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

where the last relation is obtained by resorting to the explicit representation (11.16), and then by
evaluating the restriction of the quadruple integral to each simplex of the type ft (1) > t (2) >
t (3) > t (4) g, where is a permutation of the set f1; 2; 3; 4g.
12 Further readings
The content of the previous sections is mainly related to the papers [57] and [60], dealing
with one- and multi-dimensional upper bounds in the Gaussian and Gamma approximations of
functionals of Gaussian …elds. In the following list, we shall provide a short description of some
further works that have been written on related subjects.
In [58] it is described how one can once again combine Stein’s method with Malliavin
calculus, in order to detect optimal rates of convergence for sequences of functionals of
Gaussian …elds. Given a sequence fF n g of such functionals and given N N (0; 1), we say
that a sequence of positive numbers ' (n) & 0 provides an optimal rate of convergence, if
there exists a constant 0 < c < 1, such that
c < d (F n; N)
' (n)
1
for n large enough (where d is some suitable distance between the law of F n and the law
of N). Proposition 11.1 gives an example of such a situation.
In [64] one can …nd applications of (7.30) to the estimation of densities and tail probabilities
associated with smooth functionals of Gaussian processes. In particular, a new formula for
the density of a regular functional “with full support”is derived. Part of the computations
performed in this paper are related to the theory developed by Ch. Stein in [89, Chapter
VI].
The paper [97] contains new estimates for tail bounds based on (7.30). The results are
also related to polymer models.
The paper [62] contains the proof of new second-order Poincaré inequalities, involving the
operator norm of the second derivative of a given smooth random variable. This gives a
generalization of a class of inequalities proved by Chatterjee in [9].
In [70] one can …nd an extension of the theory developed in [57] to the case of the Gaussian
approximation of functionals of Poisson random measures. This is based on an appropriate
version of Malliavin calculus on the Poisson space, and is related to the work by Decreusefond
and Savy [17].
The paper [61] provides an extension of the Malliavin/Stein approach to deal with functionals
of in…nite Rademacher sequences. The necessary discrete Malliavin operators are
discussed in [79].
60