On Complete Convergence for Arrays of Rowwise Mixing Random Variables and Its Applications

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Volume 2010, Article ID 769201,12pages 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,jglin@seu.edu.cn

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

Academic Editor: Soo Hak Sung

Copyrightq2010 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{Xn;n≥ 1}be a sequence of random variables defined on this space.

Definition 1.1. The sequence{Xn;n≥1}is said to beρ-mixing if

ρn sup k≥1

sup X∈L2_Fk₁_{, Y}_∈_L2_F∞

nk

⎧ ⎪ ⎨ ⎪ ⎩

|EXY−EXEY|

EX−EX2_E_Y₋_EY2

⎫ ⎪ ⎬ ⎪

⎭−→0 1.1

asn → ∞, whereFn_mdenotes theσ-field generated by{Xi;m≤i≤n}.

Theρ-mixing random variables were first introduced by Kolmogorov and Rozanov

1. The limiting behavior ofρ-mixing random variables is very rich, for example, these in the study by Ibragimov2, Peligrad3, and Bradley4for central limit theorem; Peligrad

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9for almost sure invariance principle; Peligrad10, Shao11and Liang and Yang12for 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 Kuczmaszewska16forρ-mixing andρ-mixing sequences, Kuczmaszewska17for negatively associated sequence, and Baek and Park18 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 Kuczmaszewska16. 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 variableXif there exists a constantC, such that

P{|Xni|> x} ≤CP{C|X|> x} 1.2

for allx≥0,i≥1 andn≥1.

Definition 1.3. A real-valued functionlx, positive and measurable onA,∞for someA >0, is said to be slowly varying if

lim x→ ∞

lλx

lx 1 for eachλ >0. 1.3

Throughout the sequel,C will represent a positive constant although its value may change from one appearance to the next;xindicates the maximum integer not larger than

x;IBdenotes the indicator function of the setB. The following lemmas will be useful in our study.

Lemma 1.4Shao11. Let{X_n;n≥1}be a sequence ofρ-mixing random variables withEX_i0

andE|Xi|q <∞for someq≥2. Then there exists a positive constantK Kq, ρ·depending only

onqandρ·such that for anyn≥1

Emax

1≤i≤n

i

j1 Xj

q

≤K

⎛

⎝_n2/q_exp

⎛ ⎝_Klogn

i0 ρ2i

⎞ ⎠_max

1≤i≤n

E|Xi|2

q/2

nexp

⎛ ⎝_Klogn

i0

ρ2/q₂i

⎞ ⎠_max

1≤i≤nE|Xi| q

⎞ ⎠_.

1.4

Lemma 1.5Sung19. Let{X_n;n≥1}be a sequence of random variables which is stochastically dominated by a random variableX. For anyα >0andb >0, the following statement holds:

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Lemma 1.6Zhou20. Iflx>0is a slowly varying function asx → ∞, then

im_n₁nsln≤Cms1_l_m_for_{s >}₋₁_,

ii∞_n_mnsln≤Cms1_l_m_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. Letρnibe the mixing coeﬃcient defined inDefinition 1.1for thenth row of an array{X_ni;i≥1, n≥1}, that is, for the sequence

Xn1, Xn2, . . . , n≥1.

Now, we state our main result.

Theorem 2.1. Let{X_ni;i≥ 1, n ≥1}be an array of rowwiseρ-mixing random variables satisfying

sup_n∞_i₁ρ_n2/q2i<∞for someq≥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 some0< t <2and anyε >0the following conditions are fulfilled:

a∞_n₁c_nbn

i1P{|aniXni| ≥εb 1/t n }<∞,

b∞_n₁cnbn−q/t1max1≤i≤bn|ani|qE|Xni|qI|aniXni|< εb1n/t<∞,

c∞_n₁cnbn−q/tq/2max1≤i≤bn|ani|2E|Xni|2I|aniXni|< εb1n/t

q/2 <∞,

then

∞

n1 cnP

⎧ ⎨ ⎩1max≤i≤bn

i

j1

anjXnj−anjEXnjIanjXnj< εb1n/t

> εb1n/t

⎫ ⎬

⎭<∞. 2.1

Remark 2.2. Theorem 2.1extends some results of Kuczmaszewska17to the case of arrays of rowwiseρ-mixing sequence and generalizes the results of Kuczmaszewska16to 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–iiiin Theorem 2.1 of Kuczmaszewska16. Our conditions are also diﬀerent from those of Theorem 2.1 in the study by Kuczmaszewska17:q ≥ 2 is only required inTheorem 2.1, notq >2 in Theorem 2.1 of Kuczmaszewska17; the powers of

bninbandcofTheorem 2.1are−q/t1 and−q/tq/2, respectively, not−q/tin Theorem 2.1 of Kuczmaszewska17.

