Strong Laws of Large Numbers for Arrays of Rowwise NA and LNQD Random Variables

Abstract

Some strong laws of large numbers and strong convergence properties for arrays of rowwise negatively associated and linearly negative quadrant dependent random variables are obtained. The results obtained not only generalize the result of Hu and Taylor to negatively associated and linearly negative quadrant dependent random variables, but also improve it.

1. Introduction

Let {𝑋_𝑛}_𝑛∈ℕ be a sequence of independent distributed random variables. The Marcinkiewicz-Zygmund strong law of large numbers (SLLN) provides that1𝑛^𝑛1/𝛼_𝑖=11𝑋_𝑖−𝐸𝑋_𝑖⟶0a.s.for1≤𝛼<2,𝑛^𝑛1/𝛼_𝑖=1𝑋_𝑖⟶0a.s.for0<𝛼<1as𝑛⟶∞(1.1) if and only if 𝐸|𝑋|^𝛼<∞. The case 𝛼=1 is due to Kolmogorov. In the case of independence (but not necessarily identically distributed), Hu and Taylor [1] proved the following strong law of large numbers.

Theorem 1.1. Let {𝑋_𝑛𝑖,1≤𝑖≤𝑛,𝑛≥1} be a triangular array of rowwise independent random variables. Let {𝑎_𝑛}_𝑛∈ℕ be a sequence of positive real numbers such that 0<𝑎_𝑛↑∞. Let 𝜓(𝑡) be a positive, even function such that 𝜓(𝑡)/|𝑡|^𝑝 is an increasing function of |𝑡| and 𝜓(𝑡)/|𝑡|^𝑝+1 is a decreasing function of |𝑡|, respectively, that is, 𝜓(𝑡)|𝑡|^𝑝↑,𝜓(𝑡)|𝑡|^𝑝+1↓,as|𝑡|↑,(1.2) for some positive integer 𝑝. If 𝑝≥2 and 𝐸𝑋_𝑛𝑖=0,_∞_𝑛𝑛=1_𝑖=1𝐸𝜓||𝑋^𝑛𝑖||𝜓||𝑎_𝑛||<∞,_∞_𝑛=1^𝑛_𝑖=1𝐸𝑋^𝑛𝑖𝑎_𝑛²^2𝑘<∞,(1.3) where 𝑘 is a positive integer, then 1𝑎_{𝑛𝑛𝑖=1}𝑋_𝑛𝑖⟶0a.s.(1.4)

Definition 1.2 (cf. [2]). A finite family of random variables {𝑋_𝑛}_𝑛∈ℕ is said to be negatively associated (NA, in short) if, for any disjoint subsets 𝐴 and 𝐵 of {1,2,…,𝑛} and any real coordinate-wise nondecreasing functions 𝑓 on on ℝ^𝐴 and 𝑔 on ℝ^𝐵, Cov𝑓𝑋_𝑖,𝑖∈𝐴,𝑔𝑌_𝑗,𝑗∈𝐵≤0,(1.5) whenever the covariance exists. An infinite family of random variables is NA if every finite subfamily is NA. This concept was introduced by Joag-Dev and Proschan [2].

Definition 1.3 (cf. [3, 4]). Two random variables 𝑋 and 𝑌 are said to be negative quadrant dependent (NQD, in short) if, for any 𝑥,𝑦∈ℝ, 𝑃(𝑋<𝑥,𝑌<𝑦)≤𝑃(𝑋<𝑥)𝑃(𝑌<𝑦).(1.6) A sequence {𝑋_𝑛}_𝑛∈ℕ of random variables is said to be pairwise NQD if 𝑋_𝑖 and 𝑋_𝑗 are NQD for all 𝑖,𝑗∈ℕ⁺ and 𝑖≠𝑗.

