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

10.3 A non-central limit theorem (with bounds)
We now present (without proofs) two statements concerning the Gamma approximation of multiple
integrals of even order q 2. The …rst result, which is taken from [57], provides an explcit
representation for the quantities appearing on the RHS of (9.11) and (9.12).
Proposition 10.4 (See [57]) Let q 2 be an even integer, and let G = I q (g), where g 2 H q .
Then,
where
E[(2 + 2G hDG; DL 1 Gi H ) 2 ] = E[(2 + 2G q 1 kDGk 2 H )2 ] (10.15)
(2 q! kgk 2 H q)2 +
+q 2 X
1
c q =
(q=2)!
q 1
q=2 1
r2f1;:::;q 1g
r6=q=2
4
2
=
(q=2)!
q 1 4
(2q 2r)!(r 1)! 2 kg r gk 2 +
r 1
H 2(q r)
q
q=2
+4q! c 1
q g e q=2 g g 2 H q ;
2
: (10.16)
The next statement, which is a main result of [56], contains a “non-central” analogous of
Theorem 10.1. Recall the de…nition of the centered Gamma random variables F (), > 0,
given in (8.2).
Theorem 10.3 (See [56]) Fix > 0, as well as an even integer q 2. De…ne c q as in (10.16).
Then, for any sequence ff k g k1 H q verifying
lim q!kf kk 2 H
= lim E I
k!1 n q (f k ) 2 = Var (F ()) = 2; (10.17)
k!1
the following six conditions are equivalent:
(i) lim k!1 E[I q (f k ) 3 ] = E[F () 3 ] = 8 and lim k!1 E[I q (f k ) 4 ] = E[F () 4 ] = 48 + 12 2 ;
(ii) lim k!1 E[I q (f k ) 4 ] 12E[I q (f k ) 3 ] = 12 2 48;
(iii) lim k!1 kf k
e q=2 f k c q f k k H q = 0 and lim k!1 kf k
e p f k k H 2(q p) = 0, for every
p = 1; :::; q 1 such that p 6= q=2;
(iv) lim k!1 kf k
e q=2 f k c q f k k H q = 0 and lim k!1 kf k p f k k H 2(q p) = 0, for every
p = 1; :::; q 1 such that p 6= q=2;
(v) as k ! 1, kD[I q (f k )]k 2 H
2qI q (f k ) ! 2q in L 2 ;
(vi) as k ! 1, the sequence fI q (f k )g k1 converges in distribution to F ().
Remark. In [56], Theorem 10.3 is not proved with Stein’s method, but rather by implementing
the “di¤erential approach” initiated by Nualart and Ortiz-Latorre in [66]. However,
it is not di¢ cult to see that (9.11), (9.12) and (10.15) can be combined in order to deduce an
alternate proof of the implications (iv) ) (v) ) (vi).
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