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

(i) For every 1 i d, F (n)
i
converges in distribution to a centered Gaussian random variable
with variance C(i; i).
h i
(ii) For every 1 i d, E (F (n)
i
) 4 ! 3C(i; i) 2 .
(iii) For every 1 i d and every 1 r q i
1, kf (n)
i
r f (n)
i
k H 2(q i r) ! 0.
(iv) The vector F (n) converges in distribution to a d-dimensional Gaussian vector N d (0; C).
Moreover, if C(i; j) = ij , where ij is the Kronecker symbol, then either one of conditions
(i)–(iv) above is equivalent to the following:
(v) For every 1 i d, kDF (n)
i
k 2 H
L 2 ! q i .
Remark. The crucial implication in the statement of Theorem 10.2 is (i) ) (iv), yielding
that, for random vectors composed of chaotic random variables and verifying the asymptotic
covariance condition (10.14), componentwise convergence in distribution towards a Gaussian
vector always implies joint convergence. This fact is extremely useful for applications: see for
instance [3], [33], [48], [54] and [55].
We conclude this section by pointing out the remarkable fact that, for vectors of multiple
Wiener-Itô integrals of arbitrary length, the Wasserstein distance metrizes the weak convergence
towards a Gaussian vector with positive de…nite covariance. Once again, this result is not trivial,
since the topology induced by the Wasserstein distance is stronger than the topology of weak
convergence.
Proposition 10.3 (See [60]) Fix d 2, let C be a positive de…nite d d symmetric matrix,
and let 1 q 1 : : : q d . Consider vectors
F (n) = (F (n)
1 ; : : : ; F (n)
d
) = (I q1 (f (n)
1 ); : : : ; I qd (f (n)
d
)); n 1;
with f (n)
i
2 H q i
for every i = 1 : : : ; d. Assume moreover that F (n) satis…es condition (10.14).
Then, as n ! 1, the following three conditions are equivalent:
(a) d W (F (n) ; Z) ! 0.
(b) For every 1 i d, q 1
i
kDF (n)
i
k 2 H
L 2 ! C(i; i) and, for every 1 i 6= j d,
hDF i ; DL 1 F j i H = q 1
j
hDF i ; DF j i H
L 2 ! C(i; j):
(c) F (n) converges in distribution to Z N d (0; C).
Proof. Since convergence in the Wasserstein distance implies convergence in distribution,
the implication (a) ! (c) is trivial. The implication (b) ! (a) is a consequence of relation
(10.10). Now assume that (c) is veri…ed, that is, F (n) converges in law to Z N d (0; C) as n
goes to in…nity. By Theorem 10.2 we have that, for any i 2 f1; : : : ; dg and r 2 f1; : : : ; q i 1g,
kf (n)
i
r f (n)
i
k H 2(q i r) !
n!1
0:
By combining Corollary 10.2 with Lemma 10.1, one therefore easily deduces that, since (10.14)
is in order, condition (b) must necessarily be satis…ed.
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