Definition 1.4 (cf. [5]). A sequence {𝑋_𝑛}_𝑛∈ℕ of random variables is said to be linearly negative quadrant dependent (LNQD, in short) if, for any disjoint subsets 𝐴,𝐵∈ℤ⁺ and positive 𝑟^𝑗𝑠, _𝑘_∈𝐴𝑟_𝑘𝑋_𝑘,_𝑗∈𝐵𝑟_𝑗𝑋_𝑗areNQD.(1.7)

Remark 1.5. It is easily seen that if {𝑋_𝑛}_𝑛∈ℕ is a sequence of LNQD random variables, then {𝑎𝑋_𝑛+𝑏}_𝑛∈ℕ is still a sequence of LNQD random variables, where 𝑎 and 𝑏 are real numbers.

The NA property has aroused wide interest because of numerous applications in reliability theory, percolation theory, and multivariate statistical analysis. In the past decades, a lot of effort was dedicated to proving the limit theorems of NA random variables. A Kolmogorov-type strong law of large numbers of NA random variables was established by Matuła in [6], which is the same as I.I.D. sequence, and Marcinkiewicz-type strong law of large Numbers was obtained by Su and Wang [7] for NA random variable sequence with assumptions of identical distribution; Yang et al. [8] gave the strong law of large Numbers of a general method.

The concept of LNQD sequence was introduced by Newman [5]. Some applications for LNQD sequence have been found. See, for example, Newman [5] who established the central limit theorem for a strictly stationary LNQD process. Wang and Zhang [9] provided uniform rates of convergence in the central limit theorem for LNQD sequence. Ko et al. [10] obtained the Hoeffding-type inequality for LNQD sequence. Ko et al. [11] studied the strong convergence for weighted sums of LNQD arrays, and so forth.

The aim of this paper is to establish a strong law of large numbers for arrays of NA and LNQD random variables. The result obtained not only extends Theorem 1.1 for independent sequence above to the case of NA and LNQD random variables sequence, but also improves it.

Lemma 1.6 (cf. [12]). Let {𝑋_𝑛,𝑛≥1} be NA random variables, 𝐸𝑋_𝑛=0, 𝐸|𝑋_𝑛|^𝑞<∞, 𝑛≥1, 𝑞≥2. Then, there exists a positive constant 𝑐 such that 𝐸max_{1≤𝑖≤𝑛}|||||_𝑘_𝑖=1𝑋_𝑖|||||^𝑞≤𝑐⎡⎢⎢⎣^𝑛_𝑖=1𝐸||𝑋_𝑖||^q+^𝑛_𝑖=1𝐸𝑋_𝑖2^𝑞/2⎤⎥⎥⎦,∀𝑛≥1.(1.8)

Let 𝑐 denote a positive constant which is not necessary the same in its each appearance.

Lemma 1.7 (cf. [3, 4]). Let random variables 𝑋 and 𝑌 be NQD, then (((123)))ar𝐸𝑃ief𝑋(𝑓N𝑋𝑌Qa<≤nDd.𝑥𝐸𝑔,𝑋𝑌a𝐸re<𝑌;b𝑦o)th≤n𝑃on(𝑋dec<re𝑥a)s𝑃ing(𝑌(<or𝑦b)o;thnonincreasing)functions,then𝑓(𝑋)and𝑔(𝑌)(1.9)

Lemma 1.8. Let {𝑋_𝑛,𝑛≥1} be LNQD random variables sequences with mean zero and 0<𝐵_𝑛=∑^𝑛𝑘=1𝐸𝑋^2𝑘. Then, 𝑃||𝑆_𝑛||≥𝑥≤^𝑛_𝑘=1𝑃||𝑆_𝑘||≥𝑦+2exp𝑥𝑦−𝑥𝑦log1+𝐵𝑥𝑦_𝑛,(1.10) for any 𝑥>0, 𝑦>0.

This lemma is easily proved by following Fuk and Nagaev [13]. Here, we omit the details of the proof.

