Abstract

Linearly negative quadrant dependence is a special dependence structure. By relating such conditions to residual Cesร ro alpha-integrability assumption, as well as to strongly residual Cesร ro alpha-integrability assumption, some ๐ฟ๐‘-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 {๐‘‹๐‘›}๐‘›โˆˆโ„• of integrable random variables is defined through the condition lim๐‘Žโ†’โˆžsup๐‘›โ‰ฅ1๐ธ|๐‘‹๐‘›|๐ผ(|๐‘‹๐‘›|>๐‘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 Cesรกro uniform integrability (CUI) and obtained ๐ฟ1-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 Cesรกro 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 Cesรกro ๐›ผ-integrability (CI(๐›ผ)) and strong Cesรกro ๐›ผ-integrability (SCI(๐›ผ)) for a sequence of random variables, which are strictly weaker than CUI and SCUI, respectively. They showed that, for ๐›ผ<1/2, CI(๐›ผ) is sufficient for the WLLN to hold and SCI(๐›ผ) 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(๐›ผ) to residual Cesรกro alpha-integrability (RCI(๐›ผ), see Definition 2.1 below) and the condition of SCI(๐›ผ) to strong residual Cesรกro alpha-integrability (SRCI(๐›ผ), 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(๐›ผ) and SRCI(๐›ผ).

In this paper, we will derive some ๐ฟ๐‘-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(๐›ผ) and SRCI(๐›ผ). 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 ๐›ผโˆˆ(0,โˆž), a sequence {๐‘‹๐‘›}๐‘›โˆˆโ„• of random variables is said to be residual Cesรกro alpha-integrable (RCI(๐›ผ), in short) if sup๐‘›โ‰ฅ11๐‘›๐‘›๎“๐‘–=1๐ธ||๐‘‹๐‘–||<โˆž,lim๐‘›โ†’โˆž1๐‘›๐‘›๎“๐‘–=1๐ธ๎€ท||๐‘‹๐‘–||โˆ’๐‘–๐›ผ๎€ธ๐ผ๎€ท||๐‘‹๐‘–||>๐‘–๐›ผ๎€ธ=0.(2.1)

Clearly, {๐‘‹๐‘›} is RCI(๐›ผ) for any ๐›ผ>0 if {๐‘‹๐‘›}๐‘›โˆˆโ„• is identically distributed with ๐ธ|๐‘‹1|<โˆž, and {|๐‘‹๐‘›|๐‘}๐‘›โˆˆโ„• is RCI(๐›ผ) for any ๐›ผ>0 if {๐‘‹๐‘›}๐‘›โˆˆโ„• is stochastically dominated by a nonnegative random variable ๐‘‹ with ๐ธ๐‘‹๐‘<โˆž for some ๐‘โ‰ฅ1.

Definition 2.2. For ๐›ผโˆˆ(0,โˆž), a sequence {๐‘‹๐‘›}๐‘›โˆˆโ„• of random variables is said to be strongly residual Cesรกro alpha-integrable (SRCI(๐›ผ), in short) if sup๐‘›โ‰ฅ11๐‘›๐‘›๎“๐‘–=1๐ธ||๐‘‹๐‘–||<โˆž,โˆž๎“๐‘›=11๐‘›๐ธ๎€ท||๐‘‹๐‘›||โˆ’๐‘›๐›ผ๎€ธ๐ผ๎€ท||๐‘‹๐‘›||>๐‘›๐›ผ๎€ธ<โˆž.(2.2)

We point out that, {|๐‘‹๐‘›|๐‘}๐‘›โˆˆโ„• is SRCI(๐›ผ) for any ๐›ผ>0, provided that {๐‘‹๐‘›}๐‘›โˆˆโ„• is stochastically dominated by a nonnegative random variable ๐‘‹ with ๐ธ๐‘‹๐‘+๐›ฟ<โˆž for some ๐‘โ‰ฅ1 and ๐›ฟ>0.

