

ˆ+∞ x (1 -F ν (s)) ds = -x(1 -F ν (x)) + ˆ+∞ x sp ν (s) ds.
3. Let Fν be given by [START_REF] Kusuoka | Characterization of the convergence in total variation and extension of the fourth moment theorem to invariant measures of diffusions[END_REF]. Then we have for x ∈ (-∞, -ν),

ˆ-ν x Fν (w) dw = ˆ-ν x uq ν (u) du -x Fν (x).
Let us now obtain some estimates for the solution to the Stein's equation. The idea of the proof is based on the proof of Theorem 3.13 in [START_REF] Döbler | New developments in Stein's method with applications[END_REF].

Proposition 2. Let h : R × R n be a C 2 map with bounded partial derivatives. Let f h be the unique bounded solution to [START_REF] Döbler | New developments in Stein's method with applications[END_REF] corresponding to h. We have the following estimates :

f h ∞ C ∂h ∂x ∞ , ( 17
)
and for every j ∈ {1, ..., n},

∂f h ∂y j ∞ C ∂ 2 h ∂x∂y j ∞ and ∂f h ∂x ∞ C ∂h ∂x ∞ . ( 18
)
Proof: Let us prove the estimation [START_REF] Reinert | Three general approaches to Stein's method[END_REF]. Let x > -ν. By using Lemma 3, we have

|f h (x, y)| = F ν (x) ´x -ν up ν (u) du ˆ+∞ x ∂h ∂x (w, y)(1 -F ν (w)) dw + 1 -F ν (x) ´x -ν up ν (u) du ˆx -ν ∂h ∂x (w, y)F ν (w) dw . Note that ´x -ν up ν (u) du = -2(x + ν)p ν (x) 0. Then, via point 2. in Lemma 3, |f h (x, y)| ∂h ∂x ∞ -´x -ν up ν (u) du F ν (x) ˆ+∞ x (1 -F ν (w)) dw + (1 -F ν (x)) ˆx -ν F ν (w) dw = ∂h ∂x ∞ -´x -ν up ν (u) du F ν (x) ˆ+∞ x sp ν (s) ds -x(1 -F ν (x)) +(1 -F ν (x)) xF ν (x) - ˆx -ν sp ν (s) ds = ∂h ∂x ∞ -´x -ν up ν (u) du F ν (x) ˆ+∞ -ν sp ν (s) ds - ˆx -ν sp ν (s) ds .
By using the fact that ´+∞ -ν sp ν (s) ds = E[Z ν ] = 0, we conclude the estimation ( 17) on (-ν, +∞). Now, we proceed in the same way for x < -ν. By Lemma 3

|f h (x, y)| = 1 - ´-ν x uq ν (u) du ˆ-ν x ∂h ∂x (w, y) Fν (x) -Fν (w) dw + Fν (x) ˆ+∞ -ν ∂h ∂x (w, y)(1 -F ν (w)) dw .
Since Fν is decreasing on (-∞, -ν) and since

´-ν x uq ν (u) du = 2(x + ν)q ν (x) 0, we have

|f h (x, y)| ∂h ∂x ∞ - ´-ν x uq ν (u) du ˆ-ν x Fν (x) -Fν (w) dw + Fν (x) ˆ+∞ -ν (1 -F ν (w)) dw ∂h ∂x ∞ - ´-ν x uq ν (u) du Fν (x) -(x + ν) + ˆ+∞ -ν (1 -F ν (w)) dw - ˆ-ν x Fν (w) dw ∂h ∂x ∞ - ´-ν x uq ν (u) du -x Fν (x) - ˆ-ν x uq ν (u) du -x Fν (x) = ∂h ∂x ∞ .
We used Lemma 3 to express the integral of Fν , and the fact that ´+∞ -ν (1 -F ν (u)) du = ν. Hence, we proved the bound (17) on (-∞, -ν). Using the continuity of f h , this inequality also holds for x = -ν. So, the inequality (17) is true.

Let us now prove the inequalities [START_REF] Ch | Approximate computation of expectations[END_REF]. We start by dealing with the derivative of f h with respect to x. Since f h solves (8), we have for x = -ν and for y ∈ R n ,

∂f h ∂x (x, y) = x 2(x + ν) f h (x, y) + h(x, y) -E[h(Z ν , y)] 2(x + ν) . ( 19
)
We again separate the cases x > -ν and x < -ν.

• Assume x > -ν. By Lemma 3, point 1. and by relation (48) in the proof of this lemma, we get

xf h (x, y) + h(x, y) = x(1 -F ν (x)) ´x -ν up ν (u) du ˆx -ν ∂h ∂x (w, y)F ν (w) dw + xF ν (x) ´x -ν up ν (u) du ˆ+∞ x ∂h ∂x (w, y)(1 -F ν (w)) dw + ˆx -ν ∂h ∂x (w, y)F ν (w) dw - ˆ+∞ x ∂h ∂x (w, y)(1 -F ν (w)) dw = G(x) A ν (x) ˆx -ν ∂h ∂x (w, y)F ν (w) dw - H(x) A ν (x) ˆ+∞ x ∂h ∂x (w, y)(1 -F ν (w)) dw,
where we used the notation

     A ν (x) = ´x -ν up ν (u) du = -2(x + ν)p ν (x) 0 G(x) = A ν (x) + x(1 -F ν (x)) H(x) = -A ν (x) + xF ν (x).
Notice that