2. Main Results

Theorem 2.1. Let {𝑋_𝑛𝑖;𝑖≥1,𝑛≥1} be an array of rowwise NA random variables. Let {𝑎_𝑛}_𝑛∈ℕ be a sequence of positive real numbers such that 0<𝑎_𝑛↑∞. Let 𝜓(𝑡) be a positive, even function such that 𝜓(𝑡)/|𝑡| is an increasing function of |𝑡| and 𝜓(𝑡)/|𝑡|^𝑝 is a decreasing function of |𝑡|, respectively, that is, 𝜓(𝑡)|𝑡|↑,𝜓|𝑡|(^𝑝𝑡)↓,as|𝑡|↑(2.1) for some nonnegative integer 𝑃. If 𝑝≥2 and 𝐸𝑋_𝑛𝑖=0,_∞_𝑛𝑛=1_𝑖=1𝐸𝜓||𝑋^𝑛𝑖||𝜓||𝑎_𝑛||<∞,_∞_𝑛=1^𝑛_𝑖=1𝐸𝑋^𝑛𝑖𝑎_𝑛²^𝑣/2<∞,(2.2) where 𝑣 is a positive integer and 𝑣≥𝑝, then ^∞_𝑛=1𝑃⎛⎜⎜⎝max_{1≤𝑘≤𝑛}|||||1𝑎_{𝑛𝑘𝑖=1}𝑋_𝑛𝑖|||||>𝜀⎞⎟⎟⎠<∞,forany𝜀>0.(2.3)

