# Least Squares Estimation forα-Fractional Bridge with Discrete Observations

Authors: Guangjun Shen; Xiuwei Yin
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We consider a fractional bridge defined asd
X
t
=
-
α
(X
t
/
(
T
-
t
))
d
t
+
d
B
t
H
,
0
≤
t
<
T, where B
H is a fractional Brownian motion of Hurst parameter H
>
1
/
2 and parameter α
>
0 is unknown. We are interested in the problem of estimating the unknown parameter α
>
0. Assume that the process is observed at discrete time t
i
=
i
Δ
n
,
i
=
0
,
…
,
n, and T
n
=
n
Δ
n denotes the length of the “observation window.” We construct a least squares estimator α
^
n of α which is consistent; namely, α
^
n converges to α in probability as n
→
∞.


---

## Body

## 1. Introduction

Self-similar stochastic processes with long range dependence are of practical interest in various applications, including econometrics, internet traffic, and hydrology. These are processesX
=
{
X
t
:
t
≥
0
} whose dependence on the time parameter t is self-similar, in the sense that there exists a (self-similarity) parameter H
∈
(
0,1
) such that, for any constant c
≥
0, {
X
c
t
:
t
≥
0
} and {
c
H
X
t
:
t
≥
0
} have the same distribution. These processes are often endowed with other distinctive properties.The fractional Brownian motion (fBm) is the usual candidate to model phenomena in which the self-similarity property can be observed from the empirical data. The fBm is a suitable generalization of the standard Brownian motion, which exhibits long-range dependence and self-similarity and has stationary increments. Some surveys and complete literatures could be found in Biagini et al. [1], Hu [2], Mishura [3], and Nualart [4].Recently, Es-Sebaiy and Nourdin [5] study the asymptotic properties of a least squares estimator for the parameter α of a fractional bridge defined as
(1)
X
0
=
0
,
d
X
t
=
-
α
X
t
T
-
t
d
t
+
d
B
t
H
,
0
≤
t
<
T
,
where B
H is a fBm with Hurst parameter H
>
1
/
2 and the process X was observed continuously. In particular, when H
=
1
/
2, Barczy and Pap [6, 7] study the various problems related to the α-Wiener bridge. The parametric estimation problems for fractional diffusion processes based on continuous-time observations have been studied, for example, in Tudor and Viens [8], Hu and Nualart [9], and Belfadli et al. [10].In applications usually the process cannot be observed continuously. Only discrete-time observations are available. There exists a rich literature on the parameter estimation problem for diffusion processes driven by fBm based on discrete observations (see, e.g., Hu and Song [11], Es-Sebaiy [12]).Motivated by all these results, in this paper, we will consider theα fractional bridge (1). Assume that the process X is observed equidistantly in time with the step size t
i
=
i
Δ
n, i
=
0
,
…
,
n, and T
n
=
n
Δ
n denotes the length of the “observation window.” We also assume that T
n
+
Δ
n
=
T and Δ
n
→
0 when n
→
∞. Our goal is to study the asymptotic behavior of the least squares estimator (LSE for short) α
^
n of α based on the sampling data X
t
i, i
=
0
,
…
,
n. Our technics used in this work are inspired from Es-Sebaiy [12].The least squares estimatorα
^
n aims to minimize
(2)
α
↦
∑
i
=
1
n
∫
t
i
-
1
t
i
|
X
˙
t
+
α
X
t
i
-
1
T
-
t
i
-
1
|
2
d
t
.This is a quadratic function ofα. The minimum is achieved when
(3)
α
^
n
=
-
∑
i
=
1
n
∫
t
i
-
1
t
i
(
X
t
i
-
1
/
(
T
-
t
i
-
1
)
)
δ
H
X
t
Δ
n
∑
i
=
1
n
(
X
t
i
-
1
2
/
(
T
-
t
i
-
1
)
2
)
.By (1), we can get the following result
(4)
α
^
n
-
α
=
-
∑
i
=
1
n
M
i
Δ
n
∑
i
=
1
n
(
X
t
i
-
1
2
/
(
T
-
t
i
-
1
)
2
)
,
where M
i
=
α
(
X
t
i
-
1
/
(
T
-
t
i
-
1
)
∫
t
i
-
1
t
i
(
(
X
t
i
-
1
/
(
T
-
t
i
-
1
)
)
-
(
X
s
/
(
T
-
s
)
)
)
d
s
+
∫
t
i
-
1
t
i
(
X
t
i
-
1
/
(
T
-
t
i
-
1
)
δ
H
B
t
H, i
=
1
,
…
,
n.The paper is organized as follows. In Section2 some known results that we will use are recalled. The consistency of estimator is proved in Section 3.


