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

Theorem 11.1 Fix > 0 and q 2, and de…ne the positive constant c = c(q; ; d) :=
[2(q 1)!=] d . Then, one has that, as T ! 1,
M T (q; d) = 1 p
cT d
Z T
0
Z T
0
Z t (q; d)dt Law ! N N (0; 1); (11.12)
where t = (t 1 ; :::; t d ) and dt = dt 1 :::dt d , and there exists a …nite constant = (; q; ; d) > 0
such that, for every T > 0,
d W (M T (q; d); N) p : (11.13)
T d
Proof. By an argument similar to the one concluding the proof of Proposition 11.2, it
is enough to prove the theorem in the case d = 1. The crucial fact is that, for each T , the
random variable M T (1; q) has the form of a multiple integral, that is, M T (1; q) = I q (F T ), where
F T 2 L 2 s( q ) is given by
F T (x 1 ; :::; x q ) = 1 p
cT
Z T
0
f q
t (x 1 ; :::; x q )dt:
According to (10.3) and Proposition , both claims (11.12) and (11.13) are proved, once we show
that, as T ! 1, one has that
j1 E(M T (q; d)) 2 j 1=T; (11.14)
and also that
kF T r F T k L 2 ( 2q 2r ) = O(1=T ); 8r = 1; :::; q 1: (11.15)
In order to prove (11.14) and (11.15), for every t 1 ; t 2 0 we introduce the notation
Z
hf t1 ; f t2 i = f t1 (x)f t2 (x) dx = e (t 1+t 2 ) e 2(t 1^t 2 ) : (11.16)
R
To prove (11.14), one uses the relation (11.16) to get
E M T (1; q) 2 = q!
cT
Z T Z T
0
0
hf t1 ; f t2 i q dt 1 dt 2 = 1
1
T q (1 e qT ):
In the remaining of the proof, we will write in order to indicate a strictly positive …nite constant
independent of T , that may change from line to line. To deal with (11.15), …x r = 1; :::; q 1
and use the fact
= 1
cT
and therefore
F T r F T (w 1 ; :::; w q r ; z 1 ; :::; z q r )
Z T Z T
0
0
kF T r F T k 2 L 2 ( 2q 2r )
= T 2 Z T
0
Z T Z T Z T
0
0
0
(f t1 (w 1 ) f t1 (w q r )f t2 (z 1 ) f t2 (z q r )) hf t1 f t2 i r dt 1 dt 2 ;
hf t1 f t3 i q r hf t2 f t4 i q r hf t1 f t2 i r hf t3 f t4 i r dt 1 dt 2 dt 3 dt 4
T ; 59