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

(n)
3
q
(n)
4
6
+ ( (n)
2 1) 2 ! and
(n)
8
(n)
4
6
+ ( (n)
2 1) 2 2
! 0; (11.3)
then
P(F n z) (z)
q
! 1
p 1 z 2 e z2 2 ; as n ! 1: (11.4)
(n)
4
6
+ ( (n)
2 1) 2 3! 2
In particular, if 6= 0, there exists c 2 (0; 1) and n 0 1 such that, for any n n 0 ,
s
(n)
4
d T V (F n ; N) d Kol (F n ; N) c
6 + ((n) 2 1) 2 : (11.5)
11.1.3 A general statement
The next result provides an exhaustive solution to Problem II (and therefore to Problem I).
Proposition 11.2 For every d 1, there exist constants 0 < c(d) < C(d) < 1 and 0 < (d) <
1, depending uniquely on d, such that, for every " 2 (0; 1),
d T V [ e B d " ; N] C(d)(log 1=") d=2 (11.6)
and, for " < (d),
d T V [ e B d " ; N] c(d)(log 1=") d=2 : (11.7)
This yields that, as " ! 0, e B d "
Law
! N N (0; 1).
Proof. We denote by
e j (d; "); j = 1; 2; :::;
the sequence of the cumulants of the random variable e B d " . We deal separately with the cases
d = 1 and d 2.
(Case d = 1) As already observed, in this case, W is a standard Brownian motion on [0; 1],
so that e B 1 " takes the form e B 1 " = I 2 (f " ), where I 2 indicates a double Wiener-Itô integral with
respect to W, and
f " (x; y) = (4 log 1=") 1=2 [(x _ y _ ") 1 1]: (11.8)
The conclusion now follows from Proposition 11.1 and (2.12).
(Case d 2) In this case, e B d " has the form e B d " = I 2 (f d " ), with
f d " (x 1 ; :::; x d ; y 1 ; :::; y d ) = (4 log 1=") d=2
dY
[(x j _ y j _ ") 1 1]: (11.9)
By using an appropriate modi…cation of (2.11), one sees that the following relation holds
j=1
(2 j 1 (j 1)!) 1 e j (d; ") = [(2 j 1 (j 1)!) 1 e j (1; ")] d ;
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