## 2. Preliminaries

Recall that fBmB
H with index H
∈
(
0,1
) is a mean zero Gaussian process B
H
=
{
B
t
H
,
t
≥
0
} with B
0
H
=
0 and the covariance
(5)
R
H
(
t
,
s
)
:
=
E
(
B
t
B
s
)
=
1
2
(
t
2
H
+
s
2
H
-
|
t
-
s
|
2
H
)
for all s
,
t
≥
0. For H
=
1
/
2, B
H coincides with the standard Brownian motion B. B
H is neither a semimartingale nor a Markov process unless H
=
1
/
2, so many of the powerful techniques from stochastic analysis are not available when dealing with B
H. It is possible to construct a stochastic calculus of variations with respect to the Gaussian process B
H, which will be related to the Malliavin calculus. Some surveys and complete literatures could be found in Alòs et al. [13], Nualart [4] and the reference. We recall here the basic definitions and results of this calculus. The crucial ingredient is the canonical Hilbert space ℋ (it is also said to be reproducing kernel Hilbert space) associated with the fBm which is defined as the closure of the linear space ℰ generated by the indicator functions {
1
[
0
,
t
]
,
t
∈
[
0
,
T
]
} with respect to the scalar product
(6)
〈
1
[
0
,
t
]
,
1
[
0
,
s
]
〉
ℋ
=
R
H
(
t
,
s
)
=
1
2
(
t
2
H
+
s
2
H
-
|
t
-
s
|
2
H
)
.
The mapping 1
[
0
,
s
]
→
B
s
H can be extended to a linear isometry between ℋ and the Gaussian space associated with B
H. We will denote the isometry by φ
→
B
H
(
φ
). For 1
/
2
<
H
<
1 we denote by 𝒮 the set of smooth functionals of the form
(7)
F
=
f
(
B
H
(
φ
1
)
,
…
,
B
H
(
φ
n
)
)
,
where f
∈
C
b
∞
(
ℝ
n
) and φ
i
∈
ℋ. The Malliavin derivative of a functional F as above is given by
(8)
D
H
F
=
∑
i
=
1
n
∂
f
∂
x
i
(
B
H
(
φ
1
)
,
…
,
B
H
(
φ
n
)
)
φ
i
,
and this operator can be extended to the closure 𝔻
m
,
2 (m
≥
1) of 𝒮 with respect to the norm
(9)
∥
F
∥
m
,
2
2
≡
E
|
F
|
2
+
E
∥
D
H
F
∥
ℋ
2
+
⋯
+
E
∥
D
H
,
m
F
∥
ℋ
⊗
^
m
2
,
where ℋ
⊗
^
m denotes the m fold symmetric tensor product and the mth derivative D
H
,
m is defined by iteration. The divergence integral δ
H is the adjoint operator of D
H. Concretely, a random variable u
∈
L
2
(
Ω
,
ℋ
) belongs to the domain of the divergence operator δ
H (in symbol Dom
⁡
(
δ
H
)) if
(10)
E
|
〈
D
H
F
,
u
〉
ℋ
|
≤
c
∥
F
∥
L
2
for every F
∈
𝒮. In this case δ
H
(
u
) is given by the duality relationship
(11)
E
(
F
δ
H
(
u
)
)
=
E
〈
D
H
F
,
u
〉
ℋ
for any F
∈
𝔻
1,2, and we have the following integration by parts:
(12)
F
δ
H
(
u
)
=
δ
H
(
F
u
)
+
〈
D
H
F
,
u
〉
ℋ
for any u
∈
Dom
⁡
(
δ
H
), F
∈
𝔻
1,2 such that F
u
∈
L
2
(
Ω
,
ℋ
). It follows that
(13)
E
[
δ
H
(
u
)
2
]
=
E
∥
u
∥
ℋ
2
+
E
〈
D
H
u
,
(
D
H
u
)
*
〉
ℋ
⊗
ℋ
,
where (
D
H
u
)
* is the adjoint of D
H
u in the Hilbert space ℋ
⊗
ℋ, and
(14)
∥
u
∥
ℋ
2
=
∬
0
T
u
s
u
r
ϕ
H
(
s
,
r
)
d
s
d
r
,
where
(15)
ϕ
H
(
s
,
r
)
=
∂
2
R
H
∂
s
∂
r
(
s
,
r
)
=
H
(
2
H
-
1
)
|
s
-
r
|
2
H
-
2
≥
0
,
and, for φ
:
[
0
,
T
]
2
→
ℝ, we have(16)
∥
φ
∥
ℋ
⊗
ℋ
2
=
∫
[
0
,
T
]
4
φ
(
t
,
s
)
φ
(
t
′
,
s
′
)
ϕ
H
