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Journal of Probability and Statistics
VolumeΒ 2011, Article IDΒ 202015, 16 pages
http://dx.doi.org/10.1155/2011/202015
Research Article

Complete Convergence for Weighted Sums of Sequences of Negatively Dependent Random Variables

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

Received 30 September 2010; Accepted 21 January 2011

Academic Editor: A.Β Thavaneswaran

Copyright Β© 2011 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

Applying to the moment inequality of negatively dependent random variables the complete convergence for weighted sums of sequences of negatively dependent random variables is discussed. As a result, complete convergence theorems for negatively dependent sequences of random variables are extended.

1. Introduction and Lemmas

Definition 1.1. Random variables 𝑋 and π‘Œ are said to be negatively dependent (ND) if 𝑃(𝑋≀π‘₯,π‘Œβ‰€π‘¦)≀𝑃(𝑋≀π‘₯)𝑃(π‘Œβ‰€π‘¦)(1.1) for all π‘₯,π‘¦βˆˆR. A collection of random variables is said to be pairwise negatively dependent (PND) if every pair of random variables in the collection satisfies (1.1).
It is important to note that (1.1) implies that 𝑃(𝑋>π‘₯,π‘Œ>𝑦)≀𝑃(𝑋>π‘₯)𝑃(π‘Œ>𝑦)(1.2) for all π‘₯,π‘¦βˆˆR. Moreover, it follows that (1.2) implies (1.1), and, hence, (1.1) and (1.2) are equivalent. However, (1.1) and (1.2) are not equivalent for a collection of 3 or more random variables. Consequently, the following definition is needed to define sequences of negatively dependent random variables.

Definition 1.2. The random variables 𝑋1,…,𝑋𝑛 are said to be negatively dependent (ND) if, for all real π‘₯1,…,π‘₯𝑛, 𝑃𝑛𝑗=1𝑋𝑗≀π‘₯𝑗ξƒͺ≀𝑛𝑗=1𝑃𝑋𝑗≀π‘₯𝑗,𝑃𝑛𝑗=1𝑋𝑗>π‘₯𝑗ξƒͺ≀𝑛𝑗=1𝑃𝑋𝑗>π‘₯𝑗.(1.3) An infinite sequence of random variables {𝑋𝑛;𝑛β‰₯1} is said to be ND if every finite subset 𝑋1,…,𝑋𝑛 is ND.

Definition 1.3. Random variables 𝑋1,𝑋2,…,𝑋𝑛, 𝑛β‰₯2, are said to be negatively associated (NA) if, for every pair of disjoint subsets 𝐴1 and 𝐴2 of {1,2,…,𝑛}, 𝑓cov1𝑋𝑖;π‘–βˆˆπ΄1ξ€Έ,𝑓2𝑋𝑗;π‘—βˆˆπ΄2≀0,(1.4) where 𝑓1 and 𝑓2 are increasing in every variable (or decreasing in every variable), provided this covariance exists. A random variables sequence {𝑋𝑛;𝑛β‰₯1} is said to be NA if every finite subfamily is NA.
The definition of PND is given by Lehmann [1], the concept of ND is given by Bozorgnia et al. [2], and the definition of NA is introduced by Joag-Dev and Proschan [3]. These concepts of dependence random variables have been very useful in reliability theory and applications.
First, note that by letting 𝑓1(𝑋1,𝑋2,…,π‘‹π‘›βˆ’1)=𝐼(𝑋1≀π‘₯1,𝑋2≀π‘₯2,…,π‘‹π‘›βˆ’1≀π‘₯π‘›βˆ’1), 𝑓2(𝑋𝑛)=𝐼(𝑋𝑛≀π‘₯𝑛) and 𝑓1(𝑋1,𝑋2,…,π‘‹π‘›βˆ’1)=𝐼(𝑋1>π‘₯1,𝑋2>π‘₯2,…,π‘‹π‘›βˆ’1>π‘₯π‘›βˆ’1), 𝑓2(𝑋𝑛)=𝐼(𝑋𝑛>π‘₯𝑛), separately, it is easy to see that NA implies (1.3). Hence, NA implies ND. But there are many examples which are ND but are not NA. We list the following two examples.

Example 1.4. Let 𝑋𝑖 be a binary random variable such that 𝑃(𝑋𝑖=1)=𝑃(𝑋𝑖=0)=0.5 for 𝑖=1,2,3. Let (𝑋1,𝑋2,𝑋3) take the values (0,0,1), (0,1,0), (1,0,0), and (1,1,1), each with probability 1/4.
It can be verified that all the ND conditions hold. However, 𝑃𝑋1+𝑋3≀1,𝑋2ξ€Έ=4≀08𝑋≰𝑃1+𝑋3𝑃𝑋≀12ξ€Έ=3≀08.(1.5) Hence, 𝑋1, 𝑋2, and 𝑋3 are not NA.

In the next example 𝑋=(𝑋1,𝑋2,𝑋3,𝑋4) possesses ND, but does not possess NA obtained by Joag-Dev and Proschan [3].

Example 1.5. Let 𝑋𝑖 be a binary random variable such that 𝑃(𝑋𝑖=1)=.5 for 𝑖=1,2,3,4. Let (𝑋1,𝑋2) and (𝑋3,𝑋4) have the same bivariate distributions, and let 𝑋=(𝑋1,𝑋2,𝑋3,𝑋4) have joint distribution as shown in Table 1.
It can be verified that all the ND conditions hold. However, 𝑃𝑋𝑖𝑋=1,𝑖=1,2,3,4>𝑃1=𝑋2𝑃𝑋=13=𝑋4ξ€Έ=1,(1.6) violating NA.

tab1
Table 1

From the above examples, it is shown that ND does not imply NA and ND is much weaker than NA. In the papers listed earlier, a number of well-known multivariate distributions are shown to possess the ND properties, such as (a) multinomial, (b) convolution of unlike multinomials, (c) multivariate hypergeometric, (d) dirichlet, (e) dirichlet compound multinomial, and (f) multinomials having certain covariance matrices. Because of the wide applications of ND random variables, the notions of ND random variables have received more and more attention recently. A series of useful results have been established (cf. Bozorgnia et al. [2], Amini [4], Fakoor and Azarnoosh [5], Nili Sani et al. [6], Klesov et al. [7], and Wu and Jiang [8]). Hence, the extending of the limit properties of independent or NA random variables to the case of ND random variables is highly desirable and of considerable significance in the theory and application. In this paper we study and obtain some probability inequalities and some complete convergence theorems for weighted sums of sequences of negatively dependent random variables.

In the following, let π‘Žπ‘›β‰ͺ𝑏𝑛 (π‘Žπ‘›β‰«π‘π‘›) denote that there exists a constant 𝑐>0 such that π‘Žπ‘›β‰€π‘π‘π‘› (π‘Žπ‘›β‰₯𝑐𝑏𝑛) for sufficiently large 𝑛, and let π‘Žπ‘›β‰ˆπ‘π‘› mean π‘Žπ‘›β‰ͺ𝑏𝑛 and π‘Žπ‘›β‰«π‘π‘›. Also, let logπ‘₯ denote ln(max(e,π‘₯)) and π‘†π‘›βˆ‘ξ=𝑛𝑗=1𝑋𝑗.

