On Complete Convergence for Weighted Sums of Arrays of Dependent Random Variables

Abstract

A rate of complete convergence for weighted sums of arrays of rowwise independent random variables was obtained by Sung and Volodin (2011). In this paper, we extend this result to negatively associated and negatively dependent random variables. Similar results for sequences of 𝜑-mixing and 𝜌^∗-mixing random variables are also obtained. Our results improve and generalize the results of Baek et al. (2008), Kuczmaszewska (2009), and Wang et al. (2010).

1. Introduction

The concept of complete convergence of a sequence of random variables was introduced by Hsu and Robbins [1]. A sequence {𝑋_𝑛,𝑛≥1} of random variables converges completely to the constant 𝜃 if^∞_𝑛=1𝑃||𝑋_𝑛−𝜃||>𝜖<∞∀𝜖>0.(1.1) In view of the Borel-Cantelli lemma, this implies that 𝑋_𝑛→𝜃 almost surely. Therefore, the complete convergence is a very important tool in establishing almost sure convergence of summation of random variables as well as weighted sums of random variables. Hsu and Robbins [1] proved that the sequence of arithmetic means of independent and identically distributed random variables converges completely to the expected value if the variance of the summands is finite. Erdös [2] proved the converse. The result of Hsu-Robbins-Erdös is a fundamental theorem in probability theory and has been generalized and extended in several directions by many authors.

Ahmed et al. [3] obtained complete convergence for weighted sums of arrays of rowwise independent Banach-space-valued random elements.

We recall that the array {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} of random variables is said to be stochastically dominated by a random variable 𝑋 if𝑃||𝑋_𝑛𝑖||>𝑥≤𝐶𝑃||𝑋||>𝑥∀𝑥>0,∀𝑖≥1,𝑛≥1,(1.2) where 𝐶 is a positive constant.

Theorem 1.1 (Ahmed et al. [3]). Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise independent random elements which are stochastically dominated by a random variable 𝑋. Let {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of constants satisfying sup_𝑖≥1||𝑎_𝑛𝑖||=𝑂(𝑛^−𝛾)forsome𝛾>0,(1.3)_∞_𝑖=1||𝑎_𝑛𝑖||=𝑂(𝑛^𝛼)forsome𝛼<𝛾.(1.4) Suppose that there exists 𝛿>1 such that 1+𝛼/𝛾<𝛿≤2. Let 𝛽≠−1−𝛼 and 𝜈=max{1+(1+𝛼+𝛽)/𝛾,𝛿}. If 𝐸|𝑋|^𝜈<∞ and ∑^∞𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖→0 in probability, then ^∞_𝑛=1𝑛^𝛽𝑃‖‖‖‖^∞_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖‖‖‖‖>𝜖<∞∀𝜖>0.(1.5)

Note that there was a typographical error in Ahmed et al. [3] (the relation 𝛿>0 should be 𝛿>1). If 𝛽<−1, then the conclusion of Theorem 1.1 is immediate. Hence, Theorem 1.1 is of interest only for 𝛽≥−1.

Baek et al. [4] extended Theorem 1.1 to negatively associated random variables.

Theorem 1.2 (Baek et al. [4]). Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise negatively associated random variables which are stochastically dominated by a random variable 𝑋. Let {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of constants satisfying (1.3) and (1.4). Suppose that there exists 𝛿>0 such that 1+𝛼/𝛾<𝛿≤2. Let 𝛽≥−1 and 𝜈=max{1+(1+𝛼+𝛽)/𝛾,𝛿}. If 𝐸𝑋_𝑛𝑖=0, for all 𝑖≥1 and 𝑛≥1, and 𝐸||𝑋𝐸||||l𝑋og||||𝑋||<∞,for1+𝛼+𝛽=0,_𝜈<∞,for1+𝛼+𝛽>0,(1.6) then ^∞_𝑛=1𝑛^𝛽𝑃⎛⎜⎜⎝|||||^∞_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖|||||>𝜖⎞⎟⎟⎠<∞∀𝜖>0.(1.7)

Sung and Volodin [5] improved Theorem 1.1 as follows.