The condition of SRCI(๐›ผ) is a โ€œstrongโ€ version of the condition of RCI(๐›ผ). Moreover, for any ๐›ผ>0, RCI(๐›ผ) is strictly weaker than CI(๐›ผ), thereby weaker than CUI, while SRCI(๐›ผ) is strictly weaker than SCI(๐›ผ), 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 ๐‘‹ and ๐‘Œ are said to be negative quadrant dependent (NQD, in short) if for any ๐‘ฅ,๐‘ฆโˆˆโ„, ๐‘ƒ(๐‘‹<๐‘ฅ,๐‘Œ<๐‘ฆ)โ‰ค๐‘ƒ(๐‘‹<๐‘ฅ)๐‘ƒ(๐‘Œ<๐‘ฆ).(2.3) A sequence {๐‘‹๐‘›}๐‘›โˆˆโ„• of random variables is said to be pairwise NQD if ๐‘‹๐‘– and ๐‘‹๐‘— are NQD for all ๐‘–,๐‘—โˆˆโ„•+ and ๐‘–โ‰ ๐‘—.

Definition 2.4 (cf. Newman [9]). A sequence{๐‘‹๐‘›}๐‘›โˆˆโ„• of random variables is said to be linearly negative quadrant dependent (LNQD, in short) if for any disjoint subsets ๐ด,๐ตโˆˆโ„ค+ and positive ๐‘Ÿ๎…ž๐‘—๐‘ , ๎“๐‘˜โˆˆ๐ด๐‘Ÿ๐‘˜๐‘‹๐‘˜,๎“๐‘—โˆˆ๐ต๐‘Ÿ๐‘—๐‘‹๐‘—areNQD.(2.4)

Remark 2.5. It is easily seen that if {๐‘‹๐‘›}๐‘›โˆˆโ„• is a sequence of LNQD random variables, then {๐‘Ž๐‘‹๐‘›+๐‘}๐‘›โˆˆโ„• is still a sequence of LNQD random variables, where ๐‘Ž and ๐‘ 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 โ€œโ‰ช" means โ€œ๐‘‚.โ€

Lemma 2.6 (cf. Lehmann [8]). Let random variables ๐‘‹ and ๐‘Œ be NQD. Then(1)๐ธ๐‘‹๐‘Œโ‰ค๐ธ๐‘‹๐ธ๐‘Œ; (2)๐‘ƒ(๐‘‹<๐‘ฅ,๐‘Œ<๐‘ฆ)โ‰ค๐‘ƒ(๐‘‹<๐‘ฅ)๐‘ƒ(๐‘Œ<๐‘ฆ); (3)If ๐‘“ and ๐‘” are both nondecreasing (or both nonincreasing) functions, then ๐‘“(๐‘‹) and ๐‘”(๐‘Œ) are NQD.

Lemma 2.7 (cf. Hu et al. [14]). Let {๐‘‹๐‘›}๐‘›โˆˆโ„• be a LNQD sequence of random variables with ๐ธ๐‘‹๐‘›=0. Assume that there exists a ๐‘>2 satisfying ๐ธ|๐‘‹๐‘–|๐‘<โˆž for every ๐‘–โ‰ฅ1. Then, there exists a positive constant ๐‘ such that ๐ธโŽ›โŽœโŽœโŽmax1โ‰ค๐‘˜โ‰ค๐‘›|||||๐‘˜๎“๐‘–=1๐‘‹๐‘–|||||โŽžโŽŸโŽŸโŽ ๐‘๎ƒฉโ‰ค๐‘๐ด(๐‘)๐‘›๎“๐‘–=1๎€ท๐ธ||๐‘‹๐‘–||๐‘๎€ธ2/๐‘๎ƒช๐‘/2,โˆ€๐‘›โ‰ฅ1,(2.5) where ๐ด(๐‘)=๐ด๐‘,๐‘/2 is a positive constant depending only on ๐‘.

It is easily seen that when ๐‘=2, the above equation still holds true.

Lemma 2.8. Let {๐‘‹๐‘›}๐‘›โˆˆโ„• be LNQD random variables sequences with mean zero. Then for 1<๐‘<2, there exists a positive constant ๐‘ such that ๐ธ๎‚ตmax1โ‰ค๐‘–โ‰ค๐‘›||๐‘†๐‘–||๐‘๎‚ถ๎ƒฉโ‰ค๐‘๐‘›๎“๐‘–=1๎€ท๐ธ||๐‘‹๐‘–||๐‘๎€ธ1/๐‘๎ƒช๐‘,โˆ€๐‘›โ‰ฅ1.(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 {๐‘‹๐‘˜}๐‘˜โˆˆโ„•๐‘‘ be a centered LNQD random field. Then for any ๐‘>1, there exists a positive constant ๐‘ such that ๐ธ|||||๐‘›๎“๐‘–=1๐‘‹๐‘–|||||๐‘โ‰ค๐‘๐‘›๎“๐‘–=1๐ธ||๐‘‹๐‘–||๐‘,(2.7) for all ๐‘›โ‰ฅ1.