Lemma 1.6 (see [2]). Let 𝑋1,…,𝑋𝑛 be ND random variables and {𝑓𝑛;𝑛β‰₯1} a sequence of Borel functions all of which are monotone increasing (or all are monotone decreasing). Then {𝑓𝑛(𝑋𝑛);𝑛β‰₯1} is still a sequence of ND r. v. ’s.

Lemma 1.7 (see [2]). Let 𝑋1,…,𝑋𝑛 be nonnegative r. v. ’s which are ND. Then 𝐸𝑛𝑗=1𝑋𝑗ξƒͺ≀𝑛𝑗=1𝐸𝑋𝑗.(1.7) In particular, let 𝑋1,…,𝑋𝑛 be ND, and let 𝑑1,…,𝑑𝑛 be all nonnegative (or non-positive) real numbers. Then 𝐸exp𝑛𝑗=1𝑑𝑗𝑋𝑗≀ξƒͺξƒͺ𝑛𝑗=1𝐸𝑑expj𝑋𝑗.(1.8)

Lemma 1.8. Let {𝑋𝑛;𝑛β‰₯1} be an ND sequence with 𝐸𝑋𝑛=0 and 𝐸|𝑋𝑛|𝑝<∞,𝑝β‰₯2. Then for 𝐡𝑛=βˆ‘π‘›π‘–=1𝐸𝑋2𝑖, 𝐸||𝑆𝑛||𝑝≀𝑐𝑝𝑛𝑖=1𝐸||𝑋𝑖||𝑝+𝐡𝑛𝑝/2ξƒ°,𝐸(1.9)max1≀𝑖≀𝑛||𝑆𝑖||𝑝≀c𝑝log𝑝𝑛𝑛𝑖=1𝐸||𝑋𝑖||𝑝+𝐡𝑛𝑝/2ξƒ°,(1.10) where 𝑐𝑝>0 depends only on 𝑝.

Remark 1.9. If {𝑋𝑛;𝑛β‰₯1} is a sequence of independent random variables, then (1.9) is the classic Rosenthal inequality [9]. Therefore, (1.9) is a generalization of the Rosenthal inequality.

Proof of Lemma 1.8. Let π‘Ž>0, π‘‹ξ…žπ‘–=min(𝑋𝑖,π‘Ž),andπ‘†ξ…žπ‘›=βˆ‘π‘›π‘–=1π‘‹ξ…žπ‘–. It is easy to show that {π‘‹ξ…žπ‘–;𝑖β‰₯1} is a negatively dependent sequence by Lemma 1.6. Noting that (eπ‘₯βˆ’1βˆ’π‘₯)/π‘₯2 is a nondecreasing function of π‘₯ on R and that πΈπ‘‹ξ…žπ‘–β‰€πΈπ‘‹π‘–=0, π‘‘π‘‹ξ…žπ‘–β‰€π‘‘π‘Ž, we have 𝐸e𝑑𝑋′𝑖=1+π‘‘πΈπ‘‹ξ…žπ‘–ξƒ©e+πΈπ‘‘π‘‹β€²π‘–βˆ’1βˆ’π‘‘π‘‹ξ…žπ‘–π‘‘2π‘‹π‘–ξ…ž2𝑑2π‘‹π‘–ξ…ž2ξƒͺξ€·e≀1+π‘‘π‘Žξ€Έπ‘Žβˆ’1βˆ’π‘‘π‘Žβˆ’2πΈπ‘‹π‘–ξ…ž2ξ€·e≀1+π‘‘π‘Žξ€Έπ‘Žβˆ’1βˆ’π‘‘π‘Žβˆ’2𝐸𝑋2𝑖e≀expξ€½ξ€·π‘‘π‘Žξ€Έπ‘Žβˆ’1βˆ’π‘‘π‘Žβˆ’2𝐸𝑋2𝑖.(1.11) Here the last inequality follows from 1+π‘₯≀eπ‘₯, for all π‘₯∈R.
Note that 𝐡𝑛=βˆ‘π‘›π‘–=1𝐸𝑋2𝑖 and {π‘‹ξ…žπ‘–;𝑖β‰₯1} is ND, we conclude from the above inequality and Lemma 1.7 that, for any π‘₯>0 and β„Ž>0, we get eβˆ’β„Žπ‘₯𝐸eβ„Žπ‘†β€²π‘›ξ‚=eβˆ’β„Žπ‘₯𝐸𝑛𝑖=1eβ„Žπ‘‹β€²π‘–ξƒͺ≀eπ‘›βˆ’β„Žπ‘₯𝑖=1𝐸eβ„Žπ‘‹β€²π‘–ξ‚ξ€½ξ€·e≀expβˆ’β„Žπ‘₯+β„Žπ‘Žξ€Έπ‘Žβˆ’1βˆ’β„Žπ‘Žβˆ’2𝐡𝑛.(1.12)
Letting β„Ž=ln((π‘₯π‘Ž)/𝐡𝑛+1)/π‘Ž>0, we get ξ€·eβ„Žπ‘Žξ€Έπ‘Žβˆ’1βˆ’β„Žπ‘Žβˆ’2𝐡𝑛=π‘₯π‘Žβˆ’π΅π‘›π‘Ž2ξ‚΅lnπ‘₯π‘Žπ΅π‘›ξ‚Άβ‰€π‘₯+1π‘Ž.(1.13) Putting this one into (1.12), we get furthermore eβˆ’β„Žπ‘₯𝐸eβ„Žπ‘†β€²π‘›ξ‚ξ‚»π‘₯≀expπ‘Žβˆ’π‘₯π‘Žξ‚΅lnπ‘₯π‘Žπ΅π‘›+1ξ‚Άξ‚Ό.(1.14) Putting π‘₯/π‘Ž=𝑑 into the above inequality, we get 𝑃𝑆𝑛≀β‰₯π‘₯𝑛𝑖=1𝑃𝑋𝑖𝑆>π‘Ž+π‘ƒξ…žπ‘›ξ€Έβ‰€β‰₯π‘₯𝑛𝑖=1𝑃𝑋𝑖>π‘Ž+eβˆ’β„Žπ‘₯𝐸eβ„Žπ‘†β€²π‘›β‰€π‘›ξ“π‘–=1𝑃𝑋𝑖>π‘₯𝑑π‘₯+expπ‘‘βˆ’π‘‘ln2𝑑𝐡𝑛=+1𝑛𝑖=1𝑃𝑋𝑖>π‘₯𝑑+e𝑑π‘₯1+2π‘‘π΅π‘›ξ‚Άβˆ’π‘‘.(1.15) Letting βˆ’π‘‹π‘– take the place of 𝑋𝑖 in the above inequality, we can get π‘ƒξ€·βˆ’π‘†π‘›ξ€Έξ€·π‘†β‰₯π‘₯=π‘ƒπ‘›ξ€Έβ‰€β‰€βˆ’π‘₯𝑛𝑖=1π‘ƒξ‚€βˆ’π‘‹π‘–>π‘₯𝑑+e𝑑π‘₯1+2π‘‘π΅π‘›ξ‚Άβˆ’π‘‘=𝑛𝑖=1𝑃𝑋𝑖<βˆ’π‘₯𝑑+e𝑑π‘₯1+2π‘‘π΅π‘›ξ‚Άβˆ’π‘‘.(1.16) Thus 𝑃||𝑆𝑛||𝑆β‰₯π‘₯=𝑃𝑛𝑆β‰₯π‘₯+π‘ƒπ‘›ξ€Έβ‰€β‰€βˆ’π‘₯𝑛𝑖=1𝑃||𝑋𝑖||<π‘₯𝑑+2e𝑑π‘₯1+2π‘‘π΅π‘›ξ‚Άβˆ’π‘‘.(1.17)
Multiplying (1.17) by 𝑝π‘₯π‘βˆ’1, letting 𝑑=𝑝, and integrating over 0<π‘₯<+∞, according to 𝐸||𝑋||π‘ξ€œ=𝑝0+∞π‘₯π‘βˆ’1𝑃||𝑋||ξ€Έβ‰₯π‘₯dπ‘₯,(1.18) we obtain 𝐸||𝑆𝑛||π‘ξ€œ=𝑝0+∞π‘₯π‘βˆ’1𝑃||𝑆𝑛||ξ€Έβ‰₯π‘₯dπ‘₯≀𝑝𝑛𝑖=1ξ€œ0+∞π‘₯π‘βˆ’1𝑃||𝑋𝑖||β‰₯π‘₯𝑝dπ‘₯+2𝑝eπ‘ξ€œ0+∞π‘₯π‘βˆ’1ξ‚΅π‘₯1+2π‘π΅π‘›ξ‚Άβˆ’π‘dπ‘₯=𝑝𝑛𝑝+1𝑖=1𝐸||𝑋𝑖||𝑝+𝑝e𝑝𝑝𝐡𝑛𝑝/2ξ€œ0+βˆžπ‘’π‘/2βˆ’1(1+𝑒)𝑝d𝑒=𝑝𝑛𝑝+1𝑖=1𝐸||𝑋𝑖||𝑝+𝑝𝑝/2+1e𝑝𝐡𝑝2,𝑝2𝐡𝑛𝑝/2,(1.19) where ∫𝐡(𝛼,𝛽)=10π‘₯π›Όβˆ’1(1βˆ’π‘₯)π›½βˆ’1∫dπ‘₯=0+∞π‘₯π›Όβˆ’1(1+π‘₯)βˆ’(𝛼+𝛽)dπ‘₯,𝛼,𝛽>0 is Beta function. Letting 𝑐𝑝=max(𝑝𝑝+1,𝑝1+𝑝/2e𝑝𝐡(𝑝/2,𝑝/2)),we can deduce (1.9) from (1.19). From (1.9), we can prove (1.10) by a similar way of Stout's paper [10, Theorem  2.3.1].

