# A Note on the Rate of Strong Convergence for Weighted Sums of Arrays of Rowwise Negatively Orthant Dependent Random Variables

Authors: Qingxia Zhang; Dingcheng Wang
Journal: Discrete Dynamics in Nature and Society
 (2014)

---

## Abstract

Let{
X
n
i
;
i
≥
1
,
n
≥
1
} be an array of rowwise negatively orthant dependent (NOD) random variables. The authors discuss the rate of strong convergence for weighted sums of arrays of rowwise NOD random variables and solve an open problem posed by Huang and Wang (2012).


---

## Body

## 1. Introduction

Firstly, let us recall the definitions of negatively associated (NA) random variables and NOD random variables as follows.Definition 1.
A finite collection of random variables{
X
i
;
1
≤
i
≤
n
} is said to be NA if for every pair of disjoint subsets A
1 and A
2 of {
1,2
,
…
,
n
},
(1)
Cov
(
f
1
(
X
i
,
i
∈
A
1
)
,
f
2
(
X
j
,
j
∈
A
2
)
)
≤
0
,
whenever f
1 and f
2 are nondecreasing functions such that the covariance exists. An infinite collection of random variables {
X
i
;
i
≥
1
} is NA if every finite subcollection is NA.An array of random variables{
X
n
i
;
i
≥
1
,
n
≥
1
} is called rowwise NA random variables if for every n
≥
1
,
{
X
n
i
;
i
≥
1
} is a sequence of NA random variables.Definition 2.
A finite collection of random variables{
X
i
;
1
≤
i
≤
n
} is said to be NOD if
(2)
P
(
X
1
≤
x
1
,
X
2
≤
x
2
,
…
,
X
n
≤
x
n
)
≤
∏
j
=
1
n
P
(
X
j
≤
x
j
)
,
P
(
X
1
>
x
1
,
X
2
>
x
2
,
…
,
X
n
>
x
n
)
≤
∏
j
=
1
n
P
(
X
j
>
x
j
)
,
for all x
1
,
x
2
,
…
,
x
n
∈
R. An infinite collection of random variables {
X
i
;
i
≥
1
} is said to be NOD if every finite subcollection is NOD.An array of random variables{
X
n
i
;
i
≥
1
,
n
≥
1
} is called rowwise NOD random variables if for every n
≥
1
,
{
X
n
i
;
i
≥
1
} is a sequence of NOD random variables.The concepts of NA and NOD random variables were introduced by Joag-Dev and Proschan [1]. Obviously, independent random variables are NOD, and NA implies NOD from the definition of NA and NOD, but NOD does not imply NA. So, NOD is much weaker than NA. Because of the wide applications of NOD random variables, the notion of NOD random variables has been received more and more attention recently. Many applications have been found. We can refer to Volodin [2], Asadian et al. [3], Amini et al. [4, 5], Kuczmaszewska [6], Zarei and Jabbari [7], Wu and Zhu [8], Wu [9], Sung [10], Wang et al. [11], Huang and Wang [12], and so forth. Hence, it is very significant to study limit properties of this wider NOD random variables in probability theory and practical applications.Let{
X
n
;
n
≥
1
} be a sequence of independent and identically distributed (i.i.d.) random variables and let {
a
n
i
;
i
≥
1
,
n
≥
1
} be an array of real constants. As Bai and Cheng [13] remarked, many useful linear statistics, for example, least-squares estimators, nonparametric regression function estimators, and jackknife estimates, are based on weighted sums of i.i.d. random variables. In this respect, the strong convergence for weighted sums ∑
i
=
1
n
a
n
i
X
i has been studied by many authors (see, e.g., Bai and Cheng [13]; Cuzick [14]; Sung [15]; Tang [16]; etc.).Cai [17] proved the following complete convergence result for weighted sums of NA random variables.Theorem A.