(
t
,
t
′
)
ϕ
H
(
s
,
s
′
)
d
t
d
s
d
t
′
d
s
′
.
We denote by |
ℋ
| the subspace of ℋ, which is defined as the set of measurable functions f on [
0
,
T
] with
(17)
∥
f
∥
|
ℋ
|
2
:
=
∬
0
T
|
f
(
s
)
|
|
f
(
r
)
|
ϕ
H
(
s
,
r
)
d
s
d
r
<
∞
.Note that, ifφ
,
ψ
∈
|
ℋ
|, then
(18)
E
B
H
(
φ
)
B
H
(
ψ
)
=
∬
0
T
φ
(
s
)
ψ
(
r
)
ϕ
H
(
s
,
r
)
d
s
d
r
.It follows actually from Pipiras and Taqqu [14] that the space |
ℋ
|
⊂
ℋ is a Banach space for the norm ∥
·
∥
|
ℋ
|. Moreover,
(19)
L
2
(
[
0
,
T
]
)
⊂
L
1
/
H
(
[
0
,
T
]
)
⊂
|
ℋ
|
⊂
ℋ
.
If u
∈
𝔻
1,2
(
|
ℋ
|
), u
∈
Dom
⁡
(
δ
H
), then we have (Nualart [4])
(20)
E
(
δ
H
(
u
)
)
2
≤
C
H
(
E
∥
u
∥
|
ℋ
|
2
+
E
∥
D
H
(
u
)
∥
|
ℋ
|
⊗
|
ℋ
|
2
)
and if φ
:
[
0
,
T
]
2
→
ℝ, then
(21)
∥
φ
∥
|
ℋ
|
⊗
|
ℋ
|
2
=
∫
[
0
,
T
]
4
|
φ
(
t
,
s
)
|
|
φ
(
t
′
,
s
′
)
|
ϕ
H
(
t
,
t
′
)
ϕ
H
(
s
,
s
′
)
d
t
d
s
d
t
′
d
s
′
.
As a consequence, we have
(22)
E
(
δ
H
(
u
)
)
2
≤
C
H
(
E
∥
u
∥
L
1
/
H
(
[
0
,
T
]
)
2
+
E
∥
D
H
(
u
)
∥
L
1
/
H
(
[
0
,
T
]
2
)
2
)
.For everyn
≥
1, let ℋ
n be the nth Wiener chaos of B
H, that is, the closed linear subspace of L
2
(
Ω
) generated by the random variables {
H
n
(
B
H
(
h
)
)
,
h
∈
ℋ
,
∥
h
∥
ℋ
=
1
}, where H
n is the nth Hermite polynomial. The mapping I
n
(
h
⊗
n
)
=
n
!
H
n
(
B
H
(
h
)
) provides a linear isometry between the symmetric tensor product ℋ
⊙
n (equipped with the modified norm ∥
·
∥
ℋ
⊙
n
=
(
1
/
n
!
)
∥
·
∥
ℋ
⊗
n) and ℋ
n. For every f
,
g
∈
ℋ
⊙
n the following multiplication formula holds
(23)
E
(
I
n
(
f
)
I
n
(
g
)
)
=
n
!
〈
f
,
g
〉
ℋ
⊗
n
.Letf
,
g
:
[
0
,
T
]
→
ℝ be Hölder continuous functions of orders α
∈
(
0,1
) and β
∈
(
0,1
) with α
+
β
>
1. Young proved that the Riemann-Stieltjes integral (so-called Young integral) ∫
0
T
‍
f
s
d
g
s exists. Moreover, if α
=
β
∈
(
1
/
2
,
1
) and F
:
ℝ
2
→
ℝ is a function of class 𝒞
1, the integrals ∫
0
·
‍
(
∂
F
/
∂
f
)
(
f
u
,
g
u
)
d
f
u and ∫
0
·
‍
(
∂
F
/
∂
g
)
(
f
u
,
g
u
)
d
g
u exist in the Young sense and the following change of variables formula holds:
(24)
F
(
f
t
,
g
t
)
=
F
(
f
0
,
g
0
)
+
∫
0
t
∂
F
∂
f
(
f
u
,
g
u
)
d
f
u
+
∫
0
t
∂
F
∂
g
(
f
u
,
g
u
)
d
g
u
,
t
∈
[
0
,
T
]
.As a consequence, ifH
∈
(
1
/
2
,
1
) and (u
t, t
∈
[
0
,
T
]) is a process with Hölder paths of order α
<
(
1
-
H
,
1
), the integral ∫
0
T
u
s
d
B
s
H is well defined as Young integral. Suppose that, for any t
∈
[
0
,
T
], u
t
∈
𝔻
1,2
(
|
ℋ
|
), and
(25)
∬
0
T
|
D
s
u
t
|
|
t
-
s
|
2
H
-
2
d
s
d
t
<
∞
a
.
s
.
Then, following from Alòs and Nualart [15], we have
(26)
∫
0
t
u
s
d
B
s
H
=
∫
0
t
u
s
δ
H
B
s
H
+
H
(
2
H
-
1
)
×
∬
0
t
D
s
u
r
|
r
-
s
|
2
H
-
2
d
r
d
s
.
In particular, when φ is a nonrandom Hölder continuous function of order α
∈
(
1
-
H
,
1
), we have
(27)
∫
0
t
φ
s
d
B
s
H
=
∫
0
t
φ
s
δ
H
B
s
H
=
B
H
(
φ
)
.
In addition, for all φ
,
ψ
∈
|
ℋ
|,
(28)
E
(
∫
0
T
φ
s
d
B
s
H
∫
0
T
ψ
s
d
B
s
H
)
=
H
(
2
H
-
1
)
∬
0
T
φ
u
ψ
v
|
u
-
v
|
2
H
-
2
d
u
d
v
.