Lemma 1.10. Let {𝑋𝑛;𝑛β‰₯1} be a sequence of ND random variables. Then there exists a positive constant 𝑐 such that, for any π‘₯β‰₯0 and all 𝑛β‰₯1, ξ‚΅ξ‚΅1βˆ’π‘ƒmax1β‰€π‘˜β‰€π‘›||π‘‹π‘˜||>π‘₯ξ‚Άξ‚Ά2π‘›ξ“π‘˜=1𝑃||π‘‹π‘˜||ξ€Έξ‚΅>π‘₯≀𝑐𝑃max1β‰€π‘˜β‰€π‘›||π‘‹π‘˜||ξ‚Ά.>π‘₯(1.20)

Proof. Let π΄π‘˜=(|π‘‹π‘˜|>π‘₯) and 𝛼𝑛⋃=1βˆ’π‘ƒ(π‘›π‘˜=1π΄π‘˜)=1βˆ’π‘ƒ(max1β‰€π‘˜β‰€π‘›|π‘‹π‘˜|>π‘₯). Without loss of generality, assume that 𝛼𝑛>0. Note that {𝐼(π‘‹π‘˜>π‘₯)βˆ’πΈπΌ(π‘‹π‘˜>π‘₯);π‘˜β‰₯1} and {𝐼(π‘‹π‘˜<βˆ’π‘₯)βˆ’πΈπΌ(π‘‹π‘˜<βˆ’π‘₯);π‘˜β‰₯1} are still ND by Lemma 1.6. Using (1.9), we get πΈξƒ©π‘›ξ“π‘˜=1ξ€·πΌπ΄π‘˜βˆ’πΈπΌπ΄π‘˜ξ€Έξƒͺ2=πΈπ‘›ξ“π‘˜=1𝐼(π‘‹π‘˜>π‘₯)βˆ’πΈπΌ(π‘‹π‘˜>π‘₯)ξ€Έ+𝐼(π‘‹π‘˜<βˆ’π‘₯)βˆ’πΈπΌ(π‘‹π‘˜<βˆ’π‘₯)ξ€Έξƒͺ2≀2πΈπ‘›ξ“π‘˜=1𝐼(π‘‹π‘˜>π‘₯)βˆ’πΈπΌ(π‘‹π‘˜>π‘₯)ξ€Έξƒͺ2+2πΈπ‘›ξ“π‘˜=1𝐼(π‘‹π‘˜<βˆ’π‘₯)βˆ’πΈπΌ(π‘‹π‘˜<βˆ’π‘₯)ξ€Έξƒͺ2β‰€π‘π‘›ξ“π‘˜=1π‘ƒξ€·π΄π‘˜ξ€Έ.(1.21) Combining with the Cauchy-Schwarz inequality, we obtain π‘›ξ“π‘˜=1π‘ƒξ€·π΄π‘˜ξ€Έ=π‘›ξ“π‘˜=1π‘ƒξƒ©π΄π‘˜,π‘›ξšπ‘—=1𝐴𝑗ξƒͺ=π‘›ξ“π‘˜=1πΈξ‚€πΌπ΄π‘˜πΌβ‹ƒπ‘›π‘—=1𝐴𝑗=πΈπ‘›ξ“π‘˜=1ξ€·πΌπ΄π‘˜βˆ’πΈπΌπ΄π‘˜ξ€Έξƒͺ𝐼⋃𝑛𝑗=1𝐴𝑗+π‘›ξ“π‘˜=1π‘ƒξ€·π΄π‘˜ξ€Έπ‘ƒξƒ©π‘›ξšπ‘—=1𝐴𝑗ξƒͺβ‰€βŽ›βŽœβŽœβŽπΈξƒ©π‘›ξ“π‘˜=1ξ€·πΌπ΄π‘˜βˆ’πΈπΌπ΄π‘˜ξ€Έξƒͺ2𝐸𝐼⋃𝑛𝑗=1π΄π‘—βŽžβŽŸβŽŸβŽ 1/2+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘›ξ“π‘˜=1π‘ƒξ€·π΄π‘˜ξ€Έβ‰€ξƒ©π‘ξ€·1βˆ’π›Όπ‘›ξ€Έπ›Όπ‘›π›Όπ‘›π‘›ξ“π‘˜=1π‘ƒξ€·π΄π‘˜ξ€Έξƒͺ1/2+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘›ξ“π‘˜=1π‘ƒξ€·π΄π‘˜ξ€Έβ‰€12𝑐1βˆ’π›Όπ‘›ξ€Έπ›Όπ‘›+π›Όπ‘›π‘›ξ“π‘˜=1π‘ƒξ€·π΄π‘˜ξ€Έξƒͺ+ξ€·1βˆ’π›Όπ‘›ξ€Έπ‘›ξ“π‘˜=1π‘ƒξ€·π΄π‘˜ξ€Έ.(1.22) Thus 𝛼2π‘›π‘›ξ“π‘˜=1π‘ƒξ€·π΄π‘˜ξ€Έξ€·β‰€π‘1βˆ’π›Όπ‘›ξ€Έ,(1.23) that is, ξ‚΅ξ‚΅1βˆ’π‘ƒmax1β‰€π‘˜β‰€π‘›||π‘‹π‘˜||>π‘₯ξ‚Άξ‚Ά2π‘›ξ“π‘˜=1𝑃||π‘‹π‘˜||ξ€Έξ‚΅>π‘₯≀𝑐𝑃max1β‰€π‘˜β‰€π‘›||π‘‹π‘˜||ξ‚Ά.>π‘₯(1.24)

