Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 372390, 13 pages
doi:10.1155/2010/372390
Research Article
On Complete Convergence for Weighted Sums of
ϕ-Mixing Random Variables
Wang Xuejun, Hu Shuhe, Yang Wenzhi, and Shen Yan
School of Mathematical Science, Anhui University, Hefei 230039, China
Correspondence should be addressed to Hu Shuhe, hushuhe@263.net
Received 4 February 2010; Revised 8 May 2010; Accepted 11 June 2010
Academic Editor: Andrei Volodin
Copyright q 2010 Wang Xuejun et al. 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.
Some results on complete convergence for weighted sums ni1 ani Xi are presented, where {Xn ,
n ≥ 1} is a sequence of ϕ-mixing random variables and {ani , n ≥ 1, i ≥ 1} is an array of constants.
They generalize the corresponding results for i.i.d sequence to the case of ϕ-mixing sequence.
1. Introduction
Let {Xn , n ≥ 1} be a sequence of random variables defined on a fixed probability space
Ω, F, P . Let n and m be positive integers. Write Fnm σXi , n ≤ i ≤ m. Given σ-algebras
B, R in F, let
ϕB, R sup
|P B | A − P B|.
A∈B,B∈R,P A>0
1.1
Define the ϕ-mixing coefficients by
∞
,
ϕn sup ϕ F1k , Fkn
n ≥ 0.
1.2
k≥1
Definition 1.1. A random variable sequence {Xn , n ≥ 1} is said to be a ϕ-mixing random
variable sequence if ϕn ↓ 0 as n → ∞.
ϕ-mixing random variables were introduced by Dobrushin 1 and many applications
have been found. See, for example, Dobrushin 1, Utev 2, and Chen 3 for central limit
2
Journal of Inequalities and Applications
theorem, Herrndorf 4 and Peligrad 5 for weak invariance principle, Sen 6, 7 for weak
convergence of empirical processes, Shao 8 for almost sure invariance principles, Hu and
Wang 9 for large deviations, and so forth. When these are compared with the corresponding
results of independent random variable sequences, there still remains much to be desired.
Throughout the paper, let IA be the indicator function of the set A. We assume that
φx is a positive increasing function on 0, ∞ satisfying φx ↑ ∞ as x → ∞ and ψx is the
inverse function of φx. Since φx ↑ ∞, it follows that ψx ↑ ∞. For easy notation, we let
φ0 0 and ψ0 0. an Obn denotes that there exists a positive constant C such that
|an /bn | ≤ C. C denotes a positive constant which may be different in various places.
Let {X, Xn , n ≥ 1} be a sequence of i.i.d. random variables and let {ani , n ≥ 1, i ≥ 1}
be an array of constants. The almost sure limiting behavior of weighted sums ni1 ani Xi was
studied by many authors; see, for example, Choi and Sung 10, Cuzick 11, Wu 12, and
Sung 13, 14, and so forth.
The main purpose of this paper is to extend the complete convergence for weighted
n
sums i1 ani Xi of i.i.d. random variables to the case of ϕ-mixing random variables.
Definition 1.2. A sequence {Xn , n ≥ 1} of random variables is said to be stochastically
dominated by a random variable X if there exists a positive constant C, such that
P |Xn | > x ≤ CP |X| > x
1.3
for all x ≥ 0 and n ≥ 1.
Definition 1.3. A double array {ani , n ≥ 1, i ≥ 1} of real numbers is said to be a Toeplitz array
if limn → ∞ ani 0 for each i ≥ 1 and
∞
|ani | ≤ C
1.4
i1
for all n ≥ 1, where C is a positive constant.
Lemma 1.4. 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
where C is a positive constant.
Lemma 1.5 cf. 15, Lemma 1.2.8. Let {Xn , n ≥ 1} be a sequence of ϕ-mixing random variables.
∞
, p ≥ 1, q ≥ 1, and 1/p 1/q 1. Then
Let X ∈ Lp F1k , Y ∈ Lq Fkn
1/p 1/p 1/q
E|X|p
E|Y |q
.
|EXY − EXEY | ≤ 2 ϕn
Lemma 1.6 cf. 8, Lemma 2.2. Let {Xn , n ≥ 1} be a ϕ-mixing sequence. Put Ta n Suppose that there exists an array {Ca,n } of positive numbers such that
ETa2 n ≤ Ca,n
for every a ≥ 0, n ≥ 1.
1.6
an
ia1
Xi .