Let{
X
,
X
n
;
n
≥
1
} be a sequence of identically distributed NA random variables, and let {
a
n
i
;
1
≤
i
≤
n
,
n
≥
1
} be an array of real constants satisfying
(3)
A
α
=
limsup
⁡
n
⟶
∞
A
α
,
n
<
∞
,
A
α
,
n
=
1
n
∑
i
=
1
n
|
a
n
i
|
α
,
for some 0
<
α
≤
2. Suppose that E
X
=
0 when 1
<
α
≤
2. If
(4)
E
{
exp
⁡
(
h
|
X
|
γ
)
}
<
∞
for
some
h
>
0
,
γ
>
0
,
then, for b
n
=
n
1
/
α
(
log
⁡
⁡
n
)
1
/
γ,
(5)
∑
n
=
1
∞
1
n
P
(
max
⁡
1
≤
j
≤
n
⁡
|
∑
i
=
1
j
a
n
i
X
i
|
>
ε
b
n
)
<
∞
∀
ε
>
0
.Wang et al. [11] extended the above result of Cai [17] to arrays of rowwise NOD random variables as follows.Theorem B.
Let{
X
n
i
;
i
≥
1
,
n
≥
1
} be an array of rowwise NOD random variables which is stochastically dominated by a random variable X and let {
a
n
i
;
1
≤
i
≤
n
,
n
≥
1
} be an array of real constants. Assume that there exist some δ with 0
<
δ
<
1 and some α with 0
<
α
<
2 such that ∑
i
=
1
n
|
a
n
i
|
α
=
O
(
n
δ
) and assume further that E
X
n
i
=
0 if 1
<
α
<
2. If for some h
>
0 and γ
>
0 such that (4), then
(6)
∑
n
=
1
∞
n
α
p
-
2
P
(
max
⁡
1
≤
j
≤
n
⁡
|
∑
i
=
1
j
a
n
i
X
n
i
|
>
ε
b
n
)
<
∞
∀
ε
>
0
,
where p
≥
1
/
α and b
n
=
n
1
/
α
(
log
⁡
⁡
n
)
1
/
γ.Recently, Huang and Wang [12] partially extended the corresponding theorems of Cai [17] and Wang et al. [11] to NOD random variables under a mild moment condition.Theorem C.
Let{
X
n
;
n
≥
1
} be a sequence of NOD random variables which is stochastically dominated by a random variable X and let {
a
n
i
;
i
≥
1
,
n
≥
1
} be a triangular array of real constants such that a
n
i
=
0 for i
>
n. Let
(7)
A
β
=
limsup
⁡
n
⟶
∞
A
β
,
n
<
∞
;
A
β
,
n
=
n
-
1
∑
i
=
1
n
|
a
n
i
|
β
,
where β
=
max
⁡
(
α
,
γ
) for some 0
<
α
≤
2, γ
>
0, and α
≠
γ. Assume that E
X
n
=
0 for 1
<
α
≤
2 and E
|
X
|
β
<
∞. Then,
(8)
∑
n
=
1
∞
1
n
P
(
|
∑
i
=
1
n
a
n
i
X
i
|
>
ε
b
n
)
<
∞
,
where b
n
=
n
1
/
α
(
log
⁡
⁡
n
)
1
/
γ.As Huang and Wang [12] pointed out, Theorem C partially extends only the case of α
>
γ of Theorems A and B. They left an open problem whether the case of α
=
γ of Theorem C holds for NOD random variables.The main purpose of this paper is to further study strong convergence for weighted sums of NOD random variables and to obtain the rate of strong convergence for weighted sums of arrays of rowwise NOD random variables under a suitable moment condition. We solve the above problem posed by Huang and Wang [12].We will use the following concept in this paper.Definition 3.
An array of random variables{
X
n
i
;
i
≥
1
,
n
≥
1
} is said to be stochastically dominated by a random variable X if there exists a positive constant C such that
(9)
P
(
|
X
n
i
|
>
t
)
≤
C
P
(
|
X
|
>
t
)
,
for all t
≥
0
,
i
≥
1, and n
≥
1.


## 2. Main Results

Now, we will present the main results of this paper; the detailed proofs will be given in the next section.Theorem 4.
