R E S E A R C H
Open Access
On complete moment convergence for
arrays of rowwise pairwise negatively
quadrant dependent random variables
Meimei Ge
1, Zexing Dai
1and Yongfeng Wu
1,2**Correspondence:[email protected] 1School of Mathematics and
Finance, Chuzhou University, Chuzhou, China
2College of Mathematics and
Computer Science, Tongling University, Tongling, China
Abstract
In this paper, we establish some results on the complete moment convergence for weighted sums of pairwise negatively quadrant dependent (PNQD) random variables. The obtained results improve the corresponding ones of Ko (Stoch. Int. J. Probab. Stoch. Process. 85:172–180,2013).
MSC: 60F15
Keywords: Pairwise negatively quadrant dependent; Complete moment
convergence; Weighted sums
1 Introduction
The following concept of PNQD random variables was introduced by Lehmann [2].
Definition 1.1 A sequence{Xn,n≥1}of random variables is said to be pairwise nega-tively quadrant dependent (PNQD) if for anyri,rjandi=j,
P(Xi>ri,Xj>rj)≤P(Xi>ri)P(Xj>rj).
Negative quadrant dependence is shown to be a stronger notion of dependence than negative correlation but weaker than negative association. The convergence properties of NQD random sequences have been studied in many papers. We refer to Wu [3] for Kolmogorov-type three-series theorem, Matula [4] for the Kolmogorov-type strong law of large numbers, Jabbari [5] for the almost sure limit theorems for weighted sums of pairwise NQD random variables under some fragile conditions, Li and Yang [6], Wu [7], and Xu and Tang [8] for strong convergence, Gan and Chen [9] for complete convergence and complete moment convergence, Wu and Guan [10] for a mean convergence theorem and weak laws of large numbers for dependent random variables, and so on.
The concept of complete convergence of a sequence of random variables was first given by Hsu and Robbins [11].
Definition 1.2 A sequence of random variables{Un,n∈N}is said to converge completely to a constantaif for anyε> 0,
∞
n=1
P|Un–a|>ε
<∞.
By the Borel–Cantelli lemma this result implies thatUn→aalmost surely. Therefore, the complete convergence is a very important tool in establishing the almost sure conver-gence of sums of random variables and weighted sums of random variables.
Recently, Ko [1] proved the following complete convergence theorem for arrays of PNQD random variables.
Theorem A Let{Xnj, 1≤j≤bn,n≥1}be an array of rowwise and PNQD random
vari-ables with mean zero, and let {anj,j≥1,n≥1} be an array of positive numbers. Let
{bn,n≥1}be a nondecreasing sequence of positive numbers.Assume that,for some0 <t< 2
and allε> 0,
∞
n=1
cn(log2bn)2 bn
j=1
P|anjXnj| ≥εb
1
t n
<∞ (1.1)
and
∞
n=1
cnb –2t
n (log2bn)2 bn
j=1
a2njE(Xnj)2I
|anjXnj|<εb
1
t n
<∞. (1.2)
Then
∞
n=1
cnP
max
1≤k≤bn k
j=1
anjXnj–anjEXnjI
|anjXnj|<εb
1
t n
≥εb
1
t
n <∞. (1.3)
Chow [12] was the first who showed the complete moment convergence for a sequence of independent and identically distributed random variables by generalizing the result of Baum and Katz [13]. The concept of complete moment convergence is as follows.
Definition 1.3 Let{Zn,n≥1}be a sequence of random variables, and letan> 0,bn> 0, andq> 0. If for anyε> 0,
∞
n=1
anE
b–1n |Zn|–ε q
+<∞,
then this is called the complete moment convergence.
su-peradditive dependent (NSD) random variables. Wu et al. [17] for arrays of rowwise END random variables, Wu [7] for negatively associated random variables, Wu et al. [18] for weighted sums of weakly dependent random variables, Wang et al. [19] for double indexed randomly weighted sums and its applications, Wu and Wang [20] for a class of dependent random variables, and so forth.