Theorem 1.3 (Sung and Volodin [5]). Suppose that 𝛽≥−1. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise independent random elements which are stochastically dominated by a random variable 𝑋. Let {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of constants satisfying (1.3) and (1.4). Assume that ∑^∞𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖→0 in probability. If 𝐸𝐸||𝑋||𝑋||||log||𝑋||<∞,for1+𝛼+𝛽=0,_{1+(1+𝛼+𝛽)/𝛾}<∞,for1+𝛼+𝛽>0,(1.8) then (1.5) holds.

In this paper, we extend Theorem 1.3 to negatively associated and negatively dependent random variables. We also obtain similar results for sequences of 𝜑-mixing and 𝜌^∗-mixing random variables. Our results improve and generalize the results of Baek et al. [4], Kuczmaszewska [6], and Wang et al. [7].

Throughout this paper, the symbol 𝐶 denotes a positive constant which is not necessarily the same one in each appearance. It proves convenient to define log𝑥=max{1,ln𝑥}, where ln𝑥 denotes the natural logarithm.

2. Preliminaries

In this section, we present some background materials which will be useful in the proofs of our main results.

The following lemma is well known, and its proof is standard.

Lemma 2.1. Let {𝑋_𝑛,𝑛≥1} be a sequence of random variables which are stochastically dominated by a random variable 𝑋. For any 𝛼>0 and 𝑏>0, the following statements hold:(i)𝐸|𝑋_𝑛|^𝛼𝐼(|𝑋_𝑛|≤𝑏)≤𝐶{𝐸|𝑋|^𝛼𝐼(|𝑋|≤𝑏)+𝑏^𝛼𝑃(|𝑋|>𝑏)},(ii)𝐸|𝑋_𝑛|^𝛼𝐼(|𝑋_𝑛|>𝑏)≤𝐶𝐸|𝑋|^𝛼𝐼(|𝑋|>𝑏).

Lemma 2.2 (Sung [8]). Let 𝑋 be a random variable with 𝐸|𝑋|^𝑟<∞ for some 𝑟>0. For any 𝑡>0, the following statements hold:(i)∑^∞𝑛=1𝑛^{−1−𝑡𝛿}𝐸|𝑋|^𝑟+𝛿𝐼(|𝑋|≤𝑛^𝑡)≤𝐶𝐸|𝑋|^𝑟 for any 𝛿>0,(ii)∑^∞𝑛=1𝑛^{−1+𝑡𝛿}𝐸|𝑋|^𝑟−𝛿𝐼(|𝑋|>𝑛^𝑡)≤𝐶𝐸|𝑋|^𝑟 for any 𝛿>0 such that 𝑟−𝛿>0,(iii)∑^∞𝑛=1𝑛^{−1+𝑡𝑟}𝑃(|𝑋|>𝑛^𝑡)≤𝐶𝐸|𝑋|^𝑟.

The Rosenthal-type inequality plays an important role in establishing complete convergence. The Rosenthal-type inequalities for sequences of dependent random variables have been established by many authors.

The concept of negatively associated random variables was introduced by Alam and Saxena [9] and carefully studied by Joag-Dev and Proschan [10]. A finite family of random variables {𝑋_𝑖,1≤𝑖≤𝑛} is said to be negatively associated if for every pair of disjoint subsets 𝐴 and 𝐵 of {1,2,…,𝑛},Cov𝑓₁𝑋_𝑖,𝑖∈𝐴,𝑓₂𝑋_𝑗,𝑗∈𝐵≤0,(2.1) whenever 𝑓₁ and 𝑓₂ are coordinatewise increasing and the covariance exists. An infinite family of random variables is negatively associated if every finite subfamily is negatively associated.

The following lemma is a Rosenthal-type inequality for negatively associated random variables.