## 3. Asymptotic Behavior of the Least Squares Estimator

Throughout this paper we assumeH
∈
(
1
/
2
,
1
). We will study (1) driven by a fractional Brownian motion B
H with Hurst parameter H and α
>
0 being the unknown parameter to be estimated for discretely observed X. It is readily checked that we have the following explicit expression for X
t:
(29)
X
t
=
(
T
-
t
)
α
∫
0
t
(
T
-
s
)
-
α
d
B
s
H
,
0
≤
t
<
T
,
where the integral can be understood as Young integral. In order to study the asymptotic behavior of the least squares estimator, let us introduce the following processes:
(30)
A
t
:
=
∫
0
t
(
T
-
s
)
-
α
d
B
s
H
,
0
≤
t
<
T
.
Hence, we have
(31)
X
t
=
(
T
-
t
)
α
A
t
,
0
≤
t
<
T
.
For simplicity, we assume that the notation a
n
⊵
b
n means that there exists positive constants C
=
C
H
,
α
>
0 (depending only on H, α and its value may differ from line to line) so that
(32)
sup
⁡
n
≥
1
|
a
n
|
|
b
n
|
<
C
<
∞
.
We firstly give the following lemmas.Lemma 1.
Letα
>
0, 1
/
2
<
H
<
1. Then
(33)
∫
0
T
n
X
s
T
-
s
d
B
s
H
=
∫
0
T
n
X
s
T
-
s
δ
H
B
s
H
+
β
n
,
where
(34)
β
n
=
H
(
2
H
-
1
)
∫
0
T
n
∫
0
r
‍
(
T
-
r
)
α
-
1
(
T
-
s
)
-
α
(
r
-
s
)
2
H
-
2
d
s
d
r
,
lim
⁡
n
→
∞
β
n
=
H
B
(
α
,
2
H
-
1
)
T
2
H
-
1
.Proof.
By (26), we have
(35)
∫
0
T
n
X
s
T
-
s
d
B
s
H
=
∫
0
T
n
X
s
T
-
s
δ
H
B
s
H
+
H
(
2
H
-
1
)
×
∬
0
T
n
D
s
H
X
r
T
-
r
|
s
-
r
|
2
H
-
2
d
r
d
s
=
∫
0
T
n
X
s
T
-
s
δ
H
B
s
H
+
H
(
2
H
-
1
)
×
∫
0
T
n