1.7
Journal of Inequalities and Applications
3
Then for every q ≥ 2, there exists a constant C depending only on q and ϕ· such that
q
E maxTa j 1≤j≤n
≤C
q/2
Ca,n
E
max |Xi |
q
a1≤i≤an
1.8
for every a ≥ 0 and n ≥ 1.
1/2
Lemma 1.7. Let {Xn , n ≥ 1} be a sequence of ϕ-mixing random variables satisfying ∞
n <
n1 ϕ
q
∞. q ≥ 2. Assume that EXn 0 and E|Xn | < ∞ for each n ≥ 1. Then there exists a constant C
depending only on q and ϕ· such that
⎡
aj q ⎞
q/2 ⎤
an
an
q
2
⎦
Xi ⎠ ≤ C⎣
E|Xi | EXi
E⎝max
1≤j≤n
ia1
ia1
ia1
⎛
1.9
for every a ≥ 0 and n ≥ 1. In particular, one has
⎛
⎡
j
⎞
q/2 ⎤
n
n
q
⎦
E⎝max Xi ⎠ ≤ C⎣ E|Xi |q EXi2
1≤j≤n
i1
i1
i1
1.10
for every n ≥ 1.
Proof. By Lemma 1.5, we can see that
E
an
2
Xi
≤
ia1
an
ia1
≤
an
ia1
a1≤i<j≤an
n−1 an−k
2
EXi2 2
ϕ1/2 k EXi2 EXki
1.11
k1 ia1
≤
1/2 1/2
EXj2
ϕ1/2 j − i EXi2
EXi2 4
∞
1 4 ϕ1/2 k
k1
an
.
EXi2 Ca,n ,
ia1
which implies 1.7. By Lemma 1.6, we can get the desired result 1.9 immediately. The proof
is complete.
Lemma 1.8. Assume that the inverse function ψx of φx satisfies
ψn
n
1
On.
ψi
i1
1.12
If Eφ|X| < ∞, then
∞
1
E|X|I |X| > ψn < ∞.
ψn
n1
1.13
4
Journal of Inequalities and Applications
Proof. The proof is similar to that of Lemma 1 by Sung 14. So we omit it.
2. Main Results and Their Proofs
Theorem 2.1. Let {X, Xn , n ≥ 1} be a sequence of identically distributed ϕ-mixing random variables
1/2
n < ∞, EX 0, EX 2 < ∞, and Eφ|X| < ∞. Assume that the inverse
satisfying ∞
n1 ϕ
function ψx of φx satisfies 1.12. Let {ani , n ≥ 1, i ≥ 1} be an array of constants such that
i max1≤i≤n |ani | O1/ψn;
ii ni1 a2ni Olog−1−α n for some α > 0.
Then for any ε > 0,
∞
−1
n P
n1
j
max ani Xi > ε < ∞.
1≤j≤n
i1
2.1
Proof. For each n ≥ 1, denote
n
Xj
A
n i1
Xj I Xj ≤ ψn ,
n
Xi Xi
n
Tj
j i1
n
|Xi | ≤ ψn ,
i1
n
ani Xi
BA
n
− Eani Xi
n i1
n
Xi / Xi
1 ≤ j ≤ n,
,
j
max ani Xi > ε .
1≤j≤n
i1
n
|Xi | > ψn ,
i1
En 2.2
It is easy to check that
j
ani Xi i1
j
j
ani Xi I |Xi | ≤ ψn ani Xi I |Xi | > ψn
i1
n
Tj
i1
j
j
n
Eani Xi ani Xi I |Xi | > ψn ,
i1
i1
j
n n maxT Eani Xi > ε En B
1≤j≤n j
i1
En En A En B j
n n maxTj > ε − max Eani Xi B.
1≤j≤n
1≤j≤n
i1
⊂
2.3
Journal of Inequalities and Applications
5
Therefore
P En ≤ P
j
n n maxTj > ε − max Eani Xi P B
1≤j≤n
1≤j≤n
i1
j
n
n n maxTj > ε − max Eani Xi P |Xi | > ψn .
1≤j≤n
1≤j≤n
i1
i1
≤P
2.4
Firstly, we will show that
j
n max Eani Xi −→ 0,
1≤j≤n
i1
as n −→ ∞.
2.5
It follows from Lemma 1.8 and Kronecker’s lemma that
n
1 E|X|I |X| > ψi −→ 0,
ψn i1
as n −→ ∞.