H ′ (x) = F ν (x) and H(-ν) = 0. Hence, H(x) = ´x -ν F ν (u) du. We also have G(x) = ´+∞ x (1 -F ν (u)) du, since G ′ (x) = 1 -F ν (x), G(-ν) = -ν and ´+∞ -ν (1 -F ν (u)) du = ν.
Therefore, we have the following bound for ∂f h ∂x :

∂f h ∂x (x, y) ∂h ∂x ∞ 2 ´x -ν F ν (u) du ´+∞ x (1 -F ν (u)) du (x + ν)|A ν (x)| .
To conclude, we need to prove that the function S ν defined by

S ν (x) := 2 ´x -ν F ν (u) du ´+∞ x (1 -F ν (u)) du (x + ν)|A ν (x)| (20)
is bounded on (-ν, +∞). We will prove that S ν is bounded on R + and admits a finite limit at -ν. For x > 0, we use Lemma 3 to have

ˆ+∞ x (1 -F ν (u)) du = 2(x + ν)p ν (x) -x(1 -F ν (x)) 2(x + ν)p ν (x).
Consequently, for every x > 0, we have (by using that F ν is increasing and

´∞ -ν (1 -F (u))du = ν), S ν (x) 2(x + ν)F ν (x) • 2(x + ν)p ν (x) 2(x + ν) 2 p ν (x) = 2F ν (x) 2.
Now, for x going to -ν, notice first that :

lim x→-ν S ν (x) = 2ν lim x→-ν ´x -ν F ν (u) du 2(x + ν)p ν (x) ,
Since both numerator and denominator go to zero, we apply l'Hôpital's rule several times to conclude. We use the relation [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF] to differentiate in [START_REF] Üstünel | On independence and conditioning on Wiener space[END_REF].

lim x→-ν S ν (x) = 2ν lim x→-ν F ν (x) (2 -x)(x + ν)p ν (x) = 4ν 2 + ν lim x→-ν F ν (x) 2(x + ν)p ν (x) = 4ν 2 + ν lim x→-ν p ν (x) -xp ν (x) = 4 2 + ν .
Hence, S ν is continuous, bounded on R + and admits a finite limit at -ν, so S ν is bounded in (-ν, +∞) and so we conclude that

∀x > -ν, ∂f h ∂x (x, y) S ν ∞ ∂h ∂x ∞ .
• For x < -ν. We follow the same computations. We get

xf h (x, y) + h(x, y) = G(x) B ν (x) ˆ+∞ x ∂h ∂x (w, y) [1 -F ν (w)] dw - x B ν (x) ˆ-ν x ∂h ∂x (w, y) Fν (w) dw,
where we denoted

B ν (x) = ´-ν x uq ν (u) du = 2(x + ν)q ν (x) G(x) = x Fν (x) -B ν (x).
Then, we have G(x) =

´-ν

x Fν (u) du. Set for all x < -ν :

R ν (x) := ´-ν x Fν (u) du ´+∞ x [1 -F ν (w)] dw -x (x + ν)B ν (x) = -2x
´-ν

x Fν (u) du

2q ν (x)(x + ν) 2 .
Then we have by [START_REF] Tudor | Multidimensional Stein method and quantittative asymptotic independence[END_REF], for every x < -ν and for every y ∈ R n ,

∂f h ∂x (x, y) R ν (x) ∂h ∂x ∞ .
We need to prove that R ν is bounded. Here, we will prove that R ν admits finite limits at -∞ and at -ν. We will use again the fundamental equality [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF]. In -ν :

lim x→-ν R ν (x) = 2ν lim x→-ν ´-ν x Fν (u) du 2(x + ν) 2 q ν (x) = 2ν lim x→-ν -Fν (x) (2 -x)(x + ν)q ν (x) = 4ν ν + 2 lim x→-ν -Fν (x) 2(x + ν)q ν (x) = 4ν ν + 2 lim x→-ν q ν (x) -xq ν (x) = 4 2 + ν .
In -∞, we proceed in a similar way,

lim x→-∞ R ν (x) = -2 lim x→-∞ ´-ν x Fν (u) du 2(x + ν)q ν (x) = -2 lim x→-∞ Fν (x) xq ν (x) = -4 lim x→-∞ Fν (x) 2(x + ν)q ν (x) = -4 lim x→-∞ q ν (x) xq ν (x) = 0.
This proves that R ν is bounded on (-∞, -ν), and so that for every x < -ν, y ∈ R n ,

∂f h ∂x (x, y) R ν ∞ ∂h ∂x ∞ .
Since f h is C 1 with respect to x, this holds on -ν by continuity. Hence, we proved that

∂f h ∂x ∞ max { S ν ∞ , R ν ∞ } ∂h ∂x ∞
and the first inequality in ( 18) is done.

To deal with the second bound in [START_REF] Ch | Approximate computation of expectations[END_REF], by differentiating the Stein's equation ( 8) with respect to y j , we observe that we have in fact

∂f h ∂y j = f ∂h ∂y j .
Since h is C 2 , the estimation made in [START_REF] Reinert | Three general approaches to Stein's method[END_REF] directly yields the second estimation of (18).

Multidimensional Stein-Malliavin calculus for the Gamma distribution

We use the multidimensional Stein's equation and the properties of its solution proven in Proposition 2 in combination with the techniques of Malliavin calculus in order to obtain bounds for the distance between the probability distributions of random vectors. Actually, we will evaluate the second Wasserstein distance between the law of an arbitrary random vector (X, Y) in R × R n and the random vector (Z ν , Y), where Z ν ∼ F (ν) and Z ν is independent by Y.