Lemma 2.3 (Shao [11]). Let {𝑋_𝑛,𝑛≥1} be a sequence of negatively associated random variables with 𝐸𝑋_𝑛=0 and 𝐸|𝑋_𝑛|^𝑞<∞ for some 𝑞≥2 and all 𝑛≥1. Then there exists a constant 𝐶>0 depending only on 𝑞 such that 𝐸⎛⎜⎜⎝max_{1≤𝑗≤𝑛}|||||_𝑗_𝑖=1𝑋_𝑖|||||^𝑞⎞⎟⎟⎠≤𝐶⎧⎪⎨⎪⎩^𝑛_𝑖=1𝐸||𝑋_𝑖||^𝑞+^𝑛_𝑖=1𝐸𝑋^2𝑖^𝑞/2⎫⎪⎬⎪⎭.(2.2)

The concept of negatively dependent random variables was given by Lehmann [12]. A finite family of random variables {𝑋₁,…,𝑋_𝑛} is said to be negatively dependent (or negatively orthant dependent) if for each 𝑛≥2, the following two inequalities hold:𝑃𝑋₁≤𝑥₁,…,𝑋_𝑛≤𝑥_𝑛≤^𝑛_𝑖=1𝑃𝑋₁𝑃𝑋_𝑖≤𝑥_𝑖,>𝑥₁,…,𝑋_𝑛>𝑥_𝑛≤^𝑛_𝑖=1𝑃𝑋_𝑖>𝑥_𝑖,(2.3) for all real numbers 𝑥₁,…,𝑥_𝑛. An infinite family of random variables is negatively dependent if every finite subfamily is negatively dependent.

Obviously, negative association implies negative dependence, but the converse is not true.

The following lemma is a Rosenthal-type inequality for negatively dependent random variables.

Lemma 2.4 (Asadian et al. [13]). Let {𝑋_𝑛,𝑛≥1} be a sequence of negatively dependent random variables with 𝐸𝑋_𝑛=0 and 𝐸|𝑋_𝑛|^𝑞<∞ for some 𝑞≥2 and all 𝑛≥1. Then there exists a constant 𝐶>0 depending only on 𝑞 such that 𝐸|||||^𝑛_𝑖=1𝑋_𝑖|||||^𝑞≤𝐶⎧⎪⎨⎪⎩^𝑛_𝑖=1𝐸||𝑋_𝑖||^𝑞+^𝑛_𝑖=1𝐸𝑋^2𝑖^𝑞/2⎫⎪⎬⎪⎭.(2.4)

For a sequence {𝑋_𝑛,𝑛≥1} of random variables defined on a probability space (Ω,ℱ,𝑃), let ℱ^𝑚𝑛 denote the 𝜎-algebra generated by the random variables 𝑋_𝑛,𝑋_𝑛+1,…,𝑋_𝑚. Define the 𝜑-mixing coefficients by𝜑(𝑛)=sup_𝑘≥1sup||𝑃(𝐵∣𝐴)−𝑃(𝐵)||,𝐴∈ℱ^𝑘1,𝑃(𝐴)≠0,𝐵∈ℱ^∞𝑘+𝑛.(2.5) The sequence {𝑋_𝑛,𝑛≥1} is called 𝜑-mixing (or 𝜙-mixing) if 𝜑(𝑛)→0 as 𝑛→∞.

For any 𝑆⊂ℕ, let ℱ_𝑆=𝜎(𝑋_𝑖,𝑖∈𝑆). Define the 𝜌^∗-mixing coefficients by𝜌^∗(𝑛)=supcorr(𝑓,𝑔),(2.6) where the supremum is taken over all 𝑆,𝑇⊂ℕ with dist(𝑆,𝑇)≥𝑛, and all 𝑓∈𝐿₂(ℱ_𝑆) and 𝑔∈𝐿₂(ℱ_𝑇). The sequence {𝑋_𝑛,𝑛≥1} is called 𝜌^∗-mixing (or 𝜌̃-mixing) if there exists 𝑘∈ℕ such that 𝜌^∗(𝑘)<1.

Note that if {𝑋_𝑛,𝑛≥1} is a sequence of independent random variables, then 𝜑(𝑛)=0 and 𝜌^∗(𝑛)=0 for all 𝑛≥1.

The following lemma is a Rosenthal-type inequality for 𝜑-mixing random variables.

