Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 379743, 12 pages
doi:10.1155/2009/379743
Research Article
Moment Inequality for ϕ-Mixing Sequences and
Its Applications
Wang Xuejun, Hu Shuhe, Shen Yan, and Yang Wenzhi
School of Mathematical Science, Anhui University, Hefei, Anhui 230039, China
Correspondence should be addressed to Hu Shuhe, hushuhe@263.net
Received 11 April 2009; Accepted 21 September 2009
Recommended by Sever Silvestru Dragomir
Firstly, the maximal inequality for ϕ-mixing sequences is given. By using the maximal inequality,
we study the convergence properties for ϕ-mixing sequences. The Hájek-Rényi-type inequality,
strong law of large numbers, strong growth rate and the integrability of supremum for ϕ-mixing
sequences are obtained.
Copyright q 2009 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.
1. Introduction
Let {Xn , n ≥ 1} be a random variable sequence defined on a fixed probability space Ω, F, P and Sn ni1 Xi for each n ≥ 1. 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
k≥1
n ≥ 0.
1.2
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 → ∞.
The concept of ϕ-mixing random variables was introduced by Dobrushin 1 and
many applications have been found. See, for example, Dobrushin 1, Utev 2, and Chen 3
2
Journal of Inequalities and Applications
for central limit theorem, Herrndorf 4 and Peligrad 5 for weak invariance principle, Sen
6, 7 for weak convergence of empirical processes, Iosifescu 8 for limit theorem, Peligrad
9 for Ibragimov-Iosifescu conjecture, Shao 10 for almost sure invariance principles, Hu
and Wang 11 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. The main purpose of this paper is to study the maximal inequality for ϕ-mixing
sequences, by which we can get the Hájek-Rényi-type inequality, strong law of large numbers,
strong growth rate, and the integrability of supremum for ϕ-mixing sequences.
Throughout the paper, C denotes a positive constant which may be different in various
places. The main results of this paper depend on the following lemmas.
Lemma 1.2 see Lu and Lin 12. 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 1.3 see Shao 10. 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
for every a ≥ 0, n ≥ 1.
ETa2 n ≤ Ca,n
1.3
an
ia1
Xi .
1.4
Then for every q ≥ 2, there exists a constant C depending only on q and ϕ· such that
q
q/2
max |Xi |q
E maxTa j ≤ C Ca,n E
1≤j≤n
a1≤i≤an
1.5
for every a ≥ 0 and n ≥ 1.
Lemma 1.4 see Hu et al. 13. Let b1 , b2 , . . . be a nondecreasing unbounded sequence of positive
numbers and let α1 , α2 , . . . be nonnegative numbers. Let r and C be fixed positive numbers. Assume
that for each n ≥ 1,
r
n
E max|Sl | ≤ C αl ,
1≤l≤n
∞
αl
l1
1.6
l1
blr
< ∞,
1.7
then
lim
Sn
n → ∞ bn
0 a.s.,
1.8
and with the growth rate
βn
Sn
O
a.s.,
bn
bn
1.9
Journal of Inequalities and Applications
3
where
βn max bk vkδ/r ,
∀0 < δ < 1,
1≤k≤n
vn ∞
αk
br
kn k
,
r n
Sl αl
E max ≤ 4C
r < ∞,
b
1≤l≤n bl
l1 l
r ∞
Sl αl
≤ 4C
E sup r < ∞.
b
b
l
l≥1
l1 l
βn
0,
n → ∞ bn
lim
1.10
If further we assume that αn > 0 for infinitely many n, then
r ∞
Sl αl
≤ 4C
E sup < ∞.
βr
l≥1 βl
l1 l
1.11
Lemma 1.5 see Fazekas and Klesov 14 and Hu 15. Let b1 , b2 , . . . be a nondecreasing
unbounded sequence of positive numbers and let α1 , α2 , . . . be nonnegative numbers. Denote Λk α1 α2 · · · αk for k ≥ 1. Let r be a fixed positive number satisfying 1.6. If
∞
Λl
l1
1
1
− r
blr bl1
< ∞,
Λn
is bounded,
bnr
1.12
1.13
then 1.8–1.11 hold.
2. Maximal Inequality for ϕ-Mixing Sequences
Theorem 2.1. Let {Xn , n ≥ 1} be a sequence of ϕ-mixing random variables satisfying
∞ 1/2
n < ∞. Assume that EXn 0 and EXn2 < ∞ for each n ≥ 1. Then there exists a
n1 ϕ
constant C depending only on ϕ· such that for any n ≥ 1 and a ≥ 0
E
⎛
an
ia1
2
Xi
≤C
an
EXi2 ,
2.1
ia1
aj 2 ⎞
an
E⎝max
Xi ⎠ ≤ C
EXi2 .
