Strong Convergence Properties for Asymptotically Almost Negatively Associated Sequence

Abstract

By applying the moment inequality for asymptotically almost negatively associated (in short AANA) random sequence and truncated method, we get the three series theorems for AANA random variables. Moreover, a strong convergence property for the partial sums of AANA random sequence is obtained. In addition, we also study strong convergence property for weighted sums of AANA random sequence.

1. Introduction

A finite family of random variables {𝑋_𝑘,1≤𝑘≤𝑛,𝑛≥2} is said to be negatively associated (in short NA) if for every pair of disjoint subsets 𝐴₁,𝐴₂ of {1,2,…,𝑛}Cov𝑓𝑋_𝑖∶𝑖∈𝐴₁,𝑔𝑋_𝑗∶𝑗∈𝐴₂≤0, (1.1) whenever 𝑓,𝑔 are coordinate-wise nondecreasing such that the covariance exists. An infinite sequence of random variables {𝑋_𝑛,𝑛≥1} is said to be NA if every finite subfamily is NA.

The notion of NA was first introduced by Block et al. (1982) [1]. Joag-Dev and Proschan (1983) [2] showed that many well-known multivariate distributions possess the NA property. By inspecting the proof of maximal inequality for NA random variables in Matuła [3], Chandra and Ghosal discovered that one can also allow negative correlations provided they are small. Primarily motivated by this, Chandra and Ghosal [4, 5] introduced the following dependence.

Definition 1.1. A sequence {𝑋_𝑛,𝑛≥1} of random variables is said to be asymptotically almost negatively associated, if there exists a nonnegative sequence 𝑞(𝑛)→0 as 𝑛→∞ such that Cov𝑓𝑋_𝑛,𝑔𝑋_𝑛+1,𝑋_𝑛+2,…,𝑋_𝑛+𝑘≤𝑞(𝑛)Var𝑓𝑋_𝑛Var𝑔𝑋_𝑛+1,𝑋_𝑛+2,…,𝑋_𝑛+𝑘^1/2, (1.2) for all 𝑛,𝑘≥1 and for all coordinatewise nondecreasing continuous functions 𝑓 and 𝑔 whenever the variances exit.
Obviously, the family of AANA sequences contain NA (in particular, independent) sequences (with 𝑞(𝑛)=0,𝑛≥1) and some more sequences of random variables which are not much deviated from being NA. An example of an AANA sequence which is not NA was introduced by Chandra and Ghosal [4].
Since the notion of AANA sequence was introduced by Chandra and Ghosal [4], the AANA properties have 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 AANA random variables; we can refer to [4–10]. Hence, extending the limit properties of AANA random variables has very important significance in the theory and application.
In this paper, we mainly study the strong convergence property for the partial sums of AANA random variables; furthermore the strong convergence property for weighted sums of AANA random variables is also obtained.
Throughout the paper, let 𝐼(𝐴) be the indicator function of the set 𝐴, and let 𝑋^𝑐=−𝑐𝐼(𝑋<−𝑐)+𝑋𝐼(|𝑋|≤𝑐)+𝑐𝐼(𝑋>𝑐) for some 𝑐>0. The 𝑎_𝑛=𝑂(𝑏_𝑛) denotes that there exits a positive constant 𝐶 such that |𝑎_𝑛/𝑏_𝑛|≤𝐶. The symbol 𝐶 represents a positive constant which may be different in various places. The main results of this paper are dependent on the following lemmas.

Lemma 1.2 (Yuan and An [6]). Let {𝑋_𝑛,𝑛≥1} be a sequence of AANA random variables with mixing coefficients {𝑞(𝑛),𝑛≥1}, and let 𝑓₁,𝑓₂,… be all nondecreasing (or nonincreasing) functions; then {𝑓_𝑛(𝑋_𝑛),𝑛≥1} is still a sequence of AANA random variables with mixing coefficients {𝑞(𝑛),𝑛≥1}.

Lemma 1.3 (Wang et al. [7]). For 1<𝑝≤2, let {𝑋_𝑛,𝑛≥1} be a sequence of AANA random variables with mixing coefficients {𝑞(𝑛),𝑛≥1} and 𝐸𝑋_𝑛=0 for each 𝑛≥1. If ∑^∞_𝑛=1𝑞²(𝑛)<∞, then there exists a positive constant 𝐶_𝑝 depending only on 𝑝 such that 𝐸max_{1≤𝑖≤𝑛}𝑆_𝑖^𝑝≤𝐶_𝑝^𝑛_𝑖=1𝐸𝑋_𝑖^𝑝, (1.3) for all 𝑛≥1 where 𝑆_𝑖=∑^𝑖_𝑗=1𝑋_𝑗,𝐶_𝑝=2^𝑝[2^2−𝑝𝑝+(6𝑝)^𝑝(∑^∞_𝑛=1𝑞²(𝑛))^𝑝/𝑞], and 𝑞=𝑝/(𝑝−1) is the dual number of 𝑝.