∫
0
r

(
T
-
r
)
α
-
1
(
T
-
s
)
-
α
(
r
-
s
)
2
H
-
2

d
s
d
r
=
∫
0
T
n
X
s
T
-
s
δ
H
B
s
H
+
β
n
.
On the other hand,
(36)
lim
⁡
n
→
∞
β
n
=
H
(
2
H
-
1
)
×
lim
⁡
n
→
∞
∫
0
T
n
∫
0
r
(
T
-
r
)
α
-
1
(
T
-
s
)
-
α
(
r
-
s
)
2
H
-
2
d
s
d
r
=
H
(
2
H
-
1
)
lim
⁡
n
→
∞
∫
T
-
T
n
T
∫
r
T
‍
r
α
-
1
s
-
α
(
s
-
r
)
2
H
-
2
d
s
d
r
=
H
(
2
H
-
1
)
lim
⁡
n
→
∞
∫
0
T
n
∫
r
T
n
(
T
-
T
n
+
r
)
α
-
1
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
×
(
T
-
T
n
+
s
)
-
α
(
s
-
r
)
2
H
-
2
d
s
d
r
=
H
(
2
H
-
1
)
∫
0
T
∫
r
T
‍
r
α
-
1
s
-
α
(
s
-
r
)
2
H
-
2
d
s
d
r
=
H
(
2
H
-
1
)
∫
0
T
s
-
α
∫
0
s
r
α
-
1
(
s
-
r
)
2
H
-
2
d
r
d
s
=
H
B
(
α
,
2
H
-
1
)
T
2
H
-
1
.
This completes the proof.The following Lemma2 comes from Lemma 3.2 of Es-Sebaiy and Nourdin [5].Lemma 2.
Letting0
<
α
<
H, 1
/
2
<
H
<
1, one has
(37)
E
(
X
t
T
-
t
)
2
≤
H
(
2
H
-
1
)
H
-
α
B
(
1
-
α
,
2
H
-
1
)
(
T
-
t
)
2
α
-
2
T
2
H
-
2
α
,
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
0
≤
t
<
T
.Lemma 3.
Assume1
-
H
<
α
<
H, 1
/
2
<
H
<
1, and let F
T
n
=
∫
0
T
n
‍
(
X
t
/
(
T
-
t
)
)
δ
H
B
t
H. Then
(38)
lim
⁡
n
→
∞
E
(
F
T
n
2
)
=
H
2
(
2
H
-
1
)
2
B
(
α
,
2
H
-
1
)
B
(
1
-
α
,
2
H
-
1
)
2
(
H
+
α
-
1
)
(
H
-
α
)
T
4
H
-
2
.Proof.
By the isometry property of the double stochastic integralI
2, the variance of F
T
nis given by
(39)
E
(
F
T
n
2
)
=
H
2
(
2
H
-
1
)
2
2
I
T
n
,
where
(40)
I
T
n
=
∫
[
0
,
T
n
]
4
(
T
-
t
1
)
α
-
1
(
T
-
s
1
)
-
α
(
T
-
t
2
)
α
-
1
(
T
-
s
2
)
-
α
h
h
h
h
h
h
h
×
|
s
1
-
s
2
|
2
H
-
2
|
t
1
-
t
2
|
2
H
-
2
d
s
1
d
s
2
d
t
1
d
t
2
.
Now, we study I
T
n, by setting
(41)
I
1
=
∫
[
0
,
T
n
]
2
(
T
-
s
1
)
-
α
(
T
-
s
2
)
-
α
|
s
1
-
s
2
|
2
H
-
2
d
s
1
d
s
2
,
I
2
=
∫
[
0
,
T
n
]
2
(
T
-
t
1
)
α
-
1
(
T
-
t
2
)
α
-
1
|
t
1
-
t
2
|
2
H
-
2
d
t
1
d
t
2
.
We have I
T
n
=
I
1
I
2. By (
17.40
) of Es-Sebaiy and Nourdin [5], we have
(42)
I
1
=
∫
[
0
,
T
n
]
2
(
T
-
s
1
)
-
α
(
T
-
s
2
)
-
α
|
s
1
-
s
2
|
2
H
-
2
d
s
1
d
s
2
=
B
(
1
-
α
,
2
H
-
1
)
H
-
α
T
n
2
H
-
2
α
⟶
B
(
1
-
α
,
2
H
-
1
)
H
-
α
T
2
H
-
2
α
,
n
⟶
∞
.
Similarly
(43)
I
2
⟶
B
(
α
,
2
H
-
1
)
H
+
α
-
1
T
2
H
+
2
α
-
2
.
Thus, the proof is finished.The following theorem gives the consistency of the least squares estimatorα
^
n of α.Theorem 4.
Let1
/
2
<
α
<
H
<
1. If Δ
n
→
0, T
n
=
n
Δ
n
→
T as n
→
∞, and T
n
+
Δ
n
=
T, then, one has
(44)
α
^
n
⟶
P
α
,
n
⟶
∞
,
where →
P means convergence in probability.Proof.
By (4), we have
(45)
α
^
n
-
α
=
-
(
α
/
n
)
∑
i
=
1
n
M
i
(
α
Δ
n
/
n
)
∑
i
=