2.6
By EX 0, condition i, 2.6, and ψn ↑ ∞, we can see that
j
j
n max Eani Xi max Eani Xi I |Xi | > ψn 1≤j≤n i1
1≤j≤n
i1
≤
n
E|ani Xi |I |Xi | > ψn
i1
2.7
n
≤
|ani |E|X|I |X| > ψn
i1
≤
n
1 E|X|I |X| > ψi −→ 0,
ψn i1
as n −→ ∞,
which implies 2.5. By 2.4 and 2.5, we can see that, for sufficiently large n,
j
n
n ε
max ani Xi > ε ≤ P maxTj >
P |Xi | > ψn .
1≤j≤n
1≤j≤n
2
i1
i1
P
2.8
To prove 2.1, it suffices to show that
n ε
< ∞,
n−1 P maxTj >
1≤j≤n
2
n1
∞
∞
n
n−1 P |Xi | > ψn < ∞.
n1
i1
2.9
6
Journal of Inequalities and Applications
By Markov’s inequality, Lemma 1.7, EX 2 < ∞, and condition ii, we have
∞
∞
n ε
n 2
≤ C n−1 E maxTj n−1 P maxTj >
1≤j≤n
1≤j≤n
2
n1
n1
≤C
∞
n−1
n1
C
∞
i1
n−1
n1
≤C
∞
∞
n
a2ni EX 2 I |X| ≤ ψn
2.10
i1
n−1
n1
≤C
n
n 2
Eani Xi n
a2ni
i1
n−1 log−1−α n < ∞.
n1
It follows from Eφ|X| < ∞ that
∞
∞
∞
n
n−1 P |Xi | > ψn P |X| > ψn P φ|X| > n ≤ CE φ|X| < ∞.
n1
i1
n1
2.11
n1
We complete the proof of the theorem.
1/2
Theorem 2.2. Let {Xn , n ≥ 1} be a sequence of ϕ-mixing random variables satisfying ∞
n <
n1 ϕ
∞ and let {ani , n ≥ 1, i ≥ 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 q ≥ 2, 0 < t < 2,
and for any ε > 0, the following conditions are satisfied:
bn ∞
cn P |ani Xi | ≥ εbn1/t < ∞,
n1
∞
n1
|ani |q E|Xi |q I |ani Xi | < εbn1/t < ∞,
bn
−q/t cn bn
2.13
i1
b
∞
q/2
n
−q/t 2
2
1/t
c n bn
ani EXi I |ani Xi | < εbn
< ∞,
n1
2.12
i1
2.14
i1
then
⎧
⎫
∞
⎨
⎬
#
i "
anj Xj − anj EXj I anj Xj < εbn1/t ≥ εbn1/t < ∞.
cn P max ⎩1≤i≤bn j1
⎭
n1
2.15
Proof. Note that if the series ∞
holds. Therefore, we will
n1 cn is convergent, then 2.15
consider only such sequences {cn , n ≥ 1} for which the series ∞
n1 cn is divergent.
Journal of Inequalities and Applications
7
Let
n
Yi
A
ani Xi I |ani Xi | < εbn1/t ,
i
n
Yj ,
n ≥ 1, i ≥ 1,
j1
bn '
(
n
Yi ani Xi ,
i1
S
ni BA
bn '
bn ( n
Yi /
ani Xi |ani Xi | ≥ εbn1/t ,
i1
2.16
i1
⎫
⎬
#
i "
anj Xj − anj EXj I anj Xj < εbn1/t ≥ εbn1/t .
En max ⎭
⎩1≤i≤bn j1
⎧
⎨
Therefore
⎧
⎨
⎫
⎬
#
i "
P max anj Xnj − anj EXj I anj Xj < εbn1/t ≥ εbn1/t
⎩1≤i≤bn j1
⎭
2.17
P En P En A P En B ≤ P En A P B
bn q
1/t
−q −q/t
≤
P |ani Xi | ≥ εbn ε bn E max Sni − ESni
.
1≤i≤bn
i1
n
Using the Cr inequality and Jensen’s inequality, we can estimate E|Yi
following way:
n q
− EYi | in the
n
n q
EYi − EYi ≤ C|ani |q E|Xi |q I |ani Xi | < εbn1/t .
2.18
By 2.17, 2.18, and Lemma 1.7, we can get
⎧
⎨
⎫
⎬
#
i "
P max anj Xj − anj EXj I anj Xj < εbn1/t ≥ εbn1/t
⎩1≤i≤bn j1
⎭
≤C
bn bn
−q/t P |ani Xi | ≥ εbn1/t Cbn
|ani |q E|Xi |q I |ani Xi | < εbn1/t
i1
−q/t
Cbn
i1
b
n
q/2
a2ni EXi2 I
|ani Xi | < εbn1/t
.
i1
Therefore, we can conclude that 2.15 holds by 2.12, 2.13, 2.14, and 2.19.