In this work, we improve TheoremAfrom complete convergence to complete moment convergence for PNQD random variables under some stronger conditions. In addition, we obtain some much stronger conclusions under the same conditions of the corresponding theorems in Ko [1].
Throughout this paper, the symbolCalways stands for a generic positive constant which may differ from one place to another. ByI(A) we denote the indicator function of a setA. We also denotex+=xI(x≥0).
2 Main results
Now we state the main results of this paper. The proofs are given in next section.
Theorem 2.1 Let{Xnj, 1≤j≤bn,n≥1} be an array of rowwise and PNQD random
variables with mean zero,and let{anj,j≥1,n≥1}be an array of positive numbers.Let
{bn,n≥1}be a nondecreasing sequence of positive numbers.Assume that,for some0 <t< 2
and allε> 0,
∞
n=1
cnb –1t
n (log2bn)2 bn
j=1
|anj|E|Xnj|I
|anjXnj| ≥εb
1
t n
<∞ (2.1)
and
∞
n=1
cnb –2t
n (log2bn)2 bn
j=1
a2njE(Xnj)2I
|anjXnj|<εb
1
t n
<∞. (2.2)
Then,for allε> 0,
∞
n=1
cnE
b– 1
t n max
1≤k≤bn
k
j=1
anjXnj–anjEXnjI
|anjXnj|<εb
1
t n
–ε
+
<∞. (2.3)
Remark2.1 LetHnj= k
j=1(anjXnj–anjEXnjI[|anjXnj|<εb
1
t
n]). Note that
∞
n=1
cnE
b– 1
t n max
1≤k≤bn|
Hnj|–ε
+
=
∞
n=1
cn ∞
0
P
b– 1
t n max
1≤j≤bn|
Hnj|>ε+u
du
≥
∞
n=1
cn ε
0
P
max
1≤j≤bn|
Hnj|> (ε+u)b –1t n
du
≥
∞
n=1
cnP
max
1≤j≤bn|
Hnj|> 2εb –1t n
.
Theorem 2.2 Let{Xnj, 1≤j≤bn,n≥1} be an array of rowwise and PNQD random
variables with mean zero,and let{anj,j≥1,n≥1}be an array of positive numbers.Let
{bn,n≥1}be a nondecreasing sequence of positive numbers. Assume that,for some
se-quence{λn,n≥1}with0 <λn≤1,we have E|Xnj|1+λn<∞for1≤j≤bn,n≥1.If for some
sequence{cn,n≥1}of positive real numbers and0 <t< 2,
∞
n=1
cn(log2bn)2
b 1
t n
–1–λnbn
j=1
E|anjXnj|1+λn<∞, (2.4)
then for anyε> 0,
∞
n=1
cnE
b– 1
t n max
1≤k≤bn
k
j=1
anjXnj –ε
+
<∞. (2.5)
Remark2.2 Noting that the conditions of Theorem2.2are the same as in Theorem 3.2 in Ko [1], we have
∞
n=1
cnE
b– 1
t n max
1≤k≤bn
k
j=1
anjXnj –ε
+
= ∞
0
P
b– 1
t n max
1≤k≤bn
k
j=1
anjXnj >ε+u
du
≥
∞
n=1
cn ε
0
P
max
1≤k≤bn k
j=1
anjXnj
> (ε+u)b –1t n
du
≥
∞
n=1
cnP
max
1≤k≤bn k
j=1
anjXnj > 2εb
–1t n
.
Therefore (2.5) is much stronger than (3.8) of Theorem 3.2 in Ko [1]. To sum up, Theo-rem2.2improves Theorem 3.2 in Ko [1].