1
n
(
X
t
i
-
1
2
/
(
T
-
t
i
-
1
)
2
)
.
Letting0
<
ɛ
<
1, we obtain
(46)
P
(
|
α
^
n
-
α
|
>
ɛ
)
=
P
(
|
(
α
/
n
)
∑
i
=
1
n
M
i
(
α
Δ
n
/
n
)
∑
i
=
1
n
(
X
t
i
-
1
2
/
(
T
-
t
i
-
1
)
2
)
|
>
ɛ
)
≤
P
(
|
α
n
∑
i
=
1
n
M
i
|
>
ɛ
(
1
-
ɛ
)
)
+
P
(
|
α
Δ
n
n
∑
i
=
1
n
X
t
i
-
1
2
(
T
-
t
i
-
1
)
2
-
1
|
>
ɛ
)
:
=
B
1
(
n
)
+
B
2
(
n
)
.
First, we considering the term B
1
(
n
), we have
(47)
B
1
(
n
)
=
P
(
|
α
n
∑
i
=
1
n
M
i
|
>
ɛ
(
1
-
ɛ
)
)
≤
P
(
|
α
n
∑
i
=
1
n
[
M
i
-
∫
t
i
-
1
t
i
X
t
i
-
1
T
-
t
i
-
1
δ
H
B
t
H
]
|
>
1
3
ɛ
(
1
-
ɛ
)
)
+
P
(
|
α
n
∑
i
=
1
n
∫
t
i
-
1
t
i
‍
(
X
t
i
-
1
T
-
t
i
-
1
-
X
t
T
-
t
)
δ
H
B
t
H
|
>
1
3
ɛ
(
1
-
ɛ
)
)
+
P
(
|
α
n
∫
0
T
n
X
t
T
-
t
δ
H
B
t
H
|
>
1
3
ɛ
(
1
-
ɛ
)
)
:
=
B
1,1
(
n
)
+
B
1,2
(
n
)
+
B
1,3
(
n
)
.
For the term B
1,1
(
n
), using Lemma 2, we obtain
(48)
∑
i
=
1
n
E
|
[
M
i
-
∫
t
i
-
1
t
i
X
t
i
-
1
T
-
t
i
-
1
δ
H
B
t
H
]
|
≤
α
∑
i
=
1
n
(
E
(
X
t
i
-
1
T
-
t
i
-
1
)
2
)
1
/
2
h
h
h
h
h
h
h
×
∫
t
i
-
1
t
i
(
E
(
X
t
i
-
1
T
-
t
i
-
1
-
X
t
T
-
t
)
2
)
1
/
2
d
t
⊵
∑
i
=
1
n
(
T
-
t
i
-
1
)
α
-
1
∫
t
i
-
1
t
i
(
E
(
X
t
i
-
1
T
-
t
i
-
1
-
X
t
T
-
t
)
2
)
1
/
2
d
t
≤
Δ
n
α
-
1
∑
i
=
1
n
∫
t
i
-
1
t
i
(
E
(
X
t
i
-
1
T
-
t
i
-
1
-
X
t
T
-
t
)
2
)
1
/
2
d
t
≤
Δ
n
α
-
1
[
∑
i
=
1
n
∫
t
i
-
1
t
i
(
E
(
X
t
i
-
1
T
-
t
i
-
1
)
2
)
1
/
2
d
t
h
h
h
h
h
h
h
h
+
∑
i
=
1
n
∫
t
i
-
1
t
i
(
E
(
X
t
T
-
t
)
2
)
1
/
2
d
t
∑
i
=
1
n
∫
t
i
-
1
t
i
(
E
(
X
t
i
-
1
T
-
t
i
-
1
)
2
)
1
/
2
d
t
]
≤
Δ
n
α
-
1
[
∑
i
=
1
n
∫
t
i
-
1
t
i
(
E
(
X
t
i
-
1
T
-
t
i
-
1
)
2
)
1
/
2
d
t
h
h
h
h
h
h
h
h
+
∑
i
=
1
n
∫
t
i
-
1
t
i
(
E
(
X
t
i
T
-
t
i
)
2
)
1
/
2
d
t
]
⊵
n
Δ
n
2
α
-
1
.
So, we get
(49)
α
n
∑
i
=
1
n
E
|
[
M
i
-
∫
t
i
-
1
t
i
X
t
i
-
1
T
-
t
i
-
1
δ
H
B
t
H
]
|
⊵
Δ
n
2
α
-
1
.
Hence,
(50)
B
1,1
(
n
)
⊵
Δ
n
2
α
-
1
ɛ
(
1
-
ɛ
)
.
For the term B
1,2
(
n
), it follows the fact that, for 0
≤
t
<
T,
(51)
X
t
i
-
1
T
-
t
i
-
1
-
X
t
T
-
t
=
-
[
(
(
T
-
t
)
α
-
1
-
(
T
-
t
i
-
1
)
α
-
1
)
A
t
i
-
1
h
h
h
h
h
+
(
T
-
t
)
α
-
1
(
A
t
-
A
t
i
-
1
)
]
.
We have
(52)
E
|
∑
i
=
1
n
‍
∫
t
i
-
1
t
i
(
X
t
i
-
1
T
-
t
i
-
1
-
X
t
T
-
t
)
δ
H
B
t
H
|
≤
E
|
∑
i
=
1
n
∫
t
i
-
1
t
i
‍
(
(
T
-
t
)
α
-
1
-
(
T
-
t
i
-
1
)
α
-
1
)
A
t
i
-
1
δ
H
B
t
H
|
+
E
|
∑
i
=
1
n
∫
t
i
-
1
t
i
‍
(
T
-
t
)
α
-
1
(
A
t
-
A
t
i
-
1
)
δ
H
B
t
H
|
.
Using inequality (22) and E
A
t
=
0, D
s
H
A
t
=
(
T
-
s
)
-
α
1
[
0
,
t
]
(
s
), we have
(53)
E
|
∑
i
=
1
n
‍
∫
t
i
-
1
t
i
‍
(
(
T
-
t
)
α
-
1
-
(
T
-
t
i
-
1
)
α
-
1
)
A
t
i
-
1
δ
H
B
t
H
|
=

