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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 735973, 10 pages
http://dx.doi.org/10.1155/2012/735973
Research Article

Limiting Behavior of the Maximum of the Partial Sum for Linearly Negative Quadrant Dependent Random Variables under Residual CesΓ ro Alpha-Integrability Assumption

College of Science, Guilin University of Technology, Guilin 541004, China

Received 13 September 2011; Accepted 29 December 2011

Academic Editor: XianhuaΒ Tang

Copyright Β© 2012 Jiangfeng Wang and Qunying Wu. 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.

Abstract

Linearly negative quadrant dependence is a special dependence structure. By relating such conditions to residual CesΓ ro alpha-integrability assumption, as well as to strongly residual CesΓ ro alpha-integrability assumption, some 𝐿𝑝-convergence and complete convergence results of the maximum of the partial sum are derived, respectively.

1. Introduction

The classical notion of uniform integrability of a sequence {𝑋𝑛}π‘›βˆˆβ„• of integrable random variables is defined through the condition limπ‘Žβ†’βˆžsup𝑛β‰₯1𝐸|𝑋𝑛|𝐼(|𝑋𝑛|>π‘Ž)=0. Landers and Rogge [1] proved that the uniform integrability condition is sufficient in order that a sequence of pairwise independent random variables verifies the weak law of large numbers (WLLNs). Chandra [2] weakened the assumption of uniform integrability to CesΓ‘ro uniform integrability (CUI) and obtained 𝐿1-convergence for pairwise independent random variables.

Chandra and Goswami [3] improved the above-mentioned result of Landers and Rogge [1]. They showed that for a sequence of pairwise independent random variables, CUI is sufficient for the WLLN to hold and strong CesΓ‘ro uniform integrability (SCUI) is sufficient for the strong law of large numbers (SLLNs) to hold. Landers and Rogge [4] obtained a slight improvement over the results of Chandra [2] and Chandra and Goswami [3] for the case of nonnegative random variables. They showed that, in this case, the condition of pairwise independence can be replaced by the weaker assumption of pairwise nonpositive correlation.

Chandra and Goswami [5] introduced a new set of conditions called CesΓ‘ro 𝛼-integrability (CI(𝛼)) and strong CesΓ‘ro 𝛼-integrability (SCI(𝛼)) for a sequence of random variables, which are strictly weaker than CUI and SCUI, respectively. They showed that, for 𝛼<1/2, CI(𝛼) is sufficient for the WLLN to hold and SCI(𝛼) is sufficient for the SLLN to hold for a sequence of pairwise independent random variables, which are improvements over the results of Landers and Rogge [4] and the earlier results.

Chandra and Goswami [6] relaxed the condition of CI(𝛼) to residual CesΓ‘ro alpha-integrability (RCI(𝛼), see Definition 2.1 below) and the condition of SCI(𝛼) to strong residual CesΓ‘ro alpha-integrability (SRCI(𝛼), see Definition 2.3 below) and significantly improved the results of Chandra and Goswami [5].

Recently, Yuan and Wu [7] discussed some limiting behaviors of the maximum of partial sum for asymptotically negatively associated random variables when such random variables are subject to RCI(𝛼) and SRCI(𝛼).

In this paper, we will derive some 𝐿𝑝-convergence and complete convergence of the maximum of partial sum for linearly negative quadrant dependent random variables when such random variables are subject to RCI(𝛼) and SRCI(𝛼). These results generalize previous work in the literature.

2. Preliminaries

First let us specify the two special kinds of uniform integrability we are dealing with in the subsequent sections, which were introduced by Chandra and Goswami [6].

Definition 2.1. For π›Όβˆˆ(0,∞), a sequence {𝑋𝑛}π‘›βˆˆβ„• of random variables is said to be residual CesΓ‘ro alpha-integrable (RCI(𝛼), in short) if sup𝑛β‰₯11𝑛𝑛𝑖=1𝐸||𝑋𝑖||<∞,limπ‘›β†’βˆž1𝑛𝑛𝑖=1𝐸||𝑋𝑖||βˆ’π‘–π›Όξ€ΈπΌξ€·||𝑋𝑖||>𝑖𝛼=0.(2.1)

Clearly, {𝑋𝑛} is RCI(𝛼) for any 𝛼>0 if {𝑋𝑛}π‘›βˆˆβ„• is identically distributed with 𝐸|𝑋1|<∞, and {|𝑋𝑛|𝑝}π‘›βˆˆβ„• is RCI(𝛼) for any 𝛼>0 if {𝑋𝑛}π‘›βˆˆβ„• is stochastically dominated by a nonnegative random variable 𝑋 with 𝐸𝑋𝑝<∞ for some 𝑝β‰₯1.

