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Volume 2010, Article ID 769201, 12 pagesdoi:10.1155/2010/769201 Research Article On Complete Convergence for Arrays of Rowwise ρ-Mixing Random Variables and Its Applications Xing-cai Zho

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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 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,jglin@seu.edu.cn

Received 15 May 2010; Revised 23 August 2010; Accepted 21 October 2010

Academic Editor: Soo Hak Sung

Copyrightq 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 rowwiseρ-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 n;n ≥ 1} is said to be ρ-mixing if

ρn sup

X∈L2Fk

1, Y∈L2 F ∞

nk

⎧

⎪

|EXY − EXEY|

EX − EX2EY − EY2

⎫

⎪

⎪−→ 0 1.1

asn → ∞, where F n

mdenotes theσ-field generated by {X i;m ≤ i ≤ n}.

Theρ-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 Ibragimov2, 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

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2 Journal of Inequalities and Applications

9 for almost sure invariance principle; Peligrad 10, Shao 11 and Liang and Yang 12 for convergence rate; Shao11, 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 Kuczmaszewska16

sequences, Kuczmaszewska17 for negatively associated sequence, and Baek and Park 18 for negatively dependent sequence In the paper, we study the complete convergence for arrays of rowwiseρ-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 {X ni;i ≥ 1, n ≥ 1} of random variables is said to be stochastically

dominated by a random variableX if there exists a constant C, such that

P{|X ni | > x} ≤ CP{C|X| > x} 1.2 for allx ≥ 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

lx 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

on q and ρ· such that for any n ≥ 1

E max

1≤i≤n

i

j1

X j

q

≤ K

⎛

⎝n2/qexp

⎛

⎝Klogn

i0

ρ2i⎞

⎠max

1≤i≤n

E|X i|2q/2

n exp

⎛

⎝Klogn

i0

ρ2/q

2i⎞

⎠max

1≤i≤nE|X i|q

⎞

⎠.

1.4

Lemma 1.5 Sung 19 Let {X n;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|X n|α I|X n | ≤ b ≤ CE|X| α I|X| ≤ b b α P{|X| > b}. 1.5

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Lemma 1.6 Zhou 20 If lx > 0 is a slowly varying function as x → ∞, then

im n1 n s ln ≤ Cm s1 lm for s > −1,

ii∞nm n s ln ≤ Cm s1 lm for s < −1.

This paper is organized as follows InSection 2, we give the main result and its proof

A few applications of the main result are provided inSection 3

2 Main Result and Its Proof

This paper studies arrays of rowwiseρ-mixing sequence Let ρ n i be the mixing coeﬃcient

defined inDefinition 1.1for thenth row of an array {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

supn∞

i1 ρ2/q

n 2i < ∞ 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 real numbers If for some 0 < t < 2 and any ε > 0 the following conditions are fulfilled:

a∞

n1 c nbn i1 P{|a ni X ni | ≥ εb1/t

n } < ∞,

b∞n1 c n b −q/t1 n max1≤i≤bn |a ni|q E|X ni|q I|a ni X ni | < εb1/t

n < ∞,

c∞n1 c n b −q/tq/2 n max1≤i≤bn |a ni|2E|X ni|2I|a ni X ni | < εb1/t

n q/2 < ∞, then

∞

n1

c n P

⎧

⎨

⎩1≤i≤bmaxn

i

j1

a nj X nj − a nj EX nj Ia

nj X nj < εb1/t

n

> εb1n /t

⎫

⎬

⎭ < ∞. 2.1

Remark 2.2. Theorem 2.1extends some results of Kuczmaszewska17 to the case of arrays of rowwiseρ-mixing sequence and generalizes the results of Kuczmaszewska 16 to the case of maximum weighted sums

Remark 2.3. Theorem 2.1firstly 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 conditionsi–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 inTheorem 2.1, notq > 2 in Theorem 2.1 of Kuczmaszewska 17; the powers of

b ninb and c ofTheorem 2.1are−q/t1 and −q/tq/2, respectively, not −q/t in Theorem

2.1 of Kuczmaszewska 17

Now, we give the proof ofTheorem 2.1

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Proof The conclusion of the theorem is obvious if∞

n1 c nis convergent Therefore, we will consider that only∞

n1 c nis divergent Let

Y nj a nj X nj Ia

nj X nj < εb1/t

n

, T nii

j1

Y nj , S nii

j1

a nj X nj ,

A bn

i1

{a ni X ni Y ni }, B bn

i1

{a ni X ni / Y ni }.