2.19
8
Journal of Inequalities and Applications
Theorem 2.3. Let 1 ≤ p ≤ 2 and let {Xn , n ≥ 1} be a sequence of ϕ-mixing random variables
1/2
n < ∞, EXn 0, and E|Xn |p < ∞ for n ≥ 1. Let {ani , n ≥ 1, i ≥ 1} be an
satisfying ∞
n1 ϕ
array of real numbers satisfying the following condition:
n
|ani |p E|Xi |p O nδ
as n −→ ∞
2.20
i1
for some 0 < δ ≤ 2/q and q > 2. Then for any ε > 0 and αp ≥ 1,
⎛
⎞
∞
i
αp−2 ⎝
α
n
P max anj Xj ≥ εn ⎠ < ∞.
1≤i≤n n1
j1
2.21
Proof. Take cn nαp−2 , bn n, and 1/t α in Theorem 2.2. By 2.20 we have
bn ∞
∞
∞
n
|ani |p E|Xi |p
cn P |ani Xi | ≥ εbn1/t ≤ C nαp−2
≤
C
n−2δ < ∞,
nαp
n1
i1
n1
i1
n1
bn
∞
∞
∞
n
−q/t c n bn
n−2 |ani |p E|Xi |p ≤ C n−2δ < ∞,
|ani |q E|Xi |q I |ani Xi | < εbn1/t ≤
n1
∞
n1
i1
b
n
−q/t c n bn
n1
a2ni EXi2 I
q/2
|ani Xi | <
εbn1/t
≤C
i1
∞
n1
≤C
∞
n1
i1
αp−2−αpq/2
n
n1
n
q/2
p
p
|ani | E|Xi |
i1
nαp−2−αpq/2δq/2 ≤ C
∞
nαp1−q/2−1 < ∞
n1
2.22
following from δq/2 ≤ 1. By the assumption EXn 0 for n ≥ 1 and 2.20 we get
n i
1
α
anj EXj I anj Xj < εn ≤ 1
anj EXj I anj Xj < εnα max
α
α
n 1≤i≤n j1
n j1
n 1
anj EXj I anj Xj ≥ εnα α
n j1
≤
n 1 anj p EXj p ≤ Cnδ−αp −→ 0,
nαp j1
as n −→ ∞
2.23
following from δ < 1 and αp ≥ 1. We get the desired result by Theorem 2.2 immediately. The
proof iscompleted.
Journal of Inequalities and Applications
9
Theorem 2.4. Let 1 ≤ p ≤ 2 and let {Xn , n ≥ 1} be a sequence of ϕ-mixing random variables
1/2
n < ∞, EXn 0, and E|Xn |p < ∞ for n ≥ 1. Assume that the random
satisfying ∞
n1 ϕ
variables are stochastically dominated by a random variable X such that E|X|p < ∞ and let {ani ,
n ≥ 1, i ≥ 1} be an array of real numbers satisfying the following condition:
n
|ani |p O nδ
as n −→ ∞
2.24
i1
for some 0 < δ ≤ 2/q and q > 2. Then for any ε > 0 and αp ≥ 1, 2.21 holds.
Proof. The proof is similar to that of Theorem 2.3. We only need to note that
p
E|Xn | )∞
tp dP |Xn | ≤ t
0
−
)∞
tp dP |Xn | > t
0
− lim t P |Xn | > t p
t→∞
0p
≤ Cp
)∞
)∞
)∞
P |Xn | > tdtp
0
2.25
tp−1 P |Xn | > tdt
0
tp−1 P |X| > tdt
0
CE|X|p < ∞
for each n ≥ 1.
1/2
Theorem 2.5. Let {Xn , n ≥ 1} be a sequence of ϕ-mixing random variables satisfying ∞
n <
n1 ϕ
∞ and let {ani , n ≥ 1, i ≥ 1} be a Toeplitz array. Assume that the random variables are stochastically
dominated by a random variable X. If for some 0 < t < 2 and δ > 1/t,
sup|ani | O n1/t−δ ,
E|X|β < ∞,
2.26
⎞
i
P ⎝max anj Xj ≥ εn1/t ⎠ < ∞.
