Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 769201, 12 pages
doi:10.1155/2010/769201
Research Article
On Complete Convergence for Arrays of Rowwise
ρ-Mixing Random Variables and Its Applications
Xing-cai Zhou
1, 2
and Jin-guan Lin
1
1
Department of Mathematics, Southeast University, Nanjing 210096, China
2
Department of Mathematics and Computer Science, Tongling University, Tongling, Anhui 244000, China
Correspondence should be addressed to Jin-guan Lin, [email protected]
Received 15 May 2010; Revised 23 August 2010; Accepted 21 October 2010
Academic Editor: Soo Hak Sung
Copyright q 2010 X c. Zhou and J g. Lin. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
We give out a general method to prove the complete convergence for arrays of rowwise ρ-mixing
random variables and to present some results on complete convergence under some suitable
conditions. Some results generalize previous known results for rowwise independent random
variables.
1. Introduction
Let {Ω, F,P} be a probability space, and let {X
n
; n ≥ 1} be a sequence of random variables
defined on this space.
Definition 1.1. The sequence {X
E
X − EX
2
E
Y − EY
2
⎫
⎪
⎬
⎪
⎭
−→ 0 1.1
as n →∞, where F
n
m
denotes the σ-field generated by {X
i
; m ≤ i ≤ n}.
The ρ-mixing random variables were first introduced by Kolmogorov and Rozanov
1. The limiting behavior of ρ-mixing random variables is very rich, for example, these in
the study by Ibragimov 2, Peligrad 3,andBradley4 for central limit theorem; Peligrad
5 and Shao 6, 7 for weak invariance principle; Shao 8 for complete convergence; Shao
2 Journal of Inequalities and Applications
9 for almost sure invariance principle; Peligrad 10,Shao11 and Liang and Yang 12 for
convergence rate; Shao 11, for the maximal inequality, and so forth.
1.2
for all x ≥ 0, i ≥ 1andn ≥ 1.
Definition 1.3. A real-valued function lx, positive and measurable on A, ∞ for some A>0,
is said to be slowly varying if
lim
x →∞
l
λx
l
x
1 for each λ>0.
1.3
Throughout the sequel, C will represent a positive constant although its value may
change from one appearance to the next; x indicates the maximum integer not larger than
x; IB denotes the indicator function of the set B.
The following lemmas will be useful in our study.
Lemma 1.4 Shao 11. Let {X
n
; n ≥ 1} be a sequence of ρ-mixing random variables with EX
i
0
and E|X
i
|
q
< ∞ for some q ≥ 2. Then there exists a positive constant K Kq, ρ· depending only
log n
i0
ρ
2
i
⎞
⎠
max
1≤i≤n
E
|
X
i
|
2
q/2
n exp
⎛
⎝
K
log n
|
α
I
|
X
n
|
≤ b
≤ C
E
|
X
|
α
I
|
X
|
≤ b
b
α
P
{|
X
|
defined in Definition 1.1 for the nthrowofanarray{X
ni
; i ≥ 1,n ≥ 1}, that is, for the sequence
X
n1
,X
n2
, ,n≥ 1.
Now, we state our main result.
Theorem 2.1. Let {X
ni
; i ≥ 1,n ≥ 1} be an array of rowwise ρ-mixing random variables satisfying
sup
n
∞
i1
ρ
2/q
n
2
i
< ∞ for some q ≥ 2, and let {a
ni
; i ≥ 1,n ≥ 1} be an array of real numbers. Let
{b
n
; n ≥ 1} be an increasing sequence of positive integers, and let {c
n
; n ≥ 1} be a sequence of positive
n
|a
ni
|
q
E|X
ni
|
q
I|a
ni
X
ni
| <εb
1/t
n
< ∞,
c
∞
n1
c
n
b
−q/tq/2
n
max
1≤i≤b
n
|a
i
j1
a
nj
X
nj
− a
nj
EX
nj
I
a
nj
X
nj
<εb
1/t
∞
n1
c
n
is convergent. Therefore, we will
consider that only
∞
n1
c
n
is divergent. Let
Y
nj
a
nj
X
nj
I
a
nj
X
nj
<εb
X
ni
Y
ni
}
,B
b
n
i1
{
a
ni
X
ni
/
Y
ni
}
.
2.2
Note that
P
max
1≤i≤b
n
|
S
ni
|
S
ni
− ET
ni
|
>εb
1/t
n
B
≤ P
max
1≤i≤b
n
|
T
ni
− ET
ni
|
>εb
1/t
n
b
ni
− ET
ni
|
>εb
1/t
n
< ∞.