2.2

Note that

P

max

1≤i≤bn |S ni − ET ni | > εb1/t

n

 P

max

n

A

 P

max

n

B

≤ Pmax

1≤i≤bn |T ni − ET ni | > εb1/t

n

bn

i1

P|a ni X ni | > εb1/t

n

.

2.3

Bya it is enough to prove that for all ε > 0

∞

n1

c n P

max

n

< ∞. 2.4

By Markov inequality and Lemma 1.4, and note that the assumption supn∞

i1 ρ2/q

n 2i < ∞ for some q ≥ 2, we get

P

max

n

≤ Cb −q/t n Emax

1≤i≤bn |T ni − ET ni|q

≤ Cb −q/t n

⎧

⎨

⎩b nexp

⎛

⎝Klogbn

i0

ρ2/q n

2i⎞

⎠max

1≤i≤bn E|a ni X ni|q

× I|a ni X ni | < εb1/t

n

 K exp

⎛

⎝Klogbn

i1

ρ n2i⎞

⎠

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b nmax

1≤i≤bn E|a ni X ni|2I|a ni X ni | < εb1/t

q/2⎫⎬

⎭

≤ Cb −q/t1 n max

1≤i≤bn |a ni|q E|X ni|q I|a ni X ni | < εb1/t

n

 Cb n −q/tq/2

max

1≤i≤bn |a ni|2E|X ni|2I|a ni X ni | < εb1/t

q/2

.

2.5 Fromb, c, and 2.5, we see that 2.4 holds

3 Applications

supn∞

i1 ρ2/q

n 2i < ∞ for some q ≥ 2, EX ni 0, and E|X ni|p < ∞ for all n ≥ 1, i ≥ 1, and

1≤ p ≤ 2 Let {a ni;i ≥ 1, n ≥ 1} be an array of real numbers satisfying the condition

max

1≤i≤n|a ni|p E|X ni|p On 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

⎧

⎨

⎩max1≤i≤n

i

j1

a nj X nj

> εn α

⎫

⎬

⎭ < ∞. 3.2

Proof Put c n n αp−2,b n n, and 1/t α inTheorem 2.1 By3.1, we get

∞

n1

c nbn

i1

P|a ni X ni | ≥ εb1/t

n

≤ C∞

n1

n αp−2 n

i1

n −αp |a ni|p E|X ni|p ≤ C∞

n1

n−1max

1≤i≤n|a ni|p E|X ni|p ≤ C∞

n1

n −2ν < ∞,

∞

n1

c n b −q/t1 n max

1≤i≤bn E|a ni X ni|q I|a ni X ni | < εb1/t

n

≤ C∞

n1

n αp−2 n −αq1 n αq−pmax

1≤i≤n|a ni|p E|X ni|p ≤ C∞

n1

n −2ν < ∞,

∞

n1

c n b −q/tq/2 n

max

1≤i≤bn E|a ni X ni|2I|a ni X ni | < εb1/t

n

q/2

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≤ C∞

n1

n αp−2 n −αqq/2 n α2−pq/2

max

1≤i≤n|a ni|p E|X ni|pq/2

≤ C∞

n1

n αp1−q/2νq/2−1−1 < ∞

3.3 following fromνq/2−1 < 0 By the assumption EX ni 0 for n ≥ 1, i ≥ 1 and by 3.1, we have

n −αmax

1≤i≤n

i

j1

a nj EX nj I a nj X nj < εn α!

≤ Cn −α n

j1

a nj EX nj I a nj X nj < εn α!

≤ Cn −α n

j1

a nj EX nj I a nj X nj ≥ εn α!

