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

We also set
D 1;2 =
1\
D k;2 (7.9)
k=1
Remark. One can actually de…ne more general domains D k;p , p 1, as the closure of D k
in L p ( (X)). See [65, pp. 26–27].
The following result provides an important characterization of D k;2 , namely that F 2 D k;2
if and only if the norms of its chaotic projections decrease su¢ cienlty fast. The proof can be
found in [65, Proposition 1.2.2], and uses the representation (6.5).
Proposition 7.2 Fix k 1. A random variable F 2 L 2 ( (X)) with a chaotic representation
(6.6) is an element of D k;2 if and only if the kernels ff q g verify
1X
q k q! kf q k 2 H
< 1, (7.10)
q
q=1
and in this case
E D k F 2 = X
1 (q) k
q! kf q k 2
H k H
, q
q=k
where (q) k
= q (q 1) (q k + 1) is the Pochammer symbol.
Note that the previous result implies that random variables belonging to a …nite sum of
Wiener chaoses are in D k;2 , for every k 1 (they are actually in D k;p , for every k; p 1)
7.1.2 The case of Gaussian measures
We now focus on the case where H = L 2 (Z; Z; ), so that each symmetric power H q can be
identi…ed with the space of symmetric functions L 2 s ( q ), and the integrals I q (f), f 2 L 2 s ( q ), are
just (multiple) Wiener-Itô integrals of f with respect to the Gaussian measure G (A) = X (1 A ),
where (A) < 1:
As already observed in this case the derivative D k F of F 2 D k;2 takes the form of a stochastic
process
(z 1 ; :::; z k ) 7! D k z 1 ;:::;z k
F ,
verifying moreover
2
E Dz ZZ k 1 ;:::;z k
F (dz1 ; :::; dz k ) < 1:
k
The following statement provides a neat algorithm allowing to deduce the explicit form of the
…rst derivative of a general random variable in D 1;2 : This result can be proved by …rst focusing
on smooth random variables of the type (4.11), and then by using (7.3) as well as a density
argument. Note that a similar statement (that one can deduce by recursion –Exercise!) holds
also for derivatives of order greater than 1.
27