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

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 𝑌 are said to be negatively dependent (ND) if 𝑃(𝑋≤𝑥,𝑌≤𝑦)≤𝑃(𝑋≤𝑥)𝑃(𝑌≤𝑦)(1.1) for all 𝑥,𝑦∈R. 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>𝑥,𝑌>𝑦)≤𝑃(𝑋>𝑥)𝑃(𝑌>𝑦)(1.2) for all 𝑥,𝑦∈R. 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,…,𝑋_𝑛 are said to be negatively dependent (ND) if, for all real 𝑥₁,…,𝑥_𝑛, 𝑃^𝑛_𝑗=1𝑋_𝑗≤𝑥_𝑗≤^𝑛_𝑗=1𝑃_𝑛𝑃𝑋_𝑗≤𝑥_𝑗,_𝑗=1𝑋_𝑗>𝑥_𝑗≤^𝑛_𝑗=1𝑃𝑋_𝑗>𝑥_𝑗.(1.3) An infinite sequence of random variables {𝑋_𝑛;𝑛≥1} is said to be ND if every finite subset 𝑋₁,…,𝑋_𝑛 is ND.

Definition 1.3. Random variables 𝑋₁,𝑋₂,…,𝑋_𝑛, 𝑛≥2, are said to be negatively associated (NA) if, for every pair of disjoint subsets 𝐴₁ and 𝐴₂ of {1,2,…,𝑛}, cov𝑓₁𝑋_𝑖;𝑖∈𝐴₁,𝑓₂𝑋_𝑗;𝑗∈𝐴₂≤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 {𝑋_𝑛;𝑛≥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,…,𝑋_𝑛−1)=𝐼_{(𝑋1≤𝑥1,𝑋2≤𝑥2,…,𝑋𝑛−1≤𝑥𝑛−1)}, 𝑓₂(𝑋_𝑛)=𝐼_{(𝑋𝑛≤𝑥𝑛)} and 𝑓₁(𝑋₁,𝑋₂,…,𝑋_𝑛−1)=𝐼_{(𝑋1>𝑥1,𝑋2>𝑥2,…,𝑋𝑛−1>𝑥𝑛−1)}, 𝑓₂(𝑋_𝑛)=𝐼_{(𝑋𝑛>𝑥𝑛)}, 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 𝑋_𝑖 be a binary random variable such that 𝑃(𝑋_𝑖=1)=𝑃(𝑋_𝑖=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,𝑋₂≤0=48≰𝑃𝑋¹+𝑋₃≤1𝑃𝑋₂≤0=38.(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 𝑋_𝑖 be a binary random variable such that 𝑃(𝑋_𝑖=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𝑋_𝑖=1,𝑖=1,2,3,4>𝑃𝑋₁=𝑋₂=1𝑃𝑋₃=𝑋₄=1,(1.6) violating NA.

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 𝑎_𝑛≪𝑏_𝑛 (𝑎_𝑛≫𝑏_𝑛) denote that there exists a constant 𝑐>0 such that 𝑎_𝑛≤𝑐𝑏_𝑛 (𝑎_𝑛≥𝑐𝑏_𝑛) for sufficiently large 𝑛, and let 𝑎_𝑛≈𝑏_𝑛 mean 𝑎_𝑛≪𝑏_𝑛 and 𝑎_𝑛≫𝑏_𝑛. Also, let log𝑥 denote ln(max(e,𝑥)) and 𝑆_𝑛=∑^𝑛𝑗=1𝑋_𝑗.

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

Lemma 1.7 (see [2]). Let 𝑋₁,…,𝑋_𝑛 be nonnegative r. v. ’s which are ND. Then 𝐸^𝑛_𝑗=1𝑋_𝑗≤^𝑛_𝑗=1𝐸𝑋_𝑗.(1.7) In particular, let 𝑋₁,…,𝑋_𝑛 be ND, and let 𝑡₁,…,𝑡_𝑛 be all nonnegative (or non-positive) real numbers. Then 𝐸exp^𝑛_𝑗=1𝑡_𝑗𝑋_𝑗≤^𝑛_𝑗=1𝐸exp𝑡_j𝑋_𝑗.(1.8)

Lemma 1.8. Let {𝑋_𝑛;𝑛≥1} be an ND sequence with 𝐸𝑋_𝑛=0 and 𝐸|𝑋_𝑛|^𝑝<∞,𝑝≥2. Then for 𝐵_𝑛=∑^𝑛𝑖=1𝐸𝑋^2𝑖, 𝐸||𝑆_𝑛||^𝑝≤𝑐_𝑝^𝑛_𝑖=1𝐸||𝑋_𝑖||^𝑝+𝐵^𝑛𝑝/2𝐸max_{1≤𝑖≤𝑛},(1.9)||𝑆_𝑖||^𝑝≤c^𝑝log^𝑝𝑛^𝑛_𝑖=1𝐸||𝑋_𝑖||^𝑝+𝐵^𝑛𝑝/2,(1.10) where 𝑐_𝑝>0 depends only on 𝑝.