Definition 2.2. For π›Όβˆˆ(0,∞), a sequence {𝑋𝑛}π‘›βˆˆβ„• of random variables is said to be strongly residual CesΓ‘ro alpha-integrable (SRCI(𝛼), in short) if sup𝑛β‰₯11𝑛𝑛𝑖=1𝐸||𝑋𝑖||<∞,βˆžξ“π‘›=11𝑛𝐸||𝑋𝑛||βˆ’π‘›π›Όξ€ΈπΌξ€·||𝑋𝑛||>𝑛𝛼<∞.(2.2)

We point out that, {|𝑋𝑛|𝑝}π‘›βˆˆβ„• is SRCI(𝛼) for any 𝛼>0, provided that {𝑋𝑛}π‘›βˆˆβ„• is stochastically dominated by a nonnegative random variable 𝑋 with 𝐸𝑋𝑝+𝛿<∞ for some 𝑝β‰₯1 and 𝛿>0.

The condition of SRCI(𝛼) is a β€œstrong” version of the condition of RCI(𝛼). Moreover, for any 𝛼>0, RCI(𝛼) is strictly weaker than CI(𝛼), thereby weaker than CUI, while SRCI(𝛼) is strictly weaker than SCI(𝛼), thereby much weaker than SCUI.

Next, we turn our attention to the dependence structure for random variables. For our purpose, we have to mention a special kind of dependence, namely, negative quadrant dependence.

Definition 2.3 (cf. Lehmann [8]). Two random variables 𝑋 and π‘Œ are said to be negative quadrant dependent (NQD, in short) if for any π‘₯,π‘¦βˆˆβ„, 𝑃(𝑋<π‘₯,π‘Œ<𝑦)≀𝑃(𝑋<π‘₯)𝑃(π‘Œ<𝑦).(2.3) A sequence {𝑋𝑛}π‘›βˆˆβ„• of random variables is said to be pairwise NQD if 𝑋𝑖 and 𝑋𝑗 are NQD for all 𝑖,π‘—βˆˆβ„•+ and 𝑖≠𝑗.

Definition 2.4 (cf. Newman [9]). A sequence{𝑋𝑛}π‘›βˆˆβ„• of random variables is said to be linearly negative quadrant dependent (LNQD, in short) if for any disjoint subsets 𝐴,π΅βˆˆβ„€+ and positive π‘Ÿξ…žπ‘—π‘ , ξ“π‘˜βˆˆπ΄π‘Ÿπ‘˜π‘‹π‘˜,ξ“π‘—βˆˆπ΅π‘Ÿπ‘—π‘‹π‘—areNQD.(2.4)

Remark 2.5. It is easily seen that if {𝑋𝑛}π‘›βˆˆβ„• is a sequence of LNQD random variables, then {π‘Žπ‘‹π‘›+𝑏}π‘›βˆˆβ„• is still a sequence of LNQD random variables, where π‘Ž and 𝑏 are real numbers.

The concept of LNQD sequence was introduced by Newman [9]. Some applications for LNQD sequence have been found; see, for example, the work by Newman [9] who established the central limit theorem for a strictly stationary LNQD process. Wang and Zhang [10] provided uniform rates of convergence in the central limit theorem for LNQD sequence. Ko et al. [11] obtained the Hoeffding-type inequality for LNQD sequence. Ko et al. [12] studied the strong convergence for weighted sums of LNQD arrays. Fu and Wu [13] studied the almost sure central limit theorem for LNQD sequences, and so forth. We note that β€œβ‰ͺ" means β€œπ‘‚.”