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 {๐‘Ž๐‘›}๐‘›โˆˆโ„• and {๐‘๐‘›}๐‘›โˆˆโ„• of nonnegative real numbers, if sup๐‘›โ‰ฅ1๐‘›๐‘›โˆ’1๎“๐‘–=1๐‘Ž๐‘–<โˆž,โˆž๎“๐‘›=1๐‘๐‘›<โˆž,(2.8) then ๐‘›๎“๐‘–=1๐‘Ž๐‘–๐‘๐‘–โ‰ค๎ƒฉsup๐‘šโ‰ฅ1๐‘š๐‘šโˆ’1๎“๐‘–=1๐‘Ž๐‘–๎ƒช๐‘›๎“๐‘–=1๐‘๐‘–<โˆž,(2.9) for every ๐‘›โ‰ฅ1.

3. Residual Cesรกro Alpha-Integrability and ๐ฟ๐‘-Convergence of the Maximum of the Partial Sum

Let ๐‘>1, and let โ„Ž(๐‘ฅ) be a strictly positive function defined on (1,+โˆž). In this section, we discuss ๐ฟ๐‘-convergence of the form of ๐‘›โˆ’โ„Ž(๐‘)max1โ‰ค๐‘–โ‰ค๐‘›|๐‘†๐‘–โˆ’๐ธ๐‘†๐‘–| for a LNQD sequence {๐‘‹๐‘›}๐‘›โˆˆโ„• of random variables, provided that {|๐‘‹๐‘›|๐‘}๐‘›โˆˆโ„• is RCI(๐›ผ) for an appropriate condition.

Our first result is dealing with the case 1<๐‘<2.

Theorem 3.1. Let 1<๐‘<2, and let {๐‘‹๐‘›}๐‘›โˆˆโ„• be a LNQD sequence of random variables. If {|๐‘‹๐‘›|๐‘}๐‘›โˆˆโ„• is RCI(๐›ผ) for some ๐›ผโˆˆ(0,1/(2โˆ’๐‘)), then ๐‘›โˆ’1max1โ‰ค๐‘–โ‰ค๐‘›||๐‘†๐‘–โˆ’๐ธ๐‘†๐‘–||โŸถ0in๐ฟ๐‘.(3.1)