Lemma 2.5 (Wang et al. [7]). Let {𝑋_𝑛,𝑛≥1} be a sequence of 𝜑-mixing random variables with 𝐸𝑋_𝑛=0 and 𝐸|𝑋_𝑛|^𝑞<∞ for some 𝑞≥2 and all 𝑛≥1. Assume that ∑^∞𝑛=1𝜑^1/2(𝑛)<∞. Then there exists a constant 𝐶>0 depending only on 𝑞 and 𝜑(⋅) such that 𝐸⎛⎜⎜⎝max_{1≤𝑗≤𝑛}|||||_𝑗_𝑖=1𝑋_𝑖|||||^𝑞⎞⎟⎟⎠≤𝐶⎧⎪⎨⎪⎩^𝑛_𝑖=1𝐸||𝑋_𝑖||^𝑞+^𝑛_𝑖=1𝐸𝑋^2𝑖^𝑞/2⎫⎪⎬⎪⎭.(2.7)

The following lemma is a Rosenthal-type inequality for 𝜌^∗-mixing random variables.

Lemma 2.6 (Utev and Peligrad [14]). Let {𝑋_𝑛,𝑛≥1} be a sequence of random variables with 𝐸𝑋_𝑛=0 and 𝐸|𝑋_𝑛|^𝑞<∞ for some 𝑞≥2 and all 𝑛≥1. If 𝜌^∗(𝑘)<1 for some 𝑘, then there exists a constant 𝐶>0 depending only on 𝑞,𝑘, and 𝜌^∗(𝑘) such that 𝐸⎛⎜⎜⎝max_{1≤𝑗≤𝑛}|||||_𝑗_𝑖=1𝑋_𝑖|||||^𝑞⎞⎟⎟⎠≤𝐶⎧⎪⎨⎪⎩^𝑛_𝑖=1𝐸||𝑋_𝑖||^𝑞+^𝑛_𝑖=1𝐸𝑋^2𝑖^𝑞/2⎫⎪⎬⎪⎭.(2.8)

3. Main Results

In this section, we extend Theorem 1.3 to negatively associated and negatively dependent random variables. We also obtain similar results for sequences of 𝜑-mixing and 𝜌^∗-mixing random variables.

The following theorem extends Theorem 1.3 to negatively associated random variables.

Theorem 3.1. Suppose that 𝛽≥−1. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise negatively associated random variables which are stochastically dominated by a random variable 𝑋. Let {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of constants satisfying (1.3) and (1.4). If 𝐸𝑋_𝑛𝑖=0 for all 𝑖≥1 and 𝑛≥1, and (1.8) holds, then ^∞_𝑛=1𝑛^𝛽𝑃⎛⎜⎜⎝sup_𝑗≥1|||||_𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖|||||>𝜖⎞⎟⎟⎠<∞∀𝜖>0.(3.1)