Proof of Theorem 2.1. For all 𝑖≥1, let𝑋^𝑖(𝑛)=−𝑎_𝑛𝐼(𝑋_𝑛𝑖<−𝑎_𝑛)+𝑋_𝑛𝑖𝐼(|𝑋_𝑛𝑖|≤𝑎_𝑛)+𝑎_𝑛𝐼(𝑋_𝑛𝑖>𝑎_𝑛), 𝑇^𝑗(𝑛)=(1/𝑎_𝑛)∑^𝑗𝑖=1(𝑋^𝑖(𝑛)−𝐸𝑋^𝑖(𝑛)), then, for all 𝜀>0, 𝑃⎛⎜⎜⎝max_{1≤𝑘≤𝑛}|||||1𝑎_{𝑛𝑘𝑖=1}𝑋_𝑛𝑖|||||>𝜀⎞⎟⎟⎠≤𝑃max_{1≤𝑗≤𝑛}||𝑋_𝑛𝑗||>𝑎_𝑛+𝑃⎛⎜⎜⎝max_{1≤𝑗≤𝑛}||𝑇^𝑗(𝑛)||>𝜀−max_{1≤𝑗≤𝑛}|||||1𝑎_{𝑛𝑗𝑖=1}𝐸𝑋^𝑖(𝑛)|||||⎞⎟⎟⎠.(2.4) First, we show that max_{1≤𝑗≤𝑛}|||||1𝑎_{𝑛𝑗𝑖=1}𝐸𝑋^𝑖(𝑛)|||||⟶0,as𝑛⟶∞.(2.5) In fact, by 𝐸𝑋_𝑛𝑖=0, 𝜓(𝑡)/|𝑡|↑ as |𝑡|↑ and Σ^∞𝑛=1Σ^𝑛𝑖=1𝐸(𝜓(|𝑋_𝑛𝑖|)/𝜓(𝑎_𝑛))<∞, then max_{1≤𝑗≤𝑛}|||||1𝑎_{𝑛𝑗𝑖=1}𝐸𝑋^𝑖(𝑛)|||||≤max_{1≤𝑗≤𝑛}1𝑎_𝑛⎛⎜⎜⎝|||||^𝑗_𝑖=1𝐸𝑋_𝑛𝑖𝐼||𝑋_𝑛𝑖||≤𝑎_𝑛|||||+|||||^𝑗_𝑖=1≤max_{1≤𝑗≤𝑛}𝐸𝑎_𝑛𝐼||𝑋_𝑛𝑖||>𝑎_𝑛|||||⎞⎟⎟⎠1𝑎_𝑛⎛⎜⎜⎝^𝑗_𝑖=1||𝐸𝑋_𝑛𝑖𝐼||𝑋_𝑛𝑖||≤𝑎_𝑛||+|||||^𝑗_𝑖=1=max_{1≤𝑗≤𝑛}𝐸𝑎_𝑛𝐼||𝑋_𝑛𝑖||>𝑎_𝑛|||||⎞⎟⎟⎠1𝑎_𝑛⎛⎜⎜⎝^𝑗_𝑖=1||𝐸𝑋_𝑛𝑖𝐼||𝑋_𝑛𝑖||>𝑎_𝑛||+|||||^𝑗_𝑖=1≤2_𝑛𝐸𝑎_𝑛𝐼||𝑋_𝑛𝑖||>𝑎_𝑛|||||⎞⎟⎟⎠_𝑖=1𝐸||𝑋_𝑛𝑖||𝐼||𝑋_𝑛𝑖||>𝑎_𝑛𝑎_𝑛≤2^𝑛_𝑖=1𝐸𝜓||𝑋_𝑛𝑖||𝐼||𝑋_𝑛𝑖||>𝑎_𝑛≤2_𝑛𝜓𝑎_𝑛_𝑖=1𝐸𝜓||𝑋_𝑛𝑖||𝜓𝑎_𝑛⟶0,as𝑛⟶∞.(2.6) From (2.4) and (2.5), it follows that, for 𝑛 sufficiently large, 𝑃⎛⎜⎜⎝max_{1≤𝑘≤𝑛}|||||1𝑎_{𝑛𝑘𝑖=1}𝑋_𝑛𝑖|||||>𝜀⎞⎟⎟⎠≤_𝑛_𝑗=1𝑃||𝑋_𝑛𝑗||>𝑎_𝑛+𝑃max_{1≤𝑗≤𝑛}||𝑇^𝑗(𝑛)||>𝜀2.(2.7) Hence, we need only to prove that 𝐼=∶^∞_𝑛𝑛=1_𝑗=1𝐼𝐼=∶_∞𝑃||𝑋_𝑛𝑗||>𝑎_𝑛<∞,_𝑛=1𝑃max_{1≤𝑗≤𝑛}||𝑇^𝑗(𝑛)||>𝜀2<∞.(2.8) From the fact that Σ^∞𝑛=1Σ^𝑛𝑖=1𝐸(𝜓(|𝑋_𝑛𝑖|)/𝜓(𝑎_𝑛))<∞, it follows easily that 𝐼=^∞_𝑛𝑛=1_𝑗=1𝑃||𝑋_𝑛𝑗||>𝑎_𝑛≤^∞_𝑛𝑛=1_𝑗=1𝐸𝜓||𝑋^𝑛𝑗||𝜓𝑎_𝑛<∞.(2.9) By 𝑣≥𝑝 and 𝜓(𝑡)/|𝑡|^𝑝↓ as |𝑡|↑, then 𝜓(𝑡)/|𝑡|^𝑣↓ as |𝑡|↑.
By the Markov inequality, Lemma 1.6, and ∑^∞𝑛=1(∑^𝑛𝑖=1𝐸(𝑋_𝑛𝑖/𝑎_𝑛)²)^𝑣/2<∞, we have 𝐼𝐼=^∞_𝑛=1𝑃max_{1≤𝑗≤𝑛}||𝑇^𝑗(𝑛)||>𝜀2≤^∞_𝑛=12𝜀^−𝑣𝐸max_{1≤𝑗≤𝑛}||𝑇^𝑗(𝑛)||^𝑣≤𝑐𝜀2^∞−𝑣_𝑛=11𝑎^𝑣𝑛⎡⎢⎢⎣^𝑛_𝑗=1𝐸||𝑋^𝑗(𝑛)||²^𝑣/2+^𝑛_𝑗=1𝐸||𝑋^𝑗(𝑛)||^𝑣≤𝑐_∞⎤⎥⎥⎦_𝑛=11𝑎^𝑣𝑛𝑛_𝑗=1𝐸||𝑋^𝑗(𝑛)||^𝑣+𝑐^∞_𝑛=11𝑎^𝑣𝑛^𝑛_𝑗=1𝐸||𝑋^𝑗(𝑛)||²^𝑣/2≤𝑐^∞_𝑛=11𝑎^𝑣𝑛𝑛_𝑗=1𝐸||𝑋_𝑛𝑗||^𝑣𝐼||𝑋_𝑛𝑗||≤𝑎_𝑛+𝐼+𝑐^∞_𝑛=11𝑎^𝑣𝑛^𝑛_𝑗=1𝐸||𝑋^𝑗(𝑛)||²^𝑣/2≤𝑐^∞_𝑛𝑛=1_𝑖=1𝐸𝜓||𝑋^𝑛𝑖||𝜓𝑎_𝑛+𝑐^∞_𝑛=11𝑎^𝑣𝑛^𝑛_𝑗=1𝐸||𝑋^𝑗(𝑛)||²^𝑣/2≤𝑐^∞_𝑛𝑛=1_𝑖=1𝐸𝜓||𝑋^𝑛𝑖||𝜓𝑎_𝑛+𝑐^∞_𝑛=1^𝑛_𝑗=1𝐸𝑋^𝑛𝑖𝑎_𝑛²^𝑣/2<∞.(2.10) Now we complete the proof of Theorem 2.1.