Proof of Theorem 3.1. Let ๐‘Œ๐‘›=โˆ’๐‘›๐›ผ๐ผ(๐‘‹๐‘›<โˆ’๐‘›๐›ผ)+๐‘‹๐‘›๐ผ(|๐‘‹๐‘›๐‘–|โ‰ค๐‘›๐›ผ)+๐‘›๐›ผ๐ผ(๐‘‹๐‘›>๐‘›๐›ผ),๐‘›โ‰ฅ1, and define, for each ๐‘›โ‰ฅ1, ๐‘๐‘›=๐‘‹๐‘›โˆ’๐‘Œ๐‘›, ๐‘†๐‘›(1)=โˆ‘๐‘›๐‘–=1๐‘Œ๐‘–, and ๐‘†๐‘›(2)=โˆ‘๐‘›๐‘–=1๐‘๐‘–. It is easy to see that |๐‘Œ๐‘›|=min{|๐‘‹๐‘›|,๐‘›๐›ผ}, |๐‘๐‘›|=(|๐‘‹๐‘›|โˆ’๐‘›๐›ผ)๐ผ(|๐‘‹๐‘›|>๐‘›๐›ผ), and ||๐‘๐‘›||๐‘โ‰ค๎€ท||๐‘‹๐‘›||๐‘โˆ’๐‘›๐›ผ๎€ธ๐ผ๎€ท||๐‘‹๐‘›||๐‘>๐‘›๐›ผ๎€ธ(3.2) for all ๐‘>1. Note that, for each ๐‘›โ‰ฅ1, ๐‘Œ๐‘› and ๐‘๐‘› are monotone transformations of the initial variable ๐‘‹๐‘›. This implies that LNQD assumption is preserved by this construction in view of Lemma 2.6. Precisely, {๐‘Œ๐‘›โˆ’๐ธ๐‘Œ๐‘›}๐‘›โˆˆโ„• and {๐‘๐‘›โˆ’๐ธ๐‘๐‘›}๐‘›โˆˆโ„• are also LNQD sequences of zero mean random variables.
For our purpose, it suffices to prove ๐‘›โˆ’1max1โ‰ค๐‘–โ‰ค๐‘›||๐‘†๐‘–(1)โˆ’๐ธ๐‘†๐‘–(1)||โŸถ0in๐ฟ2,๐‘›(3.3)โˆ’1max1โ‰ค๐‘–โ‰ค๐‘›||๐‘†๐‘–(2)โˆ’๐ธ๐‘†๐‘–(2)||โŸถ0in๐ฟ๐‘.(3.4)
Using Lemma 2.8, the Hรถlder inequality, relation (3.2), and the second condition in (2.1) of the RCI(๐›ผ) property of the sequence {|๐‘‹๐‘›|๐‘}๐‘›โˆˆโ„•, we obtain ๐‘›โˆ’๐‘๐ธ๎‚ตmax1โ‰ค๐‘–โ‰ค๐‘›||๐‘†๐‘–(2)โˆ’๐ธ๐‘†๐‘–(2)||๐‘๎‚ถโ‰ช๐‘›โˆ’๐‘๎ƒฉ๐‘›๎“๐‘–=1๎€ท๐ธ||๐‘๐‘–โˆ’๐ธ๐‘๐‘–||๐‘๎€ธ1/๐‘๎ƒช๐‘โ‰ช๐‘›๐‘›โˆ’1๎“๐‘–=1๐ธ||๐‘๐‘–โˆ’๐ธ๐‘๐‘–||๐‘โ‰ช๐‘›๐‘›โˆ’1๎“๐‘–=1๐ธ||๐‘๐‘–||๐‘โ‰ค๐‘›๐‘›โˆ’1๎“๐‘–=1๐ธ๎€ท||๐‘‹๐‘–||๐‘โˆ’๐‘–๐›ผ๎€ธ๐ผ๎€ท||๐‘‹๐‘–||๐‘>๐‘–๐›ผ๎€ธโŸถ0.(3.5) This proves (3.4). To verify (3.3), using Lemma 2.7, we have ๐‘›โˆ’2๐ธ๎‚ตmax1โ‰ค๐‘–โ‰ค๐‘›||๐‘†๐‘–(1)โˆ’๐ธ๐‘†๐‘–(1)||2๎‚ถโ‰ช๐‘›โˆ’2๎ƒฉ๐‘›๎“๐‘–=1๐ธ๎€ท๐‘Œ๐‘–โˆ’๐ธ๐‘Œ๐‘–๎€ธ2๎ƒชโ‰ช๐‘›๐‘›โˆ’2๎“๐‘–=1๐ธ๐‘Œ2๐‘–โ‰ค๐‘›๐‘›โˆ’2+(2โˆ’๐‘)๐›ผ๎“๐‘–=1๐ธ||๐‘‹๐‘–||๐‘โ‰ค๐‘›โˆ’1+(2โˆ’๐‘)๐›ผโ‹…sup๐‘›โ‰ฅ1๎ƒฉ๐‘›๐‘›โˆ’1๎“๐‘–=1๐ธ||๐‘‹๐‘–||๐‘๎ƒช.(3.6) Using the first condition of (2.1) of the RCI(๐›ผ) property of the sequence {|๐‘‹๐‘›|๐‘}๐‘›โˆˆโ„•, the last expression above clearly goes to 0 as ๐‘›โ†’โˆž, from 1<๐‘<2 and ๐›ผ<1/(2โˆ’๐‘), thus completing the proof.

Remark 3.2. Let 1<๐‘<2, and let {๐‘‹๐‘›}๐‘›โˆˆโ„• be a LNQD sequence of random variables. If {|๐‘‹๐‘›|๐‘}๐‘›โˆˆโ„• is RCI(๐›ผ) for some ๐›ผโˆˆ(0,1/๐‘), then ๐‘›โˆ’1/๐‘(๐‘†๐‘›โˆ’๐ธ๐‘†๐‘›)โ†’0in๐ฟ๐‘.

