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

obtained by juxtaposing n copies of the kernel f " . By plugging (2.8) into (2.11), and after some
lengthy (but standard) computations, one obtains that, as " ! 0,
eB" n c n log 1 1
n
2
, for every n 3, (2.12)
"
where c n > 0 is a …nite constant independent of ". This yields (2.10) and therefore (2.5). The
implication (2.12) ) (2.10) ) (2.5) is a typical application of the method of cumulants to the
proof of Central Limit Theorems (CLTs) for functionals of Gaussian …elds. In particular, one
should note that (2.11) can be equivalently expressed in terms of diagram formulae.
In the following list we pinpoint some of the main disadvantages of this approach. As already
discussed, all these di¢ culties disappear when using the Stein/Malliavin techniques developed
in Section 9 below.
D1 Formulae (2.11) and (2.12) characterize the speed of convergence to zero of the cumulants
of B e " . However, there is no way to deduce from (2.12) an estimate for quantities of the
type d eB" ; N , where d indicates some distance between the law of B" e and the law of
N (d can be for instance the total variation distance, or the Wasserstein distance – see
Section 8.1 below) 2 .
D2 Relations (2.10) and (2.11) require that, in order to prove the CLT (2.5), one veri…es an
in…nity of asymptotic relations, each one involving the estimate of a multiple deterministic
integral of increasing order. This task can be computationally quite demanding. Here,
(2.12) is obtained by exploiting the elementary form of the kernels f " in (2.8).
D3 If one wants to apply the method of cumulants to elements of higher chaoses (for instance,
by considering functionals involving Hermite polynomials of degree greater than 3), then
one is forced to use diagram formulae that are much more involved than the Fox-Taqqu
formula (2.11). Some examples of this situation appear e.g. in [5], [27] and [46]. See [72,
Section 3 and Section 7] for an introduction to general diagram formulae for non-linear
functionals of random measures.
2.3 Random time-changes
This technique has been used in [74] and [75]; see also [71] and [96] for some analogous results
in the context of stable convergence.
Our starting point is once again formula (2.7), implying that, for each " 2 (0; 1), the random
variable B e " coincides with the value at the point t = 1 of the continuous Brownian martingale
t 7! M " t = 2
Z t Z s
0
0
f " (s; u) dW u dW s , t 2 [0; 1] . (2.13)
It is well-known that the martingale M t
" has a quadratic variation equal to
Z t
Z s
2
hM " ; M " i t
= 4 f " (s; u) dW u ds; t 2 [0; 1] :
0
0
2 This assertion is not accurate, although it is kept for dramatic e¤ect. Indeed, we will show in Section 10.2
that the combination of Stein’s method and Malliavin calculus exactly allows to deduce Berry-Esséen bounds
from estimates on cumulants.
7