Let{
X
n
i
;
i
≥
1
,
n
≥
1
} be an array of rowwise NOD random variables which is stochastically dominated by a random variable X and let {
a
n
i
;
1
≤
i
≤
n
,
n
≥
1
} be an array of real constants satisfying ∑
i
=
1
n
|
a
n
i
|
α
=
O
(
n
) for some 0
<
α
≤
2. Assume further that E
X
n
i
=
0 for 1
<
α
≤
2 and E
|
X
|
α
log
⁡
(
1
+
|
X
|
)
<
∞. Then,
(10)
∑
n
=
1
∞
n
-
1
P
(
|
∑
i
=
1
n
a
n
i
X
n
i
|
>
ε
b
n
)
<
∞
∀
ε
>
0
,
where b
n
=
n
1
/
α
(
log
⁡
⁡
n
)
1
/
α.Similar to the proof of Theorem4, we can obtain the following result for NOD random variable sequences.Corollary 5.
Let{
X
n
;
n
≥
1
} be a sequence of NOD random variables which is stochastically dominated by a random variable X and let {
a
n
i
;
1
≤
i
≤
n
,
n
≥
1
} be an array of real constants satisfying ∑
i
=
1
n
|
a
n
i
|
α
=
O
(
n
) for some 0
<
α
≤
2. Assume further that E
X
n
=
0 for 1
<
α
≤
2 and E
|
X
|
α
log
⁡
(
1
+
|
X
|
)
<
∞. Then,
(11)
∑
n
=
1
∞
n
-
1
P
(
|
∑
i
=
1
n
a
n
i
X
i
|
>
ε
b
n
)
<
∞
∀
ε
>
0
,
where b
n
=
n
1
/
α
(
log
⁡
⁡
n
)
1
/
α.Remark 6.
In Theorem4 and Corollary 5, we consider the case of α
=
γ for 0
<
α
≤
2 and obtain some strong convergence results for arrays of rowwise NOD random variables and NOD random variable sequences without assumption of identical distribution. The main result settles the open problem posed by Huang and Wang [12]. In addition, it is still an open problem whether
(12)
∑
n
=
1
∞
n
-
1
P
(
max
⁡
1
≤
j
≤
n
⁡
|
∑
i
=
1
j
a
n
i
X
n
i
|
>
ε
b
n
)
<
∞
∀
ε
>
0
holds true under the same moment condition of Theorem 4.


## 3. Proofs

In order to prove our main results, the following lemmas are needed.Lemma 7 (see Bozorgnia et al. [18]).
Let{
X
i
;
1
≤
i
≤
n
} be a sequence of NOD random variables, and let {
f
i
;
1
≤
i
≤
n
} be a sequence of Borel functions all of which are monotone nondecreasing (or all are monotone nonincreasing). Then, {
f
i
(
X
i
)
;
1
≤
i
≤
n
} is a sequence of NOD random variables.Lemma 8 (see Asadian et al. [3]).
LetM
≥
2 and let {
X
n
;
n
≥
1
} be a sequence of NOD random variables with E
X
n
=
0 and E
|
X
n
|
M
<
∞ for all n
≥
1. Then, there exists a positive constant C
=
C
(
M
) depending only on M such that, for all n
≥
1,
(13)
E
(
|
∑
i
=
1
n
X
i
|
M
)
≤
C
[
∑
i
=
1
n
E
|
X
i
|
M
+
(
∑
i
=
1
n
E
X
i
2
)
M
/
2
]
.Lemma 9.
Let{
X
n
;
n
≥
1
} be a sequence of random variables which is stochastically dominated by a random variable X. For any u
>
0 and t
>
0, the following two statements hold:
(14)
E
|
X
n
i
|
u
I
(
|
X
n
i
|
≤
t
)
≤
C
1
(
E
|
X
|
u
I
(
|
X
|
≤
t
)
+
t
u
P
(
|
X
|
>
t
)
)
,
(15)
E
|
X
n
i
|
u
I
(
|
X
n
i
|
>
t
)
≤
C
2
E
|
X
|
u
I
(
|
X
|
>
t
)
,
where C
1 and C
2 are positive constants.Lemma 10 (see Sung [15]).