Let us first introduce the distances that we use in the sequel. Let H the set of functions h : R n+1 -→ R which are C 1 such that every partial derivative of h are Lipschitz with

h Lip + h ′ Lip 1,
where

h Lip := sup x,y∈R n+1
x =y

|h(x)h(y)|

xy and h ′ Lip := max

1 j n+1 sup x,y∈R n+1 x =y |∂ j h(x) -∂ j h(y)| x -y .
Then, for every integrable random vectors X and Y of R n+1 , we define the second Wasserstein distance defined by

d 2 (P X , P Y ) := sup h∈H |E[h(X)] -E[h(Y)]| . ( 21
)
(By convention, if X or Y are not integrable, we set it equal to ∞: we cannot measure anything with this distance). We also recall the classical Wasserstein distance. Let

A = {h : R n → R, h is Lipschitz continuous with h Lip 1}.
Then the Wasserstein distance between the probability distributions of X and Y is defined by

d W (P X , P Y ) := sup h∈A |E[h(X)] -E[h(Y)]| . ( 22
)
We can show the following fact: the convergence for the distance d 2 implies the convergence for the classical Wasserstein distance.

Lemma 4. For every integrable random vectors X and Y, we have

d W (X, Y) 32I n √ π d 2 (X, Y),
where

I n := 1 √ (2π) n ´Rn u e -u 2 2
du.

Proof: Let h be a 1-Lipschitzian map on R n . Then by Rademacher's theorem, h is Lebesgue almost-everywhere differentiable. We can consider a version of its partial derivatives. For σ > 0, we consider the n-dimensional Gaussian kernel with variance σ 2 given by

G σ (x) := 1 (2πσ 2 ) n exp -x 2 2σ 2 ,
and the following approximation of h: h σ := h * G σ . In a first step, let us prove that there exists a constant a σ such that a σ h σ ∈ H. The function h σ is C ∞ and its first partial derivatives are given by

∂ i h σ = ∂ i h * G σ . Hence, we conclude that ∂ j h σ ∞ h Lip 1.
Concerning the second derivative of h σ , we have:

∂ 2 i,j h σ (x) = |∂ i h * ∂ j G σ (x)| ≤ h Lip σ 2 ˆRn |u j | exp -u 2 2σ 2 du (2πσ 2 ) n = 2 π h Lip σ ≤ 2 πσ 2 .
This implies that for every σ ∈ 0, 2 π , πσ 2 2 h σ ∈ H.

Let us next evaluate the difference between h and h σ . We can write

|h(x) -h σ (x)| = ˆRn (h(x) -h(y))G σ (x -y) dy (2πσ 2 ) n ≤ h Lip ˆRn u G σ (u) du (2πσ 2 ) n ≤ σ n+1 h Lip (2πσ 2 ) n ˆRn u e -u 2 2 du ≤ 1 (2π) n ˆRn u e -u 2 2
du σ := I n σ.

Therefore, for every σ ∈ 0, 2 π , we have

|E[h(X)] -E[h(Y)]| |E[h(X)] -E[h σ (X)]| + |E[h(Y)] -E[h σ (Y)]| + |E[h σ (X)] -E[h σ (Y)]| 2 h -h σ ∞ + σ π 2 d 2 (X, Y ) 2I n σ + 1 σ 2 π d 2 (X, Y).
The left hand side, seen as a function of σ, can be optimized on 0, π 2 by taking

σ 0 = 2 π d 2 (X, Y) 2I n ∈ 0, π 2 .
Note that since (I n ) n is increasing, we have indeed σ 0

d 2 (X,Y) 2 π 2
, since the distance is lower than 1. This value of σ yields to

d W (X, Y) 2 8I n √ π d 2 (X, Y),
which is the desired conclusion.

Let us state and prove the main result of this section.

Theorem 1. Let X a centered random variable and Y

= (Y 1 , • • • , Y n ) a centered random vector of R n such that X ∈ D 1,2 and Y j ∈ D 1,2 for every j ∈ {1, • • • , n}. Then there exists a constant C = C(ν) > 0 such that d 2 P (X,Y) , F (ν) ⊗ P Y (23) ≤ C E 2(X + ν) -D(-L) -1 X, DX + C n j=1 E D(-L) -1 X, DY j .
If moreover we suppose that X ∈ D 1,4 and Y j ∈ D 1,4 for every j ∈ {1, • • • , n}, then

d 2 P (X,Y) , F (ν) ⊗ P Y (24) ≤ C E 2(X + ν) -D(-L) -1 X, DX 2 1 2 + C n j=1 E D(-L) -1 X, DY j 2 1 2
.