Lemma 1.4 (Wu [11]). Let {𝑋_𝑛,𝑛≥1} be a sequence of random variables. For each 𝑛≥1, there exists a random variable 𝑋 such that 𝑃𝑋_𝑛≥𝑥≤𝐶𝑃(|𝑋|≥𝑥) (1.4) then, for any 𝑟>0,𝑥>0, the following two statements hold: 𝐸𝑋_𝑛^𝑟𝐼𝑋_𝑛≤𝑥≤𝐶𝐸|𝑋|^𝑟𝐼(|𝑋|≤𝑥)+𝑥^𝑟𝑃𝑋|,𝐸𝑋(|>𝑥)_𝑛𝑟𝐼𝑋_𝑛>𝑥≤𝐶𝐸|𝑋|^𝑟𝐼(|𝑋|>𝑥). (1.5)

Lemma 1.5 (Sung [12]). Let 𝜙(𝑥) be a positive increasing function on (0,+∞) satisfying 𝜙(𝑥)↑∞ as 𝑛→∞, and let 𝜓(𝑥) be the inverse function of 𝜙(𝑥). If 𝜓(𝑥) and 𝜙(𝑥) satisfy, respectively, 𝜓(𝑛)^𝑛_𝑖=11𝜓(𝑖)=𝑂(𝑛),𝐸𝜙(|𝑋|)<∞, (1.6) then ^∞_𝑖=11𝜓(𝑛)𝐸|𝑋|𝐼|𝑋|>𝜓(𝑛)<∞. (1.7)

2. Strong Convergence for the Partial Sums of AANA Random Variables

Theorem 2.1. Let {𝑋_𝑛,𝑛≥1} be a sequence of AANA random variables with ∑^∞_𝑛=1𝑞²(𝑛)<∞, if the following assumptions holds: ^∞_𝑛=1𝑃𝑋_𝑛>𝑐<∞,^∞_𝑛=1𝐸𝑋^𝑐_𝑛<∞,^∞_𝑛=1Var𝑋^𝑐_𝑛<∞; (2.1) then ∑^∞_𝑛=1𝑋_𝑛 almost surely convergence.

Remark 2.2. The proof of Theorem 2.1 is similar to the proof of Theorem  4.3.4 in [11], and by Lemmas 1.2 and 1.3, we omit it.

Theorem 2.3. Let {𝑋_𝑛,𝑛≥1} be a sequence of AANA random variables with ∑^∞_𝑛=1𝑞²(𝑛)<∞.
Assume that {𝑔_𝑛(𝑥),𝑛≥1} is a sequence of even functions in 𝑅¹, for each 𝑛≥1, 𝑔_𝑛(𝑥) is a positive nondecreasing function in (0,+∞) and satisfies one of the following conditions:(i)for 𝑥∈(0,1] there exists a constant 𝛼>0 such that 𝑔_𝑛(𝑥)≥𝛼𝑥;(ii)for 𝑥∈(0,1], there exists a constant 𝑟∈(1,2] and 𝛼>0 such that 𝑔_𝑛(𝑥)≥𝛼𝑥^𝑟; however, for 𝑥∈(1,∞),𝑔_𝑛(𝑥)≥𝛼𝑥, furthermore assume that 𝐸𝑋_𝑛=0, for each 𝑛≥1.
Let {𝑎_𝑛,𝑛≥1} be a constant sequence satisfying 0<𝑎_𝑛↑∞ such that ^∞_𝑛=1𝐸𝑔_𝑛𝑋_𝑛𝑎_𝑛<∞, (2.2) then ∑^∞_𝑛=1(𝑋_𝑛/𝑎_𝑛) almost surely convergence, and further it follows from the “Kronecker lemma” that 𝑎_𝑛^𝑛−1_𝑘=1𝑋_𝑘→0a.s.,as𝑛→∞. (2.3)

