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 Zhou1, 2 and Jin-guan Lin1
1
2
Department of Mathematics, Southeast University, Nanjing 210096, China
Department of Mathematics and Computer Science, Tongling University, Tongling, Anhui 244000, China
Correspondence should be addressed to Jin-guan Lin, jglin@seu.edu.cn
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 {Xn ; n ≥ 1} be a sequence of random variables
defined on this space.
Definition 1.1. The sequence {Xn ; n ≥ 1} is said to be ρ-mixing if
ρn sup
k≥1
⎧
⎪
⎨
⎫
⎪
⎬
|EXY − EXEY |
−→ 0
sup
⎪
⎪
∞ ⎩
X∈L2 F1k , Y ∈L2 Fnk
EX − EX2 EY − EY 2 ⎭
1.1
n
as n → ∞, where Fm
denotes the σ-field generated by {Xi ; 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, and Bradley 4 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, Shao 11 and Liang and Yang 12 for
convergence rate; Shao 11, for the maximal inequality, and so forth.
For arrays of rowwise independent random variables, complete convergence has
been extensively investigated see, e.g., Hu et al. 13, Sung et al. 14, and Kruglov
et al. 15. Recently, complete convergence for arrays of rowwise dependent random
variables has been considered. We refer to Kuczmaszewska 16 for ρ-mixing and ρ-mixing
sequences, Kuczmaszewska 17 for negatively associated sequence, and Baek and Park 18
for negatively dependent sequence. In the paper, we study the complete convergence for
arrays of rowwise ρ-mixing sequence under some suitable conditions using the techniques
of Kuczmaszewska 16, 17. We consider the case of complete convergence of maximum
weighted sums, which is different from Kuczmaszewska 16. Some results also generalize
some previous known results for rowwise independent random variables.
Now, we present a few definitions needed in the coming part of this paper.
Definition 1.2. An array {Xni ; i ≥ 1, n ≥ 1} of random variables is said to be stochastically
dominated by a random variable X if there exists a constant C, such that
P {|Xni | > x} ≤ CP {C|X| > x}
1.2
for all x ≥ 0, i ≥ 1 and n ≥ 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→∞
lλx
1 for each λ > 0.
lx
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 {Xn ; n ≥ 1} be a sequence of ρ-mixing random variables with EXi 0
and E|Xi |q < ∞ for some q ≥ 2. Then there exists a positive constant K Kq, ρ· depending only
on q and ρ· such that for any n ≥ 1
E max
1≤i≤n
i
q
Xj
⎛
⎛
≤ K ⎝n
2/q
⎞
log n q/2
i
exp⎝K
ρ 2 ⎠max E|Xi |2
j1
i0
⎛
n exp⎝K
log n
i0
1≤i≤n
⎞
⎞
q⎠
2/q
i ⎠
2
max E|Xi | .
ρ
1.4
1≤i≤n
Lemma 1.5 Sung 19. Let {Xn ; n ≥ 1} be a sequence of random variables which is stochastically
dominated by a random variable X. For any α > 0 and b > 0, the following statement holds:
E|Xn |α I|Xn | ≤ b ≤ C E|X|α I|X| ≤ b bα P {|X| > b} .
1.5
Journal of Inequalities and Applications
3
Lemma 1.6 Zhou 20. If lx > 0 is a slowly varying function as x → ∞, then
i
ii
m
n1
∞
ns ln ≤ Cms1 lm for s > −1,
nm
ns ln ≤ Cms1 lm for s < −1.
This paper is organized as follows. In Section 2, we give the main result and its proof.
A few applications of the main result are provided in Section 3.
2. Main Result and Its Proof
This paper studies arrays of rowwise ρ-mixing sequence. Let ρn i be the mixing coefficient
defined in Definition 1.1 for the nth row of an array {Xni ; i ≥ 1, n ≥ 1}, that is, for the sequence
Xn1 , Xn2 , . . . , n ≥ 1.
Now, we state our main result.