Proof: Let suppose first that h ∈ C 2 (R n ×R) with h Lip + h ′ Lip 1.
Let f h be the corresponding solution to the Stein's equation [START_REF] Döbler | New developments in Stein's method with applications[END_REF]. Then

ˆ(-ν,+∞)×R n h(x, y) dP (X,Y) (x, y) - ˆ(-ν,+∞)×R n h(x, y) dP Zν ⊗ dP Y (x, y) = ˆ(-ν,+∞)×R n h(x, y) dP (X,Y) (x, y) - ˆRn E[h(Z ν , y)] dP Y (y) = E [h(X, Y) -E [h(Z ν , Y)]] .
So, by the Stein's equation ( 8), we have

ˆ(-ν,+∞)×R n h(x, y) dP (X,Y) (x, y) - ˆ(-ν,+∞)×R n h(x, y) dP Zν ⊗ dP Y (x, y) = 2E (X + ν) ∂f h ∂x (X, Y) -E [Xf h (X, Y)] . ( 25
)
Since X is centered, we can write X = δ D(-L) -1 X , and by plugging this into (25)

ˆ(-ν,+∞)×R n h(x, y) dP (X,Y) (x, y) - ˆ(-ν,+∞)×R n h(x, y) dP Zν ⊗ dP Y (x, y) = 2E (X + ν) ∂f h ∂x (X, Y) -E δ D(-L) -1 X f h (X, Y) = 2E (X + ν) ∂f h ∂x (X, Y) -E D(-L) -1 X, Df h (X, Y) = 2E (X + ν) ∂f h ∂x (X, Y) -E ∂f h ∂x (X, Y) D(-L) -1 X, DX - n j=1 E ∂f h ∂y j (X, Y) D(-L) -1 X, DY j . Since h Lip + h ′ Lip 1, we have ∂f h ∂x ∞ C and ∂f h ∂y j ∞
1 by using Lemma 3. So, we obtain for a generic constant C

ˆ(-ν,+∞)×R n h(x, y) dP (X,Y) (x, y) - ˆ(-ν,+∞)×R n h(x, y) dP X ⊗ dP Y (x, y) CE (X + ν) -D(-L) -1 X, DX + C n j=1 E D(-L) -1 X, DY j .
To conclude the bound (23), we need to have the above inequality for every h ∈ H. If h is one of those functions, we approach it by the sequence

h k (x, y) := E h x + N √ k , y + N √ k ,
where N ∼ N (0, 1) is independent of N ∼ N n (0, I n ). Then (h k ) k uniformly converges to h and we still have

h k Lip + h ′ k Lip
1. Consequently, we have:

ˆ(-ν,+∞)×R n h(x, y) dP (X,Y) (x, y) - ˆ(-ν,+∞)×R n h(x, y) dP Zν ⊗ dP Y (x, y) 2 h -h k ∞ + CE (X + ν) -D(-L) -1 X, DX + C n j=1 E D(-L) -1 X, DY j .
Taking the limit as k → ∞, (23) is obtained. For (24), it suffices to notice that if X, Y j ∈ D 1,4 , then the scalar products D(-L) -1 X, DX and D(-L) -1 X, DY j are in L 2 , so that we can apply Cauchy-Schwarz's inequality.

Asymptotic independence on Wiener chaos

We now focus on random variables in Wiener chaos and we give an asymptotic variant of the result proven in Theorem 1. We will consider a sequence (X k , k ≥ 1) that converges in distribution when k → ∞ to the centered Gamma law F (ν) and a sequence of random vectors (Y k , k ≥ 1) converging in law as k → ∞ to an arbitrary random vector Y. We assume that for each k ≥ 1, X k and the components of Y k belong to a Wiener chaos of fixed order. Under some pretty natural assumptions, we deduce, by using Theorem 1 and the properties of random variables in Wiener chaos, that the random sequence ((X k , Y k ), k ≥ 1) converges in law to (Z ν , Y), where Z ν follows the centered Gamma distribution with parameter ν > 0 and Z ν and Y are independent. This means that the sequences (X k , k ≥ 1) and (Y k , k ≥ 1) are asymptotically independent. We obtain bounds to quantify this asymptotic independence under the d 2 -distance defined by (21).

Let us first recall, that if a sequence (X k , k ≥ 1) in the qth Wiener chaos converges to F (ν), then the order q of the chaos must be an even integer. Indeed, if q is odd then we have E X 3 k = 0 for any k 1 and it contradicts the fact that

E Z 3 ν = 8ν > 0 if Z ν ∼ F (ν)
. Before stating our result, let us recall the following criterion for the Gamma approximation on Wiener chaos. We refer to [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF] for its proof. Theorem 2. Let (X k , k ≥ 1) be a sequence of random variables such that for every k 1, X k = I q (f k ) where q is an even integer and

f k ∈ H ⊙q . Assume that E X 2 k ---→ k→∞ 2ν.
Then the following are equivalent:

1. The sequence (X k , k ≥ 1) converges in distribution to F (ν).

2. For every p ∈ {1, ..., q -1} with p = q 2 ,

f k ⊗ p f k 2 H ⊗2q-2p ---→ k→∞ 0 and f k ⊗ q 2 f k -c q f k 2 H ⊗q ---→ k→∞ 0,
where c q > 0 is an explicit constant.

3. As k → ∞, DX k 2 H -2qX k -2qν ---→ k→∞ 0 in L 2 (Ω).
Other equivalent conditions to 1.-3. (not needed in our work) are stated and proven in [START_REF] Nourdin | Non-central convergence of multiple integrals[END_REF].