2. Main Results and the Proofs

The concept of complete convergence of a sequence of random variables was introduced by Hsu and Robbins [11] as follows. A sequence {π‘Œπ‘›;𝑛β‰₯1} of random variables converges completely to the constant 𝑐 if βˆ‘βˆžπ‘›=1𝑃(|π‘‹π‘›βˆ’π‘|>πœ€)<∞, for all πœ€>0. In view of the Borel-Cantelli lemma, this implies that π‘Œπ‘›β†’0 almost surely. Therefore, complete convergence is one of the most important problems in probability theory. Hsu and Robbins [11] proved that the sequence of arithmetic means of independent and identically distributed (i.i.d.) random variables converges completely to the expected value if the variance of the summands is finite. Baum and Katz [12] proved that if {𝑋,𝑋𝑛;𝑛β‰₯1} is a sequence of i.i.d. random variables with mean zero, then 𝐸|𝑋|𝑝(𝑑+2)<∞(1≀𝑝<2,𝑑β‰₯βˆ’1) is equivalent to the condition that βˆ‘βˆžπ‘›=1π‘›π‘‘βˆ‘π‘ƒ(𝑛1=1|𝑋𝑖|/𝑛1/𝑝>πœ€)<∞, for all πœ€>0. Recent results of the complete convergence can be found in Li et al. [13], Liang and Su [14], Wu [15, 16], and Sung [17].

In this paper we study the complete convergence for negatively dependent random variables. As a result, we extend some complete convergence theorems for independent random variables to the negatively dependent random variables without necessarily imposing any extra conditions.

Theorem 2.1. Let {𝑋,𝑋𝑛;𝑛β‰₯1} be a sequence of identically distributed ND random variables and {π‘Žπ‘›π‘˜;1β‰€π‘˜β‰€π‘›,𝑛β‰₯1} an array of real numbers, and let π‘Ÿ>1, 𝑝>2. If, for some 2β‰€π‘ž<𝑝, ξ€½||π‘Žπ‘(𝑛,π‘š+1)=β™―π‘˜β‰₯1;π‘›π‘˜||β‰₯(π‘š+1)βˆ’1/π‘ξ€Ύβ‰ˆπ‘šπ‘ž(π‘Ÿβˆ’1)/𝑝,𝑛,π‘šβ‰₯1,(2.1)𝐸𝑋=0for1β‰€π‘ž(π‘Ÿβˆ’1),(2.2)π‘›ξ“π‘˜=1π‘Ž2π‘›π‘˜β‰ͺ𝑛𝛿2for2β‰€π‘ž(π‘Ÿβˆ’1)andsome0<𝛿<𝑝,(2.3) then, for π‘Ÿβ‰₯2, 𝐸||𝑋||𝑝(π‘Ÿβˆ’1)<∞(2.4) if and only if βˆžξ“π‘›=1π‘›π‘Ÿβˆ’2π‘ƒβŽ›βŽœβŽœβŽmax1β‰€π‘˜β‰€π‘›|||||π‘˜ξ“π‘–=1π‘Žπ‘›π‘–π‘‹π‘–|||||>πœ€π‘›1/π‘βŽžβŽŸβŽŸβŽ <∞,βˆ€πœ€>0.(2.5) For 1<π‘Ÿ<2, (2.4) implies (2.5), conversely, and (2.5) and π‘›π‘Ÿβˆ’2𝑃(max1β‰€π‘˜β‰€π‘›|π‘Žπ‘›π‘˜π‘‹π‘˜|>𝑛1/𝑝) decreasing on 𝑛 imply (2.4).

For 𝑝=2, π‘ž=2, we have the following theorem.

Theorem 2.2. Let {𝑋,𝑋𝑛;𝑛β‰₯1} be a sequence of identically distributed ND random variables and {π‘Žπ‘›π‘˜;1β‰€π‘˜β‰€π‘›,𝑛β‰₯1} an array of real numbers, and let π‘Ÿ>1. If ξ€½||π‘Žπ‘(𝑛,π‘š+1)=β™―π‘˜;π‘›π‘˜||β‰₯(π‘š+1)βˆ’1/2ξ€Ύβ‰ˆπ‘šπ‘Ÿβˆ’1,𝑛,π‘šβ‰₯1,(2.6)𝐸𝑋=0,1≀2(π‘Ÿβˆ’1),π‘›ξ“π‘˜=1||π‘Žπ‘›π‘˜||2(π‘Ÿβˆ’1)=𝑂(1),(2.7) then, for π‘Ÿβ‰₯2, 𝐸||𝑋||2(π‘Ÿβˆ’1)||𝑋||log<∞(2.8) if and only if βˆžξ“π‘›=1π‘›π‘Ÿβˆ’2π‘ƒβŽ›βŽœβŽœβŽmax1β‰€π‘˜β‰€π‘›|||||π‘˜ξ“π‘–=1π‘Žπ‘›π‘–π‘‹π‘–|||||>πœ€π‘›1/2⎞⎟⎟⎠<∞,βˆ€πœ€>0.(2.9) For 1<π‘Ÿ<2, (2.8) implies (2.9), conversely, and (2.9) and π‘›π‘Ÿβˆ’2𝑃(max1β‰€π‘˜β‰€π‘›|π‘Žπ‘›π‘˜π‘‹π‘˜|>𝑛1/2) decreasing on 𝑛 imply (2.8).

Remark 2.3. Since NA random variables are a special case of ND r. v. 's, Theorems 2.1 and 2.2 extend the work of Liang and Su [14, Theorem  2.1].

Remark 2.4. Since, for some 2β‰€π‘žβ‰€π‘, βˆ‘π‘˜βˆˆπ‘|π‘Žπ‘›π‘˜|π‘ž(π‘Ÿβˆ’1)β‰ͺ1 as π‘›β†’βˆž implies that ξ€½||π‘Žπ‘(𝑛,π‘š+1)=β™―π‘˜β‰₯1;π‘›π‘˜||β‰₯(π‘š+1)βˆ’1/𝑝β‰ͺπ‘šπ‘ž(π‘Ÿβˆ’1)/𝑝asπ‘›βŸΆβˆž,(2.10) taking π‘Ÿ=2, then conditions (2.1) and (2.6) are weaker than conditions (2.13) and (2.9) in Li et al. [13]. Therefore, Theorems 2.1 and 2.2 not only promote and improve the work of Li et al. [13, Theorem 2.2] for i.i.d. random variables to an ND setting but also obtain their necessities and relax the range of π‘Ÿ.