Remark 1.9. If {𝑋_𝑛;𝑛≥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 𝑎>0, 𝑋^𝑖=min(𝑋_𝑖,𝑎),and𝑆^𝑛=∑^𝑛𝑖=1𝑋^𝑖. It is easy to show that {𝑋^𝑖;𝑖≥1} is a negatively dependent sequence by Lemma 1.6. Noting that (e^𝑥−1−𝑥)/𝑥² is a nondecreasing function of 𝑥 on R and that 𝐸𝑋^𝑖≤𝐸𝑋_𝑖=0, 𝑡𝑋^𝑖≤𝑡𝑎, we have 𝐸e^{𝑡𝑋′𝑖}=1+𝑡𝐸𝑋^𝑖+𝐸e^{𝑡𝑋′𝑖}−1−𝑡𝑋^𝑖𝑡²𝑋^𝑖2𝑡²𝑋^𝑖2≤1+e_𝑡𝑎−1−𝑡𝑎𝑎⁻²𝐸𝑋^𝑖2≤1+e^𝑡𝑎−1−𝑡𝑎𝑎⁻²𝐸𝑋^2𝑖≤expe^𝑡𝑎−1−𝑡𝑎𝑎⁻²𝐸𝑋^2𝑖.(1.11) Here the last inequality follows from 1+𝑥≤e^𝑥, for all 𝑥∈R.
Note that 𝐵_𝑛=∑^𝑛𝑖=1𝐸𝑋^2𝑖 and {𝑋^𝑖;𝑖≥1} is ND, we conclude from the above inequality and Lemma 1.7 that, for any 𝑥>0 and ℎ>0, we get e^−ℎ𝑥𝐸e^{ℎ𝑆′𝑛}=e^−ℎ𝑥𝐸^𝑛_𝑖=1e^{ℎ𝑋′𝑖}≤e^{𝑛−ℎ𝑥}_𝑖=1≤exp−ℎ𝑥+e_ℎ𝑎𝐸e^{ℎ𝑋′𝑖}−1−ℎ𝑎𝑎⁻²𝐵_𝑛.(1.12)
Letting ℎ=ln((𝑥𝑎)/𝐵_𝑛+1)/𝑎>0, we get e^ℎ𝑎−1−ℎ𝑎𝑎⁻²𝐵_𝑛=𝑥𝑎−𝐵^𝑛𝑎²ln𝑥𝑎𝐵_𝑛+1≤𝑥𝑎.(1.13) Putting this one into (1.12), we get furthermore e^−ℎ𝑥𝐸e^{ℎ𝑆′𝑛}≤exp𝑥𝑎−𝑥𝑎ln𝐵𝑥𝑎_𝑛+1.(1.14) Putting 𝑥/𝑎=𝑡 into the above inequality, we get 𝑃𝑆_𝑛≥𝑥≤^𝑛_𝑖=1≤𝑃𝑋_𝑖>𝑎+𝑃𝑆^𝑛≥𝑥_𝑛_𝑖=1𝑃𝑋_𝑖>𝑎+e^−ℎ𝑥𝐸e^{ℎ𝑆′𝑛}≤^𝑛_𝑖=1𝑃𝑋_𝑖>𝑥𝑡+exp𝑡−𝑡ln𝑡𝑥𝐵²_𝑛=+1_𝑛_𝑖=1𝑃𝑋_𝑖>𝑥𝑡+e^𝑡1+𝑡𝑥𝐵²_𝑛^−𝑡.(1.15) Letting −𝑋_𝑖 take the place of 𝑋_𝑖 in the above inequality, we can get 𝑃−𝑆_𝑛≥𝑥=𝑃𝑆_𝑛≤−𝑥≤^𝑛_𝑖=1𝑃−𝑋_𝑖>𝑥𝑡+e^𝑡1+𝑡𝑥𝐵²_𝑛^−𝑡=^𝑛_𝑖=1𝑃𝑋_𝑖<−𝑥𝑡+e^𝑡1+𝑡𝑥𝐵²_𝑛^−𝑡.(1.16) Thus 𝑃||𝑆_𝑛||≥𝑥=𝑃𝑆_𝑛≥𝑥+𝑃𝑆_𝑛≤−𝑥≤^𝑛_𝑖=1𝑃||𝑋_𝑖||<𝑥𝑡+2e^𝑡1+𝑡𝑥𝐵²_𝑛^−𝑡.(1.17)
Multiplying (1.17) by 𝑝𝑥^𝑝−1, letting 𝑡=𝑝, and integrating over 0<𝑥<+∞, according to 𝐸||𝑋||^𝑝=𝑝^0+∞𝑥^𝑝−1𝑃||𝑋||≥𝑥d𝑥,(1.18) we obtain 𝐸||𝑆_𝑛||^𝑝≤=𝑝𝑝^0+∞_𝑛𝑥^𝑝−1𝑃||𝑆_𝑛||≥𝑥d𝑥_𝑖=1^0+∞𝑥^𝑝−1𝑃||𝑋_𝑖||≥𝑥𝑝d𝑥+2𝑝e^𝑝^0+∞𝑥^𝑝−11+𝑝𝑥𝐵²_𝑛^−𝑝=𝑝_𝑛𝑝+1d𝑥_𝑖=1𝐸||𝑋_𝑖||^𝑝+𝑝e^𝑝𝑝𝐵_𝑛^𝑝/2^0+∞𝑢^𝑝/2−1(1+𝑢)^𝑝=𝑝_𝑛𝑝+1d𝑢_𝑖=1𝐸||𝑋_𝑖||^𝑝+𝑝^𝑝/2+1e^𝑝𝐵𝑝2,2𝑝𝐵^𝑛𝑝/2,(1.19) where 