Proof. Since 𝑎_𝑛𝑖=𝑎^+𝑛𝑖−𝑎^−𝑛𝑖, we may assume that 𝑎_𝑛𝑖≥0. For 𝑖≥1 and 𝑛≥1, define 𝑋_𝑛𝑖′=𝑋_𝑛𝑖𝐼||𝑋_𝑛𝑖||≤𝑛^𝛾+𝑛^𝛾𝐼𝑋_𝑛𝑖>𝑛^𝛾−𝑛^𝛾𝐼𝑋_𝑛𝑖<−𝑛^𝛾,𝑋^{𝑛𝑖}=𝑋_𝑛𝑖−𝑋^𝑛𝑖.(3.2) Then {𝑋^𝑛𝑖,𝑖≥1,𝑛≥1} is still an array of rowwise negatively associated random variables. Moreover, {𝑎_𝑛𝑖𝑋^𝑛𝑖,𝑖≥1,𝑛≥1} is also an array of rowwise negatively associated random variables. Since 𝐸𝑋_𝑛𝑖=0 for all 𝑖≥1 and 𝑛≥1, it suffices to show that 𝐼₁=∶^∞_𝑛=1𝑛^𝛽𝑃⎛⎜⎜⎝sup_𝑗≥1|||||_𝑗_𝑖=1𝐼𝑎_𝑛𝑖𝑋_𝑛𝑖′−𝐸𝑋_𝑛𝑖′|||||>𝜖⎞⎟⎟⎠<∞,₂=∶^∞_𝑛=1𝑛^𝛽𝑃⎛⎜⎜⎝sup_𝑗≥1|||||_𝑗_𝑖=1𝑎_𝑛𝑖𝑋^{𝑛𝑖}−𝐸𝑋^{𝑛𝑖}|||||>𝜖⎞⎟⎟⎠<∞.(3.3) We will prove (3.3) with three cases.
Case 1 (1+(1+𝛼+𝛽)/𝛾=1(i.e.,1+𝛼+𝛽=0)). For 𝐼₁, we get by Markov’s inequality, Lemmas 2.1–2.3, (1.3), and (1.4) that 𝐼₁≤𝜖^∞−2_𝑛=1𝑛^𝛽𝐸sup_𝑗≥1|||||_𝑗_𝑖=1𝑎_𝑛𝑖(𝑋_𝑛𝑖′−𝐸𝑋_𝑛𝑖′)|||||²≤𝐶^∞_𝑛=1𝑛^𝛽∞_𝑖=1≤𝐶_∞||𝑎_𝑛𝑖||²𝐸||𝑋_𝑛𝑖′||²(byLemma2.3)_𝑛=1𝑛^𝛽∞_𝑖=1||𝑎_𝑛𝑖||²≤𝐶_∞𝐸||𝑋||²𝐼||𝑋||≤𝑛^𝛾+𝑛^2𝛾𝑃||𝑋||>𝑛^𝛾(byLemma2.1)_𝑛=1≤𝐶𝐸||𝑋𝑛||^𝛽𝑛^−𝛾𝑛^𝛼𝐸_{1+(1+𝛼+𝛽)/𝛾}||𝑋||²𝐼||𝑋||≤𝑛^𝛾+𝑛^2𝛾𝑃||𝑋||>𝑛^𝛾(by(1.3)and(1.4))<∞.(3.4) The fifth inequality follows from Lemma 2.2.
For 𝐼₂, we get by Markov’s inequality, stochastic domination, and (1.4) that 𝐼₂≤𝜖^∞−1_𝑛=1𝑛^𝛽𝐸sup_𝑗≥1|||||_𝑗_𝑖=1≤2𝜖_∞−1𝑎_𝑛𝑖𝑋^{𝑛𝑖}−𝐸𝑋^{𝑛𝑖}|||||_𝑛=1𝑛^𝛽∞_𝑖=1≤𝐶_∞||𝑎_𝑛𝑖||𝐸||𝑋^{𝑛𝑖}||_𝑛=1𝑛^𝛽∞_𝑖=1≤𝐶_∞||𝑎_𝑛𝑖||𝐸||𝑋||𝐼||𝑋||>𝑛^𝛾_𝑛=1=𝐶_∞𝑛^𝛽𝑛^𝛼𝐸||𝑋||𝐼||𝑋||>𝑛^𝛾_𝑛=1𝑛^∞−1_𝑖=𝑛=𝐶_∞𝐸||𝑋||𝐼𝑖^𝛾<||𝑋||≤(𝑖+1)^𝛾_𝑖=1𝐸||𝑋||𝐼𝑖^𝛾<||𝑋||≤(𝑖+1)^𝛾^𝑖_𝑛=1𝑛⁻¹≤𝐶𝐸||𝑋||log||𝑋||<∞.