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 1/๐‘› into 1/๐‘›1/๐‘.

Next we consider the case ๐‘โ‰ฅ2.

Theorem 3.3. Let ๐‘โ‰ฅ2, and let {๐‘‹๐‘›}๐‘›โˆˆโ„• be a LNQD sequence of random variables. If {๐‘‹๐‘›}๐‘›โˆˆโ„• satisfies sup๐‘›โ‰ฅ11๐‘›๐‘›๎“๐‘–=1๐ธ||๐‘‹๐‘–||p<โˆž,(3.7) then for any ๐›ฟ>1/2๐‘›โˆ’๐›ฟmax1โ‰ค๐‘–โ‰ค๐‘›||๐‘†๐‘–โˆ’๐ธ๐‘†๐‘–||โŸถ0in๐ฟ๐‘.(3.8)

Proof of Theorem 3.3. By Lemma 2.7 and the Hรถlder inequality, ๐ธ๎‚ต๐‘›โˆ’๐›ฟmax1โ‰ค๐‘–โ‰ค๐‘›||๐‘†๐‘–โˆ’๐ธ๐‘†๐‘–||๎‚ถ๐‘โ‰ช๐‘›โˆ’๐‘๐›ฟ๎ƒฉ๐‘›๎“๐‘–=1๎€ท๐ธ||๐‘‹๐‘–||๐‘๎€ธ2/๐‘๎ƒช๐‘/2โ‰ค๐‘›๐‘›โˆ’๐‘๐›ฟ+(๐‘/2)โˆ’1๎“๐‘–=1๐ธ||๐‘‹๐‘–||๐‘โ‰ค๐‘›โˆ’๐‘๐›ฟ+(๐‘/2)โ‹…sup๐‘›โ‰ฅ11๐‘›๐‘›๎“๐‘–=1๐ธ||๐‘‹๐‘–||๐‘โŸถ0.(3.9) The proof is completed.

4. Strongly Residual Cesรกro Alpha-Integrability and Complete Convergence of the Maximum of the Partial Sum

A sequence of random variables {๐‘‹๐‘›}๐‘›โˆˆโ„• is said to converge completely to a constant ๐‘Ž if for any ๐œ€>0, โˆž๎“๐‘›=1๐‘ƒ๎€ท||๐‘‹๐‘›||๎€ธโˆ’๐‘Ž>๐œ€<โˆž.(4.1) In this case we write ๐‘‹๐‘›โ†’๐‘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(๐›ผ) is a strong version of the condition of RCI(๐›ผ). In this section, we will show that each of the theorems in the previous section has a corresponding โ€œstrongโ€ analogue in the sense of complete convergence.

Theorem 4.1. Let 1<๐‘<2, and let {๐‘‹๐‘›}๐‘›โˆˆโ„• be a LNQD sequence of random variables. If {|๐‘‹๐‘›|๐‘}๐‘›โˆˆโ„• is SRCI(๐›ผ) for some ๐›ผโˆˆ(0,1/(2โˆ’๐‘)), then ๐‘›โˆ’1max1โ‰ค๐‘–โ‰ค๐‘›||๐‘†๐‘–โˆ’๐ธ๐‘†๐‘–||โŸถ0completely.(4.2)