Corollary 2.3 Let{Xnj, 1≤j≤bn,n≥1}be an array of rowwise PNQD random variables,
and let{anj,j≥1,n≥1}be an array of positive numbers.Let h(x) > 0be a slowly varying
function as x→ ∞,and letα> 12 andαr≥1.Suppose that,for0 <t< 2,the following conditions hold for anyε> 0:
∞
n=1
nαr–2–1t(log
2n)2h(n) n
j=1
|anj|E|Xnj|I
|anjXnj| ≥εn
1
t<∞ (2.6)
and
∞
n=1
nαr–2–2t(log
2n)2h(n) n
j=1
a2njE(Xnj)2I
|anjXnj|<εn
1
Then,for allε> 0,
∞
n=1
nαr–2h(n)E
n–1t max 1≤k≤n
k
j=1
anjXnj–anjEXnjI
|anjXnj|<εn
1
t
–ε
+
<∞. (2.8)
Theorem 2.4 Let{Xnj,j≥1,n≥1}be an array of rowwise identically distributed PNQD
random variables with EX11= 0,and let h(x) > 0be a slowly varying function as x→ ∞.If
E|X11|(αr+2)th(|X11|t) <∞forα>12,αr≥1,and0 <t< 2,then
∞
n=1
nαr–2h(n)E
n–1t max 1≤k≤bn
k
j=1
Xnj –ε
+
<∞. (2.9)
Remark2.3 Noting that the conditions of Theorem2.2are the same as in Theorem 3.4 in Ko [1], we have
∞
n=1
nαr–2h(n)E
n–1t max 1≤k≤bn
k
j=1
Xnj –ε
+
=
∞
n=1
nαr–2h(n) ∞
0
P
n–1t max 1≤k≤bn
k
j=1
Xnj >ε+u
du
≥
∞
n=1
nαr–2h(n) ε
0
P
max
1≤k≤bn
k
j=1
Xnj
> (ε+u)n
1
t
du
≥
∞
n=1
nαr–2h(n)P
max
1≤k≤bn
k
j=1
Xnj > 2εn
1
t
.
Therefore (2.9) is much stronger than (3.13) of Theorem 3.4 in Ko [1]. Theorem2.4 im-proves Theorem 3.4 in Ko [1].
Corollary 2.5 Let {Xnj, 1≤j≤bn,n≥1} be an array of rowwise and PNQD random
variables with mean zero,and let{anj,j≥1,n≥1}be an array of positive numbers.Let
{bn,n≥1}be a nondecreasing sequence of positive numbers,and let{cn,n≥1}be a
se-quence of positive numbers.Assume that,for allε> 0,
∞
n=1
cn(log2bn)2 bn
j=1
|anj|E|Xnj|I
|anjXnj| ≥εlog2bn
<∞ (2.10)
and
∞
n=1
cn bn
j=1
a2njE(Xnj)2I
|anjXnj|<εlog2bn
<∞. (2.11)
Then,for allε> 0,
∞
n=1
cnE
(log2bn)–1 max 1≤k≤bn
k
j=1
anjXnj–anjEXnjI
|anjXnj|<εlog2bn
<∞. (2.12)
Corollary 2.6 Let{Xnj, 1≤j≤bn,n≥1}be an array of rowwise and PNQD random
vari-ables with mean zero and finite variances.Let{anj,j≥1,n≥1} be an array of positive
numbers satisfying
n
j=1
a2njE(Xnj)2=O
nδ as n→ ∞ (2.13)
for some0 <δ< 1.Then,for allε> 0andα> 0,
∞
n=1
n2(α–1)E
n–α(log2n)–1 max
1≤k≤bn
k
j=1
anjXnj –ε
+
<∞. (2.14)
Remark2.4 Note that
∞
n=1
n2(α–1)E
n–α(log2n)–1 max
1≤k≤bn
k
j=1
anjXnj –ε
+
=
∞
n=1
n2(α–1)
∞
0
P
n–α(log2n)–1 max
1≤k≤bn
k
j=1
anjXnj >ε+u
du
≥
∞
n=1
n2(α–1)
ε
0
P
n–α(log2n)–1 max
1≤k≤bn
k
j=1
anjXnj
> (ε+u)nαlog2n
du
≥
∞
n=1
n2(α–1)P
max
1≤k≤bn
k
j=1
anjXnj
> 2εnαlog2n
.
Therefore (2.14) is much stronger than (3.18) of Corollary 3.6 in Ko [1].
3 The proofs
To prove our results, we need some lemmas. The first one is the basic property for PNQD random variables, which can be referred to Lehmann [2].