Proof of Theorem 2.1. Equation (2.4)β‡’(2.5). To prove (2.5) it suffices to show that βˆžξ“π‘›=1π‘›π‘Ÿβˆ’2π‘ƒβŽ›βŽœβŽœβŽmax1β‰€π‘˜β‰€π‘›|||||π‘˜ξ“π‘–=1π‘ŽΒ±π‘›π‘–π‘‹π‘–|||||>πœ€π‘›1/π‘βŽžβŽŸβŽŸβŽ <∞,βˆ€πœ€>0,(2.11) where π‘Ž+𝑛𝑖=max(π‘Žπ‘›π‘–,0) and π‘Žβˆ’π‘›π‘–=max(βˆ’π‘Žπ‘›π‘–,0). Thus, without loss of generality, we can assume that π‘Žπ‘›π‘–>0 for all 𝑛β‰₯1,𝑖≀𝑛. For 0<𝛼<1/𝑝 small enough and sufficiently large integer 𝐾, which will be determined later, let 𝑋(1)𝑛𝑖=βˆ’π‘›π›ΌπΌ(π‘Žπ‘›π‘–π‘‹π‘–<βˆ’π‘›π›Ό)+π‘Žπ‘›π‘–π‘‹π‘–πΌ(π‘Žπ‘›π‘–|𝑋𝑖|≀𝑛𝛼)+𝑛𝛼𝐼(π‘Žπ‘›π‘–π‘‹π‘–>𝑛𝛼),𝑋(2)𝑛𝑖=ξ€·π‘Žπ‘›π‘–π‘‹π‘–βˆ’π‘›π›Όξ€ΈπΌ(𝑛𝛼<π‘Žπ‘›π‘–π‘‹π‘–<πœ€π‘›1/𝑝/𝐾),𝑋(3)𝑛𝑖=ξ€·π‘Žπ‘›π‘–π‘‹π‘–+𝑛𝛼𝐼(βˆ’πœ€π‘›1/𝑝/𝐾<π‘Žπ‘›π‘–π‘‹π‘–<βˆ’π‘›π›Ό),𝑋(4)𝑛𝑖=π‘Žπ‘›π‘–π‘‹π‘›π‘–βˆ’π‘‹(1)π‘›π‘–βˆ’π‘‹(2)π‘›π‘–βˆ’π‘‹(3)𝑛𝑖=ξ€·π‘Žπ‘›π‘–π‘‹π‘–+𝑛𝛼𝐼(π‘Žπ‘›π‘–π‘‹π‘–β‰€βˆ’πœ€π‘›1/𝑝/𝐾)+ξ€·π‘Žπ‘›π‘–π‘‹π‘–βˆ’π‘›π›Όξ€ΈπΌ(π‘Žπ‘›π‘–π‘‹π‘–β‰₯πœ€π‘›1/𝑝/𝐾),𝑆(𝑗)π‘›π‘˜=π‘˜ξ“π‘–=1𝑋(𝑗)𝑛𝑖,𝑗=1,2,3,4;1β‰€π‘˜β‰€π‘›,𝑛β‰₯1.(2.12) Thus π‘†π‘›π‘˜βˆ‘ξ=π‘˜π‘–=1π‘Žπ‘›π‘–π‘‹π‘–=βˆ‘4𝑗=1𝑆(𝑗)π‘›π‘˜. Note that ξ‚΅max1β‰€π‘˜β‰€π‘›||π‘†π‘›π‘˜||>4πœ€π‘›1/π‘ξ‚ΆβŠ†4ξšπ‘—=1ξ‚΅max1β‰€π‘˜β‰€π‘›||𝑆(𝑗)π‘›π‘˜||>πœ€π‘›1/𝑝.(2.13) So, to prove (2.5) it suffices to show that 𝐼𝑗=βˆžξ“π‘›=1π‘›π‘Ÿβˆ’2𝑃max1β‰€π‘˜β‰€π‘›||𝑆(𝑗)π‘›π‘˜||>πœ€π‘›1/𝑝<∞,𝑗=1,2,3,4.(2.14) For any π‘žξ…ž>π‘ž, 𝑛𝑖=1π‘Žπ‘žβ€²(π‘Ÿβˆ’1)𝑛𝑖=βˆžξ“π‘—=1(𝑗+1)βˆ’1β‰€π‘Žπ‘π‘›π‘–<π‘—βˆ’1π‘Žπ‘žβ€²(π‘Ÿβˆ’1)π‘›π‘–β‰€βˆžξ“π‘—=1(𝑗+1)βˆ’1β‰€π‘Žπ‘π‘›π‘–<π‘—βˆ’1π‘—βˆ’π‘žβ€²(π‘Ÿβˆ’1)/𝑝β‰ͺβˆžξ“π‘—=1(𝑁(𝑛,𝑗+1)βˆ’π‘(𝑛,𝑗))π‘—βˆ’π‘žβ€²(π‘Ÿβˆ’1)/𝑝β‰ͺβˆžξ“π‘—=1𝑁𝑗(𝑛,𝑗)βˆ’π‘žβ€²(π‘Ÿβˆ’1)/π‘βˆ’(𝑗+1)βˆ’π‘žβ€²(π‘Ÿβˆ’1)/𝑝β‰ͺβˆžξ“π‘—=1π‘—βˆ’1βˆ’(π‘žβ€²βˆ’π‘ž)(π‘Ÿβˆ’1)/𝑝<∞.(2.15)
Now, we prove that π‘›βˆ’1/𝑝max1β‰€π‘˜β‰€π‘›||𝐸𝑆(1)π‘›π‘˜||⟢0,π‘›βŸΆβˆž.(2.16)
(i) For 0<π‘ž(π‘Ÿβˆ’1)<1, taking π‘ž<π‘žξ…ž<𝑝 such that 0<π‘žξ…ž(π‘Ÿβˆ’1)<1, by (2.4) and (2.15), we get π‘›βˆ’1/𝑝max1β‰€π‘˜β‰€π‘›||𝐸𝑆(1)π‘›π‘˜||β‰€π‘›π‘›βˆ’1/𝑝𝑖=1𝐸||π‘Žπ‘›π‘–π‘‹π‘–||𝐼(|π‘Žπ‘›π‘–π‘‹π‘–|≀𝑛𝛼)+𝑛𝛼𝑃||π‘Žπ‘›π‘–π‘‹π‘–||>π‘›π›Όξ€Έξ€Έβ‰€π‘›βˆ’1/𝑝𝑛𝑖=1𝐸||π‘Žπ‘›π‘–π‘‹π‘–||π‘žβ€²(π‘Ÿβˆ’1)||π‘Žπ‘›π‘–π‘‹π‘–||1βˆ’π‘žβ€²(π‘Ÿβˆ’1)𝐼(|π‘Žπ‘›π‘–π‘‹π‘–|≀𝑛𝛼)+π‘›π›Όβˆ’π›Όπ‘žβ€²π‘›(π‘Ÿβˆ’1)𝑖=1𝐸||π‘Žπ‘›π‘–π‘‹π‘–||π‘žβ€²(π‘Ÿβˆ’1)ξƒͺβ‰ͺπ‘›βˆ’1/𝑝+π›Όβˆ’π›Όπ‘žβ€²(π‘Ÿβˆ’1)⟢0,π‘›βŸΆβˆž.