E
|
∫
0
T
n

∑
i
=
1
n
(
(
T
-
t
)
α
-
1
-
(
T
-
t
i
-
1
)
α
-
1
)

A
t
i
-
1
1
(
t
i
-
1
,
t
i
]
(
t
)
δ
H
B
t
H
|
≤
(
×
A
t
i
-
1
1
(
t
i
-
1
,
t
i
]
(
t
)
δ
H
B
t
H
∫
0
T
n
∑
i
=
1
n
(
(
T
-
t
)
α
-
1
-
(
T
-
t
i
-
1
)
α
-
1
)
|
2
E
|
∫
0
T
n
∑
i
=
1
n
(
(
T
-
t
)
α
-
1
-
(
T
-
t
i
-
1
)
α
-
1
)
h
h
h
h
h
h
h
h
h
h
h
×
A
t
i
-
1
1
(
t
i
-
1
,
t
i
]
(
t
)
δ
H
B
t
H
∫
0
T
n
∑
i
=
1
n
(
(
T
-
t
)
α
-
1
-
(
T
-
t
i
-
1
)
α
-
1
)
|
2
)
1
/
2
≤
C
H
(
∬
0
T
n
|
∑
i
=
1
n
‍
(
(
T
-
t
)
α
-
1
-
(
T
-
t
i
-
1
)
α
-
1
)
h
h
h
h
h
h
h
h
h
h
h
h
h
×
D
s
H
A
t
i
-
1
1
(
t
i
-
1
,
t
i
]
(
t
)
∬
0
T
n
|
∑
i
=
1
n
‍
(
(
T
-
t
)
α
-
1
-
(
T
-
t
i
-
1
)
α
-
1
)
|
1
/
H
d
s
d
t
∬
0
T
n
|
∑
i
=
1
n
‍
(
(
T
-
t
)
α
-
1
-
(
T
-
t
i
-
1
)
α
-
1
)
)
H
=
C
H
(
∬
0
T
n
∑
i
=
1
n
|
(
(
T
-
t
)
α
-
1
-
(
T
-
t
i
-
1
)
α
-
1
)
(
T
-
s
)
-
α
|
1
/
H
h
h
h
h
h
h
h
h
h
h
h
h
×
1
(
t
i
-
1
,
t
i
]
(
t
)
1
[
0
,
t
i
-
1
)
(
s
)
d
s
d
t
∬
0
T
n
∑
i
=
1
n
|
(
(
T
-
t
)
α
-
1
-
(
T
-
t
i
-
1
)
α
-
1
)
(
T
-
s
)
-
α
|
1
/
H
)
H
=
C
H
(
∑
i
=
1
n
∫
t
i
-
1
t
i
(
(
T
-
t
)
α
-
1
-
(
T
-
t
i
-
1
)
α
-
1
)
1
/
H
d
t
h
h
h
h
h
h
h
h
h
×
∫
0
t
i
-
1
(
T
-
s
)
-
α
/
H
d
s
∑
)
H
≤
C
H
T
H
-
α
(
∑
i
=
1
n
∫
t
i
-
1
t
i
Δ
n
(
α
-
1
)
/
H
d
t
)
H
≤
C
H
T
H
-
α
n
Δ
n
H
+
α
-
1
.
On the other hand,
(54)
E
|
∑
i
=
1
n
‍
∫
t
i
-
1
t
i
(
T
-
t
)
α
-
1
(
A
t
-
A
t
i
-
1
)
δ
H
B
t
H
|
=
E
|
∫
0
T
n
∑
i
=
1
n
(
T
-
t
)
α
-
1
(
A
t
-
A
t
i
-
1
)
1
(
t
i
-
1
,
t
i
]
(
t
)
δ
H
B
t
H
|
≤
C
H
(
∬
0
T
n
|
∑
i
=
1
n
(
T
-
t
)
α
-
1
D
s
H
(
A
t
-
A
t
i
-
1
)
h
h
h
h
h
h
h
h
h
h
h
h
×
1
(
t
i
-
1
,
t
i
]
(
t
)
∬
0
T
n
|
∑
i
=
1
n
(
T
-
t
)
α
-
1
D
s
H
(
A
t
-
A
t
i
-
1
)
|
1
/
H
d
s
d
t
∬
0
T
n
|
∑
i
=
1
n
(
T
-
t
)
α
-
1
D
s
H
(
A
t
-
A
t
i
-
1
)
)
H
≤
C
H
(
∬
0
T
n
∑
i
=
1
n
‍
|
(
T
-
t
)
α
-
1
D
s
H
(
A
t
-
A
t
i
-
1
)
|
1
/
H
h
h
h
h
h
h
h
h
h
h
h
h
×
1
(
t
i
-
1
,
t
i
]
(
t
)
d
s
d
t
∬
0
T
n
∑
i
=
1
n
‍
)
H
=
C
H
(
∬
0
T
n
∑
i
=
1
n
‍
(
(
T
-
t
)
α
-
1
(
T
-
s
)
-
α
)
H
1
[
t
i
-
1
,
t
]
(
s
)
h
h
h
h
h
h
h
h
h
h
h
h
×
1
(
t
i
-
1
,
t
i
]
(
t
)
d
s
d
t
∬
0
T
n
∑
i
=
1
n
‍
)
H
=
C
H
(
∑
i
=
1
n
∫
t
i
-
1
t
i
‍
(
T
-
t
)
(
α
-
1
)
/
H
d
t
∫
t
i
-
1
t
(
T
-
s
)
-
α
/
H
d
s
)
H
≤
C
H
(
∑
i
=
1
n
∫
t
i
-
1
t
i
(
T
-
t
)
(
α
-
1
)
/
H
d
t
∫
t
i
-
1
t
(
T
-
t
n
)
-
α
/
H
d
s
)
H
≤
C
H
(
n
Δ
n