LetX be a random variable and let {
a
n
i
;
1
≤
i
≤
n
,
n
≥
1
} be an array of real constants satisfying ∑
i
=
1
n
|
a
n
i
|
α
=
O
(
n
) for some α
>
0. Let b
n
=
n
1
/
α
(
log
⁡
⁡
n
)
1
/
γ for some γ
>
0. Then,
(16)
∑
n
=
1
∞
n
-
1
∑
i
=
1
n
P
(
|
a
n
i
X
|
>
b
n
)
≤
{
C
E
|
X
|
α
,
for
α
>
γ
,
C
E
|
X
|
α
log
⁡
(
1
+
|
X
|
)
,
for
α
=
γ
,
C
E
|
X
|
γ
,
for
α
<
γ
.Lemma 11 (see Sung [19]).
LetX be a random variable and let {
a
n
i
;
1
≤
i
≤
n
,
n
≥
1
} be an array of real constants satisfying a
n
i
=
0 or |
a
n
i
|
>
1 and ∑
i
=
1
n
|
a
n
i
|
α
=
O
(
n
) for some α
>
0. Let b
n
=
n
1
/
α
(
log
⁡
⁡
n
)
1
/
α. If q
>
α, then
(17)
∑
n
=
1
∞
n
-
1
b
n
-
q
∑
i
=
1
n
E
|
a
n
i
X
|
q
I
(
|
a
n
i
X
|
≤
b
n
)
≤
C
E
|
X
|
α
log
⁡
(
1
+
|
X
|
)
.Throughout this paper, letI
(
A
) be the indicator function of the set A. C denotes a positive constant, which may be different in various places and a
n
=
O
(
b
n
) stands for a
n
≤
C
b
n.Proof of Theorem4.
Without loss of generality, suppose that∑
i
=
1
n
|
a
n
i
|
α
≤
C
n and a
n
i
≥
0, for all 1
≤
i
≤
n
,
n
≥
1. For fixed n
≥
1, define
(18)
X
i
(
n
)
=
-
b
n
I
(
a
n
i
X
n
i
<
-
b
n
)
+
a
n
i
X
n
i
I
(
|
a
n
i
X
n
i
|
≤
b
n
)
+
b
n
I
(
a
n
i
X
n
i
>
b
n
)
,
i
≥
1
,
T
n
(
n
)
=
∑
i
=
1
n
(
X
i
(
n
)
-
E
X
i
(
n
)
)
.
Denote
(19)
A
=
⋂
i
=
1
n
‍
(
a
n
i
X
n
i
=
X
i
(
n
)
)
,
B
=
A
-
=
⋃
i
=
1
n
‍
(
a
n
i
X
n
i
≠
X
i
(
n
)
)
=
⋃
i
=
1
n
‍
(
|
a
n
i
X
n
i
|
>
b
n
)
,
E
n
=
(
|
∑
i
=
1
n
a
n
i
X
n
i
|
>
ε
b
n
)
.
It is easily seen that, for all ε
>
0,
(20)
E
n
=
E
n
A
⋃
‍
E
n
B
⊂
(
|
∑
i
=
1
n
‍
X
i
(
n
)
|
>
ε
b
n
)
⋃
(
⋃
i
=
1
n
‍
|
a
n
i
X
n
i
|
>
b
n
)
,
which implies that
(21)
P
(
E
n
)
≤
P
(
|
∑
i
=
1
n
‍
X
i
(
n
)
|
>
ε
b
n
)
+
P
(
⋃
i
=
1
n
‍
|
a
n
i
X
n
i
|
>
b
n
)
≤
P
(
|
T
n
(
n
)
|
>
ε
b
n
-
|
∑
i
=
1
n
E
X
i
(
n
)
|
)
+
∑
i
=
1
n
P
(
|
a
n
i
X
n
i
|
>
b
n
)
.
First, we will prove that
(22)
b
n
-
1
|
∑
i
=
1
n
E
X
i
(
n
)
|
⟶
0
,
as
n
⟶
∞
.
