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

Lemma 8.1
(i) Let W be a random variable. Then, W Law = N N (0; 1) if, and only if,
E[f 0 (W ) W f(W )] = 0; (8.4)
for every continuous and piecewise continuously di¤erentiable function f verifying the
relation Ejf 0 (Z)j < 1.
(ii) If g(x) = 1 ( 1;z] (x), z 2 R, then (8.3) admits a solution f which is bounded by p 2=4,
piecewise continuously di¤erentiable and such that kf 0 k 1 1.
(iii) If g is bounded by 1/2, then (8.3) admits a solution f which is bounded by p =2, Lebesgue
a.e. di¤erentiable and such that kf 0 k 1 2.
(iv) If g is absolutely continuous with bounded derivative, then (8.3) has a solution f which is
twice di¤erentiable and such that kf 0 k 1 kg 0 k 1 and kf 00 k 1 2kg 0 k 1 .
We also recall the relation:
2d T V (X; Y ) = supfjE(u(X)) E(u(Y ))j : kuk 1 1g: (8.5)
Note that point (ii) and (iii) (via (8.5)) imply the following bounds on the Kolmogorov and total
variation distance between Z and an arbitrary random variable Y :
d Kol (Y; Z) sup
f2F Kol
jE(f 0 (Y ) Y f(Y ))j (8.6)
d T V (Y; Z) sup
f2F T V
jE(f 0 (Y ) Y f(Y ))j (8.7)
where F Kol and F T V are, respectively, the class of piecewise continuously di¤erentiable functions
that are bounded by p 2=4 and such that their derivative is bounded by 1, and the class of
piecewise continuously di¤erentiable functions that are bounded by p =2 and such that their
derivative is bounded by 2.
Analogously, by using (v) along with the relation khk L = kh 0 k 1 , one obtains
d W (Y; Z) sup
f2F W
jE(f 0 (Y ) Y f(Y ))j; (8.8)
where F W is the class of twice di¤erentiable functions, whose …rst derivative is bounded by 1
and whose second derivative is bounded by 2.
Centered Gamma distribution. Let F () be as in (8.2). Consider a real-valued function
g : R ! R such that the expectation E[g(F ())] exists. The Stein equation associated with g
and F () is:
g(x) E[g(F ())] = 2(x + )f 0 (x) xf(x); x 2 ( ; +1): (8.9)
The following statement collects some slight variations around results proved by Stein [89],
Diaconis and Zabell [20], Luk [40], Schoutens [85] and Pickett [94]. See also [80]. It is the
“Gamma counterpart”of Lemma 8.1.
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