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

(ii) Consider the element e (1; 7; 7). In (1; 7; 7) the number 1 is not repeated and 7 is repeated
twice, hence e (1; 7; 7) is associated with the multiindex 2 A 1;3 such that 1 = 1, 7 = 2 and
j = 0 for every j 6= 1; 7, that is, = (1; 0; 0; 0; 0; 0; 2; 0; 0; :::).
(iii) The multindex = (1; 2; 2; 0; 5; 0; 0; 0; :::) is associated with the element of H 10 given
by e (1; 2; 2; 3; 3; 5; 5; 5; 5; 5).
In what follows, given 2 A 1;q (q 1), we shall write e () in order to indicate the element
of H q uniquely associated with .
De…nition 6.2 For every h 2 H, we set I1
X (h) = I 1 (h) = X (h). Now …x an orthonormal basis
fe j : j 1g of H: for every q 2 and every h 2 H q such that
h =
X
c e ()
2A 1;q
(with convergence in H q , endowed with the inner product h; i H q), we set
Iq X (h) = I q (h) = X Y
c H j (X (e j )) , (6.5)
2A 1;q j
where the products only involve the non-zero terms of each multiindex , and H m indicates the
mth Hermite polynomial . For q 1, the collection of all random variables of the type I q (h),
h 2 H q , is called the qth Wiener chaos associated with X and is denoted by C q (X). One
sets conventionally C 0 (X) = R.
Examples. (i) If h = e (), where = (1; 1; 0; 0; 0; :::) 2 A 1;2 , then
I 2 (h) = H 1 (X (e 1 )) H 1 (X (e 2 )) = X (e 1 ) X (e 2 ) .
(ii) If = (1; 0; 1; 2; 0; :::) 2 A 1;4 , then
I 4 (h) = H 1 (X (e 1 )) H 1 (X (e 3 )) H 2 (X (e 4 ))
= X (e 1 ) X (e 3 ) X (e 4 ) 2 1
= X (e 1 ) X (e 3 ) X (e 4 ) 2 X (e 1 ) X (e 3 ) .
(iii) If = (3; 1; 1; 0; 0; :::) 2 A 1;5 , then
I 5 (h) = H 3 (X (e 1 )) H 1 (X (e 2 )) H 1 (X (e 3 ))
= X (e 1 ) 3 3X (e 1 ) X (e 2 ) X (e 3 )
= X (e 1 ) 3 X (e 2 ) X (e 3 ) 3X (e 1 ) X (e 2 ) X (e 3 ) .
The following result collects some well-known facts concerning Wiener chaos and isonormal
Gaussian processes. In particular: the …rst point characterizes the operators Iq
X as isomorphisms;
the third point is an equivalent of the chaotic representation property for Gaussian measures, as
stated in formula (5.12); the fourth point establishes a formal relation between random variables
of the type Iq
X (h) and the multiple Wiener-Itô integrals introduced in Section 4.2 (see [65, Ch.
1] for proofs and further discussions of all these facts).
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