Actually, for 0
<
α
≤
1, by (14) of Lemma 9, Markov inequality, and E
|
X
|
α
log
⁡
(
1
+
|
X
|
)
<
∞, we have that
(23)
b
n
-
1
|
∑
i
=
1
n
E
X
i
(
n
)
|
≤
C
b
n
-
1
∑
i
=
1
n
|
E
X
i
(
n
)
|
≤
C
b
n
-
1
∑
i
=
1
n
E
|
a
n
i
X
n
i
|
I
(
|
a
n
i
X
n
i
|
≤
b
n
)
+
C
∑
i
=
1
n
P
(
|
a
n
i
X
n
i
|
>
b
n
)
≤
C
b
n
-
1
∑
i
=
1
n
(
E
|
a
n
i
X
|
I
(
|
a
n
i
X
|
≤
b
n
)
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
+
b
n
P
(
|
a
n
i
X
|
>
b
n
)
)
+
C
∑
i
=
1
n
P
(
|
a
n
i
X
|
>
b
n
)
≤
C
b
n
-
1
∑
i
=
1
n
(
E
|
a
n
i
X
|
I
(
|
a
n
i
X
|
≤
b
n
)
)
+
C
∑
i
=
1
n
P
(
|
a
n
i
X
|
>
b
n
)
≤
C
b
n
-
α
∑
i
=
1
n
(
E
|
a
n
i
X
|
α
I
(
|
a
n
i
X
|
≤
b
n
)
)
+
C
b
n
-
α
∑
i
=
1
n
E
|
a
n
i
X
|
α
≤
C
b
n
-
α
∑
i
=
1
n
|
a
n
i
|
α
E
|
X
|
α
+
C
b
n
-
α
∑
i
=
1
n
|
a
n
i
|
α
E
|
X
|
α
≤
C
(
log
⁡
⁡
n
)
-
1
E
|
X
|
α
⟶
0
as
n
⟶
∞
.
Next, for 1
<
α
≤
2, by E
X
n
i
=
0, (15) of Lemmas 9 and 10, Markov inequality, and E
|
X
|
α
log
⁡
(
1
+
|
X
|
)
<
∞, we also have that
(24)
b
n
-
1
|
∑
i
=
1
n
E
X
i
(
n
)
|
≤
C
∑
i
=
1
n
P
(
|
a
n
i
X
n
i
|
>
b
n
)
+
C
b
n
-
1
|
∑
i
=
1
n
E
a
n
i
X
n
i
I
(
|
a
n
i
X
n
i
|
>
b
n
)
|
≤
C
∑
i
=
1
n
P
(
|
a
n
i
X
|
>
b
n
)
+
C
b
n
-
1
∑
i
=
1
n
E
|
a
n
i
X
n
i
|
I
(
|
a
n
i
X
n
i
|
>
b
n
)
≤
C
b
n
-
α
∑
i
=
1
n
E
|
a
n
i
X
|
α
+
C
b
n
-
1
∑
i
=
1
n
E
|
a
n
i
X
|
I
(
|
a
n
i
X
|
>
b
n
)
≤
C
b
n
-
α
∑
i
=
1
n
E
|
a
n
i
X
|
α
+
C
b
n
-
α
∑
i
=
1
n
E
|
a
n
i
X
|
α
I
(
|
a
n
i
X
|
>
b
n
)
≤
C
b
n
-
α
∑
i
=
1
n
E
|
a
n
i
X
|
α
+
C
b
n
-
α
∑
i
=
1
n
E
|
a
n
i
X
|
α
≤
C
(
log
⁡
⁡
n
)
-
1
E
|
X
|
α
⟶
0
as
n
⟶
∞
.
From the above statements, we can get (22) immediately. Hence, for n large enough,
(25)
P
(
E
n
)
≤
P
(
|
T
n
(
n
)
|
>
ε
b
n
2
)
.
To prove (10), it is sufficient to show that
(26)
I
≜
∑
n
=
1
∞
n
-
1
P
(
|
T
n
(
n
)
|
>
ε
b
n
2
)
<
∞
,
J
≜
∑
n
=
1
∞
n
-
1
∑
i
=
1
n
‍
P
(
|
a
n
i
X
n
i
|
>
b
n
)
<
∞
.