2.4
By Markov inequality and Lemma 1.4, and note that the assumption
sup
n
∞
i1
ρ
2/q
n
2
i
< ∞ for some q ≥ 2, we get
P
max
1≤i≤b
n
|
T
ni
⎝
K
log b
n
i0
ρ
2/q
n
2
i
⎞
⎠
max
1≤i≤b
n
E
|
a
ni
X
ni
|
q
× I
b
n
max
1≤i≤b
n
E
|
a
ni
X
ni
|
2
I
|
a
ni
X
ni
|
<εb
1/t
n
q/2
⎫
⎬
⎭
−q/tq/2
n
max
1≤i≤b
n
|
a
ni
|
2
E
|
X
ni
|
2
I
|
a
ni
X
ni
|
<εb
1/t
n
q/2
a
ni
|
p
E
|
X
ni
|
p
O
n
ν−1
, as n −→ ∞ ,
3.1
for some 0 <ν<2/q. Then for any ε>0 and αp ≥ 1
∞
n1
n
αp−2
P
⎧
⎨
⎩
max
1≤i≤n
n,and1/t α in Theorem 2.1.By3.1,weget
∞
n1
c
n
b
n
i1
P
|
a
ni
X
ni
|
≥ εb
1/t
n
≤ C
∞
n1
n
αp−2
n
ni
|
p
≤ C
∞
n1
n
−2ν
< ∞,
∞
n1
c
n
b
−q/t1
n
max
1≤i≤b
n
E
|
a
ni
X
ni
|
q
I
X
ni
|
p
≤ C
∞
n1
n
−2ν
< ∞,
∞
n1
c
n
b
−q/tq/2
n
max
1≤i≤b
n
E
|
a
ni
X
ni
|
a
ni
|
p
E
|
X
ni
|
p
q/2
≤ C
∞
n1
n
αp1−q/2νq/2−1−1
< ∞
3.3
following from νq/2−1 < 0. By the assumption EX
ni
0forn ≥ 1, i ≥ 1andby3.1, we have
n
−α
max
1≤i≤n
−α
n
j1
a
nj
EX
nj
I
a
nj
X
nj
<εn
α
≤ Cn
−α
n
j1
p
E
X
nj
p
≤Cn
−αp1
max
1≤j≤n
|a
nj
|
p
E|X
nj
|
p
≤Cn
−αpν
−→ 0, as n −→ ∞ ,
3.4
because ν<1andαp ≥ 1. Thus, we complete the proof of the theorem.
Theorem 3.2. Let {X
ni
p
O
n
ν−1
, as n −→ ∞ ,
3.5
for some 0 <ν<2/q. Then for any ε>0 and αp ≥ 1 3.2 holds.
Theorem 3.3. Let {X
ni
,n ≥ 1,i ≥ 1} be an array of rowwise ρ-mixing random variables satisfying
sup
n
∞
i1
ρ
2/q
n
2
i
< ∞ for some q ≥ 2 and EX
ni
0 for all n ≥ 1,i ≥ 1. Let the random variables in
each row be stochastically dominated by a random variable X, and let {a
ni
; i ≥ 1,n ≥ 1} be an array of
real numbers. If for some 0 <t<2, ν>1/2
sup
i
j1
a
nj
X
nj
>εn
1/t
⎫
⎬
⎭
< ∞. 3.7
Proof. Take c
n
1andb
n
n for n ≥ 1. Then we see that a and b are satisfied. Indeed,
taking q ≥ max2, 1 2/ν,byLemma 1.5 and 3.6,weget
∞
a
ni
X
ni
|
≥ εn
1/t
≤ C
∞
n1
n
i1
P
{|
X
|
≥ Cn
ν
}
C
∞
n1
n
∞
kn
k 1
ν
≤ CE
|
X
|
2/ν
< ∞,
∞
n1
c
n
b
−q/t1
n
max
1≤i≤b
n
|
a
ni
|
q
E
|
X
ni
X
|
q
I
|
a
ni
X
|
<εn
1/t
n
q/t
|
a
ni
|
q
P
|
a
ni
X
|
≥ εn
1/t
ni
X
|
≥ εn
1/t
≤ C
∞
n1
n
−12/ν/t1
sup
i≥1
|
a
ni
|
12/ν
E
|
X
|
12/ν
C
∞
n1
∞
n1
n
−ν−1
< ∞ 3.8
In order to prove that c holds, we consider the following two cases.
