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

so that the conclusion derives once again from Proposition 11.1 and (2.12).
Remark. The example developed in this section shows how the Malliavin/Stein approach
can overcome most of the di¢ culties D1 – D5, that were pointed out at the end of Sections
2.2 and 2.3 in connection with the method of cumulants and with random time-changes. In
particular, one has that
Stein’s method allows in this case to deduce exact rates of convergence in the sense of
the total variation (but also Kolmogorov and Wasserstein) distance. This successfully
addresses D1 and D5.
Law
The convergence B e "
d ! N is now implied by the simple condition (n)
4 ! 0. Moreover,
to obtain lower bounds one must merely verify the three relations at (11.2) and (11.3).
This eliminates the di¢ culty pointed out in D2.
Finally, our techniques allow to deal directly with quadratic functionals of a Brownian
sheet, without making use of any underlying martingale structure. This overcomes the
drawbacks of random time-changes described at D4:
In the next section, we will describe a situation involving non-quadratic transformations
(thus adressing point D3 in the above quoted list).
11.2 Hermite functionals of Ornstein-Uhlenbeck sheets
Let d 1. Let G be a centered Gaussian measure over R d , with control given by the Lebesgue
measure (dx 1 ; :::; dx d ) = dx 1 dx d . Fix > 0, and, for every t = (t 1 ; :::; t d ) 2 R d +, de…ne the
d-variate Ornstein-Uhlenbeck kernel
dY
f t (x) = (2) d 2 expf (t i x i )g1 fxi t i g; x = (x 1 ; :::; x d ) 2 R d : (11.10)
j=1
For every …xed q 2, we consider the qth tensor power of f t , denoted by f q
t , which is a function
on R dq . Now write
Z t (1; d) = I 1 (f t ) , t 2 R d +
h
Note that, for every t = (t 1 ; :::; t d ) ; s = (s 1 ; :::; s d ), one has that E Z t (1; d) 2i = R ft 2 (x)dx =
1 and, more generally,
E [Z t (1; d) Z s (1; d)] =
dY
expf jt j s j jg: (11.11)
j=1
In view of (11.11), the process t 7! Z t (d; 1) is called an Ornstein-Uhlenbeck sheet with d
parameters. It follows form (5.11) that
Z t (q; d) , I q (f q
t ) = H q (Z t (d; 1)) ;
where H q is the qth Hermite polynomial. The main result of this section is the following CLT
for linear functionals of Z (q; d)
58