6.1 Vectors of single and double integrals
The following statement corresponds to Theorem 3.3, in the special case F = F
1
, . . . , F
d
= I
1
g
1
, . . . , I
1
g
m
, I
2
h
1
, . . . , I
2
h
n
. 32
The proof, which is based on a direct computation of the general bounds proved in Theorem 3.3, serves as a further illustration in a simpler setting of the techniques used throughout the paper.
Some of its applications will be illustrated in Section 6.2.
Proposition 6.1. Fix integers n, m ≥ 1, let d = n+m, and let C be a d ×d nonnegative definite matrix.
Let X ∼ N
d
0, C. Assume that the vector in 32 is such that 1. the function g
i
belongs to L
2
µ ∩ L
3
µ, for every 1 ≤ i ≤ m, 2. the kernel h
i
∈ L
2 s
µ
2
1 ≤ i ≤ n is such that: a h
i
1
⋆
1 2
h
i
2
∈ L
2
µ
1
, for 1 ≤ i
1
, i
2
≤ n, b h
i
∈ L
4
µ
2
and c the functions |h
i
1
| ⋆
1 2
|h
i
2
|, |h
i
1
| ⋆
2
|h
i
2
| and |h
i
1
| ⋆
1
|h
i
2
| are well defined and finite for every value of their arguments and for every 1
≤ i
1
, i
2
≤ n, d every pair h
i
, h
j
verifies Assumption C, that in this case is equivalent to requiring that Z
Z
ÈZ
Z
µd ah
2 i
z, ah
2 j
z, aµdz ∞. Then,
d
3
F, X ≤ 1
2 p
S
1
+ S
2
+ S
3
+ S
4
≤ 1
2 p
S
1
+ S
5
+ S
6
+ S
4
where S
1
=
m
X
i
1
,i
2
=1
Ci
1
, i
2
− 〈g
i
1
, g
i
2
〉
L
2
µ 2
S
2
=
n
X
j
1
, j
2
=1
Cm + j
1
, m + j
2
− 2〈h
j
1
, h
j
2
〉
L
2
µ
2
2
+ 4kh
j
1
⋆
1 2
h
j
2
k
2 L
2
µ
+ 8kh
j
1
⋆
1 1
h
j
2
k
2 L
2
µ
2
S
3
=
m
X
i=1 n
X
j=1
2Ci, m + j
2
+ 5kg
i
⋆
1 1
h
j
k
2 L
2
µ
S
4
= m
2 m
X
i=1
kg
i
k
3 L
3
µ
+ 8n
2 n
X
j=1
kh
j
k
L
2
µ
2
kh
j
k
2 L
4
µ
2
+ p
2 kh
j
1
⋆
1
h
j
1
k
L
2
µ
3
S
5
=
n
X
j
1
, j
2
=1
Cm + j
1
, m + j
2
− 2〈h
j
1
, h
j
2
〉
L
2
µ
2
2
+ 4kh
j
1
⋆
1
h
j
1
k
L
2
µ
3
× kh
j
2
⋆
1
h
j
2
k
L
2
µ
3
+8kh
j
1
⋆
1 1
h
j
1
k
L
2
µ
2
× kh
j
2
⋆
1 1
h
j
2
k
L
2
µ
2
S
6
=
m
X
i=1 n
X
j=1
2Ci, m + j
2
+ 5kg
i
k
2 L
2
µ
× kh
j
⋆
1 1
h
j
k
L
2
µ
2
1514
Proof. Assumptions 1 and 2 in the statement ensure that each integral appearing in the proof is well-defined, and that the use of Fubini arguments is justified. In view of Theorem 4.2, our strategy