Lemma 3.1 Let{Xn,n≥1}be a sequence of PNQD random variables,and let{fn,n≥1}
be a sequence of nondecreasing functions.Then{fn(Xn),n≥1}is still a sequence of PNQD
random variables.
The next lemma comes from Wu [3] and plays an essential role to prove the result of the paper.
Lemma 3.2 Let{Xn,n≥1}be a sequence of PNQD random variables with mean zero and
finite second moments.Then
E max
1≤k≤n
k
j=1
Xj 2
≤C(log2n)2 n
j=1
A positive measurable function h(x)on[a,∞)for some a> 0is said to be slowly varying as x→ ∞if
lim
x→∞
h(λx)
h(x) = 1 for eachλ> 0. (3.2)
The last lemma can be found in Wu [21].
Lemma 3.3 If h(x) > 0is a slowly varying function as x→ ∞,then
(i) limx→∞sup2k≤x<2k+1h(x)/h
2k= 1,and
(ii) c12kh(ε2k)≤k
j=12jh(ε2j)≤c22kh(ε2k)
for all r> 0,ε> 0,and positive integers k and some positive constants c1and c2.
Proof of Theorem2.1 LetSj= k
j=1(anjXnj–anjEXnjI[|anjXnj|<εb
1
t
n]). For any fixedε> 0,
∞
n=1
cnE
b– 1
t n max
1≤j≤bn
|Sj|–ε
+
=
∞
n=1
cn ∞
0
Pb– 1
t n max
1≤j≤bn|
Sj|–ε>u
du
≤ε
∞
n=1
cnP
max
1≤j≤bn|
Sj|>εb
1
t n
+
∞
n=1
cn ∞
ε
P
max
1≤j≤bn|
Sj|>ub
1
t n
du
=:I1+I2.
Obviously, we haveI1<∞by TheoremA. Hence we need only to proveI2<∞. Clearly,
P
max
1≤j≤bn|
Sj|>ub
1
t n
=P
max
1≤j≤bn|
Sj|>ub
1
t n,
bn
j=1
|anjXnj| ≥ub
1
t n
+P
max
1≤j≤bn|
Sj|>ub
1
t n,
bn
j=1
|anjXnj|<ub
1
t n
≤
bn
j=1
P|anjXnj| ≥ub
1
t n
+P
max
1≤k≤bn
k
j=1
anjXnjI
|anjXnj|<εb
1
t n
–anjEXnjI
|anjXnj|<εb
1
t n
>ub
1
t n
.
Then we can get
I2≤
∞
n=1
cn bn
j=1 ∞
ε
P|anjXnj|>ub
1
t n
du
+
∞
n=1
cn ∞
ε
P
max
1≤k≤bn k
j=1
anjXnjI
|anjXnj|<ub
1
–anjEXnjI
|anjXnj|<ub
1 t n >ub 1 t n
=:I3+I4.
Firstly, we will prove thatI3<∞. Noting that ∞
ε
P|anjXnj| ≥ub
1
t n
du≤b– 1
t
n E|anjXnj|I
|anjXnj| ≥εb
1
t n
,
by (2.1) we have
I3≤
∞ n=1 cn bn j=1 ∞ ε
P|anjXnj| ≥ub
1 t n du ≤ ∞ n=1
cnb –1t n
bn
j=1
E|anjXnj|I
|anjXnj| ≥εb
1
t n
<∞.
To prove thatI4<∞, let
Ynk= –ub
1
t nI
anjXnj< –ub
1
t n
+anjXnjI
|anjXnj| ≤ub
1 t n +ub 1 t nI
anjXnj>ub
1
t n
,
Znk= –ub
1
t nI
anjXnj< –ub
1 μ n +ub 1 t nI
anjXnj>ub
1 t n . We have P max
1≤k≤bn k j=1
anjXnjI
|anjXnj| ≤εb
1
t n
–anjEXnjI
|anjXnj| ≤εb
1 t n >ub 1 t n =P max
1≤k≤bn k j=1
(Ynk–EYnk–Znk+EZnk) >ub 1 t n .
