5.3 Central limit theorems with contraction conditions
We will now deduce the announced CLTs for sequences of vectors of the type F
n
= F
n 1
, . . . , F
n d
:= I
q
1
f
n 1
, . . . , I
q
d
f
n d
, n
≥ 1. 28
As already discussed, our results should be compared with other central limit results for multiple stochastic integrals in a Gaussian or Poisson setting – see e.g. [9, 11, 13, 14, 18, 20]. The following
statement, which is a genuine multi-dimensional generalization of Theorem 5.1 in [17], is indeed one of the main achievements of the present article.
Theorem 5.8 CLT for chaotic vectors. Fix d ≥ 2, let X ∼ N 0, C, with
C = {Ci, j : i, j = 1, . . . , d}
a d × d nonnegative definite matrix, and fix integers q
1
, . . . , q
d
≥ 1. For any n ≥ 1 and i = 1, . . . , d, let f
n i
belong to L
2 s
µ
q
i
. Define the sequence {F
n
: n ≥ 1}, according to 28 and suppose that
lim
n →∞
E [F
n i
F
n j
] = 1
q
j
=q
i
q
j
× lim
n →∞
〈 f
n i
, f
n j
〉
L
2
µ
qi
= Ci, j, 1
≤ i, j ≤ d. 29
Assume moreover that the following Conditions 1–4 hold for every k = 1, ..., d: 1. For every n, the kernel f
n k
satisfies Assumptions A and B. 2. For every l = 1, ..., d and every n, the kernels f
n k
and f
n l
satisfy Assumption C. 3. For every r = 1, . . . , q
k
and every l = 1, . . . , r ∧ q
k
− 1, one has that k f
n k
⋆
l r
f
n k
k
L
2
µ
2qk−r−l
→ 0, as n
→ ∞. 4. As n
→ ∞, R
Z
qk
d µ
q
k
f
n k
4
→ 0. Then, F
n
converges to X in distribution as n → ∞. The speed of convergence can be assessed by
combining the estimates of Proposition 5.5 and Proposition 5.6 either with Theorem 3.3 when C is positive definite or with Theorem 4.2 when C is merely nonnegative definite.
Remark 5.9. 1. For every f
∈ L
2 s
µ
q
, q ≥ 1, one has that k f ⋆
q
f k
2 L
2
µ
q
= Z
Z
q
d µ
q
f
4
. 2. When q
i
6= q
j
, then F
n i
and F
n j
are not in the same chaos, yielding that Ci, j = 0 in formula 29. In particular, if Conditions 1-4 of Theorem 5.8 are verified, then F
n i
and F
n j
are asymptotically independent.
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3. When specializing Theorem 5.8 to the case q
1
= ... = q
d
= 1, one obtains a set of con- ditions that are different from the ones implied by Corollary 4.3.
First observe that, if q
1
= ... = q
d
= 1, then Condition 3 in the statement of Theorem 5.8 is immaterial. As a consequence, one deduces that F
n
converges in distribution to X , provided that 29 is verified and
k f
n
k
L
4
µ
→ 0. The L
4
norms of the functions f
n
appear due to the use of Cauchy-Schwarz inequality in the proof of Proposition 5.6.
Proof of Theorem 5.8. By Theorem 4.2, d
3
F
n
, X ≤
d 2
v u
u t
d
X
i, j=1
E [Ci, j − 〈DF
n i
, −DL
−1
F
n j
〉
L
2
µ 2
] 30
+ 1
4 Z
Z
µdzE
d
X
i=1
|D
z
F
n i
|
2 d
X
i=1
|D
z
L
−1
F
n i
|
,
31 so that we need only show that, under the assumptions in the statement, both 30 and 31 tend
to 0 as n → ∞.
On the one hand, we take a = Ci, j in Proposition 5.5. In particular, we take a = 0 when q
i
6= q
j
. Admitting Condition 3 , 4 and 29, line 30 tends to 0 is a direct consequence of Proposition 5.5.
On the other hand, under Condition 3 and 4, Proposition 5.6 shows that 31 converges to 0. This concludes the proof and the above inequality gives the speed of convergence.
If the matrix C is positive definite, then one could alternatively use Theorem 3.3 instead of Theorem 4.2 while the deduction remains the same.
Remark 5.10. Apart from the asymptotic behavior of the covariances 29 and the presence of Assumption C, the statement of Theorem 5.8 does not contain any requirements on the joint distri-
bution of the components of F
n
. Besides the technical requirements in Condition 1 and Condition 2, the joint convergence of the random vectors F
n
only relies on the ‘one-dimensional’ Conditions 3 and 4, which are the same as condition II and III in the statement of Theorem 5.1 in [17]. See
also Remark 3.5.
6 Examples
In what follows, we provide several explicit applications of the main estimates proved in the paper. In particular:
• Section 6.1 focuses on vectors of single and double integrals. • Section 6.2 deals with three examples of continuous-time functionals of Ornstein-Uhlenbeck
Lévy processes.
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6.1 Vectors of single and double integrals
Kategori