is to study the quantities in line 15 and line 16 separately. On the one hand, we know that: for 1
≤ i ≤ m, 1 ≤ j ≤ n, D
z
I
1
g
i
· = g
i
z, −D
z
L
−1
I
1
g
i
· = g
i
z D
z
I
2
h
j
·, · = 2I
1
h
j
z, ·, −D
z
L
−1
I
2
h
j
·, · = I
1
h
j
z, · Then, for any given constant a, we have:
– for 1 ≤ i ≤ m, 1 ≤ j ≤ n,
E [a − 〈D
z
I
1
g
i
1
, −D
z
L
−1
I
1
g
i
2
〉
2
] = a − 〈g
i
1
, g
i
2
〉
L
2
µ 2
;
– for 1 ≤ j
1
, j
2
≤ n, E
[a − 〈D
z
I
2
h
j
1
, −D
z
L
−1
I
2
h
j
2
〉
2
] =
a − 2〈h
j
1
, h
j
2
〉
L
2
µ
2
2
+ 4kh
j
1
⋆
1 2
h
j
2
k
2 L
2
µ
+ 8kh
j
1
⋆
1 1
h
j
2
k
2 L
2
µ
2
;
– for 1 ≤ i ≤ m, 1 ≤ j ≤ n,
E [a − 〈D
z
I
2
h
j
, −D
z
L
−1
I
1
g
i
〉
2
] = a
2
+ 4kg
i
⋆
1 1
h
j
k
2 L
2
µ
E [a − 〈D
z
I
1
g
i
, −D
z
L
−1
I
2
h
j
〉
2
] = a
2
+ kg
i
⋆
1 1
h
j
k
2 L
2
µ
. So
15 = 1
2 p
S
1
+ S
2
+ S
3
where S
1
, S
2
, S
3
are defined as in the statement of proposition. On the other hand,
2
X
i=1
|D
z
F
i
|
2
=
m
X
i=1
|g
i
z| + 2
n
X
j=1
|I
1
h
j
z, ·|
2
,
d
X
i=1
|D
z
L
−1
F
i
| =
m
X
i=1
|g
i
z| +
n
X
j=1
|I
1
h
j
z, ·|. As the following inequality holds for all positive reals a, b:
a + 2b
2
a + b ≤ a + 2b
3
≤ 4a
3
+ 32b
3
,
1515
we have, E
d
X
i=1
|D
z
F
i
|
2 d
X
i=1
|D
z
L
−1
F
i
|
= E
m
X
i=1
|g
i
z| + 2
n
X
j=1
|I
1
h
j
z, ·|
2
m
X
i=1
|g
i
z| +
n
X
j=1
|I
1
h
j
z, ·|
≤ E
4
m
X
i=1
|g
i
z|
3
+ 32
n
X
j=1
|I
1
h
j
z, ·|
3
≤ E[4m
2 m
X
i=1
|g
i
z|
3
+ 32n
2 n
X
j=1
|I
1
h
j
z, ·|
3
]. By applying the Cauchy-Schwarz inequality, one infers that
Z
Z
µdzE[|I
1
hz, ·|
3
] ≤ s
E Z
Z
µdz|I
1
hz, ·|
4
× khk
L
2
µ
2
. Notice that
E Z
Z
µdz|I
1
hz, ·|
4
= 2kh ⋆
1 2
h k
2 L
2
µ
+ khk
4 L
4
µ
2
We have 16 =
1 4
m
2
kC
−1
k
3 2
op
kCk
op
Z
Z
µdzE
d
X
i=1
|D
z
F
i
|
2 d
X
i=1
|D
z
L
−1
F
i
|
≤ kC
−1
k
3 2
op
kCk
op
m
2 m
X
i=1
kg
i
k
3 L
3
µ
+8n
2 n
X
j=1
kh
j
k
L
2
µ
2
kh
j
k
2 L
4
µ
2
+ p
2 kh
j
⋆
1 2
h
j
k
L
2
µ
= kC
−1
k
3 2
op
kCk
op
S
4
We will now apply Lemma 2.9 to further assess some of the summands appearing the definition of S
2
,S
3
. Indeed,
– for 1 ≤ j
1
, j
2
≤ n, kh
j
1
⋆
1 2
h
j
2
k
2 L
2
µ
≤ kh
j
1
⋆
1
h
j
1
k
L
2
µ
3
× kh
j
2
⋆
1
h
j
2
k
L
2
µ
3
kh
j
1
⋆
1 1
h
j
2
k
2 L
2
µ
2
≤ kh
j
1
⋆
1 1
h
j