Lemma 2.6 (cf. Lehmann [8]). Let random variables 𝑋 and π‘Œ be NQD. Then(1)πΈπ‘‹π‘Œβ‰€πΈπ‘‹πΈπ‘Œ; (2)𝑃(𝑋<π‘₯,π‘Œ<𝑦)≀𝑃(𝑋<π‘₯)𝑃(π‘Œ<𝑦); (3)If 𝑓 and 𝑔 are both nondecreasing (or both nonincreasing) functions, then 𝑓(𝑋) and 𝑔(π‘Œ) are NQD.

Lemma 2.7 (cf. Hu et al. [14]). Let {𝑋𝑛}π‘›βˆˆβ„• be a LNQD sequence of random variables with 𝐸𝑋𝑛=0. Assume that there exists a 𝑝>2 satisfying 𝐸|𝑋𝑖|𝑝<∞ for every 𝑖β‰₯1. Then, there exists a positive constant 𝑐 such that πΈβŽ›βŽœβŽœβŽmax1β‰€π‘˜β‰€π‘›|||||π‘˜ξ“π‘–=1𝑋𝑖|||||βŽžβŽŸβŽŸβŽ π‘ξƒ©β‰€π‘π΄(𝑝)𝑛𝑖=1𝐸||𝑋𝑖||𝑝2/𝑝ξƒͺ𝑝/2,βˆ€π‘›β‰₯1,(2.5) where 𝐴(𝑝)=𝐴𝑝,𝑝/2 is a positive constant depending only on 𝑝.

It is easily seen that when 𝑝=2, the above equation still holds true.

Lemma 2.8. Let {𝑋𝑛}π‘›βˆˆβ„• be LNQD random variables sequences with mean zero. Then for 1<𝑝<2, there exists a positive constant 𝑐 such that 𝐸max1≀𝑖≀𝑛||𝑆𝑖||𝑝≀𝑐𝑛𝑖=1𝐸||𝑋𝑖||𝑝1/𝑝ξƒͺ𝑝,βˆ€π‘›β‰₯1.(2.6)

This lemma is easily proved by the results of Zhang [15] and Yuan and Wu [7]. Here we omit the details of the proof.

Lemma 2.9. Let {π‘‹π‘˜}π‘˜βˆˆβ„•π‘‘ be a centered LNQD random field. Then for any 𝑝>1, there exists a positive constant 𝑐 such that 𝐸|||||𝑛𝑖=1𝑋𝑖|||||𝑝≀𝑐𝑛𝑖=1𝐸||𝑋𝑖||𝑝,(2.7) for all 𝑛β‰₯1.

This lemma is due to Zhang [15, Lemma 3.3].

Finally, we give a lemma which supplies us with the analytical part in the proofs of theorems in the subsequent sections.

Lemma 2.10 (cf. Landers and Rogge [4]). For sequences {π‘Žπ‘›}π‘›βˆˆβ„• and {𝑏𝑛}π‘›βˆˆβ„• of nonnegative real numbers, if sup𝑛β‰₯1π‘›π‘›βˆ’1𝑖=1π‘Žπ‘–<∞,βˆžξ“π‘›=1𝑏𝑛<∞,(2.8) then 𝑛𝑖=1π‘Žπ‘–π‘π‘–β‰€ξƒ©supπ‘šβ‰₯1π‘šπ‘šβˆ’1𝑖=1π‘Žπ‘–ξƒͺ𝑛𝑖=1𝑏𝑖<∞,(2.9) for every 𝑛β‰₯1.

3. Residual CesΓ‘ro Alpha-Integrability and 𝐿𝑝-Convergence of the Maximum of the Partial Sum

Let 𝑝>1, and let β„Ž(π‘₯) be a strictly positive function defined on (1,+∞). In this section, we discuss 𝐿𝑝-convergence of the form of π‘›βˆ’β„Ž(𝑝)max1≀𝑖≀𝑛|π‘†π‘–βˆ’πΈπ‘†π‘–| for a LNQD sequence {𝑋𝑛}π‘›βˆˆβ„• of random variables, provided that {|𝑋𝑛|𝑝}π‘›βˆˆβ„• is RCI(𝛼) for an appropriate condition.

Our first result is dealing with the case 1<𝑝<2.