It follows from Lemma 10 and E
|
X
|
α
log
⁡
(
1
+
|
X
|
)
<
∞ that
(27)
J
≜
∑
n
=
1
∞
n
-
1
∑
i
=
1
n
P
(
|
a
n
i
X
n
i
|
>
b
n
)
≤
C
∑
n
=
1
∞
n
-
1
∑
i
=
1
n
P
(
|
a
n
i
X
|
>
b
n
)
≤
E
|
X
|
α
log
⁡
(
1
+
|
X
|
)
<
∞
.
For fixed n
≥
1, it is easily seen that {
X
i
(
n
)
-
E
X
i
(
n
)
,
i
≥
1
,
n
≥
1
} is still a sequence of NOD random variables with mean zero by Lemma 7. Hence, it follows from (14) of Lemmas 9 and 8 and Markov inequality (for M
>
2) that
(28)
I
≜
∑
n
=
1
∞
n
-
1
P
(
|
T
n
(
n
)
|
>
ε
b
n
2
)
≤
C
∑
n
=
1
∞
n
-
1
b
n
-
M
E
(
|
T
n
(
n
)
|
M
)
≤
C
∑
n
=
1
∞
n
-
1
b
n
-
M
[
∑
i
=
1
n
E
|
X
i
(
n
)
|
M
+
(
∑
i
=
1
n
E
|
X
i
(
n
)
|
2
)
M
/
2
]
≤
C
∑
n
=
1
∞
n
-
1
b
n
-
M
∑
i
=
1
n
E
|
X
i
(
n
)
|
M
+
C
∑
n
=
1
∞
n
-
1
b
n
-
M
(
∑
i
=
1
n
E
|
X
i
(
n
)
|
2
)
M
/
2
≜
I
1
+
I
2
.
It follows from Lemma10, (14) of Lemma 9, and Markov inequality that
(29)
I
1
≜
C
∑
n
=
1
∞
‍
n
-
1
b
n
-
M
∑
i
=
1
n
‍
E
|
X
i
(
n
)
|
M
≤
C
∑
n
=
1
∞
‍
n
-
1
b
n
-
M
{
∑
i
=
1
n
|
a
n
i
|
M
E
|
X
n
i
|
M
I
(
|
a
n
i
X
n
i
|
≤
b
n
)
‍
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
l
+
∑
i
=
1
n
‍
b
n
M
P
(
|
a
n
i
X
n
i
|
>
b
n
)
}
≤
C
∑
n
=
1
∞
‍
n
-
1
b
n
-
M
{
∑
i
=
1
n
‍
|
a
n
i
|
M
E
|
X
|
M
I
(
|
a
n
i
X
|
≤
b
n
)
k
k
k
k
k
k
k
k
k
k
k
k
+
2
∑
i
=
1
n
b
n
M
P
(
|
a
n
i
X
|
>
b
n
)
‍
}
≤
C
∑
n
=
1
∞
‍
n
-
1
b
n
-
M
∑
i
=
1
n
‍
|
a
n
i
|
M
E
|
X
|
M
I
(
|
a
n
i
X
|
≤
b
n
)
+
C
∑
n
=
1
∞
‍
n
-
1
∑
i
=
1
n
‍
P
(
|
a
n
i
X
|
>
b
n
)
≜
I
11
+
I
12
.
From Lemma 10 and E
|
X
|
α
log
⁡
(
1
+
|
X
|
)
<
∞, we can obtain that
(30)
I
12
≜
C
∑
n
=
1
∞
‍
n
-
1
∑
i
=
1
n
‍
P
(
|
a
n
i
X
|
>
b
n
)
≤
E
|
X
|
α
log
⁡
(
1
+
|
X
|
)
<
∞
.