Theorem 2.1. Let {Xni ; i ≥ 1, n ≥ 1} be an array of rowwise ρ-mixing random variables satisfying
2/q i
supn ∞
i1 ρn 2 < ∞ for some q ≥ 2, and let {ani ; i ≥ 1, n ≥ 1} be an array of real numbers. Let
{bn ; n ≥ 1} be an increasing sequence of positive integers, and let {cn ; n ≥ 1} be a sequence of positive
real numbers. If for some 0 < t < 2 and any ε > 0 the following conditions are fulfilled:
a
b
c
∞
n1 cn
bn
i1
P {|ani Xni | ≥ εbn1/t } < ∞,
∞
−q/t1
max1≤i≤bn |ani |q E|Xni |q I|ani Xni |
n1 cn bn
∞
< εbn1/t < ∞,
−q/tq/2
max1≤i≤bn |ani |2 E|Xni |2 I|ani Xni |
n1 cn bn
q/2
< εbn1/t < ∞,
then
∞
n1
cn P
⎧
⎨
max
⎩1≤i≤bn
i
j1
anj Xnj − anj EXnj I anj Xnj < εbn1/t
> εbn1/t
⎫
⎬
⎭
< ∞.
2.1
Remark 2.2. Theorem 2.1 extends some results of Kuczmaszewska 17 to the case of arrays of
rowwise ρ-mixing sequence and generalizes the results of Kuczmaszewska 16 to the case of
maximum weighted sums.
Remark 2.3. Theorem 2.1 firstly gives the condition of the mixing coefficient, so the conditions
a–c do not contain the mixing coefficient. Thus, the conditions a–c are obviously
simpler than the conditions i–iii in Theorem 2.1 of Kuczmaszewska 16. Our conditions
are also different from those of Theorem 2.1 in the study by Kuczmaszewska 17: q ≥ 2 is
only required in Theorem 2.1, not q > 2 in Theorem 2.1 of Kuczmaszewska 17; the powers of
bn in b and c of Theorem 2.1 are −q/t 1 and −q/t q/2, respectively, not −q/t in Theorem
2.1 of Kuczmaszewska 17.
Now, we give the proof of Theorem 2.1.
4
Journal of Inequalities and Applications
Proof. The conclusion of the theorem is obvious if
consider that only ∞
n1 cn is divergent. Let
Ynj anj Xnj I anj Xnj < εbn1/t ,
bn
A {ani Xni Yni },
i1
∞
Tni n1 cn
i
is convergent. Therefore, we will
Ynj ,
Sni j1
i
anj Xnj ,
j1
2.2
bn
Yni }.
B {ani Xni /
i1
Note that
P max |Sni − ETni | > εbn1/t P
max |Sni − ETni | > εbn1/t
A
1≤i≤bn
1≤i≤bn
P
max |Sni − ETni | > εbn1/t
1≤i≤bn
≤ P max |Tni − ETni | > εbn1/t 1≤i≤bn
B
bn
i1
2.3
P |ani Xni | > εbn1/t .
By a it is enough to prove that for all ε > 0
∞
n1
supn
cn P max |Tni − ETni | > εbn1/t < ∞.
By
∞
Markov inequality and Lemma 1.4,
2/q i
ρ
i1 n 2 < ∞ for some q ≥ 2, we get
P max |Tni − ETni | >
1≤i≤bn
εbn1/t
2.4
1≤i≤bn
−q/t
≤ Cbn
−q/t
≤ Cbn
and
note
that
the
assumption
E max |Tni − ETni |q
1≤i≤bn
⎧
⎨
⎞
log bn 2/q
ρn 2i ⎠ max E|ani Xni |q
bn exp⎝K
⎩
⎛
i0
1≤i≤bn
⎛
⎞
log bn × I |ani Xni | < εbn1/t K exp⎝K
ρn 2i ⎠
i1
Journal of Inequalities and Applications
5
⎫
q/2 ⎬
× bn max E|ani Xni |2 I|ani Xni | < εbn1/t ⎭
1≤i≤bn
−q/t1
≤ Cbn
max |ani |q E|Xni |q I |ani Xni | < εbn1/t
1≤i≤bn
−q/tq/2
max |ani |2 E|Xni |2 I|ani Xni | < εbn1/t Cbn
1≤i≤bn
q/2
.
2.5
From b, c, and 2.5, we see that 2.4 holds.