(2.17) For 1β‰€π‘ž(π‘Ÿβˆ’1), letting π‘ž<π‘žξ…ž<𝑝, by (2.2), (2.4), and (2.15), we get π‘›βˆ’1/𝑝max1β‰€π‘˜β‰€π‘›||𝐸𝑆(1)π‘›π‘˜||β‰€π‘›π‘›βˆ’1/𝑝𝑖=1𝐸||π‘Žπ‘›π‘–π‘‹π‘–||𝐼(|π‘Žπ‘›π‘–π‘‹π‘–|>𝑛𝛼)+𝑛𝛼𝑃||π‘Žπ‘›π‘–π‘‹π‘–||>π‘›π›Όξ€Έξ€Έβ‰€π‘›π‘›βˆ’1/𝑝𝑖=1𝐸||π‘Žπ‘›π‘–π‘‹π‘–||ξ‚΅||π‘Žπ‘›π‘–π‘‹π‘–||π‘›π›Όξ‚Άπ‘žβ€²(π‘Ÿβˆ’1)βˆ’1𝐼(|π‘Žπ‘›π‘–π‘‹π‘–|≀𝑛𝛼)+π‘›π›Όβˆ’π›Όπ‘žβ€²(π‘Ÿβˆ’1)𝐸||π‘Žπ‘›π‘–π‘‹π‘–||π‘žβ€²(π‘Ÿβˆ’1)ξƒͺβ‰ͺπ‘›βˆ’1/𝑝+π›Όβˆ’π›Όπ‘žβ€²(π‘Ÿβˆ’1)⟢0.(2.18) Hence, (2.16) holds. Therefore, to prove 𝐼1<∞ it suffices to prove that 𝐼1=βˆžξ“π‘›=1π‘›π‘Ÿβˆ’2𝑃max1β‰€π‘˜β‰€π‘›||𝑆(1)π‘›π‘˜βˆ’πΈπ‘†(1)π‘›π‘˜||>πœ€π‘›1/𝑝<∞,βˆ€πœ€>0.(2.19) Note that {𝑋(1)𝑛𝑖;1≀𝑖≀𝑛,𝑛β‰₯1} is still ND by the definition of 𝑋(1)𝑛𝑖 and Lemma 1.6. Using the Markov inequality and Lemma 1.8, we get for a suitably large 𝑀, which will be determined later, 𝐼1β‰ͺβˆžξ“π‘›=1π‘›π‘Ÿβˆ’2βˆ’π‘€/𝑝𝐸max1β‰€π‘˜β‰€π‘›||𝑆(1)π‘›π‘˜βˆ’πΈπ‘†(1)π‘›π‘˜||𝑀β‰ͺβˆžξ“π‘›=1π‘›π‘Ÿβˆ’2βˆ’π‘€/𝑝logπ‘€π‘›βŽ‘βŽ’βŽ’βŽ£π‘›ξ“π‘–=1𝐸||𝑋(1)𝑛𝑖||𝑀+𝑛𝑖=1𝐸𝑋(1)𝑛𝑖2ξƒͺ𝑀/2⎀βŽ₯βŽ₯βŽ¦ξ‚πΌξ=11+𝐼12.(2.20) Taking 𝑀>max(2,𝑝(π‘Ÿβˆ’1)(1βˆ’π›Όπ‘žξ…ž)/(1βˆ’π›Όπ‘)), then π‘Ÿβˆ’2βˆ’π‘€/𝑝+π›Όπ‘€βˆ’π›Όπ‘žξ…ž(π‘Ÿβˆ’1)<βˆ’1, and, by (2.15), we get 𝐼11β‰€βˆžξ“π‘›=1π‘›π‘Ÿβˆ’2βˆ’π‘€/𝑝log𝑀𝑛𝑛𝑖=1𝐸||π‘Žπ‘›π‘–π‘‹π‘–||𝑀𝐼(|π‘Žπ‘›π‘–π‘‹π‘–|≀𝑛𝛼)+𝑛𝑀𝛼𝑃||π‘Žπ‘›π‘–π‘‹π‘–||>π‘›π›Όξ€Έξ‚β‰€βˆžξ“π‘›=1π‘›π‘Ÿβˆ’2βˆ’π‘€/𝑝log𝑀𝑛𝑛𝑖=1𝐸||π‘Žπ‘›π‘–π‘‹π‘–||π‘žβ€²(π‘Ÿβˆ’1)𝑛𝛼(π‘€βˆ’π‘žβ€²(π‘Ÿβˆ’1))+𝑛𝛼(π‘€βˆ’π‘žβ€²(π‘Ÿβˆ’1))𝐸||π‘Žπ‘›π‘–π‘‹π‘–||π‘žβ€²(π‘Ÿβˆ’1)β‰ͺβˆžξ“π‘›=1π‘›π‘Ÿβˆ’2βˆ’π‘€/𝑝+π›Όπ‘€βˆ’π›Όπ‘žβ€²(π‘Ÿβˆ’1)log𝑀𝑛<∞.(2.21)
(i) For π‘ž(π‘Ÿβˆ’1)<2, taking π‘ž<π‘žξ…ž<𝑝 such that π‘žξ…ž(π‘Ÿβˆ’1)<2 and taking 𝑀>max(2,2𝑝(π‘Ÿβˆ’1)/(2βˆ’2𝛼𝑝+π›Όπ‘π‘žξ…ž(π‘Ÿβˆ’1))), from (2.15) and π‘Ÿβˆ’2βˆ’π‘€/𝑝+π›Όπ‘€βˆ’π‘€π›Όπ‘žξ…ž(π‘Ÿβˆ’1)/2<βˆ’1, we have 𝐼12β‰€βˆžξ“π‘›=1π‘›π‘Ÿβˆ’2βˆ’π‘€/𝑝log𝑀𝑛𝑛𝑖=1𝐸||π‘Žπ‘›π‘–π‘‹π‘–||π‘žβ€²(π‘Ÿβˆ’1)𝑛𝛼(2βˆ’π‘žβ€²(π‘Ÿβˆ’1))𝐼(|π‘Žπ‘›π‘–π‘‹π‘–|≀𝑛𝛼)+𝑛2π›Όβˆ’π›Όπ‘žβ€²(π‘Ÿβˆ’1)𝐸||π‘Žπ‘›π‘–π‘‹π‘–||π‘žβ€²(π‘Ÿβˆ’1)𝑀/2β‰ͺβˆžξ“π‘›=1π‘›βˆ’π‘Ÿβˆ’2βˆ’π‘€/𝑝+π›Όπ‘€βˆ’π‘€π›Όπ‘žβ€²(π‘Ÿβˆ’1)/2log𝑀𝑛<∞.(2.22)
(ii) For π‘ž(π‘Ÿβˆ’1)β‰₯2, taking π‘ž<π‘žξ…ž<𝑝 and 𝑀>max(2,2𝑝(π‘Ÿβˆ’1)/(2βˆ’π‘π›Ώ)), where 𝛿 is defined by (2.3), we get, from (2.3), (2.4), (2.15), and π‘Ÿβˆ’2βˆ’π‘€/𝑝+𝛿𝑀/2<βˆ’1, 𝐼12β‰ͺβˆžξ“π‘›=1π‘›π‘Ÿβˆ’2βˆ’π‘€/𝑝log𝑀𝑛𝑛𝑖=1π‘Ž2𝑛𝑖+𝑛2π›Όβˆ’π›Όπ‘žβ€²(π‘Ÿβˆ’1)𝐸||π‘Žπ‘›π‘–π‘‹π‘–||π‘žβ€²(π‘Ÿβˆ’1)𝑀/2β‰ͺβˆžξ“π‘›=1π‘›π‘Ÿβˆ’2βˆ’π‘€/𝑝+𝛿𝑀/2log𝑀𝑛<∞.(2.23) Since 𝑛𝑖=1𝑋(2)𝑛𝑖>πœ€π‘›1/𝑝ξƒͺ=𝑛𝑖=1ξ€·π‘Žπ‘›π‘–π‘‹π‘–βˆ’π‘›π›Όξ€ΈπΌ(𝑛𝛼<π‘Žπ‘›π‘–π‘‹π‘–<πœ€π‘›1/𝑝/𝐾)>πœ€π‘›1/𝑝ξƒͺβŠ†ξ€·thereatleastexist𝐾indicesπ‘˜suchthatπ‘Žπ‘›π‘˜π‘‹π‘˜>𝑛𝛼,(2.24) we have 𝑃𝑛𝑖=1𝑋(2)𝑛𝑖>πœ€π‘›1/𝑝ξƒͺ≀1≀𝑖1<𝑖2<β‹―<π‘–πΎβ‰€π‘›π‘ƒξ€·π‘Žπ‘›π‘–1𝑋𝑖1>𝑛𝛼,π‘Žπ‘›π‘–2𝑋𝑖2>𝑛𝛼,…,π‘Žπ‘›π‘–πΎπ‘‹π‘–πΎ>𝑛𝛼.(2.25)
