5 Haar Basis
5.1 Some useful notations and some preliminary results
Recall cf. 1.4 that for any parameter α 12, the Riemann-Liouville process can be written as
R
α
t = ε
−1
t
α
Γα + 1 +
∞
X
j=0 2
j
−1
X
k=0
ε
j,k
R
α
h
j,k
t, 5.1
where R
α
is the Riemann-Liouville operator and the h
j,k
’s are the usual Haar functions, i.e. h
j,k
x = 2
j 2
n
1
[
2k 2 j+1
,
2k+1 2 j+1
x − 1
[
2k+1 2 j+1
,
2k+2 2 j+1
x o
5.2 and where the series converges almost surely uniformly in t i.e. in the sense of the norm
k · k
∞
. For any t
∈ [0, 1] and J ∈ N we set, R
α J
t := ε
−1
t
α
Γα + 1 +
J −1
X
j=0 2
j
−1
X
k=0
ε
j,k
R
α
h
j,k
t, 5.3
e σ
2 J
t := E R
α
t − R
α J
t
2
=
∞
X
j=J 2
j
−1
X
k=0
R
α
h
j,k
t
2
5.4 and
e σ
2 J
:= sup
t ∈[0,1]
e σ
2 J
t. 5.5
In order to conveniently express the coefficients R
α
h
j,k
t, for any reals ν and x we set x
ν +
= ¨
x
ν
when x 0,
else . 5.6
Then it follows from 4.1, 5.2 and 5.6 that one has for every integers j ∈ N
:= N ∪ {0} and
≤ k ≤ 2
j
− 1 and real t ∈ [0, 1], R
α
h
j,k
t = 2
j 2
Γα + 1 t
− 2k + 2
2
j+1 α
+
− 2 t
− 2k + 1
2
j+1 α
+
+ t
− 2k
2
j+1 α
+
. 5.7
Let us now give some useful lemmas. The following lemma can be proved similarly to Lemma 1 in [2], this is why we omit its proof.
Lemma 5.1. Let {ε
j,k
: j ∈ N
, 0 ≤ k ≤ 2
j
− 1} be a sequence of standard Gaussian variables. Then there exists a random variable C
1
0 of finite moment of any order such that one has almost surely, for every j
∈ N and 0
≤ k 2
j
− 1, |ε
j,k
| ≤ C
1
p 1 + j .
2707
Let us now give a lemma that allows to control the increments of the Riemann-Liouville process. This result is probably known however we will give its proof for the sake of completeness.
Lemma 5.2.
i For any α ∈ 12, 32, there is a random variable C
2
0 of finite moment of any order such that almost surely for every t
1
, t
2
∈ [0, 1], |R
α
t
1
− R
α
t
2
| ≤ C
2
|t
1
− t
2
|
α−12
p log2 +
|t
1
− t
2
|
−1
. 5.8
ii For any α 32, there is a random variable C
3
0 of finite moment of any order such that one has almost surely for every t
1
, t
2
∈ [0, 1], |R
α
t
1
− R
α
t
2
| ≤ C
3
|t
1
− t
2
|. Proof. of Lemma 5.2 Part ii is a straightforward consequence of the fact that the trajectories
of R
α
are continuously differentiable functions when α 32. Let us now prove part i. First
observe see for instance relation 7.6 in [12] that inequality 5.8 is satisfied when the Riemann- Liouville process is replaced by the fractional Brownian motion fBm B
H
t
≤t≤1
with Hurst index H :=
α − 12. Recall that for some c
α
0 again H and α are related via H = α − 12 B
H
t = Q
α
t + c
α
R
α
t, 5.9
where the process Q
α
t
t ∈[0,1]
is called the low-frequency part of fBm and up to a positive constant it is defined as
Q
α
t = Z
−∞
n t − x
α−1
− −x
α−1
o dW x.
Finally, it is well-known that the trajectories of the process Q
α
t
t ∈[0,1]
are C
∞
-functions. There- fore, it follows from 5.9 that R
α
t
≤t≤1
satisfies 5.8 as well.
Remark 5.1. It is very likely that 5.8 also holds for α = 32. But in this case our approach does
not apply. Observe that α = 32 corresponds to H = 1 and 5.9 is no longer valid.