(3.5)Case 2 (1<1+(1+𝛼+𝛽)/𝛾<2). As in Case 1, we have that 𝐼₁≤𝐶𝐸|𝑋|^{1+(1+𝛼+𝛽)/𝛾}<∞.
Similar to 𝐼₂ in Case 1, we have that 𝐼₂≤𝐶^∞_𝑛=1=𝐶_∞𝑛^𝛼+𝛽𝐸||𝑋||𝐼||𝑋||>𝑛^𝛾_𝑛=1𝑛^∞𝛼+𝛽_𝑖=𝑛=𝐶_∞𝐸||𝑋||𝐼𝑖^𝛾<||𝑋||≤(𝑖+1)^𝛾_𝑖=1𝐸||𝑋||𝐼𝑖^𝛾<||𝑋||≤(𝑖+1)^𝛾^𝑖_𝑛=1𝑛^𝛼+𝛽≤𝐶𝐸||𝑋||^{1+(1+𝛼+𝛽)/𝛾}<∞.(3.6)Case 3 (1+(1+𝛼+𝛽)/𝛾≥2). For 𝐼₁, we take 𝑡>0 sufficiently large such that (𝛾−𝛼)(1+(1+𝛼+𝛽)/𝛾+𝑡)/2>1+𝛽. Then we obtain by Markov’s inequality and Lemma 2.3 that 𝐼₁≤𝜖^{∞−1−(1+𝛼+𝛽)/𝛾−𝑡}_𝑛=1𝑛^𝛽𝐸sup_𝑗≥1|||||_𝑗_𝑖=1𝑎_𝑛𝑖(𝑋_𝑛𝑖′−𝐸𝑋_𝑛𝑖′)|||||^{1+(1+𝛼+𝛽)/𝛾+𝑡}≤𝐶^∞_𝑛=1𝑛^𝛽∞_𝑖=1𝐸||𝑎_𝑛𝑖𝑋_𝑛𝑖′||^{1+(1+𝛼+𝛽)/𝛾+𝑡}+𝐶^∞_𝑛=1𝑛^𝛽^∞_𝑖=1𝐸||𝑎_𝑛𝑖𝑋_𝑛𝑖′||²^{(1+(1+𝛼+𝛽)/𝛾+𝑡)/2}=∶𝐼₃+𝐼₄.(3.7)Similar to 𝐼₁ in Case 1, we obtain that 𝐼₃≤𝐶^∞_𝑛=1=𝐶_∞𝑛^𝛽𝑛^{−𝛾((1+𝛼+𝛽)/𝛾+𝑡)}𝑛^𝛼𝐸||𝑋||^{1+(1+𝛼+𝛽)/𝛾+𝑡}𝐼||𝑋||≤𝑛^𝛾+𝑛^{𝛾(1+(1+𝛼+𝛽)/𝛾+𝑡)}𝑃||𝑋||>𝑛^𝛾_𝑛=1𝑛^{−1−𝛾𝑡}𝐸||𝑋||^{1+(1+𝛼+𝛽)/𝛾+𝑡}𝐼||𝑋||≤𝑛^𝛾+𝐶^∞_𝑛=1≤𝐶𝐸||𝑋||_{1+(1+𝛼+𝛽)/𝛾}𝑛^{𝛼+𝛽+𝛾}𝑃||𝑋||>𝑛^𝛾<∞.(3.8)Noting 𝐸|𝑋_𝑛𝑖′|²≤𝐶𝐸|𝑋|², we obtain by (1.3) and (1.4) that 𝐼₄≤𝐶^∞_𝑛=1𝑛^𝛽𝐶𝐸||𝑋||^2∞_𝑖=1||𝑎_𝑛𝑖||²^{(1+(1+𝛼+𝛽)/𝛾+𝑡)/2}≤𝐶^∞_𝑛=1𝑛^𝛽𝐶𝐸||𝑋||²𝑛^𝛼−𝛾^{(1+(1+𝛼+𝛽)/𝛾+𝑡)/2}<∞,(3.9) since (𝛾−𝛼)(1+(1+𝛼+𝛽)/𝛾+𝑡)/2−𝛽>1. Hence, 𝐼₁<∞. As in Case 2, we obtain 𝐼₂≤𝐶𝐸|𝑋|^{1+(1+𝛼+𝛽)/𝛾}<∞.