By Lemma 1.6, {π‘Žπ‘›π‘–π‘‹π‘–;1≀𝑖≀𝑛,𝑛β‰₯1} is still ND. Hence, for π‘ž<π‘žξ…ž<𝑝 we conclude that 𝑃𝑛𝑖=1𝑋(2)𝑛𝑖>πœ€π‘›1/𝑝ξƒͺ≀1≀𝑖1<𝑖2<β‹―<𝑖𝐾𝐾≀𝑛𝑗=1π‘ƒξ‚€π‘Žπ‘›π‘–π‘—π‘‹π‘–π‘—>𝑛𝛼≀𝑛𝑖=1𝑃||π‘Žπ‘›π‘–π‘‹π‘–||>𝑛𝛼ξƒͺ𝐾≀𝑛𝑖=1π‘›βˆ’π›Όπ‘žβ€²(π‘Ÿβˆ’1)𝐸||π‘Žπ‘›π‘–π‘‹π‘–||π‘žβ€²(π‘Ÿβˆ’1)ξƒͺ𝐾β‰ͺπ‘›βˆ’π›Όπ‘žβ€²(π‘Ÿβˆ’1)𝐾,(2.26) via (2.4) and (2.15). 𝑋(2)𝑛𝑖>0 from the definition of 𝑋(2)𝑛𝑖. Hence by (2.26) and by taking 𝛼>0 and 𝐾 such that π‘Ÿβˆ’2βˆ’π›ΌπΎπ‘žξ…ž(π‘Ÿβˆ’1)<βˆ’1, we have 𝐼2=βˆžξ“π‘›=1π‘›π‘Ÿβˆ’2𝑃𝑛𝑖=1𝑋(2)𝑛𝑖>πœ€π‘›1/𝑝ξƒͺβ‰ͺβˆžξ“π‘›=1π‘›π‘Ÿβˆ’2βˆ’π›Όπ‘žβ€²(π‘Ÿβˆ’1)𝐾<∞.(2.27) Similarly, we have 𝑋(3)𝑛𝑖<0 and 𝐼3<∞.
Last, we prove that 𝐼4<∞. Let π‘Œ=K𝑋/πœ€. By the definition of 𝑋(4)𝑛𝑖 and (2.1), we have 𝑃max1β‰€π‘˜β‰€π‘›||𝑆(4)π‘›π‘˜||>πœ€π‘›1/𝑝≀𝑃𝑛𝑖=1||𝑋(4)𝑛𝑖||>πœ€π‘›1/𝑝ξƒͺξƒ©β‰€π‘ƒπ‘›ξšπ‘–=1ξ‚΅π‘Žπ‘›π‘–||𝑋𝑖||>πœ€π‘›1/𝑝𝐾ξƒͺ≀𝑛𝑖=1π‘ƒξ‚΅π‘Žπ‘›π‘–||𝑋𝑖||>πœ€π‘›1/𝑝𝐾=βˆžξ“π‘—=1(𝑗+1)βˆ’1β‰€π‘Žπ‘π‘›π‘–<π‘—βˆ’1𝑃||π‘Œ||>(𝑛𝑗)1/𝑝=βˆžξ“π‘—=1(𝑁(𝑛,𝑗+1)βˆ’π‘(𝑛,𝑗))βˆžξ“π‘™=𝑛𝑗𝑃||π‘Œ||𝑙≀𝑝=<𝑙+1βˆžξ“π‘™=𝑛[𝑙/𝑛]𝑗=1ξ€·||π‘Œ||(𝑁(𝑛,𝑗+1)βˆ’π‘(𝑛,𝑗))π‘ƒπ‘™β‰€π‘ξ€Έβ‰ˆ<𝑙+1βˆžξ“π‘™=π‘›ξ‚€π‘™π‘›ξ‚π‘ž(π‘Ÿβˆ’1)/𝑝𝑃||π‘Œ||𝑙≀𝑝.<𝑙+1(2.28) Combining with (2.15), 𝐼4β‰ˆβˆžξ“π‘›=1π‘›βˆžπ‘Ÿβˆ’2𝑙=π‘›ξ‚€π‘™π‘›ξ‚π‘ž(π‘Ÿβˆ’1)/𝑝𝑃||π‘Œ||𝑙≀𝑝=<𝑙+1βˆžξ“π‘™π‘™=1𝑛=1π‘›π‘Ÿβˆ’2βˆ’π‘ž(π‘Ÿβˆ’1)/π‘π‘™π‘ž(π‘Ÿβˆ’1)/𝑝𝑃||π‘Œ||π‘™β‰€π‘ξ€Έβ‰ˆ<𝑙+1βˆžξ“π‘™=1π‘™π‘Ÿβˆ’1𝑃||π‘Œ||𝑙≀𝑝||π‘Œ||<𝑙+1β‰ˆπΈπ‘(π‘Ÿβˆ’1)||𝑋||β‰ˆπΈπ‘(π‘Ÿβˆ’1)<∞.(2.29)
Now we prove (2.5)β‡’(2.4). Since max1≀𝑗≀𝑛||π‘Žπ‘›π‘—π‘‹π‘—||≀max1≀𝑗≀𝑛|||||𝑗𝑖=1π‘Žπ‘›π‘–π‘‹π‘–|||||+max1≀𝑗≀𝑛|||||π‘—βˆ’1𝑖=1π‘Žπ‘›π‘–π‘‹π‘–|||||,(2.30) then from (2.5) we have βˆžξ“π‘›=1π‘›π‘Ÿβˆ’2𝑃max1≀𝑗≀𝑛||π‘Žπ‘›π‘—π‘‹π‘—||>𝑛1/𝑝<∞.(2.31) Combining with the hypotheses of Theorem 2.1, 𝑃max1≀𝑗≀𝑛||π‘Žπ‘›π‘—π‘‹π‘—||>𝑛1/π‘ξ‚ΆβŸΆ0,π‘›βŸΆβˆž.(2.32) Thus, for sufficiently large 𝑛, 𝑃max1≀𝑗≀𝑛||π‘Žπ‘›π‘—π‘‹π‘—||>𝑛1/𝑝<12.(2.33) By Lemma 1.6, {π‘Žπ‘›π‘—π‘‹π‘—;1≀𝑗≀𝑛,𝑛β‰₯1} is still ND. By applying Lemma 1.10 and (2.1), we obtain π‘›ξ“π‘˜=1𝑃||π‘Žπ‘›π‘˜π‘‹π‘˜||>𝑛1/𝑝≀4𝐢𝑃max1β‰€π‘˜β‰€π‘›||π‘Žπ‘›π‘˜π‘‹π‘˜||>𝑛1/𝑝.(2.34) Substituting the above inequality in (2.5), we get βˆžξ“π‘›=1π‘›π‘›π‘Ÿβˆ’2ξ“π‘˜=1𝑃||π‘Žπ‘›π‘˜π‘‹π‘˜||>𝑛1/𝑝<∞.(2.35) So, by the process of proof of 𝐼4<∞, 𝐸||𝑋||𝑝(π‘Ÿβˆ’1)β‰ˆβˆžξ“π‘›=1π‘›π‘›π‘Ÿβˆ’2ξ“π‘˜=1𝑃||π‘Žπ‘›π‘˜π‘‹π‘˜||>𝑛1/𝑝<∞.(2.36)