𝐵(𝛼,𝛽)=∫¹⁰𝑥^𝛼−1(1−𝑥)^𝛽−1d𝑥=∫^0+∞𝑥^𝛼−1(1+𝑥)^{−(𝛼+𝛽)}d𝑥,𝛼,𝛽>0 is Beta function. Letting 𝑐_𝑝=max(𝑝^𝑝+1,𝑝^1+𝑝/2e^𝑝𝐵(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 {𝑋_𝑛;𝑛≥1} be a sequence of ND random variables. Then there exists a positive constant 𝑐 such that, for any 𝑥≥0 and all 𝑛≥1, 1−𝑃max_{1≤𝑘≤𝑛}||𝑋_𝑘||>𝑥^2𝑛_𝑘=1𝑃||𝑋_𝑘||>𝑥≤𝑐𝑃max_{1≤𝑘≤𝑛}||𝑋_𝑘||>𝑥.(1.20)

Proof. Let 𝐴_𝑘=(|𝑋_𝑘|>𝑥) and 𝛼_𝑛=1−𝑃(⋃^𝑛𝑘=1𝐴_𝑘)=1−𝑃(max_{1≤𝑘≤𝑛}|𝑋_𝑘|>𝑥). Without loss of generality, assume that 𝛼_𝑛>0. Note that {𝐼_{(𝑋𝑘>𝑥)}−𝐸𝐼_{(𝑋𝑘>𝑥)};𝑘≥1} and {𝐼_{(𝑋𝑘<−𝑥)}−𝐸𝐼_{(𝑋𝑘<−𝑥)};𝑘≥1} are still ND by Lemma 1.6. Using (1.9), we get 𝐸^𝑛_𝑘=1𝐼_𝐴𝑘−𝐸𝐼_𝐴𝑘²=𝐸^𝑛_𝑘=1𝐼_{(𝑋𝑘>𝑥)}−𝐸𝐼_{(𝑋𝑘>𝑥)}+𝐼_{(𝑋𝑘<−𝑥)}−𝐸𝐼_{(𝑋𝑘<−𝑥)}²≤2𝐸^𝑛_𝑘=1𝐼_{(𝑋𝑘>𝑥)}−𝐸𝐼_{(𝑋𝑘>𝑥)}²+2𝐸^𝑛_𝑘=1𝐼_{(𝑋𝑘<−𝑥)}−𝐸𝐼_{(𝑋𝑘<−𝑥)}²≤𝑐^𝑛_𝑘=1𝑃𝐴_𝑘.(1.21) Combining with the Cauchy-Schwarz inequality, we obtain ^𝑛_𝑘=1𝑃𝐴_𝑘=^𝑛_𝑘=1𝑃𝐴_𝑘,^𝑛_𝑗=1𝐴_𝑗=^𝑛_𝑘=1=𝐸_𝑛𝐸𝐼_𝐴𝑘𝐼_⋃^𝑛𝑗=1_𝐴𝑗_𝑘=1𝐼_𝐴𝑘−𝐸𝐼_𝐴𝑘𝐼_⋃^𝑛𝑗=1_𝐴𝑗+^𝑛_𝑘=1𝑃𝐴_𝑘𝑃^𝑛_𝑗=1𝐴_𝑗≤⎛⎜⎜⎝𝐸_𝑛_𝑘=1𝐼_𝐴𝑘−𝐸𝐼_𝐴𝑘²𝐸𝐼_⋃^𝑛𝑗=1_𝐴𝑗⎞⎟⎟⎠^1/2+1−𝛼_𝑛^𝑛_𝑘=1≤𝑐1−𝛼_𝑛𝑃𝐴_𝑘𝛼_𝑛𝛼_𝑛𝑛_𝑘=1𝑃𝐴_𝑘^1/2+1−𝛼_𝑛^𝑛_𝑘=1≤1𝑃𝐴_𝑘2𝑐1−𝛼^𝑛𝛼_𝑛+𝛼_𝑛𝑛_𝑘=1𝑃𝐴_𝑘+1−𝛼_𝑛^𝑛_𝑘=1𝑃𝐴_𝑘.(1.22) Thus 𝛼^2𝑛𝑛_𝑘=1𝑃𝐴_𝑘≤𝑐1−𝛼_𝑛,(1.23) that is, 1−𝑃max_{1≤𝑘≤𝑛}||𝑋_𝑘||>𝑥^2𝑛_𝑘=1𝑃||𝑋_𝑘||>𝑥≤𝑐𝑃max_{1≤𝑘≤𝑛}||𝑋_𝑘||>𝑥.(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 {𝑌_𝑛;𝑛≥1} of random variables converges completely to the constant 𝑐 if ∑^∞𝑛=1𝑃(|𝑋_𝑛−𝑐|>𝜀)<∞, for all 𝜀>0. In view of the Borel-Cantelli lemma, this implies that 𝑌_𝑛→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,𝑋_𝑛;𝑛≥1} is