Theorem 3.1. Let 1<𝑝<2, and let {𝑋𝑛}π‘›βˆˆβ„• be a LNQD sequence of random variables. If {|𝑋𝑛|𝑝}π‘›βˆˆβ„• is RCI(𝛼) for some π›Όβˆˆ(0,1/(2βˆ’π‘)), then π‘›βˆ’1max1≀𝑖≀𝑛||π‘†π‘–βˆ’πΈπ‘†π‘–||⟢0in𝐿𝑝.(3.1)

Proof of Theorem 3.1. Let π‘Œπ‘›=βˆ’π‘›π›ΌπΌ(𝑋𝑛<βˆ’π‘›π›Ό)+𝑋𝑛𝐼(|𝑋𝑛𝑖|≀𝑛𝛼)+𝑛𝛼𝐼(𝑋𝑛>𝑛𝛼),𝑛β‰₯1, and define, for each 𝑛β‰₯1, 𝑍𝑛=π‘‹π‘›βˆ’π‘Œπ‘›, 𝑆𝑛(1)=βˆ‘π‘›π‘–=1π‘Œπ‘–, and 𝑆𝑛(2)=βˆ‘π‘›π‘–=1𝑍𝑖. It is easy to see that |π‘Œπ‘›|=min{|𝑋𝑛|,𝑛𝛼}, |𝑍𝑛|=(|𝑋𝑛|βˆ’π‘›π›Ό)𝐼(|𝑋𝑛|>𝑛𝛼), and ||𝑍𝑛||𝑝≀||𝑋𝑛||π‘βˆ’π‘›π›Όξ€ΈπΌξ€·||𝑋𝑛||𝑝>𝑛𝛼(3.2) for all 𝑝>1. Note that, for each 𝑛β‰₯1, π‘Œπ‘› and 𝑍𝑛 are monotone transformations of the initial variable 𝑋𝑛. This implies that LNQD assumption is preserved by this construction in view of Lemma 2.6. Precisely, {π‘Œπ‘›βˆ’πΈπ‘Œπ‘›}π‘›βˆˆβ„• and {π‘π‘›βˆ’πΈπ‘π‘›}π‘›βˆˆβ„• are also LNQD sequences of zero mean random variables.
For our purpose, it suffices to prove π‘›βˆ’1max1≀𝑖≀𝑛||𝑆𝑖(1)βˆ’πΈπ‘†π‘–(1)||⟢0in𝐿2,𝑛(3.3)βˆ’1max1≀𝑖≀𝑛||𝑆𝑖(2)βˆ’πΈπ‘†π‘–(2)||⟢0in𝐿𝑝.(3.4)
Using Lemma 2.8, the HΓΆlder inequality, relation (3.2), and the second condition in (2.1) of the RCI(𝛼) property of the sequence {|𝑋𝑛|𝑝}π‘›βˆˆβ„•, we obtain π‘›βˆ’π‘πΈξ‚΅max1≀𝑖≀𝑛||𝑆𝑖(2)βˆ’πΈπ‘†π‘–(2)||𝑝β‰ͺπ‘›βˆ’π‘ξƒ©π‘›ξ“π‘–=1𝐸||π‘π‘–βˆ’πΈπ‘π‘–||𝑝1/𝑝ξƒͺ𝑝β‰ͺπ‘›π‘›βˆ’1𝑖=1𝐸||π‘π‘–βˆ’πΈπ‘π‘–||𝑝β‰ͺπ‘›π‘›βˆ’1𝑖=1𝐸||𝑍𝑖||π‘β‰€π‘›π‘›βˆ’1𝑖=1𝐸||𝑋𝑖||π‘βˆ’π‘–π›Όξ€ΈπΌξ€·||𝑋𝑖||𝑝>π‘–π›Όξ€ΈβŸΆ0.(3.5) This proves (3.4). To verify (3.3), using Lemma 2.7, we have π‘›βˆ’2𝐸max1≀𝑖≀𝑛||𝑆𝑖(1)βˆ’πΈπ‘†π‘–(1)||2ξ‚Άβ‰ͺπ‘›βˆ’2𝑛𝑖=1πΈξ€·π‘Œπ‘–βˆ’πΈπ‘Œπ‘–ξ€Έ2ξƒͺβ‰ͺπ‘›π‘›βˆ’2𝑖=1πΈπ‘Œ2π‘–β‰€π‘›π‘›βˆ’2+(2βˆ’π‘)𝛼𝑖=1𝐸||𝑋𝑖||π‘β‰€π‘›βˆ’1+(2βˆ’π‘)𝛼⋅sup𝑛β‰₯1ξƒ©π‘›π‘›βˆ’1𝑖=1𝐸||𝑋𝑖||𝑝ξƒͺ.(3.6) Using the first condition of (2.1) of the RCI(𝛼) property of the sequence {|𝑋𝑛|𝑝}π‘›βˆˆβ„•, the last expression above clearly goes to 0 as π‘›β†’βˆž, from 1<𝑝<2 and 𝛼<1/(2βˆ’π‘), thus completing the proof.

