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

where, for every v = (v 1 ; :::; v n ) 2 R n ,
Z "
1
nX
1
S(v) =
0 2 p t E @g
(v) @g ( p tv + p #
1 tV ) dt; (9.6)
@v
i=1 i @v i
so that, for instance, for N N (0; 1) and by using (8.7), Lemma 8.1 (iii), (8.5) and Cauchy-
Schwarz inequality,
d T V (Y; Z) 2E[(S(V ) 1) 2 ] 1=2 : (9.7)
We shall prove that (9.5) is a very special case of the …rst equality in (7.30), and therefore that
(9.7) is a special case of (9.3). Observe …rst that, without loss of generality, we can assume that
V i = X(h i ), where X is an isonormal process over some Hilbert space of the type H = L 2 (Z; Z; )
and fh 1 ; :::; h n g is an orthonormal system in H. Since Y = g(V 1 ; : : : ; V n ), according to (7.3)
we have that D a Y = P n @g
i=1 @x i
(V )h i (a). On the other hand, since Y is centered and square
integrable, it admits a chaotic representation of the form Y = P q1 I q( q ). This implies in
particular that D a Y = P 1
q=1 qI q 1( q (a; )). Moreover, one has that L 1 Y = P q1 1 q I q( q ),
so that D a L 1 Y = P q1 I q 1( q (a; )). Now, let T z , z 0, denote the Ornstein-Uhlenbeck
semigroup introduced in (7.21), whose action on random variables F 2 L 2 ( (X)) is given by
T z (F ) = P q0 e qz J q (F ), where J q (F ) denotes the projection of F on the qth Wiener chaos.
We can write
Z 1
0
1
2 p t T ln(1= p t) (D aY )dt =
Z 1
0
e z T z (D a Y )dz = X q1
1
q J q 1(D a Y )
= X q1
I q 1 ( q (a; )) = D a L 1 Y: (9.8)
Now recall that Mehler’s formula (7.24) implies that
T z (f(V )) = E[f(e z v + p 1 e 2z V )] ; z 0:
v=V
In particular, by applying this last relation to the partial derivatives @g
@v i
, i = 1; :::; n, we deduce
from (9.8) that
Z 1
0
1
2 p t T ln(1= p t) (D aY )dt =
nX
i=1
Z 1
1 @g
h i (a)
0 2 p t E ( p t v + p
1 t V ) dt
@v i
:
v=V
Consequently, (9.5) follows, since
* nX
hDY; DL 1 @g
nX
Y i H = (V )h i ;
@v
i=1 i
= S(V ):
i=1
Z 1
0
1 @g
2 p t E ( p t v + p
1 t V ) dt
@v i
+
h i
v=V
H
Remark. By inspection of the proof of Theorem 9.1, one sees that it is possible to re…ne
the bounds (9.1)–(9.3) and (9.4), by replacing DF; DL 1 F h DF;
H with E DL 1 F i
H
j F DF;
and Var DL 1 F h DF;
with Var E
H
DL 1 F i
H j F .
45