Proof of Theorem 4.1. For any ๐‘›โ‰ฅ1, let ๐‘š=๐‘š๐‘› be the integer such that 2๐‘šโˆ’1<๐‘›โ‰ค2๐‘š. Observe that ๐‘›โˆ’1max1โ‰ค๐‘–โ‰ค๐‘›||๐‘†๐‘–โˆ’๐ธ๐‘†๐‘–||โ‰ค๐‘›โˆ’1max1โ‰ค๐‘–โ‰ค2๐‘š||๐‘†๐‘–โˆ’๐ธ๐‘†๐‘–||โ‰ค๎€ท2๐‘šโˆ’1๎€ธโˆ’1max1โ‰ค๐‘–โ‰ค2๐‘š||๐‘†๐‘–โˆ’๐ธ๐‘†๐‘–||=2โ‹…2โˆ’๐‘šmax1โ‰ค๐‘–โ‰ค2๐‘š||๐‘†๐‘–โˆ’๐ธ๐‘†๐‘–||.(4.3) Hence it suffices to show that 2โˆ’๐‘šmax1โ‰ค๐‘–โ‰ค2๐‘š||๐‘†๐‘–โˆ’๐ธ๐‘†๐‘–||โŸถ0completely.(4.4) Let ๐‘Œ๐‘›, ๐‘๐‘›, ๐‘†๐‘›(1), and ๐‘†๐‘›(2) be defined as in the proof of Theorem 3.1. We first prove that 2โˆ’๐‘šmax1โ‰ค๐‘–โ‰ค2๐‘š|๐‘†๐‘–(2)โˆ’๐ธ๐‘†๐‘–(2)|โ†’0 completely; that is, 2โˆ’๐‘šmax1โ‰ค๐‘–โ‰ค2๐‘š|||||๐‘–๎“๐‘˜=1๎€ท๐‘๐‘˜โˆ’๐ธ๐‘๐‘˜๎€ธ|||||โŸถ0completely.(4.5) Using Lemma 2.8, the Hรถlder inequality, relation (3.2), and the second condition in (2.1) of the RCI(๐›ผ) property of the sequence {|๐‘‹๐‘›|๐‘}๐‘›โˆˆโ„•, we have โˆž๎“๐‘š=0๐ธโŽ›โŽœโŽœโŽ2โˆ’๐‘šmax1โ‰ค๐‘–โ‰ค2๐‘š|||||๐‘–๎“๐‘˜=1๎€ท๐‘๐‘˜โˆ’๐ธ๐‘๐‘˜๎€ธ|||||โŽžโŽŸโŽŸโŽ ๐‘โ‰ชโˆž๎“๐‘š=02โˆ’๐‘š๐‘โŽ›โŽœโŽœโŽ2๐‘š๎“๐‘–=1๎€ท๐ธ||๐‘๐‘–||๐‘๎€ธ1/๐‘โŽžโŽŸโŽŸโŽ ๐‘โ‰คโˆž๎“๐‘š=022โˆ’๐‘š๐‘š๎“๐‘–=1๐ธ||๐‘๐‘–||๐‘=โˆž๎“๐‘–=1๐ธ||๐‘๐‘–||๐‘๎“๐‘šโˆถ2๐‘šโ‰ฅ๐‘–2โˆ’๐‘šโ‰คโˆž๎“๐‘–=1๐‘–โˆ’1๐ธ||๐‘๐‘–||๐‘โ‰คโˆž๎“๐‘–=1๐‘–โˆ’1๐ธ๎€ท||๐‘‹๐‘–||๐‘โˆ’๐‘–๐›ผ๎€ธ๐ผ๎€ท||๐‘‹๐‘–||๐‘>๐‘–๐›ผ๎€ธ<โˆž,(4.6) which implies (4.4).
Next we show that 2โˆ’๐‘šmax1โ‰ค๐‘–โ‰ค2๐‘š|๐‘†๐‘–(1)โˆ’๐ธ๐‘†๐‘–(1)|โ†’0 completely; that is, 2โˆ’๐‘šmax1โ‰ค๐‘–โ‰ค2๐‘š|||||๐‘–๎“๐‘˜=1๎€ท๐‘Œ๐‘˜โˆ’๐ธ๐‘Œ๐‘˜๎€ธ|||||โŸถ0completely.(4.7) By Lemma 2.7 and the Hรถlder inequality, โˆž๎“๐‘š=0๐ธ๎ƒฉ2โˆ’๐‘šmax1โ‰ค๐‘–โ‰ค2๐‘š๐‘–๎“๐‘˜=1๎€ท๐‘Œ๐‘˜โˆ’๐ธ๐‘Œ๐‘˜๎€ธ๎ƒช2โ‰ชโˆž๎“๐‘š=022โˆ’2๐‘š๐‘š๎“๐‘–=1๐ธ๐‘Œ2๐‘–โ‰คโˆž๎“๐‘š=022โˆ’2๐‘š๐‘š๎“๐‘–=1๐‘–(2โˆ’๐‘)๐›ผ๐ธ||๐‘‹๐‘–||๐‘.(4.8) In view of the first condition in (2.1) of the RCI(๐›ผ) property of the sequence {|๐‘‹๐‘›|๐‘}๐‘›โˆˆโ„•, we have โˆž๎“๐‘š=0๐ธ๎ƒฉ2โˆ’๐‘šmax1โ‰ค๐‘–โ‰ค2๐‘š๐‘–๎“๐‘˜=1๎€ท๐‘Œ๐‘˜โˆ’๐ธ๐‘Œ๐‘˜๎€ธ๎ƒช2โ‰คโˆž๎“๐‘š=022โˆ’2๐‘š๐‘š๎“๐‘–=1๐‘–(2โˆ’๐‘)๐›ผ๐ธ||๐‘‹๐‘–||๐‘โ‰คโˆž๎“๐‘š=02โˆ’2๐‘šโ‹…22๐‘š(2โˆ’๐‘)๐›ผ๐‘š๎“๐‘–=1๐ธ||๐‘‹๐‘–||๐‘โ‰คโˆž๎“๐‘š=02โˆ’๐‘šโ‹…2๐‘š(2โˆ’๐‘)๐›ผโ‹…sup๐‘šโˆถ2๐‘šโ‰ฅ๐‘–12๐‘š2๐‘š๎“๐‘–=1๐ธ||๐‘‹๐‘–||๐‘.(4.9) The last series above converges since ๐›ผโˆˆ(0,1/(2โˆ’๐‘)) implies โˆ’1+(2โˆ’๐‘)๐›ผ<0, and therefore (4.7) holds. This completes the proof.