1≤i≤n n1
j1
2.27
i≥1
where β max2/δ, 1 1/δ, then for any ε > 0,
∞
⎛
10
Journal of Inequalities and Applications
Proof. Take cn 1, bn n for n ≥ 1 and q ≥ max2, 1 1/δ in Theorem 2.2. Then we can see
that 2.12 and 2.13 are satisfied. In fact, by 1.4 and 2.26 we have
bn n
∞
∞ cn P |ani Xi | ≥ εbn1/t P |ani Xi | ≥ εn1/t
n1
i1
n1 i1
≤C
n
∞ P |ani X| ≥ Cn1/t
n1 i1
≤C
∞ n
P |X| ≥ Cnδ
2.28
n1 i1
C
∞ ∞ n P Ckδ ≤ |X| < Ck 1δ
n1 kn
≤C
∞
k2 P Ckδ ≤ |X| < Ck 1δ
k1
≤ CE|X|2/δ < ∞,
and by Lemma 1.4, 1.5, and 2.26 we have
∞
|ani |q E|Xi |q I |ani Xi | < εbn1/t
bn
−q/t c n bn
n1
i1
∞
n−q/t
n1
≤C
∞
n1
≤C
∞
n1
≤C
∞
n1
≤C
∞
n
|ani |q E|Xi |q I |ani Xi | < εn1/t
i1
−q/t
n
n
q
q
|ani | E|X| I |ani X| < εn
1/t
i1
nq/t 1/t
q P |ani X| ≥ εn
|ani |
n
∞ n
n−11/δ/t |ani |11/δ E|X|11/δ C
P |ani X| ≥ εn1/t
i1
n−1/t−1 E|X|11/δ
n1 i1
n
|ani | CE|X|2/δ
i1
n−1/t−1 CE|X|2/δ < ∞.
n1
In order to prove that 2.14 holds, we should consider the following two cases.
2.29
Journal of Inequalities and Applications
11
In the case δ > 1, by Lemma 1.4, 1.5, 2.26, and Cr inequality, we have
b
∞
q/2
n
−q/t 2
2
1/t
c n bn
ani EXi I |ani Xi | < εbn
n1
i1
∞
n
q/2
−q/t
2
2
1/t
n
ani EXi I |ani Xi | < εn
n1
i1
q/2
n
∞
n
∞ 11/δ
11/δ
−q/2t−q/2δt
≤C n
E|X|
C
P |ani X| ≥ εn1/t
|ani |
n1
i1
n1 i1
q/2
∞
n
q/2 11/δ
−q/2t−q/2δt 1/δ1/t−δq/2
E|X|
n
CE|X|2/δ
≤C n
|ani |
n1
≤C
2.30
i1
∞
n−q/2t−q/2 CE|X|2/δ
n1
C
∞
n−q/211/t CE|X|2/δ < ∞.
n1
In the case 0 < δ ≤ 1, we can get
b
∞
q/2
n
−q/t 2
2
1/t
c n bn
ani EXi I |ani Xi | < εbn
n1
i1
∞
n
q/2
−q/t
2
2
1/t
n
ani EXi I |ani Xi | < εn
n1
i1
q/2
n
∞
n
∞ −q/t 1/t−δq/2
2
≤C n n
C
P |ani X| ≥ εn1/t
|ani |EX
n1
i1
2.31
n1 i1
q/2
∞
n
q/2 −q/2t−qδ/2
2
EX
CE|X|2/δ
≤C n
|ani |
n1
≤C
i1
∞
n−q/2δ1/t CE|X|2/δ < ∞.
n1
To complete the proof of the theorem, we only need to prove
i
n−1/t max anj EXj I anj Xj < εn1/t −→ 0,
1≤i≤n j1
as n −→ ∞.
2.32
12
Journal of Inequalities and Applications
Indeed, by Lemma 1.4, it follows that
i
n−1/t max anj EXj I anj Xj < εn1/t 1≤i≤n j1
≤ Cn−1/t
n
n anj E|X| C P anj X ≥ εn1/t
j1
≤ Cn−1/t C
2.33
j1
n
P anj X ≥ εn1/t −→ 0,
as n −→ ∞.
j1
Thus we get the desired result.
Acknowledgments
The authors are most grateful to the Editor Andrei I. Volodin and anonymous referee for
the careful reading of the manuscript and valuable suggestions which helped in significantly
improving an earlier version of this paper. This work was supported by the National Natural
Science Foundation of China 10871001, 60803059, Provincial Natural Science Research
Project of Anhui Colleges KJ2010A005, Talents Youth Fund of Anhui Province Universities
2010SQRL016ZD, and Youth Science Research Fund of Anhui University 2009QN011A.
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