Remark 3.2. Let 1<𝑝<2, and let {𝑋𝑛}π‘›βˆˆβ„• be a LNQD sequence of random variables. If {|𝑋𝑛|𝑝}π‘›βˆˆβ„• is RCI(𝛼) for some π›Όβˆˆ(0,1/𝑝), then π‘›βˆ’1/𝑝(π‘†π‘›βˆ’πΈπ‘†π‘›)β†’0in𝐿𝑝.

Compared with Theorem 3.1, this result, whose proof can be completed by using Lemma 2.9, drops the maximum of the partial sum at the price of enlarging 1/𝑛 into 1/𝑛1/𝑝.

Next we consider the case 𝑝β‰₯2.

Theorem 3.3. Let 𝑝β‰₯2, and let {𝑋𝑛}π‘›βˆˆβ„• be a LNQD sequence of random variables. If {𝑋𝑛}π‘›βˆˆβ„• satisfies sup𝑛β‰₯11𝑛𝑛𝑖=1𝐸||𝑋𝑖||p<∞,(3.7) then for any 𝛿>1/2π‘›βˆ’π›Ώmax1≀𝑖≀𝑛||π‘†π‘–βˆ’πΈπ‘†π‘–||⟢0in𝐿𝑝.(3.8)

Proof of Theorem 3.3. By Lemma 2.7 and the HΓΆlder inequality, πΈξ‚΅π‘›βˆ’π›Ώmax1≀𝑖≀𝑛||π‘†π‘–βˆ’πΈπ‘†π‘–||𝑝β‰ͺπ‘›βˆ’π‘π›Ώξƒ©π‘›ξ“π‘–=1𝐸||𝑋𝑖||𝑝2/𝑝ξƒͺ𝑝/2β‰€π‘›π‘›βˆ’π‘π›Ώ+(𝑝/2)βˆ’1𝑖=1𝐸||𝑋𝑖||π‘β‰€π‘›βˆ’π‘π›Ώ+(𝑝/2)β‹…sup𝑛β‰₯11𝑛𝑛𝑖=1𝐸||𝑋𝑖||π‘βŸΆ0.(3.9) The proof is completed.

4. Strongly Residual CesΓ‘ro Alpha-Integrability and Complete Convergence of the Maximum of the Partial Sum

A sequence of random variables {𝑋𝑛}π‘›βˆˆβ„• is said to converge completely to a constant π‘Ž if for any πœ€>0, βˆžξ“π‘›=1𝑃||𝑋𝑛||ξ€Έβˆ’π‘Ž>πœ€<∞.(4.1) In this case we write π‘‹π‘›β†’π‘Ž completely. This notion was given by Hsu and Robbins [16]. Note that the complete convergence implies the almost sure convergence in view of the Borel-Cantelli lemma.

The condition of SRCI(𝛼) is a strong version of the condition of RCI(𝛼). In this section, we will show that each of the theorems in the previous section has a corresponding β€œstrong” analogue in the sense of complete convergence.