a sequence of i.i.d. random variables with mean zero, then 𝐸|𝑋|^𝑝(𝑡+2)<∞(1≤𝑝<2,𝑡≥−1) is equivalent to the condition that ∑^∞𝑛=1𝑛^𝑡𝑃(∑^𝑛1=1|𝑋_𝑖|/𝑛^1/𝑝>𝜀)<∞, for all 𝜀>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,𝑋_𝑛;𝑛≥1} be a sequence of identically distributed ND random variables and {𝑎_𝑛𝑘;1≤𝑘≤𝑛,𝑛≥1} an array of real numbers, and let 𝑟>1, 𝑝>2. If, for some 2≤𝑞<𝑝, 𝑁(𝑛,𝑚+1)=♯𝑘𝐸≥𝑋1;=||𝑎0_𝑛𝑘||≥fo(r𝑚1+≤1𝑞)(𝑟^−1/𝑝−1≈),𝑚^{𝑞(𝑟−1)/𝑝},𝑛,𝑚≥1,((22..12))_𝑛_𝑘=1𝑎^2𝑛𝑘≪𝑛^𝛿for2≤𝑞(𝑟−1)andsome0<𝛿<2𝑝,(2.3) then, for 𝑟≥2, 𝐸||𝑋||^{𝑝(𝑟−1)}<∞(2.4) if and only if ^∞_𝑛=1𝑛^𝑟−2𝑃⎛⎜⎜⎝max_{1≤𝑘≤𝑛}|||||_𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖|||||>𝜀𝑛_1/𝑝⎞⎟⎟⎠<∞,∀𝜀>0.(2.5) For 1<𝑟<2, (2.4) implies (2.5), conversely, and (2.5) and 𝑛^𝑟−2𝑃(max_{1≤𝑘≤𝑛}|𝑎_𝑛𝑘𝑋_𝑘|>𝑛^1/𝑝) decreasing on 𝑛 imply (2.4).

For 𝑝=2, 𝑞=2, we have the following theorem.

Theorem 2.2. Let {𝑋,𝑋_𝑛;𝑛≥1} be a sequence of identically distributed ND random variables and {𝑎_𝑛𝑘;1≤𝑘≤𝑛,𝑛≥1} an array of real numbers, and let 𝑟>1. If 𝑁(𝑛,𝑚+1)=♯𝐸𝑘𝑋;||𝑎=_𝑛𝑘0||,≥(𝑚1+≤12)(𝑟^−1/2−1≈),_𝑛𝑚^𝑟−1,𝑛,𝑚≥1,(2.6)_𝑘=1||𝑎_𝑛𝑘||^2(𝑟−1)=𝑂(1),(2.7) then, for 𝑟≥2, 𝐸||𝑋||^2(𝑟−1)log||𝑋||<∞(2.8) if and only if ^∞_𝑛=1𝑛^𝑟−2𝑃⎛⎜⎜⎝max_{1≤𝑘≤𝑛}|||||_𝑘_𝑖=1𝑎_𝑛𝑖𝑋_𝑖|||||>𝜀𝑛_1/2⎞⎟⎟⎠<∞,∀𝜀>0.(2.9) For 1<𝑟<2, (2.8) implies (2.9), conversely, and (2.9) and 𝑛^𝑟−2𝑃(max_{1≤𝑘≤𝑛}|𝑎_𝑛𝑘𝑋_𝑘|>𝑛^1/2) 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 2≤𝑞≤𝑝, ∑_𝑘∈𝑁|𝑎_𝑛𝑘|^{𝑞(𝑟−1)}≪1 as 𝑛→∞ implies that 𝑁(𝑛,𝑚+1)=♯𝑘≥1;||𝑎_𝑛𝑘||≥(𝑚+1)^−1/𝑝≪𝑚^{𝑞(𝑟−1)/𝑝}as𝑛⟶∞,(2.10) taking 𝑟=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 𝑟.