Remark 3.2. The moment condition of Theorem 3.1 is weaker than that of Theorem 1.2. Also, the conclusion of Theorem 3.1 implies the conclusion of Theorem 1.2. Hence, Theorem 3.1 improves Theorem 1.2. Moreover, the method of the proof of Theorem 3.1 is simpler than that of the proof of Theorem 1.2.

Corollary 3.3. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise negatively associated random variables which are stochastically dominated by a random variable 𝑋. Let {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} be a Toeplitz array satisfying sup_𝑖≥1||𝑎_𝑛𝑖||=𝑂𝑛^{1/𝑡−𝛿}forsome𝑡>0,𝛿>0.(3.10) If 𝐸𝐸𝐸||𝑋||𝑋||||𝑋||l||o<g||∞𝑋,||<f∞or,0<for𝑡<t=1,1,_{1+(1−1/𝑡)/𝛿}<∞,fort>1,(3.11) then ^∞_𝑛=1𝑃⎛⎜⎜⎝max_{1≤𝑗≤𝑛}|||||_𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖|||||>𝜖𝑛_1/𝑡⎞⎟⎟⎠<∞∀𝜖>0.(3.12)

Proof. For the case 0<𝑡<1, the result can be easily proved by ^∞_𝑛=1𝑃⎛⎜⎜⎝max_{1≤𝑗≤𝑛}|||||_𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖|||||>𝜖𝑛_1/𝑡⎞⎟⎟⎠≤𝜖_∞−1_𝑛=1𝑛^−1/𝑡𝐸max_{1≤𝑗≤𝑛}|||||_𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑛𝑖≤𝜖_∞−1|||||_𝑛=1𝑛^{𝑛−1/𝑡}_𝑖=1≤𝐶𝐸||𝑋||_∞||𝑎_𝑛𝑖||𝐸||𝑋_𝑛𝑖||_𝑛=1𝑛^−1/𝑡<∞.(3.13)
For the case 𝑡≥1, we let 𝑏_𝑛𝑖=𝑎_𝑛𝑖𝑛^−1/𝑡. Observe that sup_𝑖≥1||𝑏_𝑛𝑖||=𝑂𝑛^−𝛿,^∞_𝑖=1||𝑏_𝑛𝑖||=𝑂𝑛^−1/𝑡.(3.14)
By Theorem 3.1 with 𝛼=−1/𝑡,𝛽=0,𝛾=𝛿, and 𝑎_𝑛𝑖 replaced by 𝑏_𝑛𝑖, we get that ^∞_𝑛=1𝑃⎛⎜⎜⎝max_{1≤𝑗≤𝑛}|||||_𝑗_𝑖=1𝑏_𝑛𝑖𝑋_𝑛𝑖−𝐸𝑋_𝑛𝑖|||||>𝜖⎞⎟⎟⎠<∞∀𝜖>0.(3.15) To complete the proof, we only prove that 𝐽=∶max_{1≤𝑗≤𝑛}|||||_𝑗_𝑖=1𝑏_𝑛𝑖𝐸𝑋_𝑛𝑖|||||⟶0,(3.16) but 𝐽≤∑^𝑛𝑖=1|𝑏_𝑛𝑖|𝐸|𝑋_𝑛𝑖|≤𝐶𝐸|𝑋|𝑛^−1/𝑡→0 as 𝑛→∞.

Remark 3.4. When 0<𝑡<1, Corollary 3.3 holds without negative association. Kuczmaszewska [6, Corollary  2.4], proved Corollary 3.3 under the stronger moment condition 𝐸|𝑋|^1+1/𝛿<∞.

The following theorem extends Theorem 1.3 to negatively dependent random variables.

Theorem 3.5. Suppose that 𝛽≥−1. Let {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of rowwise negatively dependent random variables which are stochastically dominated by a random variable 𝑋. Let {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of constants satisfying (1.3) and (1.4). If 𝐸𝑋_𝑛𝑖=0 for all 𝑖≥1 and 𝑛≥1, and (1.8) holds, then (1.7) holds.

Proof. The proof is the same as that of Theorem 3.1 except that we use Lemma 2.4 instead of Lemma 2.3.

If the array {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} in Theorem 3.1 is replaced by the sequence {𝑋_𝑛,𝑛≥1}, then we can extend Theorem 3.1 to 𝜑-mixing and 𝜌^∗-mixing random variables.