Theorem 4.1. Let 1<𝑝<2, and let {𝑋𝑛}π‘›βˆˆβ„• be a LNQD sequence of random variables. If {|𝑋𝑛|𝑝}π‘›βˆˆβ„• is SRCI(𝛼) for some π›Όβˆˆ(0,1/(2βˆ’π‘)), then π‘›βˆ’1max1≀𝑖≀𝑛||π‘†π‘–βˆ’πΈπ‘†π‘–||⟢0completely.(4.2)

Proof of Theorem 4.1. For any 𝑛β‰₯1, let π‘š=π‘šπ‘› be the integer such that 2π‘šβˆ’1<𝑛≀2π‘š. Observe that π‘›βˆ’1max1≀𝑖≀𝑛||π‘†π‘–βˆ’πΈπ‘†π‘–||β‰€π‘›βˆ’1max1≀𝑖≀2π‘š||π‘†π‘–βˆ’πΈπ‘†π‘–||≀2π‘šβˆ’1ξ€Έβˆ’1max1≀𝑖≀2π‘š||π‘†π‘–βˆ’πΈπ‘†π‘–||=2β‹…2βˆ’π‘šmax1≀𝑖≀2π‘š||π‘†π‘–βˆ’πΈπ‘†π‘–||.(4.3) Hence it suffices to show that 2βˆ’π‘šmax1≀𝑖≀2π‘š||π‘†π‘–βˆ’πΈπ‘†π‘–||⟢0completely.(4.4) Let π‘Œπ‘›, 𝑍𝑛, 𝑆𝑛(1), and 𝑆𝑛(2) be defined as in the proof of Theorem 3.1. We first prove that 2βˆ’π‘šmax1≀𝑖≀2π‘š|𝑆𝑖(2)βˆ’πΈπ‘†π‘–(2)|β†’0 completely; that is, 2βˆ’π‘šmax1≀𝑖≀2π‘š|||||π‘–ξ“π‘˜=1ξ€·π‘π‘˜βˆ’πΈπ‘π‘˜ξ€Έ|||||⟢0completely.(4.5) Using Lemma 2.8, the HΓΆlder inequality, relation (3.2), and the second condition in (2.1) of the RCI(𝛼) property of the sequence {|𝑋𝑛|𝑝}π‘›βˆˆβ„•, we have βˆžξ“π‘š=0πΈβŽ›βŽœβŽœβŽ2βˆ’π‘šmax1≀𝑖≀2π‘š|||||π‘–ξ“π‘˜=1ξ€·π‘π‘˜βˆ’πΈπ‘π‘˜ξ€Έ|||||βŽžβŽŸβŽŸβŽ π‘β‰ͺβˆžξ“π‘š=02βˆ’π‘šπ‘βŽ›βŽœβŽœβŽ2π‘šξ“π‘–=1𝐸||𝑍𝑖||𝑝1/π‘βŽžβŽŸβŽŸβŽ π‘β‰€βˆžξ“π‘š=022βˆ’π‘šπ‘šξ“π‘–=1𝐸||𝑍𝑖||𝑝=βˆžξ“π‘–=1𝐸||𝑍𝑖||π‘ξ“π‘šβˆΆ2π‘šβ‰₯𝑖2βˆ’π‘šβ‰€βˆžξ“π‘–=1π‘–βˆ’1𝐸||𝑍𝑖||π‘β‰€βˆžξ“π‘–=1π‘–βˆ’1𝐸||𝑋𝑖||π‘βˆ’π‘–π›Όξ€ΈπΌξ€·||𝑋𝑖||𝑝>𝑖𝛼<∞,(4.6) which implies (4.4).
Next we show that 2βˆ’π‘šmax1≀𝑖≀2π‘š|𝑆𝑖(1)βˆ’πΈπ‘†π‘–(1)|β†’0 completely; that is, 2βˆ’π‘šmax1≀𝑖≀2π‘š|||||π‘–ξ“π‘˜=1ξ€·π‘Œπ‘˜βˆ’πΈπ‘Œπ‘˜ξ€Έ|||||⟢0completely.(4.7) By Lemma 2.7 and the HΓΆlder inequality, βˆžξ“π‘š=0𝐸2βˆ’π‘šmax1≀𝑖≀2π‘šπ‘–ξ“π‘˜=1ξ€·π‘Œπ‘˜βˆ’πΈπ‘Œπ‘˜ξ€Έξƒͺ2β‰ͺβˆžξ“π‘š=022βˆ’2π‘šπ‘šξ“π‘–=1πΈπ‘Œ2π‘–β‰€βˆžξ“π‘š=022βˆ’2π‘šπ‘šξ“π‘–=1𝑖(2βˆ’π‘)𝛼𝐸||𝑋𝑖||𝑝.(4.8) In view of the first condition in (2.1) of the RCI(𝛼) property of the sequence {|𝑋𝑛|𝑝}π‘›βˆˆβ„•, we have βˆžξ“π‘š=0𝐸2βˆ’π‘šmax1≀𝑖≀2π‘šπ‘–ξ“π‘˜=1ξ€·π‘Œπ‘˜βˆ’πΈπ‘Œπ‘˜ξ€Έξƒͺ2β‰€βˆžξ“π‘š=022βˆ’2π‘šπ‘šξ“π‘–=1𝑖(2βˆ’π‘)𝛼𝐸||𝑋𝑖||π‘β‰€βˆžξ“π‘š=02βˆ’2π‘šβ‹…22π‘š(2βˆ’π‘)π›Όπ‘šξ“π‘–=1𝐸||𝑋𝑖||π‘β‰€βˆžξ“π‘š=02βˆ’π‘šβ‹…2π‘š(2βˆ’π‘)𝛼⋅supπ‘šβˆΆ2π‘šβ‰₯𝑖12π‘š2π‘šξ“π‘–=1𝐸||𝑋𝑖||𝑝.(4.9) The last series above converges since π›Όβˆˆ(0,1/(2βˆ’π‘)) implies βˆ’1+(2βˆ’π‘)𝛼<0, and therefore (4.7) holds. This completes the proof.