If ν>2, by Lemma 1.5, C
r
inequality, and 3.6, we have
∞
n1
c
n
b
−q/tq/2
n
max
1≤i≤b
n
|
a
ni
|
2
E
|
X
2
E
|
X
|
2
I
|
a
ni
X
ni
|
<εn
1/t
q/2
8 Journal of Inequalities and Applications
C
∞
n1
n
q/2
max
1≤i≤n
P
|
12/ν
q/2
C
∞
n1
n
q/2
P
{|
X
|
≥ Cn
ν
}
q/2
≤ C
∞
n1
n
−q/tq/21/t1−2/νq/2
sup
i≥1
|
∞
n1
n
−ν1q/2
E
|
X
|
12/ν
q/2
< ∞.
3.9
If 1/2 <ν≤ 2, take q>2/2ν − 1. We have that 2ν − 1q/2 > 1. Note that in this case
E|X|
2
< ∞. We have
∞
n1
c
n
b
−q/tq/2
n
max
1≤i≤b
n
−q/tq/2
max
1≤i≤n
|
a
ni
|
2
E
|
X
|
2
I
|
a
ni
X
ni
|
<εn
1/t
q/2
C
∞
E
|
X
|
2
q/2
C
∞
n1
n
q/2−νq
E
|
X
|
2
q/2
≤ C
∞
n1
n
−2ν−1q/2
E
|
X
ni
<εn
1/t
−→ 0, as n −→ ∞ . 3.11
Journal of Inequalities and Applications 9
Indeed, by Lemma 1.5, we have
n
−1/t
max
1≤i≤n
i
j1
a
nj
E
|
X
|
C
n
j1
P
a
nj
X
≥ εn
1/t
≤ Cn
−ν
E
|
X
|
CnP
2/q
n
2
i
< ∞ for some q ≥ 2, and let {a
ni
; i ≥ 1,n ≥ 1} be an array of real numbers. Let
lx > 0 be a slowly varying function as x →∞.Ifforsome0 <t<2 and real number λ, and any
ε>0 the following conditions are fulfilled:
A
∞
n1
n
λ
ln
n
i1
P{|a
ni
X
ni
|≥εn
1/t
} < ∞,
B
∞
n1
ni
|
2
I|a
ni
X
ni
| <εn
1/t
q/2
< ∞,
then
∞
n1
n
λ
l
n
P
⎧
⎨
⎩
max
1≤i≤n
>εn
1/t
⎫
⎬
⎭
< ∞. 3.13
Proof. Let c
n
n
λ
ln and b
n
n.UsingTheorem 2.1,weobtain3.13 easily.
Theorem 3.5. Let {X
ni
; i ≥ 1,n ≥ 1} be an array of rowwise ρ-mixing identically distributed random
variables satisfying
∞
i1
ρ
2/q
n
2
i
⎧
⎨
⎩
max
1≤i≤n
i
j1
X
nj
>εn
1/t
⎫
⎬
⎭
< ∞. 3.15
Proof. Put λ αp −2anda
ni
n
∞
mn
P
εm
1/t
<
|
X
11
|
≤ ε
m 1
1/t
≤ C
∞
m1
P
εm
1/t
<
|
εm
1/t
<
|
X
11
|
≤ ε
m 1
1/t
≤ CE
|
X
11
|
αpt
l
|
X
11
|
t
< ∞,
3.16
which proves that condition A is satisfied.
αp−1−q/t
l
n
n
m1
E
|
X
11
|
q
I
ε
m − 1
1/t
≤
|
X
11
|
<εm
1/t
≤ C
n
≤ C
∞
m1
m
αp−q/t
l
m
E
|
X
11
|
q
I
ε
m − 1
1/t
≤
|
X
11
n
E
|
X
11
|
2
I
|
X
11
|
<εn
1/t
q/2
≤ C
∞
n1
n
αp−2−q/tq/2
l
n
n
Journal of Inequalities and Applications 11
If αpt ≥ 2, take q>max2, 2tαp − 1/2 − t. We have αp − q/t q/2 < 1. Note that in
this case E|X
11
|
2
< ∞.Weobtain
∞
n1
n
αp−2−q/tq/2
l
n
E
|
X
11
|
2
I
|
X
11
|
<εn
nj
I
|
X
11
|
<εn
1/t
−→ 0, as n −→ ∞ . 3.20
If αpt < 1, then
n
−1/t
max
1≤i≤n
i
n
−1/t
max
1≤i≤n
i
j1
EX
nj
I
|
X
11
|
<εn
1/t
Noting that for typical slowly varying functions, lx1andlxlog x, we can get
the simpler formulas in the above theorems.
Acknowledgments
The authors thank the academic editor and the reviewers for comments that greatly improved
the paper. This work is partially supported by the Anhui P rovince College Excellent
Young Talents Fund Project of China no. 2009SQRZ176ZD and National Natural Science
Foundation of China nos. 11001052, 10871001, 10971097.
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