Theorem 3.6. Suppose that 𝛽≥−1. Let {𝑋_𝑛,𝑛≥1} be a sequence of 𝜑-mixing random variables which are stochastically dominated by a random variable 𝑋. Let {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of constants satisfying (1.3) and (1.4). Assume that ∑^∞𝑛=1𝜑^1/2(𝑛)<∞. If 𝐸𝑋_𝑛=0 for all 𝑛≥1, and (1.8) holds, then ^∞_𝑛=1𝑛^𝛽𝑃⎛⎜⎜⎝sup_𝑗≥1|||||_𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑖|||||>𝜖⎞⎟⎟⎠<∞∀𝜖>0.(3.17)

Proof. Since 𝐸𝑋_𝑛=0 for all 𝑛≥1, it suffices to show that ^∞_𝑛=1𝑛^𝛽𝑃⎛⎜⎜⎝sup_𝑗≥1|||||_𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑖𝐼||𝑋_𝑖||≤𝑛^𝛾−𝐸𝑋_𝑖𝐼||𝑋_𝑖||≤𝑛^𝛾|||||>𝜖⎞⎟⎟⎠<∞,_∞_𝑛=1𝑛^𝛽𝑃⎛⎜⎜⎝sup_𝑗≥1|||||_𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑖𝐼||𝑋_𝑖||>𝑛^𝛾−𝐸𝑋_𝑖𝐼||𝑋_𝑖||>𝑛^𝛾|||||>𝜖⎞⎟⎟⎠<∞.(3.18) The rest of the proof is the same as that of Theorem 3.1 except that we use Lemma 2.5 instead of Lemma 2.3 and it is omitted.

Remark 3.7. Can Theorem 3.6 be extended to the array {𝑋_𝑛𝑖,𝑖≥1,𝑛≥1} of rowwise 𝜑-mixing random variables? Let {𝜑_𝑛(𝑖),𝑖≥1} be the sequence of 𝜑-mixing coefficients for the 𝑛th row {𝑋_𝑛1,𝑋_𝑛2,…} of the array {𝑋_𝑛𝑖}. When we apply Lemma 2.5 to the 𝑛th row, the constant 𝐶 depends on both 𝑞 and 𝜑_𝑛(⋅). That is, the constant 𝐶 depends on 𝑛. Hence we cannot extend Theorem 3.6 to the array by using the method of the proof of Theorem 3.1.

Corollary 3.8. Let {𝑋_𝑛,𝑛≥1} be a sequence of 𝜑-mixing random variables which are stochastically dominated by a random variable 𝑋. Let {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} be a Toeplitz array satisfying (3.10). Assume that ∑^∞𝑛=1𝜑^1/2(𝑛)<∞. If (3.11) holds, then ^∞_𝑛=1𝑃⎛⎜⎜⎝max_{1≤𝑗≤𝑛}|||||_𝑗_𝑖=1𝑎_𝑛𝑖𝑋_𝑖|||||>𝜖𝑛_1/𝑡⎞⎟⎟⎠<∞∀𝜖>0.(3.19)

Proof. The proof is the same as that of Corollary 3.3 except that we use Theorem 3.6 instead of Theorem 3.1.

Remark 3.9. When 0<𝑡<1, Corollary 3.8 holds without 𝜑-mixing. Wang et al. [7, Theorem  2.5] proved Corollary 3.8 under the stronger moment condition 𝐸|𝑋|^{max{2/𝛿,1+1/𝛿}}<∞.

Theorem 3.10. Suppose that 𝛽≥−1. Let {𝑋_𝑛,𝑛≥1} be a sequence of 𝜌^∗-mixing random variables which are stochastically dominated by a random variable 𝑋. Let {𝑎_𝑛𝑖,𝑖≥1,𝑛≥1} be an array of constants satisfying (1.3) and (1.4). If 𝐸𝑋_𝑛=0 for all 𝑛≥1, and (1.8) holds, then (3.17) holds.

Proof. The proof is the same as that of Theorem 3.6 except that we use Lemma 2.6 instead of Lemma 2.5.

Remark 3.11. Likewise in Remark 3.7, we also cannot extend Theorem 3.10 to the array by using the method of the proof of Theorem 3.1.

Acknowledgments

The author would like to thank the Editor Zhenya Yan and an anonymous referee for the helpful comments. This research was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2010-0013131).