Proof of Theorem 2.1. Equation (2.4)⇒(2.5). To prove (2.5) it suffices to show that ^∞_𝑛=1𝑛^𝑟−2𝑃⎛⎜⎜⎝max_{1≤𝑘≤𝑛}|||||_𝑘_𝑖=1𝑎^±𝑛𝑖𝑋_𝑖|||||>𝜀𝑛_1/𝑝⎞⎟⎟⎠<∞,∀𝜀>0,(2.11) where 𝑎^+𝑛𝑖=max(𝑎_𝑛𝑖,0) and 𝑎^−𝑛𝑖=max(−𝑎_𝑛𝑖,0). Thus, without loss of generality, we can assume that 𝑎_𝑛𝑖>0 for all 𝑛≥1,𝑖≤𝑛. For 0<𝛼<1/𝑝 small enough and sufficiently large integer 𝐾, which will be determined later, let 𝑋𝑋^(1)𝑛𝑖_(2)𝑛𝑖=−𝑛^𝛼𝐼_{(𝑎𝑛𝑖𝑋𝑖<−𝑛𝛼)}+𝑎_𝑛𝑖𝑋_𝑖𝐼_{(𝑎𝑛𝑖|𝑋𝑖|≤𝑛𝛼)}+𝑛^𝛼𝐼_{(𝑎𝑛𝑖𝑋𝑖>𝑛𝛼)},𝑋_(3)𝑛𝑖=𝑎_𝑛𝑖𝑋_𝑖−𝑛^𝛼𝐼_(𝑛^𝛼_{<𝑎𝑛𝑖𝑋𝑖<𝜀𝑛}^1/𝑝_/𝐾),𝑋_(4)𝑛𝑖=𝑎_𝑛𝑖𝑋_𝑖+𝑛^𝛼𝐼_(−𝜀𝑛^1/𝑝_{/𝐾<𝑎𝑛𝑖𝑋𝑖<−𝑛}^𝛼₎,=𝑎_𝑛𝑖𝑋_𝑛𝑖−𝑋^(1)𝑛𝑖−𝑋^(2)𝑛𝑖−𝑋^(3)𝑛𝑖𝑆_{(𝑗)𝑛𝑘}=𝑎_𝑛𝑖𝑋_𝑖+𝑛^𝛼𝐼_{(𝑎𝑛𝑖𝑋𝑖≤−𝜀𝑛}^1/𝑝_/𝐾)+𝑎_𝑛𝑖𝑋_𝑖−𝑛^𝛼𝐼_{(𝑎𝑛𝑖𝑋𝑖≥𝜀𝑛}^1/𝑝_/𝐾),=^𝑘_𝑖=1𝑋^{(𝑗)𝑛𝑖},𝑗=1,2,3,4;1≤𝑘≤𝑛,𝑛≥1.(2.12) Thus 𝑆_𝑛𝑘=∑^𝑘𝑖=1𝑎_𝑛𝑖𝑋_𝑖=∑^4𝑗=1𝑆^{(𝑗)𝑛𝑘}. Note that max_{1≤𝑘≤𝑛}||𝑆_𝑛𝑘||>4𝜀𝑛^1/𝑝⊆⁴_𝑗=1max_{1≤𝑘≤𝑛}||𝑆^{(𝑗)𝑛𝑘}||>𝜀𝑛^1/𝑝.(2.13) So, to prove (2.5) it suffices to show that 𝐼_𝑗=^∞_𝑛=1𝑛^𝑟−2𝑃max_{1≤𝑘≤𝑛}||𝑆^{(𝑗)𝑛𝑘}||>𝜀𝑛^1/𝑝<∞,𝑗=1,2,3,4.(2.14) For any 𝑞^>𝑞, ^𝑛_𝑖=1𝑎^{𝑞′(𝑟−1)𝑛𝑖}=^∞_𝑗=1_{(𝑗+1)−1≤𝑎𝑝𝑛𝑖<𝑗−1}𝑎^{𝑞′(𝑟−1)𝑛𝑖}≤^∞_𝑗=1_{(𝑗+1)−1≤𝑎𝑝𝑛𝑖<𝑗−1}𝑗^{−𝑞′(𝑟−1)/𝑝}≪^∞_𝑗=1(𝑁(𝑛,𝑗+1)−𝑁(𝑛,𝑗))𝑗^{−𝑞′(𝑟−1)/𝑝}≪^∞_𝑗=1≪_∞𝑁(𝑛,𝑗)𝑗^{−𝑞′(𝑟−1)/𝑝}−(𝑗+1)^{−𝑞′(𝑟−1)/𝑝}_𝑗=1𝑗^{−1−(𝑞′−𝑞)(𝑟−1)/𝑝}<∞.(2.15)
Now, we prove that 𝑛^−1/𝑝max_{1≤𝑘≤𝑛}||𝐸𝑆^(1)𝑛𝑘||⟶0,𝑛⟶∞.(2.16)
(i) For 0<𝑞(𝑟−1)<1, taking 𝑞<𝑞^<𝑝 such that 0<𝑞^(𝑟−1)<1, by (2.4) and (2.15), we get 𝑛≤^−1/𝑝m𝑛ax_{1≤𝑘≤𝑛𝑛−1/𝑝}||𝐸𝑆^(1)𝑛𝑘||_𝑖=1𝐸||𝑎_𝑛𝑖≤𝑛_−1/𝑝𝑋_𝑖||𝐼_{(|𝑎𝑛𝑖𝑋𝑖|≤𝑛𝛼)}+𝑛^𝛼𝑃||𝑎_𝑛𝑖𝑋_𝑖||>𝑛^𝛼^𝑛_𝑖=1𝐸||𝑎_𝑛𝑖𝑋_𝑖||^{𝑞′(𝑟−1)}||𝑎_𝑛𝑖𝑋_𝑖||^{1−𝑞′(𝑟−1)}𝐼_{(|𝑎𝑛𝑖𝑋𝑖|≤𝑛𝛼)}+𝑛^{𝛼−𝛼𝑞′𝑛(𝑟−1)}_𝑖=1𝐸||𝑎_𝑛𝑖𝑋_𝑖||^{𝑞′(𝑟−1)}≪𝑛_{−1/𝑝+𝛼−𝛼𝑞′(𝑟−1)}⟶0,𝑛⟶∞.