Proof. For each 𝑛≥1, denote 𝑋^𝑎𝑛_𝑛≜−𝑎_𝑛𝐼(𝑋_𝑛<−𝑎_𝑛)+𝑋_𝑛𝐼(|𝑋_𝑛|≤𝑎_𝑛)+𝑎_𝑛𝐼(𝑋_𝑛>𝑎_𝑛).
By Lemma 1.2, we can see that, for fixed 𝑛≥1, {𝑋^𝑎𝑛_𝑛} is still a sequence of AANA random variables. To verity the Theorem 2.3, for 𝑐=1 we only need to prove the convergence of three series of (2.1) under condition (i) or (ii). The proof of Theorem 2.3 includes the following three steps.
(1) We prove ∑^∞_𝑛=1𝑃(|𝑋_𝑛/𝑎_𝑛|>1)<∞ under condition (i) or (ii).
For each 𝑛≥1, if 𝑔_𝑛(𝑥) satisfies condition (i), noting that 𝑔_𝑛(𝑥) is a positive nondecreasing even function in (0,+∞), it is obvious that 𝑃𝑋_𝑛𝑎_𝑛𝑋>1=𝐸𝐼^𝑛𝑎_𝑛>1≤𝛼⁻¹𝐸𝑔_𝑛𝑋_𝑛𝑎_𝑛. (2.4) By (2.2), we can get ^∞_𝑛=1𝑃𝑋_𝑛𝑎_𝑛>1≤𝛼^∞−1_𝑛=1𝐸𝑔_𝑛𝑋_𝑛𝑎_𝑛<∞. (2.5) If 𝑔_𝑛(𝑥) satisfies condition (ii), it is easy to prove that (2.5) also holds when |𝑋_𝑛|>𝑎_𝑛>0.
(2) Next we will show ∑^∞_𝑛=1𝐸|𝑋^𝑎𝑛_𝑛/𝑎_𝑛|<∞.
If 𝑔_𝑛(𝑥) satisfies condition (i), it follows that 𝐸𝑋^𝑎𝑛_𝑛𝑎_𝑛=−𝐸𝐼𝑋_𝑛<−𝑎_𝑛𝑋+𝐸^𝑛𝑎_𝑛𝐼𝑋_𝑛≤𝑎_𝑛+𝐸𝐼𝑋_𝑛>𝑎_𝑛≤𝐸𝐼𝑋_𝑛>𝑎_𝑛+𝐸𝑋_𝑛𝑎_𝑛𝐼𝑋_𝑛≤𝑎_𝑛≤𝛼⁻¹𝐸𝑔_𝑛𝑋_𝑛𝑎_𝑛+_{|𝑋𝑛|≤𝑎𝑛}𝑋_𝑛𝑎_𝑛𝑑𝑃≤2𝛼⁻¹𝐸𝑔_𝑛𝑋_𝑛𝑎_𝑛. (2.6) On the other hand, if condition (ii) holds, according to 𝐸𝑋_𝑛=0, for each 𝑛≥1, we have 𝐸𝑋^𝑎𝑛_𝑛𝑎_𝑛≤𝐸𝐼𝑋_𝑛>𝑎_𝑛+𝐸𝑋_𝑛𝑎_𝑛𝐼𝑋_𝑛≤𝑎_𝑛=𝐸𝐼𝑋_𝑛>𝑎_𝑛+𝐸𝑋_𝑛𝑎_𝑛𝐼𝑋_𝑛>𝑎_𝑛≤2𝛼⁻¹𝐸𝑔_𝑛𝑋_𝑛𝑎_𝑛. (2.7) Hence, it follows from (2.2) that ^∞_𝑛=1𝐸𝑋^𝑎𝑛_𝑛𝑎_𝑛<2𝛼^∞−1_𝑛=1𝐸𝑔_𝑛𝑋_𝑛𝑎_𝑛<∞. (2.8)
(3) Finally we prove ∑^∞_𝑛=1𝐸(𝑋^𝑎𝑛_𝑛/𝑎_𝑛)²<∞.
If 𝑔_𝑛(𝑥) satisfies condition (i), for each 𝑛≥1, it is easy to show that by the 𝐶_𝑟−inequality 𝐸𝑋^𝑎𝑛_𝑛𝑎_𝑛²=𝐸−𝐼𝑋_𝑛<−𝑎_𝑛+𝑋_𝑛𝑎_𝑛𝐼𝑋_𝑛≤𝑎_𝑛+𝐼𝑋_𝑛>𝑎_𝑛²≤3𝐸𝐼𝑋_𝑛>𝑎_𝑛+𝑋_𝑛𝑎_𝑛²𝐼𝑋_𝑛≤𝑎_𝑛≤𝐶𝛼⁻¹𝐸𝑔_𝑛𝑋_𝑛𝑎_𝑛𝑋+𝐶𝐸^𝑛𝑎_𝑛𝐼𝑋_𝑛≤𝑎_𝑛≤𝐶𝛼⁻¹𝐸𝑔_𝑛𝑋_𝑛𝑎_𝑛. (2.9) If condition (ii) holds, according to the 𝐶_𝑟-inequality, for each 𝑛≥1, we get 𝐸𝑋^𝑎𝑛_𝑛𝑎_𝑛²=𝐸−𝐼𝑋_𝑛<−𝑎_𝑛+𝑋_𝑛𝑎_𝑛𝐼𝑋_𝑛≤𝑎_𝑛+𝐼𝑋_𝑛>𝑎_𝑛²≤3𝐸𝐼𝑋_𝑛>𝑎_𝑛+𝑋_𝑛𝑎_𝑛²𝐼𝑋_𝑛≤𝑎_𝑛≤𝐶𝛼⁻¹𝐸𝑔_𝑛𝑋_𝑛𝑎_𝑛𝑋+𝐶𝐸^𝑛𝑎_𝑛^𝑟𝐼𝑋_𝑛≤𝑎_𝑛≤𝐶𝛼⁻¹𝐸𝑔_𝑛𝑋_𝑛𝑎_𝑛. (2.10) Therefore, it also follows from (2.2) that ^∞_𝑛=1𝐸𝑋^𝑎𝑛_𝑛𝑎_𝑛²<𝐶𝛼^∞−1_𝑛=1𝐸𝑔_𝑛𝑋_𝑛𝑎_𝑛<∞. (2.11) The proof of the Theorem 2.3 is completed by (2.5), (2.8), and (2.11).