We apply this theorem in our context, about asymptotic independence with a component going to a centered Gamma law. We have the following result. Theorem 3. Let n, q and q 1 , • • • , q n strictly positive integers such that q is even and q q j , for every j ∈ {1, • • • , n}. We consider a sequence (X k , k 1) = (I q (f k ), k 1) converging in distribution, as k goes to +∞, to a random variable X following F (ν). We consider a sequence of random vectors

(Y k ) k = ((Y 1 k , • • • , Y n k )) k such that Y j k = I q j (g j k )
and and there exists a random

vector Y = (Y 1 , • • • , Y n ) such that (Y k , k 1) converges in distribution to Y. We suppose that, for every j ∈ {1, • • • , n} : E X k Y j k ---→ k→∞ 0 and f k ⊗ q 2 g j k ---→ k→∞ 0. ( 26
)
Then we have the following convergence in distribution :

(X k , Y k ) (d) ----→ k→+∞ (X, Y), ( 27
)
where X ∼ F (ν) and X is independent of Y. Moreover, if we denote

θ k := P (X k ,Y k ) and η := P X ⊗ P Y ,
then we have the following estimation of the second Wasserstein distance between those two measures:

d 2 (θ k , η) C E 2(X k + ν) -D(-L) -1 X k , DX k 2 1 2 (28) + C n j=1 E D(-L) -1 X k , DY j k 2 1 2 + d 2 (P Y k , P Y ) .
Proof: For k ≥ 1, let us consider the probability measure η k := P X ⊗ P Y k = F (ν) ⊗ P Y k . Then we have for every h ∈ H (the set defined at the beginning of Section 3),

ˆR ˆRn h(x, y) dη k (x, y) - ˆR ˆRn h(x, y) dη(x, y) = ˆRn ˆR h(x, y) dP X (x) dP Y k (y) - ˆRn ˆR h(x, y) dP X (x) dP Y (y) = E ˆR h(x, Y k ) dP X (x) -E ˆR h(x, Y) dP X (x) d 2 (P Y k , P Y ),
the last inequality being true because the function y -→ ´R h(x, y) dx also belongs to

H (on R n ). Hence, d 2 (η k , η) d 2 (P Y k , P Y ). ( 29
)
Now by using the triangle inequality, (29) and the bound (24) in Theorem 1,

d 2 (θ k , η) ≤ d 2 (θ k , η k ) + d 2 (η k , η) ≤ C E 2(X k + ν) -D(-L) -1 X k , DX k 2 1 2 + C n j=1 E D(-L) -1 X k , DY j k 2 1 2 + d 2 (P Y k , P Y )
which is nothing else but (28). Let us now prove that the right-hand side of (28) converges to zero as k → ∞. For the first term, the convergence to zero comes from Theorem 2. Indeed,

E 2(X k + ν) -D(-L) -1 X k , DX k 2 = 1 q 2 E DX k 2 -2qX k -2qν 2 ---→ k→∞ 0
due to point 3. in Theorem 2. About the second term, we will use some classical computations about Wiener chaoses. We have:

E D(-L) -1 X k , DY j k 2 = 1 q 2 E DX k , DY j k 2 .
Then, by using the product formula (45), and the assumption q q j , we conclude that

E D(-L) -1 X k , DY j k 2 = 1 q 2 q j -1 r=0 (r!) 2 q j -1 r 2 q -1 r 2 (q + q j -2 -2r)! f k ⊗ r+1 g j k 2 = 1 q 2 q j r=1 ((r -1)!) 2 q j -1 r -1 2 q -1 r -1 2 (q + q j -2r)! f k ⊗ r g j k 2 . ( 30
)
Suppose first that q > q j . We estimate the norm of the symmetrical tensor product.

f k ⊗r g j k 2 f k ⊗ r g j k 2 = f k ⊗ r g j k , f k ⊗ r g j k = f k ⊗ q-r f k , g j k ⊗ q j -r g j k ≤ f k ⊗ q-r f k • g j k ⊗ q j -r g j k f k ⊗ q-r f k • g j 2 ≤ E Y j k 2 q j ! f k ⊗ q-r f k .
Hence, we obtain :

E D(-L) -1 X k , DY j k 2 E Y j k 2 q 2 q j ! q j r=1 ((r -1)!) 2 q j -1 r -1 2 q -1 r -1 2 (q + q j -2r)! f k ⊗ q-r f k .
Then, if for every j, q > q j , then by point 2. in Theorem 2 and the second condition of (26), the whole expression converges to zero as k → ∞ , and so we get (27). Suppose now that q = q j . We isolate the term r = q j = q in 30, giving :

E D(-L) -1 X k , DY j k 2 = 1 q 2 q-1 r=1 ((r -1)!) 2 q -1 r -1 4 (2(q -r))! f k ⊗ r g j k 2 + ((q -1)!) 2 f k ⊗ q g j k 2 = 1 q 2 E X k Y j k 2 + 1 q 2 q-1 r=1 ((r -1)!) 2 q -1 r -1 4 (2(q -r))! f k ⊗ r g j k 2
.

By both conditions in (26), and by point 2. of Theorem 2, we also conclude in this case that this expectation goes to zero, and so we have (27).

Remark 2. The assumption (26) is necessary to get the result in Theorem 3 and it cannot be avoided. To argue this, consider the following trivial example. Let (W (h), h ∈ H) be an isonormal process and let h

1 , h 2 ∈ H with h 1 H = h 2 H = 1 and h 1 , h 2 H = ρ ∈ (0, 1). Define X = W (h 1 ) 2 -1 = I 2 (h ⊗2 1 ) and Y = W (h 2 ).
Then X ∼ F (1), Y ∼ N (0, 1) and all the assumptions in the statement of Theorem 3,except (26), are satisfied. Notice that

h ⊗2 1 ⊗ 1 h 2 = ρh 1 and h ⊗2 1 ⊗ 1 h 2 H = ρ h 1 H = ρ = 0 so (26)
does not hold true. On the other hand, the components of the vector (X, Y ) are not independent since h ⊗2 1 ⊗ 1 h 2 does not vanish almost everywhere on H, see [START_REF] Üstünel | On independence and conditioning on Wiener space[END_REF].