For the case 𝑝β‰₯2, we have the following result.

Theorem 4.2. Let 𝑝β‰₯2, and let {𝑋𝑛}π‘›βˆˆβ„• be a LNQD sequence of random variables. If {𝑋𝑛}π‘›βˆˆβ„• satisfies sup𝑛β‰₯11𝑛𝑛𝑖=1𝐸||𝑋𝑖||𝑝<∞,(4.10) then for any 𝛿>1/2π‘›βˆ’π›Ώmax1≀𝑖≀𝑛||π‘†π‘–βˆ’πΈπ‘†π‘–||⟢0completely.(4.11)

Proof of Theorem 4.2. Let π‘šπ‘›,𝑛β‰₯1 be defined as in the proof of Theorem 4.1. Proceeding in the proof of (4.3), we see that it suffices to show that 2βˆ’π‘šπ›Ώmax1≀𝑖≀2π‘š||π‘†π‘–βˆ’πΈπ‘†π‘–||⟢0completely.(4.12) Indeed by Lemma 2.7 and the HΓΆlder inequality, βˆžξ“π‘š=0𝐸2βˆ’π‘šπ›Ώmax1≀𝑖≀2π‘š||π‘†π‘–βˆ’πΈπ‘†π‘–||𝑝β‰ͺβˆžξ“π‘š=02βˆ’π‘šπ‘π›ΏβŽ›βŽœβŽœβŽ2π‘šξ“π‘–=1𝐸||𝑋𝑖||𝑝2/π‘βŽžβŽŸβŽŸβŽ π‘/2β‰€βˆžξ“π‘–=1𝐸||𝑋𝑖||π‘ξ“π‘šβˆΆ2π‘šβ‰₯𝑖2βˆ’π‘šπ‘π›Ώβˆ’π‘š+2π‘š/π‘β‰€βˆžξ“π‘–=1π‘–βˆ’π‘π›Ώβˆ’1+𝑝/2𝐸||𝑋𝑖||𝑝.(4.13) In view of Lemma 2.10, βˆžξ“π‘–=1π‘–βˆ’π‘π›Ώβˆ’1+𝑝/2𝐸||𝑋𝑖||𝑝≀sup𝑛β‰₯11𝑛2π‘šξ“π‘–=1𝐸||𝑋𝑖||π‘βˆžξ“π‘›=1π‘›βˆ’π‘π›Ώβˆ’1+𝑝/2β‰ͺβˆžξ“π‘›=1π‘›βˆ’π‘π›Ώβˆ’1+𝑝/2<∞(4.14) from βˆ’π‘π›Ώβˆ’1+𝑝/2<βˆ’1. Therefore (4.12) holds. The proof is completed.

Acknowledgments

Supported by the National Science Foundation of China (11061012), the Guangxi China Science Foundation (2010GXNSFA013120), and Innovation Project of Guangxi Graduate Education (2010105960202M32). We are very grateful to the referees and the editors for their valuable comments and some helpful suggestions that improved the clarity and readability of the paper.