Corollary 2.4. Let {𝑋_𝑛,𝑛≥1} be a sequence of AANA random variables with ∑^∞_𝑛=1𝑞²(𝑛)<∞, and let {𝑎_𝑛,𝑛≥1} be a constant sequence satisfying 0<𝑎_𝑛↑∞. For 𝜃∈(0,1], let 𝑔_𝑛(𝑥)=|𝑥|^𝜃/(1+|𝑥|^𝜃), and if {𝑋_𝑛/𝑎_𝑛,𝑛≥1} satisfies (2.2), then 𝑎_𝑛⁻¹∑^𝑛_𝑘=1𝑋_𝑘→0 a.s., as 𝑛→∞.

Proof. It is easy to check that {𝑔_𝑛(𝑥),𝑛≥1} is a sequence of even functions in 𝑅¹, for each 𝑛≥1, 𝑔_𝑛(𝑥) is a positive nondecreasing function in(0,+∞), and the following condition holds: 𝑔_𝑛(𝑥)≥12𝑥^𝜃≥12𝑥,0<𝑥≤1,0<𝜃≤1. (2.12)

3. Strong Convergence for the Weighted Sums of AANA Random Variables

Theorem 3.1. Let {𝑋_𝑛,𝑛≥1} be a different distribution sequence of AANA random variables with ∑^∞_𝑛=1𝑞²(𝑛)<∞ and 𝐸𝑋_𝑛=0, for each 𝑛≥1. There exists a random variable 𝑋 satisfying 𝐸|𝑋|^𝑟<∞,0<𝑟≤2, such that 𝑃𝑋_𝑛>𝑥≤𝐶𝑃(|𝑋|>𝑥),𝑛≥1,𝑥>0. (3.1) Assume that the following conditions hold for the constant arrays {𝑎_𝑛𝑖,𝑛≥1,1≤𝑖≤𝑛}.
(i) max_{1≤𝑖≤𝑛}|𝑎_𝑛𝑖|=𝑂(𝜓⁻¹(𝑛)); (ii) for some constant 𝛿>0, ∑^𝑛_𝑖=1|𝑎_𝑛𝑖|^𝑟=𝑂(𝑛⁻¹log^−1−𝛿𝑛), where 𝜙(𝑥),𝜓(𝑥) satisfy Lemma 1.5; then 𝑇_𝑛=^𝑛_𝑖=1𝑎_𝑛𝑖𝑋−−→_𝑖^a.s.0,𝑛→∞. (3.2)