Proof of Theorem 2.2. Let 𝑝=2, 𝛼<1/𝑝=1/2,and𝐾>1/(2𝛼). Using the same notations and method of Theorem 2.1, we need only to give the different parts.
Letting (2.7) take the place of (2.15), similarly to the proof of (2.19) and (2.26), we obtain π‘›βˆ’1/2max1β‰€π‘˜β‰€π‘›||𝐸𝑆(1)π‘›π‘˜||β‰ͺπ‘›βˆ’1/2+π›Όβˆ’2𝛼(π‘Ÿβˆ’1)⟢0,π‘›βŸΆβˆž.(2.37) Taking 𝑀>max(2,2(π‘Ÿβˆ’1)), we have 𝐼11β‰ͺβˆžξ“π‘›=1π‘›βˆ’1βˆ’(1βˆ’2𝛼)(𝑀/2βˆ’(π‘Ÿβˆ’1))log𝑀𝑛<∞.(2.38) For π‘Ÿβˆ’1≀1, taking 𝑀>max(2,2(π‘Ÿβˆ’1)/(1βˆ’2𝛼+2𝛼(π‘Ÿβˆ’1))), we get 𝐼12β‰ͺβˆžξ“π‘›=1π‘›βˆ’1βˆ’(1βˆ’2𝛼(π‘Ÿβˆ’1)βˆ’2𝛼)𝑀/2+(π‘Ÿβˆ’1)log𝑀𝑛<∞.(2.39) For π‘Ÿβˆ’1>1, 𝐸𝑋2𝑛𝑖<∞ from (2.8). Letting 𝑀>2(π‘Ÿβˆ’1)2, by the Ḧolder inequality, 𝐼12β‰ͺβˆžξ“π‘›=1π‘›π‘Ÿβˆ’2βˆ’π‘€/2log𝑀𝑛𝑛𝑖=1π‘Ž2𝑛𝑖+𝑛2π›Όβˆ’2𝛼(π‘Ÿβˆ’1)πΈξ€·π‘Žπ‘›π‘–π‘‹π‘–ξ€Έ2(π‘Ÿβˆ’1)𝑀/2β‰ͺβˆžξ“π‘›=1π‘›π‘Ÿβˆ’2βˆ’π‘€/2logπ‘€π‘›βŽ‘βŽ’βŽ’βŽ£ξƒ©π‘›ξ“π‘–=1π‘Ž2(π‘Ÿβˆ’1)𝑛𝑖ξƒͺ1/(π‘Ÿβˆ’1)𝑛𝑖=11ξƒͺπ‘Ÿβˆ’2/(π‘Ÿβˆ’1)⎀βŽ₯βŽ₯βŽ¦π‘€/2β‰ͺβˆžξ“π‘›=1π‘›βˆ’1βˆ’π‘€/2(π‘Ÿβˆ’1)+(π‘Ÿβˆ’1)log𝑀𝑛<∞.(2.40) By the definition of 𝐾, 𝐼2β‰ͺβˆžξ“π‘›=1π‘›βˆ’1βˆ’(π‘Ÿβˆ’1)(2π›ΌπΎβˆ’1)<∞.(2.41) Similarly to the proof (2.31), we have 𝐼4β‰ͺβˆžξ“π‘™π‘™=1𝑛=1π‘›βˆ’1π‘™π‘Ÿβˆ’1𝑃||π‘Œ||𝑙≀2=<𝑙+1βˆžξ“π‘™=1π‘™π‘Ÿβˆ’1ξ‚€||π‘Œ||log𝑙𝑃𝑙≀2||π‘Œ||<𝑙+1β‰ˆπΈ2(π‘Ÿβˆ’1)||π‘Œ||||𝑋||logβ‰ˆπΈ2(π‘Ÿβˆ’1)||𝑋||log<∞.(2.42)Equation (2.9)β‡’(2.8) Using the same method of the necessary part of Theorem 2.1, we can easily get 𝐸||𝑋||2(π‘Ÿβˆ’1)||𝑋||ξ‚β‰ˆlogβˆžξ“π‘›=1π‘›π‘›π‘Ÿβˆ’2ξ“π‘˜=1𝑃||π‘Žπ‘›π‘˜π‘‹π‘˜||>𝑛1/2ξ€Έ<∞.(2.43)

Acknowledgments

The author is 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. This work was supported by the National Natural Science Foundation of China (11061012), the Support Program the New Century Guangxi China Ten-Hundred-Thousand Talents Project (2005214), and the Guangxi China Science Foundation (2010GXNSFA013120). Professor Dr. Qunying Wu engages in probability and statistics.

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