For fixedn
>
1, we divide {
a
n
i
,
1
≤
i
≤
n
} into three subsets {
a
n
i
:
|
a
n
i
|
≤
1
/
(
log
⁡
⁡
n
)
m
}, {
a
n
i
:
1
/
(
log
⁡
⁡
n
)
m
<
|
a
n
i
|
≤
1
}, and {
a
n
i
:
|
a
n
i
|
>
1
}, where m
=
(
1
/
(
M
-
α
)
). Then,
(31)
I
11
≜
C
∑
n
=
1
∞
n
-
1
b
n
-
M
∑
i
=
1
n
|
a
n
i
|
M
E
|
X
|
M
I
(
|
a
n
i
X
|
≤
b
n
)
=
C
∑
n
=
1
∞
‍
n
-
1
b
n
-
M
k
k
k
k
×
∑
i
:
|
a
n
i
|
≤
1
/
(
log
⁡
⁡
n
)
m
|
a
n
i
|
M
E
|
X
|
M
I
(
|
a
n
i
X
|
≤
b
n
)
+
C
∑
n
=
1
∞
‍
n
-
1
b
n
-
M
k
k
k
k
k
k
k
×
∑
i
:
1
/
(
log
⁡
⁡
n
)
m
<
|
a
n
i
|
≤
1
|
a
n
i
|
M
E
|
X
|
M
I
(
|
a
n
i
X
|
≤
b
n
)
+
C
∑
n
=
1
∞
‍
n
-
1
b
n
-
M
∑
i
:
|
a
n
i
|
>
1
‍
|
a
n
i
|
M
E
|
X
|
M
I
(
|
a
n
i
X
|
≤
b
n
)
≜
I
11
(
1
)
+
I
11
(
2
)
+
I
11
(
3
)
.
By Lemma 11 and E
|
X
|
α
log
⁡
(
1
+
|
X
|
)
<
∞ again, it follows that
(32)
I
11
(
3
)
≜
C
∑
n
=
1
∞
n
-
1
b
n
-
M
∑
i
:
|
a
n
i
|
>
1
|
a
n
i
|
M
E
|
X
|
M
I
(
|
a
n
i
X
|
≤
b
n
)
≤
E
|
X
|
α
log
⁡
(
1
+
|
X
|
)
<
∞
for
M
>
2
≥
α
>
0
.
Noting that ∑
i
:
|
a
n
i
|
≤
1
/
(
log
⁡
⁡
n
)
m
|
a
n
i
|
α
≤
C
n
(
log
⁡
⁡
n
)
-
m
α, for M
>
α and fixed n
>
1, we have that
(33)
I
11
(
1
)
≜
C
∑
n
=
1
∞
n
-
1
b
n
-
M
∑
i
:
|
a
n
i
|
≤
1
/
(
log
⁡
⁡
n
)
m
|
a
n
i
|
M
E
|
X
|
M
I
(
|
a
n
i
X
|
≤
b
n
)
≤
C
∑
n
=
1
∞
n
-
1
b
n
-
α
∑
i
:
|
a
n
i
|
≤
1
/
(
log
⁡
⁡
n
)
m
|
a
n
i
|
α
E
|
X
|
α
I
(
|
a
n
i
X
|
≤
b
n
)
≤
C
E
|
X
|
α
∑
n
=
1
∞
n
-
1
b
n
-
α
∑
i
:
|
a
n
i
|
≤
1
/
(
log
⁡
⁡
n
)
m
|
a
n
i
|
α
≤
C
E
|
X
|
α
∑
n
=
1
∞
n
-
1
n
-
1
(
log
⁡
⁡
n
)
-
1
n
(
log
⁡
⁡
n
)
-
m
α
<
∞
.
