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

D4 The application of the DDS Theorem and of the BDG inequality requires an explicit underlying
(Brownian) martingale structure. Although it is always possible to represent a
given Gaussian …eld in terms of a Brownian motion, this operation is often quite unnatural
and can render the asymptotic analysis very hard. For instance, what happens if one
considers quadratic functionals of a multiparameter Gaussian process, or of a Gaussian
process which is not a semimartingale (for instance, a fractional Brownian motion with
Hurst parameter H 6= 1=2)? See [71] for some further applications of random time-changes
in a general Gaussian setting.
D5 It is not clear whether this approach can be used in order to deal with expressions of the
type (2.15), when h is not Lipschitz (for instance, when h equals the indicator of a Borel
set), so that it seems di¢ cult to use these techniques in order to assess other distances,
like the total variation distance or the Kolmogorov distance.
Starting from the next section, we will describe the main objects and tools of stochastic
analysis that are involved in our techniques.
3 Gaussian measures
Let (Z; Z) be a Polish space, with Z the associated Borel -…eld, and let be a positive -…nite
measure over (Z; Z) with no atoms (that is, (fzg) = 0, for every z 2 Z). We denote by Z the
class of those A 2 Z such that (A) < 1. Note that, by -additivity, the -…eld generated by
Z coincides with Z.
De…nition 3.1 A Gaussian measure on (Z; Z) with control is a centered Gaussian family
of the type
G = fG (A) : A 2 Z g , (3.1)
verifying the relation
E [G (A) G (B)] = (A \ B) , 8A; B 2 Z . (3.2)
The Gaussian measure G is also called a white noise based on .
Remarks. (a) A Gaussian measure such as (3.1)–(3.2) always exists (just regard G as a
centered Gaussian process indexed by Z , and then apply the usual Kolmogorov criterion).
(b) Relation (3.2) implies that, for every pair of disjoint sets A; B 2 Z , the random variables
G (A) and G (B) are independent. When this property is veri…ed, one usually says that G is
a completely random measure (or, equivalently, an independently scattered random measure).
The concept of a completely random measure can be traced back to Kingman’s seminal paper
[38]. See e.g. [39], [72], [92] and [93] for a discussion around general (for instance, Poisson)
completely random measures.
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