3. Applications
Theorem 3.1. Let {Xni ; i ≥ 1, n ≥ 1} be an array of rowwise ρ-mixing random variables satisfying
2/q i
p
supn ∞
i1 ρn 2 < ∞ for some q ≥ 2, EXni 0, and E|Xni | < ∞ for all n ≥ 1, i ≥ 1, and
1 ≤ p ≤ 2. Let {ani ; i ≥ 1, n ≥ 1} be an array of real numbers satisfying the condition
max|ani |p E|Xni |p O nν−1 ,
1≤i≤n
as n −→ ∞,
3.1
for some 0 < ν < 2/q. Then for any ε > 0 and αp ≥ 1
⎧
⎨
nαp−2 P max
⎩ 1≤i≤n
n1
∞
i
anj Xnj > εnα
j1
⎫
⎬
⎭
< ∞.
3.2
Proof. Put cn nαp−2 , bn n, and 1/t α in Theorem 2.1. By 3.1, we get
∞
n1
cn
bn
i1
P |ani Xni | ≥ εbn1/t
≤C
∞
nαp−2
n1
∞
n1
−q/t1
c n bn
≤C
n
i1
n1
n−1 max|ani |p E|Xni |p ≤ C
n1
1≤i≤n
∞
nαp−2 n−αq1 nαq−p max|ani |p E|Xni |p ≤ C
1≤i≤n
∞
n1
q/2
max E|ani Xni |2 I |ani Xni | < εbn1/t
1≤i≤bn
∞
n1
1≤i≤bn
−q/tq/2
c n bn
∞
max E|ani Xni |q I |ani Xni | < εbn1/t
n1
∞
n−αp |ani |p E|Xni |p ≤ C
n−2ν < ∞,
n−2ν < ∞,
6
Journal of Inequalities and Applications
≤C
∞
αp−2 −αqq/2 α2−pq/2
n
n
n
q/2
p
p
max|ani | E|Xni |
1≤i≤n
n1
≤C
∞
nαp1−q/2νq/2−1−1 < ∞
n1
3.3
following from νq/2 − 1 < 0. By the assumption EXni 0 for n ≥ 1, i ≥ 1 and by 3.1, we have
n−α max
1≤i≤n
i
anj EXnj I anj Xnj < εnα
!
j1
n
≤ Cn−α
anj EXnj I anj Xnj < εnα
!
j1
n
≤ Cn−α
anj EXnj I anj Xnj ≥ εnα
!
3.4
j1
≤ Cn−αp
n
p
anj E Xnj
p
j1
≤ Cn−αp1 max|anj |p E|Xnj |p ≤ Cn−αpν −→ 0,
1≤j≤n
as n −→ ∞,
because ν < 1 and αp ≥ 1. Thus, we complete the proof of the theorem.
Theorem 3.2. Let {Xni ; i ≥ 1, n ≥ 1} be an array of rowwise ρ-mixing random variables satisfying
2/q i
p
supn ∞
i1 ρn 2 < ∞ for some q ≥ 2, EXni 0, and E|Xni | < ∞ for all n ≥ 1, i ≥ 1, and
1 ≤ p ≤ 2. Let the random variables in each row be stochastically dominated by a random variable X,
such that E|X|p < ∞, and let {ani ; i ≥ 1, n ≥ 1} be an array of real numbers satisfying the condition
max|ani |p O nν−1 ,
1≤i≤n
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 {Xni , n ≥ 1, i ≥ 1} be an array of rowwise ρ-mixing random variables satisfying
2/q i
supn ∞
i1 ρn 2 < ∞ for some q ≥ 2 and EXni 0 for all n ≥ 1, i ≥ 1. Let the random variables in
each row be stochastically dominated by a random variable X, and let {ani ; i ≥ 1, n ≥ 1} be an array of
real numbers. If for some 0 < t < 2, ν > 1/2
sup|ani | O n1/t−ν ,
i≥1
E|X|12/ν < ∞,
3.6
Journal of Inequalities and Applications
7
then for any ε > 0
⎧
⎨
P max
⎩ 1≤i≤n
n1
∞
i
anj Xnj > εn1/t
j1
⎫
⎬
⎭
< ∞.