1≤j≤n
ia1
ia1
2.2
4
Journal of Inequalities and Applications
In particular,
E
n
2
≤C
Xi
i1
n
EXi2 ,
j
⎞
n
2
E⎝max Xi ⎠ ≤ C EXi2 ,
1≤j≤n
i1
i1
⎛
2.3
i1
2.4
where C may be different in various places.
Proof. By Lemma 1.2 for p q 2, we can see that
E
2
an
Xi
ia1
an
ia1
≤
an
an
ia1
a1≤i<j≤an
n−1 an−k
2
EXi2 2
ϕ1/2 k EXi2 EXki
∞
1 4 ϕ1/2 k
an
2.5
k1 ia1
an
EXi2
ia1
k1
˙ C
1/2 1/2
EXj2
ϕ1/2 j − i EXi2
EXi2 4
≤
E Xi Xj
a1≤i<j≤an
ia1
≤
EXi2 2
EXi2 ,
ia1
which implies 2.1. By 2.1 and Lemma 1.3 take q 2, we can get
aj 2 ⎞
an
⎠
2
2
⎝
≤C
Xi EXi E
max Xi
E max
1≤j≤n
a1≤i≤an
ia1
ia1
⎛
≤C
an
2.6
EXi2 .
ia1
The proof is completed.
3. Hájek-Rényi-Type Inequality for ϕ-Mixing Sequences
In this section, we will give the Hájek-Rényi-type inequality for ϕ-mixing sequences, which
can be applied to obtain the strong law of large numbers and the integrability of supremum.
Theorem 3.1. Let {Xn , n ≥ 1} be a sequence of ϕ-mixing random variables satisfying
∞ 1/2
n < ∞ and let {bn , n ≥ 1} be a nondecreasing sequence of positive numbers. Then for
n1 ϕ
Journal of Inequalities and Applications
5
any ε > 0 and any integer n ≥ 1,
⎫
n Var X
⎬ 4C 1 k j
,
P max Xj − EXj ≥ ε ≤ 2
2
⎭
⎩1≤k≤n bk j1
ε
b
j1
j
⎧
⎨
3.1
where C is defined in 2.4 in Theorem 2.1.
Proof. Without loss of generality, we assume that bn ≥ 1 for all n ≥ 1. Let α define
Ai 1 ≤ k ≤ n : αi ≤ bk < αi1 .
√
2. For i ≥ 0,
3.2
For Ai /
∅, we let vi max{k : k ∈ Ai } and tn be the index of the last nonempty set Ai .
tn
Ai {1, 2, . . . , n}. It is easily seen that αi ≤ bk ≤ bvi < αi1
Obviously, Ai Aj ∅ if i / j and i0
if k ∈ Ai and {Xn −EXn , n ≥ 1} is also a sequence of ϕ-mixing random variables. By Markov’s
inequality and 2.4, we have
⎫
⎬
1 k P max Xj − EXj ≥ ε
⎭
⎩1≤k≤n bk j1
⎧
⎫
⎨
⎬
1 k max max
P
Xj − EXj ≥ ε
⎩0≤i≤tn ,Ai / ∅ k∈Ai bk j1
⎭
⎧
⎨
⎫
⎬
k ≤
≥
ε
P
max
−
EX
X
j ⎭
⎩ αi 1≤k≤vi j1 j
∅
i0,Ai /
tn
⎧
⎨1
⎧
⎨
2 ⎫
⎬
k
1
1
≤ 2
max
E
−
EX
X
j
j 2i
ε i0,Ai / ∅ α ⎩1≤k≤vi j1
⎭
tn
Now we estimate
i0 1
α
tn
≤
vi
tn
C 1 Var Xj
ε2 i0,Ai / ∅ α2i j1
≤
tn
n
1
C
Var
X
.
j
2
ε j1
α2i
∅,vi≥j
i0,Ai /
i0,Ai / ∅,vi≥j
3.3
1/α2i . Let i0 min{i : Ai /
∅, vi ≥ j}. Then bj ≤ bvi0 <
follows from the definition of vi. Therefore,
tn
∞
1
1
1
1
1
α2
4
<
<
2.
2
2
2
2i
2i
2i
0
1 − 1/α α
1 − 1/α bj
α
bj
ii0 α
i0,Ai /
∅,vi≥j
Thus, 3.1 follows from 3.3 and 3.4 immediately.