Proof. Let 𝑌_𝑖=−𝜓(𝑛)𝐼(𝑋_𝑖<−𝜓(𝑛))+𝑋_𝑖𝐼(|𝑋_𝑖|≤𝜓(𝑛))+𝜓(𝑛)𝐼(𝑋_𝑖>𝜓(𝑛)),𝑌_𝑖=𝑌_𝑖−𝐸𝑌_𝑖: 𝑇_𝑛=^𝑛_𝑖=1𝑎_𝑛𝑖𝑋_𝑖−𝑌_𝑖+^𝑛_𝑖=1𝑎_𝑛𝑖𝑌_𝑖+^𝑛_𝑖=1𝑎_𝑛𝑖𝐸𝑌_𝑖≜𝑇_𝑛1+𝑇_𝑛2+𝑇_𝑛3. (3.3) It suffices to prove that 𝑇_𝑛𝑖→0 a.s., as 𝑛→∞,𝑖=1,2,3. We will estimate each of these terms separately.
To verity 𝑇_𝑛1→0 a.s., as 𝑛→∞, we can get from (3.1) and 𝐸𝜙(|𝑋|)<∞ that ^∞_𝑛=1𝑃𝑋_𝑖≠𝑌_𝑖=^∞_𝑛=1𝑃𝑋_𝑖>𝜓(𝑛)≤𝐶^∞_𝑛=1𝑃|𝑋|>𝜓(𝑛)=𝐶^∞_𝑛=1𝑃𝜙|𝑋|𝑋>𝑛≤𝐶𝐸𝜙(||)<∞. (3.4) Hence, by the Borel-Cantelli Lemma it is obvious that 𝑇_𝑛1→0 a.s., as 𝑛→∞.
Next we will show that 𝑇_𝑛2→0， as 𝑛→∞ almost surely. For any 𝜀>0,0<𝑟≤2, note that 𝐸|𝑋|^𝑟<∞, and it follows from the Markov inequality, Lemma 1.2, Lemma 1.3, 𝐶_𝑟-inequality, and Lemma 1.5 that ^∞_𝑛=1𝑃^𝑛_𝑖=1𝑎_𝑛𝑖𝑌_𝑖>𝜀≤𝐶^∞_𝑛=1𝐸^𝑛_𝑖=1𝑎_𝑛𝑖𝑌_𝑖^𝑟≤𝐶^∞𝑛_𝑛=1𝑖=1𝐸𝑎_𝑛𝑖𝑌_𝑖^𝑟≤𝐶^∞𝑛_𝑛=1𝑖=1𝑎_𝑛𝑖^𝑟𝐸𝑋_𝑖^𝑟𝐼𝑋_𝑖≤𝜓(𝑛)+𝜓^𝑟(𝑛)𝐸𝐼𝑋_𝑖>𝜓(𝑛)≤𝐶^∞𝑛_𝑛=1𝑖=1𝑎_𝑛𝑖^𝑟𝐸|𝑋|^𝑟𝐼|𝑋|≤𝜓(𝑛)+𝜓^𝑟(𝑛)𝐸𝐼|𝑋|>𝜓(𝑛)≤𝐶^∞𝑛_𝑛=1𝑖=1𝑎_𝑛𝑖^𝑟𝐸𝑋|^𝑟𝐼|𝑋|≤𝜓(𝑛)+𝐸|𝑋|^𝑟≤𝐶^∞𝑛_𝑛=1𝑖=1𝑎_𝑛𝑖^𝑟≤𝐶^∞_𝑛=11𝑛log^1+𝛿𝑛<∞. (3.5) the last series converges using condition (ii), and by Borel-Cantelli lemma we get 𝑇_𝑛2→0 a.s., as 𝑛→∞.
Finally we will prove that 𝑇_𝑛3→0 a.s., as 𝑛→∞. Note that 𝐸𝑋_𝑛=0; for each 𝑛≥1, it is easy to show that by Lemma 1.5, Lemma 1.4, and theKronecker lemma ^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑌_𝑖≤^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑖𝐼𝑋_𝑖≤𝜓(𝑛)+^𝑛_𝑖=1𝑎_𝑛𝑖𝜓(𝑛)𝐸𝐼𝑋_𝑖(𝑛)≤>𝜓_𝑛𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑖𝐼𝑋_𝑖>𝜓(𝑛)+^𝑛_𝑖=1𝑎_𝑛𝑖𝜓(𝑛)𝐸𝐼𝑋_𝑖>𝜓(𝑛)≤𝐶^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋_𝑖𝐼𝑋_𝑖>𝜓(𝑖)≤𝐶^𝑛_𝑖=1𝐸𝑎_𝑛𝑖𝑋𝐼|𝑋|(𝑖)≤1>𝜓𝜓(𝑛)^𝑛_𝑖=1𝐸|𝑋|𝐼|𝑋|>𝜓(𝑖)→0,𝑛→∞. (3.6) The proof of Theorem 3.1 is completed.

Acknowledgments

This paper is supported by the National Natural Science Foundantion of China (10901003) and the Natural Science Foundation of Anhui Province (KJ2012ZD001, KJ2013A126, KJ2012Z233).