3.7
Proof. Take cn 1 and bn n for n ≥ 1. Then we see that a and b are satisfied. Indeed,
taking q ≥ max2, 1 2/ν, by Lemma 1.5 and 3.6, we get
∞
n1
cn
bn
i1
P |ani Xni | ≥ εbn1/t
∞
n
P |ani Xni | ≥ εn1/t ≤ C
n1 i1
C
∞
∞
n
P {|X| ≥ Cnν }
n1 i1
n
∞
P Ckν ≤ |X| < Ck 1ν
n1 kn
≤C
∞
k2 P Ckν ≤ |X| < Ck 1ν ≤ CE|X|2/ν < ∞,
k1
∞
n1
−q/t1
c n bn
≤C
max |ani |q E|Xni |q I |ani Xni | < εbn1/t
1≤i≤bn
∞
"
−q/t1
n
∞
≤C
1≤i≤n
n
∞
nq/t 1/t
P
X|
≥
εn
|a
ni
|ani |q
∞
n maxP |ani X| ≥ εn1/t
%12/ν
E|X|12/ν C
sup|ani |
i≥1
∞
nP {|X| ≥ Cnν }
n1
∞
n−ν−1 E|X|12/ν C
n1
n−ν−1 E|X|12/ν ≤ C
n1
∞
n−ν−1 < ∞
n1
In order to prove that c holds, we consider the following two cases.
If ν > 2, by Lemma 1.5, Cr inequality, and 3.6, we have
∞
n1
−q/tq/2
c n bn
≤C
∞
n1
#
1≤i≤n
n1
$
−12/ν/t1
n1
≤C
E|X| I |ani X| < εn
1/t
n−12/ν/t1 max|ani |12/ν E|X|12/ν C
n1
∞
q
1≤i≤n
n1
≤C
max|ani |
q
q/2
1/t
max |ani | E|Xni | I |ani Xni | < εbn
2
2
1≤i≤bn
q/2
n−q/tq/2 max|ani |2 E|X|2 I |ani Xni | < εn1/t
1≤i≤n
3.8
8
Journal of Inequalities and Applications
∞
C
q/2
n
1≤i≤n
n1
≤C
q/2
1/t
maxP |ani X| ≥ εn
q/2
n−q/tq/2 n1/t1−2/νq/2 max|ani |12/ν E|X|12/ν
∞
1≤i≤n
n1
∞
C
nq/2 P {|X| ≥ Cnν }q/2
n1
≤C
%12/νq/2
$
∞
−q/tq/21/t1−2/νq/2
n
E|X|12/ν
sup|ani |
i≥1
n1
∞
C
q/2
q/2
nq/2 n−12/ννq/2 E|X|12/ν
n1
≤C
q/2
n−ν1q/2 E|X|12/ν
< ∞.
∞
n1
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
−q/tq/2
c n bn
≤C
q/2
max |ani |2 E|Xni |2 I |ani Xni | < εbn1/t
1≤i≤bn
∞
q/2
n−q/tq/2 max|ani |2 E|X|2 I |ani Xni | < εn1/t
C
1≤i≤n
n1
≤C
∞
n
%q
sup|ani |
i≥1
n1
≤C
∞
nq/2 P {|X| ≥ Cnν }q/2
n1
$
−q/tq/2
∞
E|X|2
q/2
C
∞
q/2
nq/2−νq E|X|2
n1
q/2
n−2ν−1q/2 E|X|2
< ∞.
3.10
n1
The proof will be completed if we show that
n−1/t max
1≤i≤n
i
j1
anj EXnj I anj Xni < εn1/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
anj EXnj I anj Xni < εn1/t ≤ Cn−1/t
j1
n
anj E|X| C
j1
n
P
anj X ≥ εn1/t
j1
≤ Cn−ν E|X| CnP {|X| ≥ εnν }
≤ Cn−ν E|X| Cn−ν1 E|X|12/ν −→ 0,
as n −→ ∞.
3.12
Theorem 3.4. Let {Xni ; i ≥ 1, n ≥ 1} be an array of rowwise ρ-mixing random variables satisfying
2/q i
supn ∞
i1 ρn 2 < ∞ for some q ≥ 2, and let {ani ; i ≥ 1, n ≥ 1} be an array of real numbers. Let
lx > 0 be a slowly varying function as x → ∞. If for some 0 < t < 2 and real number λ, and any
ε > 0 the following conditions are fulfilled:
A
B
C
∞
n
n1
nλ ln
n1
nλ−q/t1 lnmax1≤i≤n E|ani Xni |q I|ani Xni | < εn1/t < ∞,
∞
∞
n1
i1
P {|ani Xni | ≥ εn1/t } < ∞,
nλ−q/tq/2 lnmax1≤i≤n |ani |2 E|Xni |2 I|ani Xni | < εn1/t q/2
< ∞,
then
∞
n1
nλ lnP
⎧
⎨
max
⎩ 1≤i≤n
i
anj Xnj − anj EXnj I anj Xnj < εn1/t
> εn1/t
j1
⎫
⎬
⎭
< ∞.