(2.17) For 1≤𝑞(𝑟−1), letting 𝑞<𝑞^<𝑝, by (2.2), (2.4), and (2.15), we get 𝑛≤^−1/𝑝m𝑛ax_{1≤𝑘≤𝑛𝑛−1/𝑝}||𝐸𝑆^(1)𝑛𝑘||_𝑖=1𝐸||𝑎_𝑛𝑖≤𝑛_{𝑛−1/𝑝}𝑋_𝑖||𝐼_{(|𝑎𝑛𝑖𝑋𝑖|>𝑛𝛼)}+𝑛^𝛼𝑃||𝑎_𝑛𝑖𝑋_𝑖||>𝑛^𝛼_𝑖=1𝐸||𝑎^𝑛𝑖𝑋_𝑖||||𝑎^𝑛𝑖𝑋_𝑖||𝑛^𝛼^{𝑞′(𝑟−1)−1}𝐼_{(|𝑎𝑛𝑖𝑋𝑖|≤𝑛𝛼)}+𝑛^{𝛼−𝛼𝑞′(𝑟−1)}𝐸||𝑎_𝑛𝑖𝑋_𝑖||^{𝑞′(𝑟−1)}≪𝑛_{−1/𝑝+𝛼−𝛼𝑞′(𝑟−1)}⟶0.(2.18) Hence, (2.16) holds. Therefore, to prove 𝐼₁<∞ it suffices to prove that 𝐼₁=^∞_𝑛=1𝑛^𝑟−2𝑃max_{1≤𝑘≤𝑛}||𝑆^(1)𝑛𝑘−𝐸𝑆^(1)𝑛𝑘||>𝜀𝑛^1/𝑝<∞,∀𝜀>0.(2.19) Note that {𝑋^(1)𝑛𝑖;1≤𝑖≤𝑛,𝑛≥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, 𝐼₁≪^∞_𝑛=1𝑛^{𝑟−2−𝑀/𝑝}𝐸max_{1≤𝑘≤𝑛}||𝑆^(1)𝑛𝑘−𝐸𝑆^(1)𝑛𝑘||^𝑀≪^∞_𝑛=1𝑛^{𝑟−2−𝑀/𝑝}log^𝑀𝑛⎡⎢⎢⎣^𝑛_𝑖=1𝐸||𝑋^(1)𝑛𝑖||^𝑀+^𝑛_𝑖=1𝐸𝑋^(1)𝑛𝑖²^𝑀/2=𝐼₁₁⎤⎥⎥⎦+𝐼₁₂.(2.20) Taking 𝑀>max(2,𝑝(𝑟−1)(1−𝛼𝑞^)/(1−𝛼𝑝)), then 𝑟−2−𝑀/𝑝+𝛼𝑀−𝛼𝑞^(𝑟−1)<−1, and, by (2.15), we get 𝐼₁₁≤^∞_𝑛=1𝑛^{𝑟−2−𝑀/𝑝}log^𝑀𝑛^𝑛_𝑖=1𝐸||𝑎^𝑛𝑖≤_∞𝑋_𝑖||^𝑀𝐼_{(|𝑎𝑛𝑖𝑋𝑖|≤𝑛𝛼)}+𝑛^𝑀𝛼𝑃||𝑎_𝑛𝑖𝑋_𝑖||>𝑛^𝛼_𝑛=1𝑛^{𝑟−2−𝑀/𝑝}log^𝑀𝑛^𝑛_𝑖=1𝐸||𝑎^𝑛𝑖≪_∞𝑋_𝑖||^{𝑞′(𝑟−1)}𝑛^{𝛼(𝑀−𝑞′(𝑟−1))}+𝑛^{𝛼(𝑀−𝑞′(𝑟−1))}𝐸||𝑎_𝑛𝑖𝑋_𝑖||^{𝑞′(𝑟−1)}_𝑛=1<∞.𝑛^{𝑟−2−𝑀/𝑝+𝛼𝑀−𝛼𝑞′(𝑟−1)}log^𝑀𝑛(2.21)
(i) For 𝑞(𝑟−1)<2, taking 𝑞<𝑞^<𝑝 such that 𝑞^(𝑟−1)<2 and taking 𝑀>max(2,2𝑝(𝑟−1)/(2−2𝛼𝑝+𝛼𝑝𝑞^(𝑟−1))), from (2.15) and 𝑟−2−𝑀/𝑝+𝛼𝑀−𝑀𝛼𝑞^(𝑟−1)/2<−1, we have 𝐼₁₂≤^∞_𝑛=1𝑛^{𝑟−2−𝑀/𝑝}log^𝑀𝑛^𝑛_𝑖=1𝐸||𝑎_𝑛𝑖𝑋_𝑖||^{𝑞′(𝑟−1)}𝑛^{𝛼(2−𝑞′(𝑟−1))}𝐼_{(|𝑎𝑛𝑖𝑋𝑖|≤𝑛𝛼)}+𝑛^{2𝛼−𝛼𝑞′(𝑟−1)}𝐸||𝑎_𝑛𝑖𝑋_𝑖||^{𝑞′(𝑟−1)}^𝑀/2≪^∞_𝑛=1<∞.𝑛^{−𝑟−2−𝑀/𝑝+𝛼𝑀−𝑀𝛼𝑞′(𝑟−1)/2}log^𝑀𝑛(2.22)