For the case ๐‘โ‰ฅ2, we have the following result.

Theorem 4.2. Let ๐‘โ‰ฅ2, and let {๐‘‹๐‘›}๐‘›โˆˆโ„• be a LNQD sequence of random variables. If {๐‘‹๐‘›}๐‘›โˆˆโ„• satisfies sup๐‘›โ‰ฅ11๐‘›๐‘›๎“๐‘–=1๐ธ||๐‘‹๐‘–||๐‘<โˆž,(4.10) then for any ๐›ฟ>1/2๐‘›โˆ’๐›ฟmax1โ‰ค๐‘–โ‰ค๐‘›||๐‘†๐‘–โˆ’๐ธ๐‘†๐‘–||โŸถ0completely.(4.11)

Proof of Theorem 4.2. Let ๐‘š๐‘›,๐‘›โ‰ฅ1 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 2โˆ’๐‘š๐›ฟmax1โ‰ค๐‘–โ‰ค2๐‘š||๐‘†๐‘–โˆ’๐ธ๐‘†๐‘–||โŸถ0completely.(4.12) Indeed by Lemma 2.7 and the Hรถlder inequality, โˆž๎“๐‘š=0๐ธ๎‚ต2โˆ’๐‘š๐›ฟmax1โ‰ค๐‘–โ‰ค2๐‘š||๐‘†๐‘–โˆ’๐ธ๐‘†๐‘–||๎‚ถ๐‘โ‰ชโˆž๎“๐‘š=02โˆ’๐‘š๐‘๐›ฟโŽ›โŽœโŽœโŽ2๐‘š๎“๐‘–=1๎€ท๐ธ||๐‘‹๐‘–||๐‘๎€ธ2/๐‘โŽžโŽŸโŽŸโŽ ๐‘/2โ‰คโˆž๎“๐‘–=1๐ธ||๐‘‹๐‘–||๐‘๎“๐‘šโˆถ2๐‘šโ‰ฅ๐‘–2โˆ’๐‘š๐‘๐›ฟโˆ’๐‘š+2๐‘š/๐‘โ‰คโˆž๎“๐‘–=1๐‘–โˆ’๐‘๐›ฟโˆ’1+๐‘/2๐ธ||๐‘‹๐‘–||๐‘.(4.13) In view of Lemma 2.10, โˆž๎“๐‘–=1๐‘–โˆ’๐‘๐›ฟโˆ’1+๐‘/2๐ธ||๐‘‹๐‘–||๐‘โ‰คsup๐‘›โ‰ฅ11๐‘›2๐‘š๎“๐‘–=1๐ธ||๐‘‹๐‘–||๐‘โˆž๎“๐‘›=1๐‘›โˆ’๐‘๐›ฟโˆ’1+๐‘/2โ‰ชโˆž๎“๐‘›=1๐‘›โˆ’๐‘๐›ฟโˆ’1+๐‘/2<โˆž(4.14) from โˆ’๐‘๐›ฟโˆ’1+๐‘/2<โˆ’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.