≤ Cn −αp n

j1

a nj p E X nj p

≤Cn −αp1max

1≤j≤n|a nj|p E|X nj|p ≤Cn −αpν −→0, as n −→ ∞,

3.4

becauseν < 1 and αp ≥ 1 Thus, we complete the proof of the theorem.

supn∞

i1 ρ2/q

n 2i < ∞ for some q ≥ 2, EX ni 0, and E|X ni|p < ∞ 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 {a ni;i ≥ 1, n ≥ 1} be an array of real numbers satisfying the condition

max

1≤i≤n|a ni|p On 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

supn∞

i1 ρ2/q

n 2i < ∞ 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≥1 |a ni | On1/t−ν

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then for any ε > 0

∞

n1

P

⎧

⎨

⎩max1≤i≤n

i

j1

a nj X nj

> εn1/t

⎫

⎬

⎭ < ∞. 3.7

Proof Take c n 1 and b n n for n ≥ 1 Then we see that a and b are satisfied Indeed,

takingq ≥ max2, 1 2/ν, byLemma 1.5and3.6, we get

∞

n1

c nbn

i1

P|a ni X ni | ≥ εb1/t

n

∞

n1

n

i1

P|a ni X ni | ≥ εn1/t

≤ C∞

n1

n

i1

P{|X| ≥ Cn ν}

 C∞

n1

n∞

kn

PCk ν ≤ |X| < Ck 1 ν

≤ C∞

k1

k2PCk ν ≤ |X| < Ck 1 ν

≤ CE|X|2/ν < ∞,

∞

n1

c n b −q/t1 n max

1≤i≤bn |a ni|q E|X ni|q I|a ni X ni | < εb1/t

n

≤ C∞

n1

n −q/t1max

1≤i≤n|a ni|q

"

E|X| q I|a ni X| < εn1/t

n q/t

|a ni|q P

|a ni X| ≥ εn1/t#

≤ C∞

n1

n −12/ν/t1max

1≤i≤n|a ni|12/νE|X|12/ν C∞

n1

n max

1≤i≤nP|a ni X| ≥ εn1/t

≤ C∞

n1

n −12/ν/t1

$ sup

i≥1 |a ni|

%12/ν

E|X|12/ν C∞

n1

nP{|X| ≥ Cn ν}

≤ C∞

n1

n −ν−1 E|X|12/ν C∞

n1

n −ν−1 E|X|12/ν≤ C∞

n1

n −ν−1 < ∞ 3.8

In order to prove thatc holds, we consider the following two cases

Ifν > 2, byLemma 1.5,C r inequality, and3.6, we have

∞

n1

c n b −q/tq/2 n

max

1≤i≤bn |a ni|2E|X ni|2I|a ni X ni | < εb1/t

n

q/2

≤ C∞

n1

n −q/tq/2

max

1≤i≤n|a ni|2E|X|2I|a ni X ni | < εn1/tq/2

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 C∞

n1

n q/2 max

1≤i≤nP|a ni X| ≥ εn1/tq/2

≤ C∞

n1

n −q/tq/2 n 1/t1−2/νq/2

max

1≤i≤n|a ni|12/νE|X|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 |a ni|

%12/νq/2

E|X|12/νq/2

 C∞

n1

n q/2 n −12/ννq/2

E|X|12/νq/2

≤ C∞

n1

n −ν1q/2

E|X|12/νq/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≤bn |a ni|2E|X ni|2I|a ni X ni | < εb1/t

n

q/2

≤ C∞

n1

n −q/tq/2

max

1≤i≤n|a ni|2E|X|2I|a ni X ni | < εn1/tq/2

 C∞

n1

n q/2 P{|X| ≥ Cn ν}q/2

≤ C∞

n1

n −q/tq/2

$ sup

i≥1 |a ni|

%q

E|X|2q/2

 C∞

n1

n q/2−νq

E|X|2q/2

≤ C∞

n1

n −2ν−1q/2

E|X|2q/2

The proof will be completed if we show that

n −1/tmax

1≤i≤n

i

j1

a nj EX nj Ia

nj X ni < εn1/t

−→0, as n −→ ∞. 3.11

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Indeed, byLemma 1.5, we have

n −1/tmax

1≤i≤n

i

j1

a nj EX nj Ia

nj X ni < εn1/t

≤ Cn −1/t

n

j1

a nj E|X| C n

j1

Pa

nj X ≥ εn1/t

≤ Cn −ν E|X| CnP{|X| ≥ εn ν}

≤ Cn −ν E|X| Cn −ν1 E|X|12/ν−→ 0, as n −→ ∞.

