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

Proof of Lemma 10.1 (see also [66, Lemma 2]). Without loss of generality, we can
assume that H = L 2 (Z; Z; ), where (A; Z) is a Polish space, and is a -…nite and non-atomic
measure. Thus, we can write
Z
hDF; DGi H = p q hI p 1 (f); I q 1 (g)i H
= p q I p 1 f(; z) I q 1 g(; z) (dz)
It follows that
=
E
"
a
8
><
>:
If r < p q then
Z p^q
X1
p 1
= p q r!
A
r
r=0
p^q
X1
p 1
= p q r!
r
r=0
Xp^q
p 1
= p q (r 1)!
r 1
r=1
#
1
2
q hDF; DGi H
a 2 + p 2 P p
p 1
r=1 (r 1)!2
r 1
q 1
r
Z
q 1
r
q 1
r 1
2 q 1
I p+q 2 2r f(; z)e r g(; z) (dz)
I p+q 2 2r (f e r+1 g)
I p+q
2r (f e r g):
r 1 2(p + q 2r)!kf e r gk 2 H (p+q 2r) if p < q;
(10.13)
(a p!hf; gi H p) 2 + p 2 P p 1
p 1 4(2p
r=1 (r 1)!2
r 1
2r)!kf e r gk 2 if p = q:
H (2p 2r)
kf e r gk 2 H (p+q 2r) kf r gk 2 H (p+q 2r) = hf p r f; g q r gi H 2r
If r = p < q, then
kf p r fk H 2rkg q r gk H 2r
1 2 kf p r fk 2 H 2r + kg q r gk 2 H 2r
:
kf e p gk 2 H (q p) kf p gk 2 H (q p) kfk 2 H p kg q p gk H 2p:
If r = p = q, then f e p g = hf; gi H p: By plugging these last expressions into (10.13), we deduce
immediately the desired conclusion.
The combination of the results presented in this section with Theorem 10.1 lead to the
following statement, which is a collection of the main …ndings contained in the papers by Peccati
and Tudor [73] and Nualart and Ortiz-Latorre [66].
Theorem 10.2 (See [66, 73]) Fix d 2 and let C = fC(i; j) : i; j = 1; :::; dg be a d d
positive de…nite matrix. Fix integers 1 q 1 : : : q d . For any n 1 and i = 1; : : : ; d, let f (n)
i
belong to H q i
. Assume that
F (n) = (F (n)
1 ; : : : ; F (n)
d
) = (I q1 (f (n)
1 ); : : : ; I qd (f (n)
d
)) n 1;
is such that
lim
n!1
(n)
E[F
i
F (n)
j
] = C(i; j); 1 i; j d: (10.14)
Then, as n ! 1, the following four assertions are equivalent:
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