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Proof. The conclusion of the theorem is obvious if∞_n₁cnis convergent. Therefore, we will consider that only∞_n₁c_nis divergent. Let

YnjanjXnjIanjXnj< εbn1/t

, Tni

i

j1

Ynj, Sni i

j1 anjXnj,

Abn

i1

{aniXniYni}, B bn

i1

{aniXni/Yni}.

2.2

Note that

P

max

1≤i≤bn|Sni−ETni|> εb 1/t n

P

max

1≤i≤bn|Sni−ETni|> εb 1/t

n A

P

max

1≤i≤bn|Sni−ETni|> εb 1/t

n B

≤Pmax

1≤i≤bn|Tni−ETni|> εb 1/t n

bn

i1

P|aniXni|> εb1n/t

.

2.3

Byait is enough to prove that for allε >0

∞

n1 cnP

max

<∞. 2.4

By Markov inequality and Lemma 1.4, and note that the assumption sup_n∞_i₁ρ_n2/q2i<∞for someq≥2, we get

P

max

≤Cb−nq/tEmax

1≤i≤bn|Tni−ETni|

q

≤Cb−nq/t

⎧ ⎨ ⎩bnexp

⎛

⎝_Klogbn

i0 ρ2/q

n

2i

⎞ ⎠_max

1≤i≤bnE|aniXni|

q

×I|aniXni|< εbn1/t

Kexp

⎛

⎝_Klogbn

i1 ρn

2i

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×

bnmax

1≤i≤bnE|aniXni| 2_I_|_a

niXni|< εb1n/t

q/2⎫⎬

⎭

≤Cb−q/t1

n max

1≤i≤bn|ani|

q_E_|_X ni|qI

|aniXni|< εb1n/t

Cb−q/tq/2

n

max

1≤i≤bn|ani| 2_E_|_X

ni|2I|aniXni|< εb1n/t

_q/2 .

2.5

Fromb,c, and2.5, we see that2.4holds.

3. Applications

Theorem 3.1. Let{X_ni;i≥ 1, n ≥1}be an array of rowwiseρ-mixing random variables satisfying

sup_n∞_i₁ρ_n2/q2i < ∞for some q ≥ 2,EX_ni 0, andE|X_ni|p < ∞ for alln ≥ 1, i ≥ 1, and

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

max

1≤i≤n|ani| p_E_|_X

ni|pO

nν−1_, _as _n_{−→ ∞}_,

3.1

for some0< ν <2/q. Then for anyε >0andαp≥1

∞

n1 nαp−2_P

⎧ ⎨ ⎩max1≤i≤n

i

j1 anjXnj

> εnα

⎫ ⎬

⎭<∞. 3.2

Proof. Putcnnαp−2,bnn, and 1/tαinTheorem 2.1. By3.1, we get

∞

n1 cn

bn

i1

P|aniXni| ≥εb1n/t

≤C∞

n1

nαp−2n

i1 n−αp_|_a

ni|pE|Xni|p≤C

∞

n1 n−1_max

1≤i≤n|ani| p_E_|_X

ni|p≤C

∞

n1

n−2ν_<_∞_,

∞

n1

cnb−nq/t1max

1≤i≤bnE|aniXni|

q_I_|_a

niXni|< εbn1/t

≤C∞

n1

nαp−2_n−αq1_nαq−p_max 1≤i≤n|ani|

p_E_|_X ni|p≤C

∞

n1

n−2ν_<_∞_,

∞

n1

cnb−nq/tq/2

max

1≤i≤bnE|aniXni| 2_I_|_a

niXni|< εb1n/t

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≤C∞

n1

nαp−2_n−αqq/2_nα2−pq/2

max

1≤i≤n|ani|

p_E_|_X ni|p

_q/2

≤C∞

n1

nαp1−q/2νq/2−1−1 _<_∞

3.3

following fromνq/2−1<0. By the assumptionEXni0 forn≥1,i≥1 and by3.1, we have

n−α_max

1≤i≤n

i

j1

anjEXnjI anjXnj< εnα!

≤Cn−αn j1

_a_nj_EX_nj_I _a_nj_X_nj_{< εn}α!

≤Cn−αn j1

_a_nj_EX_nj_I _a_nj_X_nj_≥_εnα!