For any reals γ 0 and t ∈ [0, 1] and for any integers j ∈ N
and 0 ≤ k ≤ 2
j
− 1 we set A
γ, j,k
t := t
− 2k + 2
2
j+1 γ
+
− 2 t
− 2k + 1
2
j+1 γ
+
+ t
− 2k
2
j+1 γ
+
. 5.10
Observe that 5.10 and 5.7 imply that R
α
h
j,k
t = 2
j 2
Γα + 1 A
α, j,k
t. 5.11
Furthermore, we denote by e k
j
t the unique integer satisfying the following property: ek
j
t 2
j
≤ t ek
j
t + 1 2
j
, 5.12
with the convention that e k
j
1 = 2
j
− 1. The following lemma allows us to estimate R
α
h
j,k
t suitably. 2708
Lemma 5.3.
i For each k ≥ ek
j
t + 1, one has A
γ, j,k
t = 0. ii There is a constant c
4
0, only depending on γ, such that the inequality |A
γ, j,k
t| ≤ c
4
2
− jγ
1 + e k
j
t − k
γ−2
5.13 holds when 0
≤ k ≤ ek
j
t. iii If
γ 6= 1, then there is a constant c
5
0, only depending on γ, such that the inequality |A
γ, j,k
t| ≥ c
5
2
− jγ
1 + e k
j
t − k
γ−2
5.14 holds when 0
≤ k ≤ ek
j
t − 2. Proof. of Lemma 5.3 Part i is a straightforward consequence of 5.6, 5.10 and 5.12, so we
will focus on parts ii and iii. Inequality 5.13 clearly holds when γ = 1, this is why we will
assume in all the sequel that γ 6= 1. Let us first show that 5.13 is satisfied when
ek
j
t − 1 ≤ k ≤ ek
j
t. 5.15
Putting together 5.12 and 5.15 one has for any l ∈ {0, 1, 2},
t −
2k + l 2
j+1 γ
+
≤ t
− ek
j
t − 1 2
j γ
≤ 2
− j−1γ
. Therefore, it follows from 5.10 and 5.15 that
|A
γ, j,k
t| ≤ 2
2 − j−1γ
≤ c
6
2
− jγ
1 + e k
j
t − k
γ−2
, where the constant c
6
= 2
2+ γ
max {1, 2
2 −γ
}. Let us now show that the inequalities 5.13 and 5.14 are verified when
≤ k ≤ ek
j
t − 2. 5.16
We denote by f
γ, j,t
the function defined for every real x ≤ k2
j
as f
γ, j,t
x = t
− x − 2
− j γ
− t
− x − 2
− j−1 γ
. By applying the Mean Value Theorem to f
γ, j,t
on the interval [
2k −1
2
j+1
,
k 2
j
], it follows that there exists a
1
∈
2k −1
2
j+1
,
k 2
j
such that A
γ, j,k
t = f
γ, j,t
k 2
j
− f
γ, j,t
2k − 1 2
j+1
= 2
− j−1
f
′ γ, j,t
a
1
= −γ 2
− j−1
t − a
1
− 2
− j γ−1
− t
− a
1
− 2
− j−1 γ−1
. 5.17
2709
Next, by applying the Mean Value Theorem to the function y
7→ t
− a
1
− y
γ−1
on the interval [2
− j−1
, 2
− j
], it follows that there exists a
2
∈ 2
− j−1
, 2
− j
such that t
− a
1
− 2
− j γ−1
− t
− a
1
− 2
− j−1 γ−1
= −γ − 1 2
− j−1
t − a
1
− a
2 γ−2
. 5.18
Observe that the inequalities 5.12,
k 2
j
a
1
+ a
2 k+1
2
j
and 5.16 imply that 1
2
j
≤ ek
j
t − k − 1 2
j
t − a
1
− a
2
ek
j
t − k − 1 + 2 2
j
≤ 3e
k
j
t − k − 1 2
j
. 5.19
Next setting c
4
= c
6
+ γ|γ − 1| max{1, 3
γ−2
} and c
5
= γ|γ − 1| min{1, 3
γ−2
} and combining 5.17 with 5.18 and 5.19, it follows that the inequalities 5.13 and 5.14 are verified when 5.16
holds.
5.2 Optimality when 1
Kategori