(ii) For 𝑞(𝑟−1)≥2, taking 𝑞<𝑞^<𝑝 and 𝑀>max(2,2𝑝(𝑟−1)/(2−𝑝𝛿)), where 𝛿 is defined by (2.3), we get, from (2.3), (2.4), (2.15), and 𝑟−2−𝑀/𝑝+𝛿𝑀/2<−1, 𝐼₁₂≪^∞_𝑛=1𝑛^{𝑟−2−𝑀/𝑝}log^𝑀𝑛^𝑛_𝑖=1𝑎^2𝑛𝑖+𝑛^{2𝛼−𝛼𝑞′(𝑟−1)}𝐸||𝑎_𝑛𝑖𝑋_𝑖||^{𝑞′(𝑟−1)}^𝑀/2≪^∞_𝑛=1<∞.𝑛^{𝑟−2−𝑀/𝑝+𝛿𝑀/2}log^𝑀𝑛(2.23) Since ^𝑛_𝑖=1𝑋^(2)𝑛𝑖>𝜀𝑛^1/𝑝=^𝑛_𝑖=1𝑎_𝑛𝑖𝑋_𝑖−𝑛^𝛼𝐼_(𝑛^𝛼_{<𝑎𝑛𝑖𝑋𝑖<𝜀𝑛}^1/𝑝_/𝐾)>𝜀𝑛^1/𝑝⊆thereatleastexist𝐾indices𝑘suchthat𝑎_𝑛𝑘𝑋_𝑘>𝑛^𝛼,(2.24) we have 𝑃^𝑛_𝑖=1𝑋^(2)𝑛𝑖>𝜀𝑛^1/𝑝≤_{1≤𝑖1<𝑖2<⋯<𝑖𝐾≤𝑛}𝑃𝑎_𝑛𝑖1𝑋_𝑖1>𝑛^𝛼,𝑎_𝑛𝑖2𝑋_𝑖2>𝑛^𝛼,…,𝑎_𝑛𝑖𝐾𝑋_𝑖𝐾>𝑛^𝛼.(2.25)
By Lemma 1.6, {𝑎_𝑛𝑖𝑋_𝑖;1≤𝑖≤𝑛,𝑛≥1} is still ND. Hence, for 𝑞<𝑞^<𝑝 we conclude that 𝑃^𝑛_𝑖=1𝑋^(2)𝑛𝑖>𝜀𝑛^1/𝑝≤_{1≤𝑖1<𝑖2<⋯<𝑖𝐾𝐾≤𝑛}_𝑗=1≤_𝑛𝑃𝑎_𝑛𝑖𝑗𝑋_𝑖𝑗>𝑛^𝛼_𝑖=1𝑃||𝑎_𝑛𝑖𝑋_𝑖||>𝑛^𝛼^𝐾≤^𝑛_𝑖=1𝑛^{−𝛼𝑞′(𝑟−1)}𝐸||𝑎_𝑛𝑖𝑋_𝑖||^{𝑞′(𝑟−1)}^𝐾≪𝑛^{−𝛼𝑞′(𝑟−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 𝑟−2−𝛼𝐾𝑞^(𝑟−1)<−1, we have 𝐼₂=^∞_𝑛=1𝑛^𝑟−2𝑃^𝑛_𝑖=1𝑋^(2)𝑛𝑖>𝜀𝑛^1/𝑝≪^∞_𝑛=1𝑛^{𝑟−2−𝛼𝑞′(𝑟−1)𝐾}<∞.(2.27) Similarly, we have 𝑋^(3)𝑛𝑖<0 and 𝐼₃<∞.
Last, we prove that 𝐼₄<∞. Let 𝑌=K𝑋/𝜀. By the definition of 𝑋^(4)𝑛𝑖 and (2.1), we have 𝑃max_{1≤𝑘≤𝑛}||𝑆^(4)𝑛𝑘||>𝜀𝑛^1/𝑝≤𝑃^𝑛_𝑖=1||𝑋^(4)𝑛𝑖||>𝜀𝑛^1/𝑝≤𝑃_𝑛_𝑖=1𝑎^𝑛𝑖||𝑋_𝑖||>𝜀𝑛^1/𝑝≤_𝑛𝐾_𝑖=1𝑃𝑎_𝑛𝑖||𝑋_𝑖||>𝜀𝑛^1/𝑝=_∞𝐾_𝑗=1_{(𝑗+1)−1≤𝑎𝑝𝑛𝑖<𝑗−1}=_∞𝑃||𝑌||>(𝑛𝑗)^1/𝑝_𝑗=1(𝑁(𝑛,𝑗+1)−𝑁(𝑛,𝑗))^∞_𝑙_=𝑛𝑗=𝑃𝑙≤||𝑌||^𝑝<𝑙+1_∞_{𝑙=𝑛[𝑙/𝑛]𝑗=1}≈(𝑁(𝑛,𝑗+1)−𝑁(𝑛,𝑗))𝑃𝑙≤||𝑌||^𝑝<𝑙+1_∞_𝑙=𝑛𝑛𝑙^{𝑞(𝑟−1)/𝑝}𝑃𝑙≤||𝑌||^𝑝<𝑙+1.(2.28) Combining with (2.15), 𝐼₄≈^∞_𝑛=1𝑛^∞𝑟−2_𝑙=𝑛𝑛𝑙^{𝑞(𝑟−1)/𝑝}=𝑃𝑙≤||𝑌||^𝑝<𝑙+1_∞_𝑙𝑙=1_𝑛=1≈𝑛^{𝑟−2−𝑞(𝑟−1)/𝑝}𝑙^{𝑞(𝑟−1)/𝑝}𝑃𝑙≤||𝑌||^𝑝<𝑙+1_∞_𝑙=1≈𝐸||𝑌𝑙||^𝑟−1𝑃_{𝑝(𝑟−1)}𝑙≤||𝑌||^𝑝<𝑙+1≈𝐸||𝑋||^{𝑝(𝑟−1)}<∞.(2.29)
Now we prove (2.5)⇒(2.4). Since max_{1≤𝑗≤𝑛}||𝑎_𝑛𝑗𝑋_𝑗||≤max_{1≤𝑗≤𝑛}|||||_𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑖|||||+max_{1≤𝑗≤𝑛}|||||_𝑗−1_𝑖=1𝑎_𝑛𝑖𝑋_𝑖|||||,(2.30) then from (2.5) we have ^∞_𝑛=1𝑛^𝑟−2𝑃max_{1≤𝑗≤𝑛}||𝑎_𝑛𝑗𝑋_𝑗||>𝑛^1/𝑝<∞.