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

By using an appropriate version of the so-called Jeulin Lemma (see [36, Lemma 1, p. 44]), one
can prove that
Z 1 Z 1
W (t1 ; :::; t d ) 2
dt 1 dt d = 1, a.s.-P.
t 1 t d
0
0
For " 2 (0; 1), we can now de…ne the random variable
Z 1 Z 1
W
B" d (t1 ; :::; t d ) 2
=
dt 1 dt 2 :
t 1 t d
"
"
A standard computation yields E B" d
= (log 1=") d , and also
Var B"
d (4 log 1=") d , as " ! 0:
By setting
eB " d , Bd " (log 1=") d
,
(4 log 1=") d 2
one can therefore state the following generalization of Problem I, as stated at the end of Section
2.1.
Problem II. Prove that, as " ! 0, e B d "
Law
! N N (0; 1).
Remark. See [21] for applications of quadratic functionals of Brownian sheets to tests of
independence.
11.1.2 Interlude: optimal rates for second chaos sequences
In order to give an exhaustive answer to Problem II, we state (without proof) a result concerning
sequences in the second Wiener chaos of a given isonormal Gaussian process X =
fX (h) : h 2 Hg. It gives a simple criterion (based on cumulants) allowing to determine whether,
for a sequence in the second chaos, the rate of convergence implied by (10.7) is optimal. Note that
the forthcoming formula (11.1) is just a rewriting of (10.7), which we added for the convenience
of the reader. We also use the notation
(z) = P [N z] , where N N (0; 1) :
Proposition 11.1 (See [58]) Let F n = I 2 (f n ), n 1, be such that f n 2 H 2 , and write
(n)
p = p (F n ), p 1. Assume that (n)
2 = E(Fn) 2 ! 1 as n ! 1. Then, as n ! 1,
Law (n)
F n ! N N (0; 1) if and only if 4 ! 0. In this case, we have moreover
s
(n)
4
d Kol (F n ; N)
6 + ((n) 2 1) 2 : (11.1)
If, in addition, we have, as n ! 1,
(n)
2 1
! 0; (11.2)
(n)
4
6
+ ( (n)
2 1) 2
56