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

matrix (i; j) = hf i ; f j i L 2 () , 1 i; j d. If B 2 Z , we write interchangeably G (B) or G (1 B )
(the two objects coincide, thanks to (4.2)). Plainly, the Gaussian family
C 1 (G) = G (f) : f 2 L 2 () (4.6)
coincides with the L 2 (P)-closed linear space generated by G. One customarily says that (4.6) is
the …rst Wiener chaos associated with G. Observe that, if fe i : i 1g is an orthonormal basis of
L 2 (), then fG (e i ) : i 1g is an i.i.d. Gaussian sequence with zero mean and common unitary
variance.
4.2 Multiple integrals
For every n 2, we write L 2 (Z n ; Z n ; n ) = L 2 ( n ) to indicate the Hilbert space of real-valued
functions that are square-integrable with respect to n . Given a function f 2 L 2 ( n ), we denote
by e f its canonical symmetrization, that is
ef (z 1 ; :::; z n ) = 1 X
f z
n! (1) ; :::z (n) , (4.7)
where the sum runs over all permutations of the set f1; :::; ng. Note that, by the triangle
inequality,
k e f k L 2 ( n ) kfk L 2 ( n ) : (4.8)
We will consider the following three subsets of L 2 ( n ) :
L 2 s (Z n ; Z n ; n ) = L 2 s ( n ) is the closed linear subspace of L 2 ( n ) composed of symmetric
functions, that is, f 2 L 2 s ( n ) if and only if: (i) f is square integrable with respect to n ,
and (ii) for d n -almost every (z 1 ; :::; z n ) 2 Z n ;
f (z 1 ; :::; z n ) = f z (1) ; :::; z (n)
,
for every permutation of f1; :::; ng.
E ( n ) is the subset of L 2 ( n ) composed of elementary functions vanishing on diagonals,
that is, f 2 E ( n ) if and only if f is a …nite linear combination of functions of the type
(z 1 ; :::; z n ) 7! 1 A1 (z 1 ) 1 A2 (z 2 ) 1 An (z n ) (4.9)
where the sets A i are pairwise disjoint elements of Z .
E s ( n ) is the subset of L 2 s ( n ) composed of symmetric elementary functions vanishing
on diagonals, that is, g 2 E s ( n ) if and only if g = e f for some f 2 E ( n ), where the
symmetrization e f is de…ned according to (4.7).
The following technical result will be used throughout the sequel.
Lemma 4.1 Fix n 2. Then, E ( n ) is dense in L 2 ( n ), and E s ( n ) is dense in L 2 s ( n ).
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