3.4
6
Journal of Inequalities and Applications
Theorem 3.2. Let {Xn , n ≥ 1} be a sequence of ϕ-mixing random variables satisfying
∞ 1/2
n < ∞ and let {bn , n ≥ 1} be a nondecreasing sequence of positive numbers. Then for
n1 ϕ
any ε > 0 and any positive integers m < n,
⎞
⎛
⎞
m Var X
n Var X
1 k 4C
j
j
⎠,
4
P ⎝ max Xj − EXj ≥ ε⎠ ≤ 2 ⎝
2
2
m≤k≤n bk
ε
b
b
m
j1
j1
jm1
j
⎛
3.5
where C is defined in 3.1.
Proof. Observe that
m
k
1 k 1 1 max Xj − EXj ≤ Xj − EXj max Xj − EXj ,
m≤k≤n bk
bm j1
m1≤k≤n bk jm1
j1
3.6
thus
⎞
1 k P ⎝ max Xj − EXj ≥ ε⎠
m≤k≤n bk
j1
⎛
⎞
⎛
⎞
k
1 m
1 ε
ε
≤ P ⎝
Xj − EXj ≥ ⎠ P ⎝ max Xj − EXj ≥ ⎠˙ I II.
m1≤k≤n bk
bm j1
2
2
jm1
⎛
3.7
For I, by Markov’s inequality and 2.3, we have
⎛
⎞2
m Var X
m
4
4C j
⎝
⎠
I≤
E
−
EX
≤
.
X
j
j
2
2
2
ε
ε 2 bm
b
m
j1
j1
3.8
For II, we will apply Theorem 3.1 to {Xmi , 1 ≤ i ≤ n − m} and {bmi , 1 ≤ i ≤ n − m}. Noting
that
k
k
1 1 max Xj − EXj max Xmj − EXmj ,
m1≤k≤n bk
1≤k≤n−m bmk j1
jm1
3.9
thus, by Theorem 3.1, we get
II ≤
4C
ε/22
n−m
Var
j1
Xmj
2
bmj
n Var X
16C j
2
.
2
ε jm1 bj
Therefore, the desired result 3.5 follows from 3.7–3.10 immediately.
3.10
Journal of Inequalities and Applications
7
Theorem 3.3. Let {Xn , n ≥ 1} be a sequence of ϕ-mixing random variables satisfying
∞ 1/2
ϕ n < ∞ and let {bn , n ≥ 1} be a nondecreasing sequence of positive numbers. Denote
n1 Tn ni1 Xi − EXi for n ≥ 1. Assume that
∞ Var X
j
j1
bj2
< ∞,
3.11
then for any r ∈ 0, 2,
r ∞ Var X
Tn 4Cr j
≤1
E sup < ∞,
2
2 − r j1
bj
n≥1 bn
3.12
where C is defined in 3.1. Furthermore, if limn → ∞ bn ∞, then
n
1
Xj − EXj 0
n → ∞ bn
j1
lim
a.s.
3.13
Proof. By the continuity of probability and Theorem 3.1, we get
r ∞ r
Tn Tn E sup P sup > t dt
0
n≥1 bn
n≥1 bn
1
∞ r
r
Tn Tn P sup > t dt P sup > t dt
0
1
n≥1 bn
n≥1 bn
∞ Tn 1/r
dt
≤1
P sup > t
1
n≥1 bn
∞
Tn ≤1
lim P max > t1/r dt
1≤n≤N bn
1 N →∞
∞ Var X ∞
j
t−2/r dt
≤ 1 4C
2
b
1
j1
j
3.14
∞ Var X
4Cr j
1
< ∞.
2
2 − r j1
bj
Observe that
∞ ∞ Tn Tn Tn >ε .
>ε
P
lim
>
ε
P
max
max
b N →∞
m≤n≤N bn m≤n≤N bn n
nm
Nm
P
3.15
8
Journal of Inequalities and Applications
By Theorem 3.2, we have that
⎛
⎞
m Var X
N Var X
Tn 4C
j
j
⎠.
4
P max > ε ≤ 2 ⎝
2
2
m≤n≤N bn
ε
bm
bj
j1
jm1
3.16
Hence, by 3.11 and Kronecker’s Lemma, it follows that
lim P
m→∞
∞ Tn >ε
0,
b n
nm
∀ε > 0,
3.17
which is equivalent to
n
1
Xj − EXj 0 a.s.
n → ∞ bn
j1
lim
3.18
So the desired results are proved.
4. Strong Law of Large Numbers and Growth Rate
for ϕ-Mixing Sequences
Theorem 4.1. Let {Xn , n ≥ 1} be a sequence of mean zero ϕ-mixing random variables satisfying
∞ 1/2
n < ∞ and let {bn , n ≥ 1} be a nondecreasing unbounded sequence of positive numbers.
n1 ϕ
Assume that
∞
EXn2
n1
bn2
< ∞,
4.1
then
lim
n→∞
Sn
0 a.s.,
bn
4.2
and with the growth rate
βn
Sn
O
a.s.,
bn
bn
4.3
Journal of Inequalities and Applications
9
where
βn max bk vkδ/2 ,
∀ 0 < δ < 1,
1≤k≤n
αk vn ∞
αk
2
kn bk
,
lim
n→∞
βn
0,
bn
4.4
k ≥ 1, where C is defined in 2.4,
2 n
Sl αl
≤4
E max < ∞,
2
1≤l≤n bl
b
l1 l
2 ∞
Sl αl
≤4
E sup < ∞.