1
k
L
2
µ
2
× kh
j
2
⋆
1 1
h
j
2
k
L
2
µ
2
;
1516
– for 1 ≤ i ≤ m, 1 ≤ j ≤ n,
kg
i
⋆
1 1
h
j
k
2 L
2
µ
≤ kg
i
k
2 L
2
µ
× kh
j
⋆
1 1
h
j
k
L
2
µ
2
by using the equality kg
k i
⋆ g
k i
k
2 L
2
µ
2
= kg
k i
k
4 L
2
µ
. Consequently,
S
2
≤
n
X
j
1
, j
2
=1
Cm + j
1
, m + j
2
− 2〈h
j
1
, h
j
2
〉
L
2
µ
2
2
+ 4kh
j
1
⋆
1
h
j
1
k
L
2
µ
3
× kh
j
2
⋆
1
h
j
2
k
L
2
µ
3
+8kh
j
1
⋆
1 1
h
j
1
k
L
2
µ
2
× kh
j
2
⋆
1 1
h
j
2
k
L
2
µ
2
= S
5
, S
3
≤
m
X
i=1 n
X
j=1
2Ci, m + j
2
+ 5kg
i
k
2 L
2
µ
× kh
j
⋆
1 1
h
j
k
L
2
µ
2
= S
6
Remark 6.2. If the matrix C is positive definite, then we have
d
2
F, X ≤ kC
−1
k
op
kCk
1 2
op
p S
1
+ S
2
+ S
3
+ p
2 π
2 kC
−1
k
3 2
op
kCk
op
S
4
≤ kC
−1
k
op
kCk
1 2
op
p S
1
+ S
5
+ S
6
+ p
2 π
2 kC
−1
k
3 2
op
kCk
op
S
4
by using Theorem 3.3. The following result can be proved by means of Proposition 6.1.
Corollary 6.3. Let d = m + n, with m, n ≥ 1 two integers . Let X
C
∼ N
d
0, C be a centered d- dimensional Gaussian vector, where C =
{Cs, t : s, t = 1, . . . , d} is a d × d nonnegative definite matrix such that
Ci, j + m = 0, ∀1 ≤ i ≤ m, 1 ≤ j ≤ n.
Assume that F
k
= F
k 1
, . . . , F
k d
:= I
1
g
k 1
, . . . , I
1
g
k m
, I
2
h
k 1
, . . . , I
2
h
k n
where for all k, the kernels g
k 1
, . . . , g
k m
and h
k 1
, . . . , h
k n
satisfy respectively the technical Conditions 1 and 2 in Proposition 6.1 . Assume moreover that the following conditions hold for each k
≥ 1: 1.
lim
k →∞
E [F
k s
F
k t
] = Cs, t, 1
≤ s, t ≤ d. or equivalently
lim
k →∞
〈g
k i
1
, g
k i
2
〉
L
2
µ
= Ci
1
, i
2
, 1
≤ i
1
, i
2
≤ m, lim
k →∞
2 〈h
k j
1
, h
k j
2
〉
L
2
µ
2
= Cm + j
1
, m + j
2
, 1
≤ j
1
, j
2
≤ n. 1517
2. For every i = 1, . . . , m and every j = 1, . . . , n, one has the following conditions are satisfied as k
→ ∞: a
kg
k i
k
3 L
3
µ
→ 0; b kh
k j
k
2 L
4
µ
2
→ 0; c
kh
k j
⋆
1 2
h
k j
k
L
2
µ
= kh
k j
⋆
1
h
k j
k
L
2
µ
3
→ 0; d
kh
k j
⋆
1 1
h
k j
k
2 L
2
µ
2
→ 0. Then F
k
→ X in law, as k → ∞. An explicit bound on the speed of convergence in the distance d
3
is provided by Proposition 6.1.
6.2 Vector of functionals of Ornstein-Uhlenbeck processes
Kategori