Examples

We illustrate the results stated in Section 4 by two examples. In these examples we consider a two -dimensional sequence of random variables which has on its first component the sequence U n defined below by (31) and which converges in law to F (1). On the second component, we first consider a fixed random variables in the second Wiener chaos and then another sequence, correlated with U n , which also converges to the centered Gamma law F (1). We obtain the joint convergence of the two-dimensional sequence to a vector with independent components and we derive the associate rate of convergence under the d 2 -distance.

Example 1

Let (h i , i ≥ 1) be orthonormal elements of the Hilbert space H. For n ≥ 2, set

U n = I 2   2 n -1 1≤i<j≤n h i ⊗hj   =: I 2 (f n ), ( 31
)
where I 2 is the multiple integral with respect to an isonormal process (W (h), h ∈ H). Then the sequence (U n , n ≥ 1) converges in distribution to the centered Gamma law F (1) (see e.g. [START_REF] Arras | A bound on the 2-Wasserstein distance between linear combinations of independent random variables[END_REF]). It follows from Section 4 in [START_REF] Azmoodeh | Optimal Gamma approximation on Wiener space[END_REF] that for n large enough,

E 2(U n + 1) -D(-L) -1 U n , DU n 2 C n . ( 32
)
In particular,

d 2 (U n , F (1)) C √ n . ( 33
)
We regard the asymptotic behavior of the two-dimensional sequence (U n , G) where

G := 2H 2 (W (h 1 )) = I 2 h ⊗2 1 .
Let us check (26). Concerning the first part of (26), we have for every n ≥ 1,

E [U n G] = 4 n -1 1≤i<j≤n h i ⊗h j , h ⊗2 1 H ⊗2 = 2 n -1 1≤i<j≤n h i ⊗ h j + h j ⊗ h i , h ⊗2 1 H ⊗2 = 4 n -1 1≤i<j≤n 1 {i=j=1} = 0. 20 Also, f n ⊗ 1 h ⊗2 1 = 2 n -1 1≤i<j≤n (h i ⊗h j ) ⊗ 1 h ⊗2 1 = 1 n -1 1≤i<j≤n (h i ⊗ h j + h j ⊗ h i ) ⊗ 1 h ⊗2 1 = 1 n -1 1≤i<j≤n (h j ⊗ h 1 )1 {i=1} + (h i ⊗ h 1 )1 {j=1} = 1 n -1 n j=2 h j ⊗ h 1 (34)
and

f n ⊗ 1 h ⊗2 1 2 H ⊗2 = 2 (n -1) 2 2 j,k n h j ⊗ h 1 , h k ⊗ h 1 H ⊗2 = 2 (n -1) 2 2 j,k n h j , h k H = 2(n -1) (n -1) 2 ---→ n→∞ 0. ( 35
)
It then follows from Theorem 3 that for n large enough,

d 2 P (Un,G) , F (1) ⊗ P G C E 2(U n + 1) -D(-L) -1 U n , DU n 2 1 2 + E D(-L) -1 U n , DG 2 1 2 . (36)
For any two multiple integrals

X = I 2 (f ), Y = I 2 (g) with f, g ∈ H ⊙2 , E D(-L) -1 X, DY 2 = E[XY ] 2 + 8 f ⊗ 1 g 2 H ⊗2 .
Thus, from (34) and ( 35),

E D(-L) -1 U n , DG 2 = 16 n -1 . ( 37
)
By plugging the estimates (32) and (37) into (36), we obtain for n sufficiently large,

d 2 P (Un,G) , F (1) ⊗ P G C √ n .

Example 2

Let H = L 2 (R + ) and for i ≥ 1 set

h i := 1 [2i,2i+1] .
Then (h i , i ≥ 1) are orthonormal elements in H. Consider the sequence (U n , n ≥ 1) given by (31). Now define for i ≥ 1,

g i := 1 [2i-1+ 1 i ,2i+ 1 i a ]
with a > 0. The family (g i , i ≥ 1) is also orthogonal in H. Notice that for every i, k 1,

h i , g k H = 1 i a 1 {i=k} .
We consider the sequence (V n , n ≥ 1) given by

V n = I 2   2 n -1 1≤i<j≤n g i ⊗g j   =: I 2 (ϕ n ). ( 38
)
We know that both (U n , n ≥ 1) (defined at previous example) and (V n , n ≥ 1) converge to the centered Gamma law F (1). We consider the two-dimensional sequence ((U n , V n ), n ≥ 1) and we prove that it converges in law to the vector (F 1 , F 2 ), where F 1 , F 2 ∼ F (1) and F 1 , F 2 are independent. We will also deduce the rate of convergence under the d 2 -distance associated to this limit theorem. We compute the quantities in (26).