Corollary 2.2. Under the conditions of Theorem 2.1, then 1𝑎_{𝑛𝑛𝑖=1}𝑋_𝑛𝑖⟶0a.s.(2.11)

Proof of Corollary 2.2. By Theorem 2.1, the proof of Corollary 2.2 is obvious.

Remark 2.3. Corollary 2.2 not only generalizes the result of Hu and Taylor to NA random variables, but also improves it.

Theorem 2.4. Let {𝑋_𝑛𝑖;𝑖≥1,𝑛≥1} be an array of rowwise LNQD random variables. Let {𝑎_𝑛}_𝑛∈ℕ be a sequence of positive real numbers such that 0<𝑎_𝑛↑∞. Let 𝜓(𝑡) be a positive, even function such that 𝜓(𝑡)/|𝑡| is an increasing function of |𝑡| and 𝜓(𝑡)/|𝑡|^𝑝 is a decreasing function of |𝑡|, respectively, that is, 𝜓(𝑡)|𝑡|↑,𝜓|𝑡|(^𝑝𝑡)↓,as|𝑡|↑(2.12) for some positive integer 𝑝. If 1<𝑝≤2 and 𝐸𝑋_𝑛𝑖=0,_∞_𝑛𝑛=1_𝑖=1𝐸𝜓||𝑋^𝑛𝑖||𝜓𝑎_𝑛<∞,(2.13) then ^∞_𝑛=1𝑃⎛⎜⎜⎝|||||1𝑎_{𝑛𝑛𝑖=1}𝑋_𝑛𝑖|||||>𝜀⎞⎟⎟⎠<∞,forany𝜀>0.(2.14)