Then we have
I4≤
∞ n=1 cn ∞ ε P max
1≤k≤bn k j=1
(Znk–EZnk) > ub 1 t n 2 du + ∞ n=1 cn ∞ ε P max
1≤k≤bn k j=1
(Ynk–EYnk) > ub 1 t n 2 du
=:I5+I6.
ForI5, by the Markov inequality and (2.1) we have
I5≤C
∞ n=1 cn bn j=1 ∞ ε
u–1E|Znk|du
≤C
∞
n=1
cnb
1 t n bn j=1 ∞ ε
P|anjXnj|>ub
1
t n
≤C
∞
n=1
cnb
1
t n
bn
j=1
b– 1
t
n E|anjXnj|I
|anjXnj|>εb
1
t n
≤C
∞
n=1
cn bn
j=1
E|anjXnj|I
|anjXnj|>ub
1
t n
<∞.
Now considerI6. By the Markov inequality and Lemma3.2we have
I6≤C
∞
n=1
cnb –2t n
∞
ε
u–2E
max
1≤k≤bn k
j=1
(Ynk–EYnk)
2
du
≤C
∞
n=1
cnb –2t
n (log2bn)2 bn
j=1 ∞
ε
u–2E|Ynk|2du
=C
∞
n=1
cnb –2t
n (log2bn)2 bn
j=1 ∞
ε
u–2|anj|2E|Xnj|2I
|anjXnj| ≤ub
1
t n
+C
∞
n=1
cn(log2bn)2 bn
j=1 ∞
ε
P|anjXnj|>ub
1
t n
du
=I7+I8.
Firstly, we will prove thatI8<∞. By (2.1) we have
I8≤C
∞
n=1
cnb –1t
n (log2bn)2 bn
j=1
|anj|E|Xnj|I
|anjXnj|>εb
1
t n
<∞.
Next, considerI7<∞. We have
I7=C
∞
n=1
cnb –2t
n (log2bn)2 bn
j=1 ∞
ε
u–2|anj|2E|Xnj|2I
|anjXnj| ≤εb
1
t n
+C
∞
n=1
cnb –2t
n (log2bn)2 bn
j=1 ∞
ε
u–2|anj|2E|Xnj|2I
εb 1
t
n<|anjXnj| ≤ub
1
t n
=:I7+I7.
By (2.2) it is easy to see that
I7≤C
∞
n=1
cnb –2t
n (log2bn)2 bn
j=1
|anj|2E|Xnj|2I
|anjXnj| ≤εb
1
t n
∞
ε
u–2du
By the Markov inequality and (2.1) we have
I7=C
∞
n=1
cnb –2t
n (log2bn)2 bn j=1 ∞ m=1 (m+1)ε
mε
u–2E|anjXnj|2I
εb 1
t
n <|anjXnj| ≤ub
1 t n du ≤C ∞ n=1
cnb –2t
n (log2bn)2 bn j=1 ∞ m=1
m–2E|anjXnj|2I
εb 1
t
n <|anjXnj| ≤(m+ 1)εb
1 t n =C ∞ n=1
cnb –2t
n (log2bn)2 bn j=1 ∞ m=1 m–2 m s=1
E|anjXnj|2I
sεb 1
t
n <|anjXnj| ≤(s+ 1)εb
1 t n =C ∞ n=1
cnb –2t
n (log2bn)2 bn j=1 ∞ s=1
E|anjXnj|2I
sεb 1
t
n <|anjXnj| ≤(s+ 1)εb
1
t n
∞
m=s
m–2
≤C
∞
n=1
cnb –2t
n (log2bn)2 bn j=1 ∞ s=1
s–1E|anjXnj|2I
sεb 1
t
n<|anjXnj| ≤(s+ 1)εb
1 t n =C ∞ n=1
cnb –2t
n (log2bn)2 bn j=1 ∞ s=1
s–1(s+ 1)2ε2b 2
t n
×E anjXnj
(s+ 1)εb 1
t n 2Isεb
1
t
n <|anjXnj| ≤(s+ 1)εb
1 t n ≤C ∞ n=1
cnb –2t
n (log2bn)2 bn j=1 ∞ s=1
s–1(s+ 1)2b 2
t n
×E anjXnj
(s+ 1)εb 1
t n Isεb
1
t
n <|anjXnj| ≤(s+ 1)εb
1 t n =C ∞ n=1
cnb –2t
n (log2bn)2 bn j=1 ∞ s=1
s–1(s+ 1)b 1
t
nE|anjXnj|I
sεb 1
t
n<|anjXnj| ≤(s+ 1)εb
1 t n ≤C ∞ n=1
cnb –2t
n (log2bn)2 bn j=1 ∞ s=1
E|anjXnj|I
sεb 1
t
n <|anjXnj| ≤(s+ 1)εb
1 t n =C ∞ n=1
cnb –2t
n (log2bn)2 bn j=1 b 1 t
nE|anjXnj|I
|anjXnj|>εb
1 t n =C ∞ n=1
cnb –1t
n (log2bn)2 n
j=1
E|anjXnj|I
|anjXnj|>εb
1
t n
<∞.
