Stein's method, Malliavin calculus and infinite-dimensional Gaussian

To prove the Wasserstein estimate (9.9), it is su¢ cient to observe that, for every globally Lipschitz
function g such that kgk L 1, there exists a family fg " : " > 0g such that:
(i) for each " > 0, the …rst and second derivatives of g " are bounded;
(ii) for each " > 0, one has that kg " k Lip kgk L ;
(iii) as " ! 0, kg " gk 1 # 0.
For instance, we can choose g " (x) = E g(x + p "S) with S N d (0; I d ).
Theorem 9.2 will be fully exploited in Section 10.2, where we will obtain bounds on the
normal approximation of random vectors woth coordinates living in a …xed Wiener chaos.
9.3 Gamma approximation
We now state a result that can be obtained by combining Malliavin calculus with the Gamma
approximations discussed in the second part of Section 8.2 (we shall use the same notation
introduced therein). The proof (left to the reader) makes use of (7.34), and of arguments
analogous to those displayed in the proof of Theorem 9.1.
Theorem 9.3 Fix > 0 and let F () have a centered Gamma distribution with parameter .
Let G 2 D 1;2 be such that E(G) = 0 and the law of G is absolutely continuous with respect to
the Lebesgue measure. Then:
d G1 (G; F ()) K 1 E[(2 + 2G hDG; DL 1 Gi H ) 2 ] 1=2 ; (9.11)
and, if 1 is an integer,
d G2 (G; F ()) K 2 E[(2 + 2G hDG; DL 1 Gi H ) 2 ] 1=2 ; (9.12)
where G 1 and G 2 are de…ned in (8.11)–(8.12), K 1 , maxf1; 1= + 2= 2 g and K 2 , maxf p 2=;
1= + 2= 2 g.
We will come back to Theorem 9.3 in Section 10.3, where we will present some characterizations
of non-central limit theorems on a …xed Wiener chaos.
10 Limit Theorems on Wiener chaos
Let X = fX (h) : h 2 Hg be an isonormal Gaussian process. In this section, we focus on the
Gaussian and Gamma approximations of (vectors of) random variables of the type F = I q (f),
where q 2 and f 2 H q . We recall that, according to the chaotic representation property
stated in Proposition 6.1-3, random variables of this form are the basic building blocks of every
square-integrable functional of X.
In order to appreciate the subtelty of the issues faced in this section, we list some well-known
properties of the laws of chaotic random variables.
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