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
2 α 1
The goal of this section is to prove the following theorem.
Theorem 5.4. Suppose 1
2 α 1. Then there is a random variable C
7
0 of finite moments of any order such that one has almost surely, for every J
∈ N, kR
α
− R
α J
k
∞
≤ C
7
2
−α−12J
p 1 + J .
In particular, this implies that in this case representation 5.1 possesses the optimal approximation rate.
Proof. of Theorem 5.4 Putting together 5.1, 5.3, Lemma 5.1, 5.11 and 5.13, one obtains that almost surely, for every t
∈ [0, 1], and every integer J ∈ N, |R
α
t − R
α J
t| ≤
∞
X
j=J 2
j
−1
X
k=0
|ε
j,k
||R
α
h
j,k
t|
≤ C
1
Γα + 1
−1
c
4 ∞
X
j=J
2
− jα−12
p j + 1
ek
j
t
X
k=0
1 + e k
j
t − k
α−2
≤ C
7
2
−Jα−12
p J + 1.
Observe that the condition 1 2 ≤ α 1 plays a crucial role in the proof of Theorem 5.4. Indeed,
one has P
ek
j
t k=0
1 + e k
j
t − k
α−2
≤ P
∞ l=1
l
α−2
∞ only when it is satisfied. 2710
5.3 Optimality when 1
α 32
The goal of this subsection is to show that the following theorem holds.
Theorem 5.5. Suppose 1 α 32. Then there is a constant c
8
0 such that for every J ∈ N one has E
kR
α
− R
α J
k
∞
≤ c
8
2
−Jα−12
p J + 1.
In particular, this implies that also in this case representation 5.1 possesses the optimal approximation rate.
First we need to prove some preliminary results.
Proposition 5.6. If 1
2 α 32, there exists a constant c
9
0 such that one has for any J ∈ N, e
σ
2 J
≤ c
2 9
2
−J2α−1
. Proof. of Proposition 5.6 It follows from 5.4, 5.11 and parts i and ii of Lemma 5.3, that
e σ
2 J
t ≤ c
2 4
Γα + 1
−2 ∞
X
j=J
2
− j2α−1 ek
j
t
X
k=0
1 + e k
j
t − k
−22−α
≤ c
2 9
2
−J2α−1
, where the constant c
2 9
= c
2 4
Γα + 1
−2
1 − 2
−2α−1
P
∞ l=1
l
−22−α
∞.
Lemma 5.7. For any α ∈ 1, 32, there exists a random variable C
10
0 of finite moment of any order such that one has almost surely for any real t
∈ [0, 1] and any integer J ∈ N, R
α J
t − R
α J
ek
J
t2
−J
≤ C
10
2
−Jα−12
p J + 1.
We refer to 5.12 for the definition of the integer e k
J
t. In order to be able to prove Lemma 5.7 we need the following lemma.
Lemma 5.8. For any real α 1, there exists a constant c
11
0 such that for all t ∈ [0, 1], J ∈ N, j
∈ N and k
∈ N satisfying
≤ j ≤ J and 0 ≤ k ≤ ek
j
t, one has
R
α
h
j,k
t − R
α
h
j,k
ek
J
t2
−J
≤ c
11
2
32−α j−J
1 + e k
j
t − k
α−3
. 5.20
Proof. Proof of Lemma 5.8 It is clear that 5.20 holds when t = e k
J
t2
−J
, so we will assume that t
6= ek
J
t2
−J
. By applying the Mean Value Theorem, it follows that there exists a ∈ ek
J
t2
−J
, t such that
R
α
h
j,k
t − R
α
h
j,k
ek
J
t2
−J
5.21 =
α2
j 2−J
Γα a
− 2k + 2
2
j+1 α−1
+
− 2 a
− 2k + 1
2
j+1 α−1
+
+ a
− 2k
2
j+1 α−1
+
. Observe that one has e
k
j
a = ek
j
t since a ∈
ek
J
t 2
J
, t ⊂
ek
j
t 2
j
,
ek
j
t+1 2
j
. Thus, putting together, 5.21, 5.10 and 5.13 in which we replace t by a and
γ by α − 1, we obtain the lemma. 2711
We are now in position to prove Lemma 5.7. Proof. of Lemma 5.7 By using Lemma 5.1 and the fact that
≤ t
α
− ek
J
t2
−J α
≤ α2
−J
, one gets that
R
α J
t − R
α J
2
−J
ek
J
t ≤ |ε
−1
| α2
−J
Γα + 1 +
C
1
p J + 1
J −1
X
j=0 2
j
−1
X
k=0
R
α
h
j,k
t − R
α
h
j,k
ek
J
t2
−J
. 5.22
On the other hand, it follows from Lemma 5.8 that
J −1
X
j=0 2
j
−1
X
k=0
R
α
h
j,k
t − R
α
h
j,k
ek
J
t2
−J
≤ c
11 J
−1
X
j=0
2
32−α j−J ek
j
t
X
k=0
1 + e k
j
t − k
α−3
≤ c
12
2
−α−12J
5.23 where c
12
= c
11
2
3 2−α
− 1
−1
P
∞ l=1
l
α−3
∞. Finally combining 5.22 with 5.23 one obtains the lemma.
Lemma 5.9. There is a random variable C
13
0 of finite moments of any order such that one has almost surely for every J
∈ N sup
t ∈[0,1]
R
α
t − R
α J
t ≤
sup
≤K2
J
, K ∈N
R
α
K2
−J
− R
α J
K2
−J
+ C
13
2
−Jα−12
p J + 1.
Proof. of Lemma 5.9 Let us fix ω. As the function t 7→ R
α
t, ω − R
α J
t, ω is continuous over the compact interval [0, 1], there exist a t
∈ [0, 1] such that sup
t ∈[0,1]
R
α
t, ω − R
α J
t, ω =
R
α
t ,
ω − R
α J
t ,
ω .
Using the triangular inequality and Lemmas 5.2 i and 5.7, it follows that R
α
t ,
ω − R
α J
t ,
ω ≤
R
α
t ,
ω − R
α
e k
J
t 2
−J
, ω
+ R
α
e k
J
t 2
−J
, ω − R
α J
e k
J
t 2
−J
, ω
+ R
α J
2
−J
ek
J
t , ω − R
α J
t ,
ω ≤ C
2
ω 2
−Jα−12
p log2 + 2
J
+ sup
≤K2
J
,K ∈N
R
α
K2
−J
, ω − R
α J
K2
−J
, ω
+ C
10
ω 2
−Jα−12
p J + 1
and thus one gets the lemma. 2712
Proof. of Theorem 5.5 Putting together Lemma 5.9, Lemma 3.2, the fact that e
σ
2 J
≥ sup
≤K2
J
,K ∈N
e σ
2 J
2
−J
K and Proposition 5.6 one obtains the theorem.
5.4 The case