References

  1. D. Landers and L. Rogge, β€œLaws of large numbers for pairwise independent uniformly integrable random variables,” Mathematische Nachrichten, vol. 130, pp. 189–192, 1987. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  2. T. K. Chandra, β€œUniform integrability in the CesΓ ro sense and the weak law of large numbers,” Sankhyā. Series A, vol. 51, no. 3, pp. 309–317, 1989. View at Google Scholar Β· View at Zentralblatt MATH
  3. T. K. Chandra and A. Goswami, β€œCesΓ ro uniform integrability and the strong law of large numbers,” Sankhyā. Series A, vol. 54, no. 2, pp. 215–231, 1992. View at Google Scholar Β· View at Zentralblatt MATH
  4. D. Landers and L. Rogge, β€œLaws of large numbers for uncorrelated CesΓ ro uniformly integrable random variables,” Sankhyā. Series A, vol. 59, pp. 301–310, 1997. View at Google Scholar
  5. T. K. Chandra and A. Goswami, β€œCesΓ ro Ξ±-integrability and laws of large numbers. I,” Journal of Theoretical Probability, vol. 16, no. 3, pp. 655–669, 2003. View at Publisher Β· View at Google Scholar
  6. T. K. Chandra and A. Goswami, β€œCesΓ ro Ξ±-integrability and laws of large numbers. II,” Journal of Theoretical Probability, vol. 19, no. 4, pp. 789–816, 2006. View at Publisher Β· View at Google Scholar
  7. D. M. Yuan and X. S. Wu, β€œLimiting behavior of the maximum of the partial sum for asymptotically negatively associated random variables under residual CesΓ ro alpha-integrability assumption,” Journal of Statistical Planning and Inference, vol. 140, no. 9, pp. 2395–2402, 2010. View at Publisher Β· View at Google Scholar
  8. E. L. Lehmann, β€œSome concepts of dependence,” Annals of Mathematical Statistics, vol. 37, pp. 1137–1153, 1966. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  9. C. M. Newman, β€œAsymptotic independence and limit theorems for positively and negatively dependent random variables,” in Inequalities in Statistics and Probability, Y. L. Tong, Ed., vol. 5 of IMS Lecture Notes Monogr. Ser., pp. 127–140, Inst. Math. Statist., Hayward, Calif, USA, 1984. View at Google Scholar
  10. J. F. Wang and L. X. Zhang, β€œA Berry-Esseen theorem for weakly negatively dependent random variables and its applications,” Acta Mathematica Hungarica, vol. 110, no. 4, pp. 293–308, 2006. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  11. M.-H. Ko, Y.-K. Choi, and Y.-S. Choi, β€œExponential probability inequality for linearly negative quadrant dependent random variables,” Korean Mathematical Society. Communications, vol. 22, no. 1, pp. 137–143, 2007. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  12. M.-H. Ko, D.-H. Ryu, and T.-S. Kim, β€œLimiting behaviors of weighted sums for linearly negative quadrant dependent random variables,” Taiwanese Journal of Mathematics, vol. 11, no. 2, pp. 511–522, 2007. View at Google Scholar Β· View at Zentralblatt MATH
  13. Y. Fu and Q. Wu, β€œAlmost sure central limit theorem for LNQD sequences,” Journal of Guilin University of Technology, vol. 30, no. 4, pp. 637–639, 2010. View at Google Scholar
  14. S. Hu, X. Li, W. Yang, and X. Wang, β€œMaximal inequalities for some dependent sequences and their applications,” Journal of the Korean Statistical Society, vol. 40, no. 1, pp. 11–19, 2011. View at Publisher Β· View at Google Scholar
  15. L.-X. Zhang, β€œA functional central limit theorem for asymptotically negatively dependent random fields,” Acta Mathematica Hungarica, vol. 86, no. 3, pp. 237–259, 2000. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH
  16. P. L. Hsu and H. Robbins, β€œComplete convergence and the law of large numbers,” Proceedings of the National Academy of Sciences of the United States of America, vol. 33, pp. 25–31, 1947. View at Publisher Β· View at Google Scholar Β· View at Zentralblatt MATH