E [U n V n ] = 4 (n -1) 2 1≤i<j≤n 1≤k<l≤n h i ⊗h j , g k ⊗g l = 1 (n -1) 2 1≤i<j≤n 1≤k<l≤n h i ⊗ h j + h j ⊗ h i , g k ⊗ g l + g l ⊗ g k = 1 (n -1) 2 1≤i<j≤n 1≤k<l≤n 2 i a j a 1 {i=k,j=l} + 1 {i=l,j=k} = 4 (n -1) 2 1≤i<j≤n 1 (ij) a ≤ 4 (n -1) 2 n i=1 1 i a 2 ≤ C n 2a ---→ n→∞ 0. ( 39
)
Next, let us calculate f n ⊗ 1 ϕ n 2 H ⊗2 . We compute first the contraction :

f n ⊗ 1 ϕ n (40) = 4 (n -1) 2 1≤i<j≤n 1≤k<l≤n (h i ⊗h j ) ⊗ 1 (g k ⊗g l ) = 1 (n -1) 2 1≤i<j≤n 1≤k<l≤n ˆR+ [h i (x)h j (u) + h j (x)h i (u)] [g k (y)g l (u) + g l (y)g k (u)] du = 1 (n -1) 2 1≤i<j≤n 1≤k<l≤n h i ⊗ g k 1 {j=l} + h i ⊗ g l 1 {j=k} + h j ⊗ g k 1 {i=l} + h j ⊗ g l 1 {i=k} = 1 (n -1) 2     1 i<j n 1 k<j n h i ⊗ g k + 1≤i<j<l≤n h i ⊗ g l + 1≤k<i<j≤n h j ⊗ g k + 1≤i<j≤n 1≤i<l≤n h j ⊗ g l     (41)
and for n large enough,

|f n ⊗ 1 ϕ n | ≤ C n 1≤i,k≤n h i ⊗ g k
and then

f n ⊗ 1 ϕ n 2 H ⊗2 C n 2 1≤i,k,i ′ ,k ′ ≤n h i ⊗ g k , h i ′ ⊗ g k ′ H ⊗2 = C n 2 1≤i,k≤n 1 i a k a ≤ C n 2a ---→ n→∞ 0.
Consequently, from (39) and (41) it follows that (26) is verified and ((U n , V n ), n ≥ 1) converges in law, as n → ∞, to F (1) ⊗ F (1) and

E D(-L) -1 U n , DV n 2 C n 2a . ( 42
)
By Theorem 3,

d 2 P (Un,Vn) , F (1) ⊗ F (1) CE 2(U n + 1) -D(-L) -1 U n , DU n H 2 1 2 + CE D(-L) -1 U n , DU n 2 H 1 2 + d 2 (P Vn , F (1)).
The estimates (32), ( 33) and (42) give, for n large,

d 2 P (Un,Vn) , F (1) ⊗ F (1) ≤ C n -1 2 + n -a .

Appendix

Here we present the basics of Malliavin calculus and the proof of Lemma 3.

Wiener-Chaos and Malliavin derivatives

Here we describe the elements from stochastic analysis that we will need in the paper. Consider H a real separable Hilbert space and (W (h), h ∈ H) an isonormal Gaussian process on a probability space (Ω, A, P), which is a centered Gaussian family of random variables such that E [W (ϕ)W (ψ)] = ϕ, ψ H . Denote by I n the multiple stochastic integral with respect to W (see [START_REF] Nualart | Malliavin Calculus and Related Topics[END_REF]). This mapping I n is actually an isometry between the Hilbert space H ⊙n (symmetric tensor product) equipped with the scaled norm

of degree n ∈ N ∀x ∈ R, H n (x) = (-1) n n! exp x 2 2 d n dx n exp - x 2 2 .
The isometry of multiple integrals can be written as follows: for m, n positive integers,

E [I n (f )I m (g)] = n! f , g H ⊗n if m = n, E [I n (f )I m (g)] = 0 if m = n. ( 43
)
It also holds that

I n (f ) = I n f
where f denotes the symmetrization of f defined by the formula

f (x 1 , . . . , x n ) = 1 n! σ∈Sn f (x σ(1) , . . . , x σ(n) ).
We recall that any square integrable random variable which is measurable with respect to the σ-algebra generated by W can be expanded into an orthogonal sum of multiple stochastic integrals

F = ∞ n=0 I n (f n ) ( 44
)
where f n ∈ H ⊙n are (uniquely determined) symmetric functions and

I 0 (f 0 ) = E [F ].
Let L be the Ornstein-Uhlenbeck operator

LF = - n≥0 nI n (f n ) if F is given by (44) and it is such that ∞ n=1 n 2 n! f n 2 H ⊗n < ∞. The pseudo-inverse of L, denoted (-L) -1 , is given by, (-L) -1 (F ) = n≥1 1 n I n (f n )
for F as in ( 44) with E[F ] = 0. For p > 1 and α ∈ R we introduce the Sobolev-Watanabe space D α,p as the closure of the set of polynomial random variables with respect to the norm

F α,p = (I -L) α 2 F L p (Ω)
where I represents the identity. We denote by D the Malliavin derivative operator that acts on smooth functions of the form F = g(W (h 1 ), . . . , W (h n )) (g is a smooth function with compact support and

h i ∈ H) DF = n i=1 ∂g ∂x i (W (h 1 ), . . . , W (h n ))h i .
The operator D is continuous from D α,p into D α-1,p (H) . We will intensively use the product formula for multiple integrals. It is well-known that for f ∈ H ⊙n and g ∈ H ⊙m

I n (f )I m (g) = n∧m r=0 r! n r m r I m+n-2r (f ⊗ r g) ( 45
)
where f ⊗ r g means the r-contraction of f and g (see e.g. Section 1.1.2 in [START_REF] Nualart | Malliavin Calculus and Related Topics[END_REF]). We also need to introduce the Skorohod integral integral (or the divergence operator), denoted by δ, which is the adjoint operator of D. Its domain is