Noting that ∑
i
:
1
/
(
log
⁡
⁡
n
)
m
<
|
a
n
i
|
≤
1
|
a
n
i
|
M
≤
C
n and m
=
1
/
(
M
-
α
), for M
>
2
,
0
<
α
≤
2, we have that
(34)
I
11
(
2
)
≜
C
∑
n
=
1
∞
n
-
1
b
n
-
M
∑
i
:
1
/
(
log
⁡
⁡
n
)
m
<
|
a
n
i
|
≤
1
|
a
n
i
|
M
E
|
X
|
M
I
(
|
a
n
i
X
|
≤
b
n
)
≤
C
∑
n
=
1
∞
b
n
-
M
E
|
X
|
M
I
(
|
X
|
≤
b
n
(
log
⁡
⁡
n
)
m
)
=
C
∑
n
=
1
∞
b
n
-
M
∑
k
=
1
n
E
|
X
|
M
I
(
(
k
-
1
)
1
/
α
(
log
⁡
⁡
(
k
-
1
)
)
m
+
1
/
α
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
k
<
|
X
|
≤
k
1
/
α
(
log
⁡
⁡
k
)
m
+
1
/
α
)
=
C
∑
k
=
1
∞
E
|
X
|
M
I
(
∑
n
=
1
∞
b
n
-
M
(
k
-
1
)
1
/
α
(
log
⁡
⁡
(
k
-
1
)
)
m
+
1
/
α
k
k
k
k
k
k
k
k
k
k
k
k
k
k
<
|
X
|
≤
k
1
/
α
(
log
⁡
⁡
k
)
m
+
1
/
α
∑
n
=
1
∞
b
n
-
M
(
k
-
1
)
1
/
α
(
log
⁡
⁡
(
k
-
1
)
)
m
+
1
/
α
)
lllll
×
∑
n
=
k
∞
n
-
M
/
α
(
log
⁡
⁡
n
)
-
M
/
α
≤
C
∑
k
=
1
∞
E
|
X
|
M
I
(
∑
n
=
1
∞
b
n
-
M
(
k
-
1
)
1
/
α
(
log
⁡
⁡
(
k
-
1
)
)
m
+
1
/
α
k
k
k
k
k
k
k
k
k
k
k
k
k
k
<
|
X
|
≤
k
1
/
α
(
log
⁡
⁡
k
)
m
+
1
/
α
∑
n
=
1
∞
b
n
-
M
(
k
-
1
)
1
/
α
(
log
⁡
⁡
(
k
-
1
)
)
m
+
1
/
α
(
k
-
1
)
1
/
α
)
kkk
×
k
1
-
M
/
α
(
log
⁡
⁡
k
)
-
M
/
α
≤
C
∑
k
=
1
∞
E
|
X
|
α
I
(
∑
n
=
1
∞
b
n
-
M
(
k
-
1
)
1
/
α
(
log
⁡
⁡
(
k
-
1
)
)
m
+
1
/
α
k
k
k
k
k
k
k
k
k
k
k
k
k
<
|
X
|
≤
k
1
/
α
(
log
⁡
⁡
k
)
m
+
1
/
α
∑
n
=
1
∞
b
n
-
M
)
≤
C
E
|
X
|
α
<
∞
.
Finally, we will prove that
(35)
I
2
≜
C
∑
n
=
1
∞
n
-
1
b
n
-
M
(
∑
i
=
1
n
E
|
X
i
(
n
)
|
2
)
M
/
2
<
∞
.
Hence, by C
r inequality, Markov inequality, Lemmas 9–11, and E
|
X
|
α
log
⁡
(
1
+
|
X
|
)
<
∞, we have that
(36)
I
2
≜
C
∑
n
=
1
∞
n
-
1
b
n
-
M
(
∑
i
=
1
n
E
|
X
i
(
n
)
|
2
)
M
/
2
=
C
∑
n
=
1
∞
n
-
1
(
∑
i
=
1
n
b
n
-
2
E
|
X
i
(
n
)
|
2
)
M
/
2
≤
C
∑
n
=
1
∞
n
-
1
(
∑
i
=
1
n
P
(
|
a
n
i
X
n
i
|
>
b
n
)
)
M
/
2
+
C
∑
n
=
1
∞
n
-
1
(
∑
i
=
1
n
b
n
-
2
E
|
a
n
i
X
n
i
|
2
(
|
a
n
i
X
n
i
|
≤
b
n
)
)
M
/
2
≤
C
(
∑
n
=
1
∞
n
-
1
∑
i
=
1
n
P
(
|
a
n
i
X
|
>
b
n
)
)
M
/
2
+
C
(
∑
n
=
1
∞
n
-
1
∑
i
=
1
n
b
n
-
2
E
|
a
n
i
X
|
2
(
|
a
n
i
X
|
≤
b
n
)
)
M
/
2
≤
C
(
E
|
X
|
α
log
⁡
(
1
+
|
X
|
)
)
M
/
2
<
∞
.
Therefore, the desired result (10) follows from the above statements. This completes the proof of Theorem 4.



---

*Source: 368702-2014-08-20.xml*