Proof of Theorem 2.4. For any 1≤𝑘≤𝑛, 𝑛≥1, let 𝑍_𝑛𝑘𝑌_𝑛𝑘=−𝑎_𝑛𝐼𝑋_𝑛𝑘<−𝑎_𝑛+𝑋_𝑛𝑘𝐼||𝑋_𝑛𝑘||≤𝑎_𝑛+𝑎_𝑛𝐼𝑋_𝑛𝑘>𝑎_𝑛,=𝑋_𝑛𝑘−𝑌_𝑛𝑘=𝑋_𝑛𝑘+𝑎_𝑛𝐼𝑋_𝑛𝑘<−𝑎_𝑛+𝑋_𝑛𝑘−𝑎_𝑛𝐼𝑋_𝑛𝑘>𝑎_𝑛.(2.15) To prove (2.14), it suffices to show that 1𝑎_{𝑛𝑛𝑘=1}1𝑍_𝑛𝑘⟶0completely,(2.16)𝑎_{𝑛𝑛𝑘=1}𝑌_𝑛𝑘1−𝐸𝑌_𝑛𝑘⟶0completely,(2.17)𝑎_{𝑛𝑛𝑘=1}𝐸𝑌_𝑛𝑘⟶0as𝑛⟶∞.(2.18)
Firstly, we prove (2.16): ^∞_𝑛=1𝑃⎛⎜⎜⎝1𝑎_𝑛|||||_𝑛_𝑘=1𝑍_𝑛𝑘|||||>𝜀⎞⎟⎟⎠≤_∞_𝑛=1𝐸||∑^𝑛𝑘=1𝑍_𝑛𝑘||𝑎_𝑛𝜀≤𝐶^∞_𝑛𝑛=1_𝑘=1𝐸||𝑋^𝑛𝑘||𝐼||𝑋_𝑛𝑘||>𝑎_𝑛𝑎_𝑛≤𝐶^∞_𝑛𝑛=1_𝑖=1𝐸𝜓||𝑋^𝑛𝑖||<∞.𝜓𝑎_𝑛(2.19)
Secondly, we prove (2.17). By Lemma 1.7, we know that {𝑌_𝑛𝑘−𝐸𝑌_𝑛𝑘,1≤𝑘≤𝑛,𝑛≥1} is an array of rowwise LNQD mean zero random variables. Let 𝐵^𝑛=∑^𝑛𝑘=1𝐸(𝑌_𝑛𝑘−𝐸𝑌_𝑛𝑘)². Take 𝑥=𝜀𝑎_𝑛, 𝑦=2𝜀𝑎_𝑛/𝑣, and 𝑣≥1. By Lemma 1.8, for all 𝜀>0, ^∞_𝑛=1𝑃⎛⎜⎜⎝1𝑎_𝑛|||||_𝑛_𝑘=1𝑌_𝑛𝑘−𝐸𝑌_𝑛𝑘|||||>𝜀⎞⎟⎟⎠≤^∞_𝑛𝑛=1_𝑘=1𝑃||𝑌_𝑛𝑘−𝐸𝑌_𝑛𝑘||>𝜀𝑎^𝑛𝜂+𝐶^∞_𝑛=1𝐵^𝑛+𝐵𝜀^𝑛2𝑎^2𝑛/𝜂^𝜂∶=𝐼₁+𝐼₂.(2.20) From (2.12), (2.13), the Markov inequality, and 𝐶_𝑟-inequality, 𝐼₁=^∞_𝑛𝑛=1_𝑘=1𝑃||𝑌_𝑛𝑘−𝐸𝑌_𝑛𝑘||>𝜀𝑎^𝑛𝜂≤𝐶^∞_𝑛𝑛=1_𝑘=1𝐸||𝑌_𝑛𝑘−𝐸𝑌_𝑛𝑘||^𝑝𝑎^𝑝𝑛≤𝐶^∞_𝑛𝑛=1_𝑘=1𝐸||𝑌_𝑛𝑘||^𝑝𝑎^𝑝𝑛≤𝐶^∞_𝑛𝑛=1_𝑘=1𝐸𝜓^𝑘||𝑌_𝑛𝑘||𝜓_𝑘𝑎_𝑛≤𝐶^∞_𝑛𝑛=1_𝑘=1𝐸𝜓^𝑘||𝑋_𝑛𝑘||𝜓_𝑘𝑎_𝑛<∞.(2.21) Note that |𝑌_𝑛𝑘|≤|𝑋_𝑛𝑘|, 𝜂≥1 and 1<𝑝≤2. From (2.12), (2.13), and the 𝐶_𝑟-inequality, 𝐼₂≤𝐶^∞_𝑛=1^𝑛_𝑘=1𝑎^𝑛−2𝐸𝑌_𝑛𝑘−𝐸𝑌_𝑛𝑘²^𝜂≤𝐶^∞_𝑛=1^𝑛_𝑘=1𝐸||𝑌_𝑛𝑘||^𝑝𝑎^𝑝𝑛^𝜂≤𝐶^∞_𝑛𝑛=1_𝑘=1𝐸||𝑌_𝑛𝑘||^𝑝𝑎^𝑝𝑛^𝜂≤𝐶^∞_𝑛𝑛=1_𝑘=1𝐸𝜓^𝑘||𝑌_𝑛𝑘||𝜓_𝑘𝑎_𝑛^𝜂≤𝐶^∞_𝑛𝑛=1_𝑘=1𝐸𝜓^𝑘||𝑋_𝑛𝑘||𝜓_𝑘𝑎_𝑛^𝜂<∞.(2.22)
Finally, we prove (2.18). For 1≤𝑘≤𝑛, 𝑛≥1, 𝐸𝑋_𝑛𝑘=0, then 𝐸𝑌_𝑛𝑘=−𝐸𝑍_𝑛𝑘. From the definition of 𝑍_𝑛𝑘 if 𝑋_𝑛𝑘>𝑎_𝑛, then 0<𝑍_𝑛𝑘=𝑋_𝑛𝑘−𝑎_𝑛<𝑋_𝑛𝑘, if 𝑋_𝑛𝑘<−𝑎_𝑛, then 𝑋_𝑛𝑘<𝑍_𝑛𝑘=𝑋_𝑛𝑘+𝑎_𝑛<0. So |𝑍_𝑛𝑘|≤|𝑋_𝑛𝑘|𝐼(|𝑋_𝑛𝑘|>𝑎_𝑛). Consequently, 1𝑎_𝑛|||||_𝑛_𝑘=1𝐸𝑌_𝑛𝑘|||||=1𝑎_𝑛|||||_𝑛_𝑘=1𝐸𝑍_𝑛𝑘|||||≤_𝑛_𝑘=1𝐸||𝑍_𝑛𝑘||𝑎_𝑛≤^𝑛_𝑘=1𝐸||𝑋_𝑛𝑘||𝐼||𝑋_𝑛𝑘||>𝑎_𝑛𝑎_𝑛≤^𝑛_𝑘=1𝐸𝜓_𝑘𝑋_𝑛𝑘𝜓_𝑘𝑎_𝑛⟶0as𝑛⟶∞.(2.23) The proof is completed.