3.12

supn∞

i1 ρ2/q

n 2i < ∞ 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 → ∞ If for some 0 < t < 2 and real number λ, and any

ε > 0 the following conditions are fulfilled:

A∞

n1 n λ lnn

i1 P{|a ni X ni | ≥ εn1/t } < ∞,

B∞

n1 n λ−q/t1 lnmax1≤i≤nE|a ni X ni|q I|a ni X ni | < εn1/t < ∞,

C∞n1 n λ−q/tq/2 lnmax1≤i≤n|a ni|2E|X ni|2I|a ni X ni | < εn1/tq/2 < ∞,

then

∞

n1

n λ lnP

⎧

⎨

⎩max1≤i≤n

i

j1

a nj X nj − a nj EX nj Ia

nj X nj < εn1/t

> εn1/t

⎫

⎬

⎭ < ∞. 3.13

Proof Let c n n λ ln and b n n UsingTheorem 2.1, we obtain3.13 easily

Theorem 3.5 Let {X ni;i ≥ 1, n ≥ 1} be an array of rowwise ρ-mixing identically distributed random variables satisfying∞

i1 ρ2/q

n 2i < ∞ 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

∞

n1

n αp−2 lnP

⎧

⎨

⎩max1≤i≤n

i

j1

X nj

> εn1/t

⎫

⎬

⎭ < ∞. 3.15

Proof Put λ αp − 2 and a ni 1 for n ≥ 1, i ≥ 1 inTheorem 3.4 To prove3.15, it is enough to note that under the assumptions ofTheorem 3.4, the conditionsA–C ofTheorem 3.4hold

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ByLemma 1.6, we obtain

∞

n1

n αp−1 lnP|X11| > εn1/t

∞

n1

n αp−1 ln∞

mn Pεm1/t < |X11| ≤ εm 11/t

≤ C∞

m1

Pεm1/t < |X11| ≤ εm 11/tm

n1

n αp−1 ln

≤ C∞

m1

m αp lmPεm1/t < |X11| ≤ εm 11/t

≤ CE|X11|αpt l|X11|t

< ∞,

3.16

which proves that conditionA is satisfied

Takingq > max2, αpt, we have αp − q/t < 0 ByLemma 1.6, we have

∞

n1

n αp−1−q/t lnE|X11|q I|X11| ≤ εn1/t

∞

n1

n αp−1−q/t lnn

m1 E|X11|q Iεm − 11/t ≤ |X11| < εm1/t

≤ C∞

m1 E|X11|q Iεm − 11/t ≤ |X11| < εm1/t∞

nm n αp−1−q/t ln

≤ C∞

m1

m αp−q/t lmE|X11|q Iεm − 11/t ≤ |X11| < εm1/t

≤ CE|X11|αpt l|X11|t

< ∞,

3.17

which proves thatB holds

In order to prove thatC holds, we consider the following two cases

Ifαpt < 2, take q > 2 We have

∞

n1

n αp−2−q/tq/2 lnE|X11|2I|X11| < εn1/tq/2

≤ C∞

n1

n αp−2−q/tq/2 lnn q/t−αpq/2

E|X11|αpt I|X11| < εn1/tq/2

≤ C∞

n1

n αp−11−q/2−1 ln < ∞.

3.18

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Proof The conclusion... InSection 2, we give the main result and its proof

A few applications of the main result are provided inSection

2 Main Result and Its Proof

This paper studies arrays. .. 2.1firstly gives the condition of the mixing coeﬃcient, so the conditions

a–c not contain the mixing coeﬃcient Thus, the conditions a–c are obviously simpler than the conditionsi–iii

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