(2.31) Combining with the hypotheses of Theorem 2.1, 𝑃max_{1≤𝑗≤𝑛}||𝑎_𝑛𝑗𝑋_𝑗||>𝑛^1/𝑝⟶0,𝑛⟶∞.(2.32) Thus, for sufficiently large 𝑛, 𝑃max_{1≤𝑗≤𝑛}||𝑎_𝑛𝑗𝑋_𝑗||>𝑛^1/𝑝<12.(2.33) By Lemma 1.6, {𝑎_𝑛𝑗𝑋_𝑗;1≤𝑗≤𝑛,𝑛≥1} is still ND. By applying Lemma 1.10 and (2.1), we obtain ^𝑛_𝑘=1𝑃||𝑎_𝑛𝑘𝑋_𝑘||>𝑛^1/𝑝≤4𝐶𝑃max_{1≤𝑘≤𝑛}||𝑎_𝑛𝑘𝑋_𝑘||>𝑛^1/𝑝.(2.34) Substituting the above inequality in (2.5), we get ^∞_𝑛=1𝑛^𝑛𝑟−2_𝑘=1𝑃||𝑎_𝑛𝑘𝑋_𝑘||>𝑛^1/𝑝<∞.(2.35) So, by the process of proof of 𝐼₄<∞, 𝐸||𝑋||^{𝑝(𝑟−1)}≈^∞_𝑛=1𝑛^𝑛𝑟−2_𝑘=1𝑃||𝑎_𝑛𝑘𝑋_𝑘||>𝑛^1/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 𝑛^−1/2max_{1≤𝑘≤𝑛}||𝐸𝑆^(1)𝑛𝑘||≪𝑛^{−1/2+𝛼−2𝛼(𝑟−1)}⟶0,𝑛⟶∞.(2.37) Taking 𝑀>max(2,2(𝑟−1)), we have 𝐼₁₁≪^∞_𝑛=1𝑛^{−1−(1−2𝛼)(𝑀/2−(𝑟−1))}log^𝑀𝑛<∞.(2.38) For 𝑟−1≤1, taking 𝑀>max(2,2(𝑟−1)/(1−2𝛼+2𝛼(𝑟−1))), we get 𝐼₁₂≪^∞_𝑛=1𝑛^{−1−(1−2𝛼(𝑟−1)−2𝛼)𝑀/2+(𝑟−1)}log^𝑀𝑛<∞.(2.39) For 𝑟−1>1, 𝐸𝑋^2𝑛𝑖<∞ from (2.8). Letting 𝑀>2(𝑟−1)², by the Hölder inequality, 𝐼₁₂≪^∞_𝑛=1𝑛^{𝑟−2−𝑀/2}log^𝑀𝑛^𝑛_𝑖=1𝑎^2𝑛𝑖+𝑛^{2𝛼−2𝛼(𝑟−1)}𝐸𝑎_𝑛𝑖𝑋_𝑖^2(𝑟−1)^𝑀/2≪^∞_𝑛=1𝑛^{𝑟−2−𝑀/2}log^𝑀𝑛⎡⎢⎢⎣^𝑛_𝑖=1𝑎^{2(𝑟−1)𝑛𝑖}^1/(𝑟−1)^𝑛_𝑖=11^{𝑟−2/(𝑟−1)}⎤⎥⎥⎦^𝑀/2≪^∞_𝑛=1𝑛^{−1−𝑀/2(𝑟−1)+(𝑟−1)}log^𝑀𝑛<∞.(2.40) By the definition of 𝐾, 𝐼₂≪^∞_𝑛=1𝑛^{−1−(𝑟−1)(2𝛼𝐾−1)}<∞.(2.41) Similarly to the proof (2.31), we have 𝐼₄≪^∞_𝑙𝑙=1_𝑛=1=𝑛⁻¹𝑙^𝑟−1𝑃𝑙≤||𝑌||²<𝑙+1_∞_𝑙=1≈𝐸𝑙||𝑌^𝑟−1||log_2(𝑟−1)𝑙𝑃𝑙≤||𝑌||²<𝑙+1≈𝐸||𝑋||_2(𝑟−1)log||𝑌||<∞.log||𝑋||(2.42)Equation (2.9)⇒(2.8) Using the same method of the necessary part of Theorem 2.1, we can easily get 𝐸||𝑋||^2(𝑟−1)log||𝑋||≈^∞_𝑛=1𝑛^𝑛𝑟−2_𝑘=1𝑃||𝑎_𝑛𝑘𝑋_𝑘||>𝑛^1/2<∞.(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.