Theorem 2.5. Let {𝑋_𝑛𝑖;𝑖≥1,𝑛≥1} be an array of rowwise LNQD random variables. Let {𝑎_𝑛}_𝑛∈ℕ, be a sequence of positive real numbers such that 0<𝑎_𝑛↑∞. Let {𝜓_𝑛(𝑡)}_𝑛∈ℕ, be a sequence of positive even functions and satisfy (2.12) for 𝑝>2. Suppose that ^∞_𝑛=1^𝑛_𝑖=1𝐸𝑋^𝑛𝑖𝑎_𝑛²^𝑣/2<∞,(2.24) where 𝑣 is a positive integer, 𝑣≥𝑝, the conditions (2.13) and (2.24) imply (2.24).

Proof of Theorem 2.5. Following the notations and the methods of the proof in Theorem 2.4, (2.16), (2.18), and 𝐼₁<∞ hold. So, we only need to show that 𝐼₂<∞. Let 𝜂>𝑣/2. By (2.24), we have 𝐼₂≤^∞_𝑛=1^𝑛_𝑘=1𝑎^𝑛−2𝐸𝑌_𝑛𝑘−𝐸𝑌_𝑛𝑘²^𝜂≤𝐶^∞_𝑛=1⎡⎢⎢⎣^𝑛_𝑘=1𝐸𝑌^2𝑛𝑘𝑎^2𝑛^𝑣/2⎤⎥⎥⎦^2𝜂/𝑣≤𝐶⎡⎢⎢⎣^∞_𝑛=1^𝑛_𝑘=1𝐸𝑋^2𝑛𝑘𝑎^2𝑛^𝑣/2⎤⎥⎥⎦^2𝜂/𝑣<∞.(2.25) The proof is completed.

Corollary 2.6. Under the conditions of Theorem 2.4 or Theorem 2.5, then 1𝑎_{𝑛𝑛𝑖=1}𝑋_𝑛𝑖⟶0a.s.(2.26)

Remark 2.7. Because of the maximal inequality of LNQD, the result of LNQD we have obtained generalizes and improves the result of Hu and Taylor.

Acknowledgments

The work is supported by the National Natural Science Foundation of China (11061012), the Guangxi China Science Foundation (2011GXNSFA018147), and the Innovation Project of Guangxi Graduate Education (2010105960202M32).