Dom(δ) = u ∈ L 2 (Ω; H) , E [| DF, u H |] ≤ C F L 2 (Ω)
and we have the duality relationship

∀F ∈ S, ∀u ∈ Dom(δ), E [F δ(u)] = E [ DF, u H ] . ( 46
)

Proof of Lemma 3

Proof: Let us first prove point 1. Concerning the denominators in [START_REF] Nourdin | Normal Approximations with Malliavin Calculus From Stein's Method to Universality[END_REF], notice that their new expressions come from the following equality easily derived by differentiating the left sides (and by being careful with the bounds which change signs) :

2(x + ν)p ν (x) = -´x -ν up ν (u) du on (-ν, +∞) 2(x + ν)q ν (x) = ´-ν x uq ν (u) du on (-∞, -ν). (47)
We will only focus on the numerator part.

• Assume x > -ν. Remember that we have, for every y ∈ R n , f h (x, y) = 1 (x + ν)p ν (x) ˆx -ν p ν (u) h(u, y) du, where h(u, y) = h(u, y) -E[h(Z ν , y)]. We will prove the expression ( 16) in two steps. First, we will use Fubini's theorem to have an expression of h in terms of ∂h ∂x and then we use again Fubini's theorem to get the desired formula for f h .

Let us Fubini's theorem for h by writing the expectation as :

By combining (47) and (49), we obtain the expression [START_REF] Pimentel | Integration by parts and the KPZ two-point function[END_REF].

• Assume x < -ν. From ( 14), we have f h (x, y) = -1 (x + ν)q ν (x) ˆ-ν x q ν (u) h(u) du.

Since the integral is on (x, -ν), we cannot use the expression of h derived on (-ν, +∞). In fact, we need to compute it again, on (-∞, -ν) this time.

Let u < -ν. We write h(u, y) = -

  y)xf (x, y) = h(x, y) -E[h(Z ν , y)],

1 - 1 - 1 -

 111 , y)p ν (z) dz. Hence, we have for every u > -ν, we can write h(u, y) = ˆ+∞ -ν p ν (z)(h(u, y)h(z, y)) dz = y) dw dz.The last line allows us to switch the integrals. We get :h(u, y) = ˆu -ν ∂h ∂x (w, y) ˆw -ν p ν (z) dz dw -F ν (w)) dw. (48)We now plug this last equality into[START_REF] Nourdin | Normal Approximations with Malliavin Calculus From Stein's Method to Universality[END_REF] to have our expression.(x + ν)p ν (x)f h (x, y) = ˆx -ν p ν (u) ˆu -ν ∂h ∂x (w, y)F ν (w) dw du -ˆx -ν p ν (u) ˆ+∞ u ∂h ∂x (w, y)(1 -F ν (w)) dw du.The first term can have its integrals switched, whereas for the second one, we need to write ν)p ν (x)f h (x, y) = ˆx -ν ∂h ∂x (w, y)F ν (w) ˆx w p ν (u) du dw -ˆx -ν ∂h ∂x (w, y)(1 -F ν (w)) ˆw -ν p ν (u) du dw -F ν (w)) ˆx -ν p ν (u) du dw.In terms of F ν , it writes(x + ν)p ν (x)f h (x, y) = ˆx -ν ∂h ∂x (w, y)F ν (w)(F ν (x) -F ν (w)) dw -ˆx -ν ∂h ∂x (w, y)(1 -F ν (w))F ν (w) dw -F ν (w))F ν (x) dw = -(1 -F ν (x)) ˆx -ν ∂h ∂x (w, y)F ν (w) dw -F ν (x)ˆ+∞x ∂h ∂x (w, y)(1 -F ν (w)) dw.

1 -( 1 -x

 11 -ν , we use Fubini's theorem to have h(u, y) =y)(1 -F ν (w)) dw.And again, we plug this expression into the expression[START_REF] Nourdin | Normal Approximations with Malliavin Calculus From Stein's Method to Universality[END_REF] of f h , we find-(x + ν)q ν (x)f h (x, y) = -F ν (w)) dw du.The minus signs cancel. For both terms, both integrals can be switched by Fubini's theorem without using any Chasles relation this time. We get(x + ν)q ν (x)f h (x, y) y)(1 -F ν (w)) dw = ˆ-ν x ∂h ∂x (w, y) Fν (x) -Fν (w) dw + Fν (x) ˆ+∞ -ν ∂h ∂x (w, y)(1 -F ν (w)) dw, so16) is proven. We now prove 2. (actually the proof is slightly adapted from[START_REF] Döbler | New developments in Stein's method with applications[END_REF]). Let x ∈ R. Thenˆx -∞ F ν (s) ds = ˆx -ν ˆs -ν p ν (u) du ds = ˆx -ν p ν (u) ˆx u ds du = xF ν (x) -ˆx -ν up ν (u) du.We proceed in a similar way for the second equality and we getˆ+∞ x F ν (s)) ds = ˆ+∞ x ˆ+∞ s p ν (u) du ds = ˆ+∞ x p ν (u) ˆu x dw du = ˆ+∞ x up ν (u)x(1 -F ν (x)).Concerning the point 3. in the statement, we use the same idea as before, uq ν (u) dux Fν (x).

  H ⊗n and the Wiener chaos of order n which is defined as the closed linear span of the random variables H n (W (h)) where h ∈ H, h H = 1 and H n is the Hermite polynomial

	1 √ n! •