(
2
H
-
1
)
/
H
)
H
≤
C
H
n
Δ
n
2
H
-
1
.
So, we get
(55)
E
|
∑
i
=
1
n
∫
t
i
-
1
t
i
‍
(
X
t
i
-
1
T
-
t
i
-
1
-
X
t
T
-
t
)
δ
H
B
t
H
|
⊵
n
Δ
n
H
+
α
-
1
.
Thus,
(56)
α
n
E
|
∑
i
=
1
n
∫
t
i
-
1
t
i
‍
(
X
t
i
-
1
T
-
t
i
-
1
-
X
t
T
-
t
)
δ
H
B
t
H
|
⊵
Δ
n
H
+
α
-
1
.
Hence
(57)
B
1,2
(
n
)
⊵
Δ
n
H
+
α
-
1
ɛ
(
1
-
ɛ
)
.
For the term B
1,3
(
n
), by setting F
T
n
=
∫
0
T
n
(
X
t
/
(
T
-
t
)
)
δ
H
B
t
H and by using Lemma 3, we get
(58)
B
1,3
(
n
)
=
P
(
|
α
n
∫
0
T
n
X
t
T
-
t
δ
H
B
t
H
|
>
1
3
ɛ
(
1
-
ɛ
)
)
≤
[
3
α
ɛ
(
1
-
ɛ
)
n
]
2
E
(
F
T
n
2
)
⊵
1
ɛ
2
(
1
-
ɛ
)
2
n
2
.
As a consequence,
(59)
B
1
(
n
)
⊵
Δ
n
2
α
-
1
ɛ
(
1
-
ɛ
)
+
Δ
n
H
+
α
-
1
ɛ
(
1
-
ɛ
)
+
1
ɛ
2
(
1
-
ɛ
)
2
n
2
.
Second, we estimate the termB
2
(
n
):
(60)
B
2
(
n
)
=
P
(
|
α
Δ
n
n
∑
i
=
1
n
(
X
t
i
-
1
T
-
t
i
-
1
)
2
-
1
|
>
ɛ
)
≤
P
(
|
α
n
∑
i
=
1
n
∫
t
i
-
1
t
i
[
(
X
t
i
-
1
T
-
t
i
-
1
)
2
-
(
X
t
T
-
t
)
2
]
d
t
|
>
ɛ
/
2
)
+
P
(
|
α
n
∫
0
T
n
(
X
t
T
-
t
)
2
d
t
-
1
|
>
ɛ
/
2
)
:
=
B
2,1
(
n
)
+
B
2,2
(
n
)
.
We firstly consider B
2,1
(
n
), since
(61)
E
|
α
n
∑
i
=
1
n
∫
t
i
-
1
t
i
[
(
X
t
i
-
1
T
-
t
i
-
1
)
2
-
(
X
t
T
-
t
)
2
]
|
d
t
≤
α
n
∑
i
=
1
n
∫
t
i
-
1
t
i
E
|
(
X
t
i
-
1
T
-
t
i
-
1
)
2
-
(
X
t
T
-
t
)
2
|
d
t
≤
α
n
∑
i
=
1
n
∫
t
i
-
1
t
i
(
E
(
X
t
i
-
1
T
-
t
i
-
1
)
2
+
E
(
X
t
T
-
t
)
2
)
d
t
≤
α
n
∑
i
=
1
n
∫
t
i
-
1
t
i
(
E
(
X
t
i
-
1
T
-
t
i
-
1
)
2
+
E
(
X
t
i
T
-
t
i
)
2
)
d
t
≤
2
α
n
∑
i
=
1
n
Δ
n
2
α
-
1
⊵
Δ
n
2
α
-
1
.
By Markov inequality, we obtain
(62)
B
2,1
(
n
)
⊵
Δ
n
2
α
-
1
ɛ
.
Now, we estimate the term B
2,2
(
n
). Applying the change of variable formula (24), we get
(63)
α
n
∫
0
T
n
(
X
t
T
-
t
)
2
d
t
-
1
=
1
n
(
α
-
(
1
/
2
)
)
×
(
X
T
n
2
Δ
n
-
∫
0
T
n
‍
X
t
T
-
t
δ
H
B
t
H
-
β
n
)
.
Hence,
(64)
B
2,2
(
n
)
≤
P
(
|
X
T
n
T
n
(
2
α
-
1
)
|
>
ɛ
6
)
+
P
(
|
1
n
(
α
-
(
1
/
2
)
)
∫
0
T
n
X
t
T
-
t
δ
B
t
H
|
>
ɛ
6
)
+
P
(
|
β
n
n
(
α
-
(
1
/
2
)
)
|
>
ɛ
6
)
.
By Markov inequality and Lemma 2, we obtain
(65)
B
2,2
(
n
)
⊵
Δ
n
2
α
ɛ
2
T
n
2
+
1
ɛ
n
2
+
1
ɛ
n
.
Therefore(66)
B
2
(
n
)
⊵
Δ
n
2
α
-
1
ɛ
+
Δ
n
2
α
ɛ
2
T
n
2
+
1
ɛ
n
2
+
1
ɛ
n
≤
Δ
n
2
α
-
1
ɛ
+
Δ
n
2
α
ɛ
2
T
n
2
+
1
ɛ
n
.
Combining (59) and (66), this completes the proof.



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*Source: 748376-2014-01-23.xml*