This completes the proof of the theorem.
Proof of Theorem2.2 We estimate
∞
n=1
cnb –1t
n (log2bn)2 bn
j=1
|anj|E|Xnj|I
|anjXnj| ≥εb
1
=ε
∞
n=1
cn(log2bn)2 bn
j=1
EanjXnj
εb 1
t n
IanjXnj
εb 1
t n
≥1
≤ε
∞
n=1
cn(log2bn)2 bn
j=1
EanjXnj
εb 1
t n
1+λnIanjXnj
εb 1
t n
≥1
≤C
∞
n=1
cn(log2bn)2
b 1
t n
–1–λn bn
j=1
E|anjXnj|1+λn<∞
and
∞
n=1
cnb –2t
n (log2bn)2 bn
j=1
a2njE(Xnj)2I
|anjXnj|<εb
1
t n
=ε2
∞
n=1
cn(log2bn)2E anjXnj
εb 1
t n
2IanjXnj
εb 1
t n
< 1
≤C
∞
n=1
cn(log2bn)2 bn
j=1
EanjXnj
εb 1
t n
1+λnIanjXnj
εb 1
t n
< 1
≤C
∞
n=1
cn(log2bn)2
b 1
t n
–1–λnbn
j=1
E|anjXnj|1+λn<∞.
Hence conditions (2.1) and (2.2) of Theorem2.1are satisfied. SinceEXnj= 0, we get
b– 1
t n max
1≤k≤bn
k
j=1
EanjXnjI
|anjXnj|<εb
1
t n
≤b– 1
t n max
1≤k≤bn k
j=1
|anj|E|Xnj|I
|anjXnj| ≥εb
1
t n
≤b– 1
t n
–1–λn bn
j=1
|anj|1+λnE|Xnj|1+λn→0 asn→ ∞,
and thus (2.5) is completed.
Proof of Corollary2.3 Letcn=nαr–2h(n) andbn=n. Then, by Theorem2.1, (2.8) is
com-pleted.
Proof of Theorem2.4 Foranj= 1,j≥1,n≥1, by Lemma3.3we have
∞
n=1
nαr–1(log2n)2h(n)E|X11|I
|X11| ≥εn
1
t
≤C
∞
n=1
2kαrk2h2kE|X11|I
|X11| ≥ε
2k1t
≤C
∞
k=1
2kαr+2h2kE|X11|I
|X11| ≥ε
2k
1
≤C
∞
m=1
E|X11|I
ε2m
1
t ≤ |X 11|<
2m+1
1
t m
j=1
2jαr+2h2j
≤C
∞
m=1
2mαr+2h2mE|X11|I
ε2m
1
t ≤ |X 11|<ε
2m+1
1
t
≤E|X11|(αr+2)th
|X11|
<∞
and
∞
n=1
nαr–1–2t(log
2n)2h(n) n
j=1
a2njE(X11)2I|X11|<εn
1
t
≤
∞
n=1
2kαr–
2
tk2h2k (2 k)1t
0
x2dF(x)
≤
∞
k=1
2kαr+2–2th2k (2k)1t
0
x2dF(x)
≤
∞
m=1
2mαr+2–2th2m (2m)1t
(2m–1)1t
x2dF(x)
=C
∞
m=1
2mαr+2–
2
t (2m)1t
(2m–1)1t
h(2×2m–1)
h(|x|t) h
|x|tx2dF(x)
≤C
∞
m=1
2mαr+2–2t (2m)1t
(2m–1)1t
h|x|tx2dF(x)
≤C
∞
m=1 (2m)1t
(2m–1)1t
|x|tαr+2h|x|tx2dF(x)
=CE|X11|(αr+2)th
|X11|t
<∞,
and thus (2.6) and (2.7) are satisfied. Then, to complete the proof, it remains to show that, for 1≤j≤n,n–1tj|EX11I[|X11|<εn1t]| →0 asn→ ∞.