≤Cn−αpn j1

_a_njp_E_X njp

≤Cn−αp1_max 1≤j≤n|anj|

p_E_|_X

nj|p≤Cn−αpν−→0, asn−→ ∞,

3.4

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

Theorem 3.2. Let{Xni;i≥ 1, n ≥1}be an array of rowwiseρ-mixing random variables satisfying sup_n∞_i₁ρn2/q2i < ∞ for someq ≥ 2, EXni 0, and E|Xni|p < ∞for all n ≥ 1, i ≥ 1, and 1≤p≤2. Let the random variables in each row be stochastically dominated by a random variableX, such thatE|X|p<∞, and let{ani;i≥1, n≥1}be an array of real numbers satisfying the condition

max

1≤i≤n|ani|

p_O_nν−1_, _as_n_{−→ ∞}_,

3.5

for some0< ν <2/q. Then for anyε >0andαp≥13.2holds.

Theorem 3.3. Let{Xni, n ≥1, i ≥1}be an array of rowwiseρ-mixing random variables satisfying sup_n∞_i₁ρn2/q2i<∞for someq≥2andEXni0for alln≥1, i≥1. Let the random variables in

each row be stochastically dominated by a random variableX, and let{a_ni;i≥1, n≥1}be an array of real numbers. If for some0< t <2,ν >1/2

sup i≥1

|ani|O

n1/t−ν_, _E_|_X_|12/ν_<_∞_,

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then for anyε >0

∞

n1 P

⎧ ⎨ ⎩max1≤i≤n

i

j1 anjXnj

> εn1/t

⎫ ⎬

⎭<∞. 3.7

Proof. Takecn 1 andbn nforn ≥ 1. Then we see thataandbare satisfied. Indeed, takingq≥max2,12/ν, byLemma 1.5and3.6, we get

∞

n1 cn

bn

i1

P|aniXni| ≥εbn1/t

∞

n1

n

i1

P|aniXni| ≥εn1/t

≤C∞

n1

n

i1

P{|X| ≥Cnν_}

C∞

n1 n∞

kn

PCkν_{≤ |}_X_|_{< C}_k₁ν

≤C∞

k1

k2_P_Ckν_{≤ |}_X_|_{< C}_k₁ν_≤_CE_|_X_|2/ν_<_∞_,

∞

n1

cnb−nq/t1max

1≤i≤bn|ani|

q_E_|_X ni|qI

≤C∞

n1

n−q/t1_max 1≤i≤n|ani|

q

"

E|X|qI|aniX|< εn1/t

nq/t

|ani|qP

|aniX| ≥εn1/t

#

≤C∞

n1

n−12/ν/t1_max 1≤i≤n|ani|

12/ν_E_|_X_|12/ν_C∞ n1

nmax

1≤i≤nP

|aniX| ≥εn1/t

≤C∞

n1

n−12/ν/t1

$

sup i≥1

|ani|

%12/ν

E|X|12/ν_C∞ n1

nP{|X| ≥Cnν_}

≤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 thatcholds, we consider the following two cases. Ifν >2, byLemma 1.5,Cr inequality, and3.6, we have

∞

n1

cnb−nq/tq/2

max

1≤i≤bn|ani| 2_E_|_X

ni|2I

q/2

≤C∞

n1

n−q/tq/2

max

1≤i≤n|ani|

2_E_|_X_|2_I_|_a

niXni|< εn1/t

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C∞

n1 nq/2

max

1≤i≤nP

|aniX| ≥εn1/t

q/2

≤C∞

n1

n−q/tq/2_n1/t1−2/νq/2_max 1≤i≤n|ani|

12/ν_E_|_X_|12/νq/2

C∞

n1

nq/2_P_{|_X_{| ≥}_Cnν_}q/2

≤C∞

n1

n−q/tq/21/t1−2/νq/2 $

sup i≥1

|ani|

%12/νq/2

E|X|12/νq/2

C∞

n1

nq/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

cnb−nq/tq/2

max

1≤i≤bn|ani| 2_E_|_X

ni|2I

q/2

≤C∞

n1

n−q/tq/2

max

1≤i≤n|ani|

2_E_|_X_|2_I_|_a

niXni|< εn1/t

q/2

C∞

n1

nq/2_P_{|_X_{| ≥}_Cnν_}q/2

≤C∞

n1

n−q/tq/2

$

sup i≥1

|ani|

%q

E|X|2q/2_C∞

n1

nq/2−νq_E_|_X_|2q/2

≤C∞

n1

n−2ν−1q/2_E_|_X_|2q/2_<_∞_. _3.10

The proof will be completed if we show that

n−1/t_max

1≤i≤n

i

j1

anjEXnjIanjXni< εn1/t

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Indeed, byLemma 1.5, we have

n−1/t_max

1≤i≤n

i

j1

anjEXnjIanjXni< εn1/t

≤Cn−1/t

n

j1

_a_nj_E_|_X_|_Cn j1

PanjX≥εn1/t

≤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 rowwiseρ-mixing random variables satisfying sup_n∞_i₁ρ_n2/q2i<∞for someq≥2, and let{a_ni;i≥1, n ≥1}be an array of real numbers. Let lx>0be a slowly varying function asx → ∞. If for some0 < t <2and real numberλ, and any ε >0the following conditions are fulfilled:

A∞_n₁nλln_in₁P{|aniXni| ≥εn1/t}<∞,

B∞_n₁nλ−q/t1_l_n_max

1≤i≤nE|aniXni|qI|aniXni|< εn1/t<∞,

C∞_n₁nλ−q/tq/2_l_n_max

1≤i≤n|ani|2E|Xni|2I|aniXni|< εn1/tq/2 <∞,

then

∞

n1

nλ_l_n_P

⎧ ⎨ ⎩max1≤i≤n

i

j1

anjXnj−anjEXnjIanjXnj< εn1/t

> εn1/t

⎫ ⎬

⎭<∞. 3.13

Proof. Letcnnλlnandbnn. UsingTheorem 2.1, we obtain3.13easily.

Theorem 3.5. Let{Xni;i≥1, n≥1}be an array of rowwiseρ-mixing identically distributed random

variables satisfying∞_i₁ρ2_n/q2i<∞for someq≥2andEX110. Letlx>0be a slowly varying

function asx → ∞. If forα >1/2,αp >1, and0< t <2

E|X11|αptl

|X11|t

<∞, 3.14

then

∞

n1

nαp−2_l_n_P

⎧ ⎨ ⎩max1≤i≤n

i

j1 Xnj

> εn1/t

⎫ ⎬

⎭<∞. 3.15

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ByLemma 1.6, we obtain

∞

n1

nαp−1_l_n_P_|_X

11|> εn1/t

∞

n1

nαp−1_l_n∞

mnP

εm1/t_<_|_X

11| ≤εm11/t

≤C∞

m1

Pεm1/t_<_|_X

11| ≤εm11/t

m

n1

nαp−1_l_n

≤C∞

m1

mαp_l_m_P_εm1/t_<_|_X

11| ≤εm11/t

≤CE|X11|αptl

|X11|t

<∞,

3.16

which proves that conditionAis satisfied.

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

∞

n1

nαp−1−q/t_l_n_E_|_X 11|qI

|X11| ≤εn1/t

∞

n1

nαp−1−q/t_l_nn

m1

E|X11|qI

εm−11/t≤ |X11|< εm1/t

≤C∞

m1

E|X11|qI

εm−11/t≤ |X11|< εm1/t

∞

nmn

αp−1−q/t_l_n

≤C∞

m1

mαp−q/t_l_m_E_|_X 11|qI

εm−11/t≤ |X11|< εm1/t

≤CE|X11|αptl

|X11|t

<∞,

3.17

which proves thatBholds.

In order to prove thatCholds, we consider the following two cases. Ifαpt <2, takeq >2. We have

∞

n1

nαp−2−q/tq/2_l_n_E_|_X 11|2I

|X11|< εn1/t

_q/2

≤C∞

n1

nαp−2−q/tq/2_l_n_nq/t−αpq/2_E_|_X 11|αptI

|X11|< εn1/t

_q/2

≤C∞

n1

nαp−11−q/2−1_l_n_<_∞_.

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Ifαpt≥2, takeq >max2,2tαp−1/2−t. We haveαp−q/tq/2<1. Note that in this caseE|X11|2 <∞. We obtain

∞

n1

nαp−2−q/tq/2_l_n_E_|_X 11|2I

|X11|< εn1/t

q/2

≤C∞

n1

nαp−2−q/tq/2_l_n_<_∞_. _3.19

The proof will be completed if we show that

n−1/t_max

1≤i≤n

i

j1 EXnjI

|X11|< εn1/t

−→0, asn−→ ∞. 3.20

Ifαpt <1, then

n−1/t_max

1≤i≤n

i

j1 EXnjI

|X11|< εn1/t

≤Cn1−αpE|X11|αpt−→0, asn−→ ∞. 3.21

Ifαpt≥1, note thatEX110, then

n−1/t_max

1≤i≤n

i

j1 EXnjI

|X11|< εn1/t

≤n−1/t1_EX 11I

|X11| ≥εn1/t≤Cn1−αpE|X11|αpt−→0, asn−→ ∞.

3.22

We complete the proof of the theorem.

Noting that for typical slowly varying functions,lx 1 andlx logx, 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 Chinano. 2009SQRZ176ZD and National Natural Science Foundation of Chinanos. 11001052, 10871001, 10971097.

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