2
l≥1 bl
l1 bl
CEXk2 ,
4.5
If further we assume that αn > 0 for infinitely many n, then
2 ∞
Sl αl
≤4
E sup < ∞.
2
l≥1 βl
l1 βl
4.6
Proof. By 2.4 in Theorem 2.1, we have
n
n
E max |Sk |2 ≤ C EXi2 αk .
1≤k≤n
i1
4.7
k1
It follows by 4.1 that
∞
αn
2
n1 bn
C
∞
EXn2
n1
bn2
< ∞.
4.8
Thus, 4.2–4.6 follow from 4.7, 4.8, and Lemma 1.4 immediately. We complete the proof
of the theorem.
Theorem 4.2. Let {Xn , n ≥ 1} be a sequence of ϕ-mixing random variables with
1 ≤ p < 2. Denote Qn max1≤k≤n EXk2 for n ≥ 1 and Q0 0. Assume that
∞
Qn
< ∞,
2/p
n1 n
∞
n1
ϕ1/2 n < ∞.
4.9
then
n
1 Xi − EXi 0 a.s.,
n → ∞ n1/p
i1
lim
4.10
10
Journal of Inequalities and Applications
and with the growth rate
n
βn
1 −
EX
O
X
i
i
n1/p i1
n1/p
4.11
a.s.,
where
βn max k1/p vkδ/2 ,
∀0 < δ < 1,
1≤k≤n
vn ∞
αk
,
2/p
kn k
βn
0,
n → ∞ n1/p
lim
αk CkQk − k − 1Qk−1 , k ≥ 1, where C is defined in 2.4,
n
S l 2
αl
≤4
E max 1/p < ∞,
2/p
1≤l≤n l
l1 l
∞
S l 2
αl
≤4
E sup 1/p < ∞.
2/p
l≥1 l
l1 l
4.12
4.13
4.14
If further we assume that αn > 0 for infinitely many n, then
2 ∞
Sl αl
< ∞.
E sup ≤4
2
l≥1 βl
l1 βl
4.15
∞
S l r
4r αl
≤1
E sup 1/p < ∞.
2/p
2 − r l1 l
l≥1 l
4.16
In addition, for any r ∈ 0, 2,
Proof. Assume that EXn 0, bn n1/p , and Λn can see that
n
l1
αl , n ≥ 1. By 2.4 in Theorem 2.1, we
⎛
2 ⎞
k
n
n
E⎝ max Xi ⎠ ≤ C EXi2 ≤ CnQn αk .
1≤k≤n
i1
i1
k1
4.17
It is a simple fact that αk ≥ 0 for all k ≥ 1. It follows by 4.9 that
∞
l1
Λl
1
1
− 2
2
bl
bl1
C
∞
l1
lQl
1
l2/p
−
∞
Ql
2C < ∞.
≤
p l1 l2/p
1
l 12/p
4.18
Journal of Inequalities and Applications
11
That is to say 1.12 holds. By Remark 2.1 in Fazekas and Klesov 14, 1.12 implies 1.13.
By Lemma 1.5, we can obtain 4.10–4.15 immetiately. By 4.14, it follows that
∞ S l r
S l r
P sup 1/p > t dt
E sup 1/p l
l
l≥1
0
l≥1
∞ Sl 1/r
dt
≤1
P sup 1/p > t
1
l≥1 l
∞
S l 2
≤ 1 E sup 1/p t−2/r dt
l
1
l≥1
≤1
4.19
∞
4r αl
< ∞.
2 − r l1 l2/p
The proof is completed.
Remark 4.3. By using the maximal inequality, we get the Hájek-Rényi-type inequality, the
strong law of large numbers and the strong growth rate for ϕ-mixing sequences. In addition,
we get some new bounds of probability inequalities for ϕ-mixing sequences, such as 3.1,
3.5, 3.12, 4.5–4.6, and 4.13–4.16.
Acknowledgments
The authors are most grateful to the Editor Sever Silvestru Dragomir, the referee Professor
Mihaly Bencze and an anonymous referee for careful reading of the manuscript and valuable
suggestions and comments which helped to significantly improve an earlier version of this
paper. This work was supported by the National Natural Science Foundation of China Grant
nos. 10871001, 60803059 and the Innovation Group Foundation of Anhui University.
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