If (αr+ 2)t< 1, then we have, asn→ ∞,
n–1tjEX 11I
|X11|<εn
1
t≤(ε)1–(αr+2)tn1–(αr+2)tE|X
11|(αr+2)t→0,
and if (αr+ 2)t≥1, then since|EX11|= 0, we have, asn→ ∞,
n–1tjEX 11I
|X11|<εn
1
t≤n1–1t–EX 11I
|X11| ≥εn
1
t
≤(ε)1–(αr+2)tn1–(αr+2)tE|X11|(αr+2)t→0.
Hence the proof of Theorem2.4is completed.
Proof of Corollary2.5 Takinglog2bninstead ofb
1
t
Proof of Corollary2.6 In Theorem2.1, letcn=n2(α–1)andb –1t
n =n–α(log2n)–1. By (2.13) we have
∞
n=1
cnb –1t
n (log2bn)2 bn
j=1
|anj|E|Xnj|I
|anjXnj| ≥εb
1
t n
=ε
∞
n=1
cn(log2bn)2 bn
j=1
EanjXnj
εb 1
t n
IanjXnj
εb 1
t n
≥1
≤ε
∞
n=1
cn(log2bn)2 bn
j=1
EanjXnj
εb 1
t n
2IanjXnj
εb 1
t n
≥1
≤C
∞
n=1
cnb –2t
n (log2bn)2 bn
j=1
E|anjXnj|2
≤C
∞
n=1
n2(α–1)n–2α(log2n)–2(log2bn)2nδ
≤C
∞
n=1
n–2+δ<∞
and
∞
n=1
cnb –2t
n (log2bn)2 bn
j=1
a2njE(Xnj)2I
|anjXnj|<εb
1
t n
≤
∞
n=1
cnb –2t
n (log2bn)2 bn
j=1
E|anjXnj|2
≤C
∞
n=1
n2(α–1)n–2α(log2n)–2(log2bn)2nδ
≤C
∞
n=1
n–2+δ<∞.
Hence conditions of (2.1) and (2.2) of Theorem2.1are satisfied. SinceEXnj= 0, by (2.13) we get
b– 1
t n max
1≤k≤bn
k
j=1
EanjXnjI
|anjXnj|<εb
1
t n
≤b– 1
t n max
1≤k≤bn k
j=1
|anj|E|Xnj|I
|anjXnj| ≥εb
1
t n
≤b– 2
t n
bn
j=1
|anj|2E|Xnj|2
≤Cn–2α+δ(log2n)–2→0 asn→ ∞,
Acknowledgements
The authors are grateful to the referee for carefully reading the manuscript and for providing some comments and suggestions, which led to improvements in this paper.
Funding
The research of M. Ge is partially supported by the NSF of Anhui Educational Committe (KJ2017B11, KJ2018A0428). The research of Z. Dai is partially supported by the NSF of Anhui Educational Committe (KJ2017B09). The research of Y. Wu is partially supported by the Natural Science Foundation of Anhui Province (1708085MA04), the Key Program in the Young Talent Support Plan in Universities of Anhui Province (gxyqZD2016316), and Chuzhou University scientific research fund (2017qd17).
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and read and approved the final manuscript.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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