3.13
Proof. Let cn nλ ln and bn n. Using Theorem 2.1, we obtain 3.13 easily.
Theorem 3.5. Let {Xni ; i ≥ 1, n ≥ 1} be an array of rowwise ρ-mixing identically distributed random
2/q i
variables satisfying ∞
i1 ρn 2 < ∞ for some q ≥ 2 and EX11 0. Let lx > 0 be a slowly varying
function as x → ∞. If for α > 1/2, αp > 1, and 0 < t < 2
E|X11 |αpt l |X11 |t < ∞,
3.14
then
⎧
⎨
nαp−2 lnP max
⎩ 1≤i≤n
n1
∞
i
j1
Xnj > εn1/t
⎫
⎬
⎭
< ∞.
3.15
Proof. Put λ αp − 2 and ani 1 for n ≥ 1, i ≥ 1 in Theorem 3.4. To prove 3.15, it is enough to
note that under the assumptions of Theorem 3.4, the conditions A–C of Theorem 3.4 hold.
10
Journal of Inequalities and Applications
By Lemma 1.6, we obtain
∞
nαp−1 lnP |X11 | > εn1/t n1
∞
nαp−1 ln
P εm1/t < |X11 | ≤ εm 11/t
mn
n1
∞
≤C
∞
P εm1/t < |X11 | ≤ εm 11/t
m1
∞
≤C
m
nαp−1 ln
n1
mαp lmP εm1/t < |X11 | ≤ εm 11/t
3.16
m1
≤ CE|X11 |αpt l |X11 |t < ∞,
which proves that condition A is satisfied.
Taking q > max2, αpt, we have αp − q/t < 0. By Lemma 1.6, we have
nαp−1−q/t lnE|X11 |q I |X11 | ≤ εn1/t
∞
n1
∞
nαp−1−q/t ln
n1
≤C
∞
n
E|X11 |q I εm − 11/t ≤ |X11 | < εm1/t
m1
E|X11 |q I εm − 11/t ≤ |X11 | < εm1/t
m1
≤C
∞
∞
nαp−1−q/t ln
3.17
nm
mαp−q/t lmE|X11 |q I εm − 11/t ≤ |X11 | < εm1/t
m1
≤ CE|X11 |αpt l |X11 |t < ∞,
which proves that B holds.
In order to prove that C holds, we consider the following two cases.
If αpt < 2, take q > 2. We have
∞
q/2
nαp−2−q/tq/2 ln E|X11 |2 I |X11 | < εn1/t
n1
≤C
∞
q/2
nαp−2−q/tq/2 lnnq/t−αpq/2 E|X11 |αpt I |X11 | < εn1/t
n1
≤C
∞
n1
nαp−11−q/2−1 ln < ∞.
3.18
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|X11 |2 < ∞. We obtain
∞
q/2
nαp−2−q/tq/2 ln E|X11 |2 I |X11 | < εn1/t
≤C
n1
∞
nαp−2−q/tq/2 ln < ∞.
n1
3.19
The proof will be completed if we show that
n−1/t max
1≤i≤n
i
EXnj I |X11 | < εn1/t −→ 0,
as n −→ ∞.
3.20
j1
If αpt < 1, then
1≤i≤n
EXnj I |X11 | < εn1/t ≤ Cn1−αp E|X11 |αpt −→ 0,
i
n−1/t max
as n −→ ∞.
3.21
j1
If αpt ≥ 1, note that EX11 0, then
n−1/t max
1≤i≤n
i
EXnj I |X11 | < εn1/t
3.22
j1
≤ n−1/t1
EX11 I |X11 | ≥ εn1/t ≤ Cn1−αp E|X11 |αpt −→ 0,
as n −→ ∞.
We complete the proof of the theorem.
Noting that for typical slowly varying functions, lx 1 and lx log 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 Province College Excellent
Young Talents Fund Project of China no. 2009SQRZ176ZD and National Natural Science
Foundation of China nos. 11001052, 10871001, 10971097.
References
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process,” Theory of Probability and Its Applications, vol. 2, pp. 222–227, 1960.
2 I. A. Ibragimov, “A note on the central limit theorem for dependent random variables,” Theory of
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