Strong Convergence Properties and Strong Stability for Weighted Sums of AANA Random Variables

Abstract

The Khintchine-Kolmogorov-type convergence theorem and three-series theorem for AANA random variables are established. By using these convergence theorems, we obtain convergence results for AANA sequences, which extend the corresponding ones for independent sequences and NA sequences. In addition, we study the strong stability for weighted sums of AANA random variables and obtain some new results, which extend some earlier ones for NA random variables.

1. Introduction

Firstly, let us recall some definitions.

Definition 1 (cf. Wu [1]). A sequence{𝑋_𝑛,𝑛≥1}of random variables is said to be stochastically dominated by a random variable𝑋if there exists a constant𝐶 such that 𝑃𝑋_𝑛>𝑥≤𝐶𝑃(|𝑋|>𝑥) (1) for all𝑥≥0and𝑛≥1.

Definition 2 (cf. Chow and Teicher [2]). A sequence{𝑌_𝑛,𝑛≥1}of random variables is said to be strongly stable if there exist two constant sequences{𝑏_𝑛,𝑛≥1}and{𝑑_𝑛,𝑛≥1}with0<𝑏_𝑛↑∞such that 𝑏_𝑛⁻¹𝑌_𝑛−𝑑_𝑛⟶0a.s. (2)

Definition 3 (cf. Wu [1]). A function𝑙(𝑥)>0(𝑥>0)is said to be quasimonotonically increasing function if there exist𝑥₀>0and constant𝐶>0with∀𝑡≥𝑥≥𝑥₀such that𝑙(𝑡)≥𝐶𝑙(𝑥). A function𝑙(𝑥)>0(𝑥>0)is said to be quasimonotonically decreasing function if there exist𝑥₀>0and constant𝐶>0with∀𝑡≥𝑥≥𝑥₀such that𝑙(𝑡)≤𝐶𝑙(𝑥).

Definition 4 (cf. Wu [1]). A real-valued function𝑙(𝑥), positive and measurable on(0,∞), is said to be slowly varying if lim_𝑥→∞𝑙(𝜆𝑥)𝑙(𝑥)=1 (3) for each𝜆>0.

Definition 5 (cf. Joag-Dev and Proschan [3]). A finite collection of random variables𝑋₁,𝑋₂,…,𝑋_𝑛is said to be negatively associated (NA, in short) if for every pair of disjoint subsets𝐴₁, 𝐴₂of{1,2,…,𝑛}, Cov𝑓𝑋_𝑖∶𝑖∈𝐴₁,𝑔𝑋_𝑗∶𝑗∈𝐴₂≤0, (4) whenever𝑓and𝑔are coordinatewise nondecreasing such that this covariance exists. An infinite sequence{𝑋_𝑛,𝑛≥1}is NA if every finite subcollection is NA.

Definition 6 (cf. Chandra and Ghosal [4]). A sequence{𝑋_𝑛,𝑛≥1}of random variables is called asymptotically almost negatively associated (AANA) if there exists a nonnegative sequence𝑞(𝑛)→0as𝑛→∞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 (5) for all𝑛,𝑘≥1and for all coordinatewise nondecreasing continuous functions𝑓and𝑔whenever the variances exist.

Obviously, the family of AANA sequences contains NA (in particular, independent) sequences (taking𝑞(𝑛)=0, 𝑛≥1) and some more sequences of random variables which do not much deviates from being NA. An example of an AANA sequence which is not NA was introduced by Chandra and Ghosal [5].

Since the concept of AANA random variables was introduced by Chandra and Ghosal [4], many applications have been found. For example, Chandra and Ghosal [4] derived the Kolmogorov-type inequality and the strong law of large numbers of Marcinkiewicz-Zygmund, Chandra and Ghosal [5] obtained the almost sure convergence of weighted averages, Ko et al. [6] studied the Hájek-Rènyi type inequality, and Wang et al. [7] established the law of the iterated logarithm for product sums. Yuan and An [8] established Rosenthal-type inequalities for maximum partial sums of AANA sequences. Wang et al. [9] studied some convergence properties for AANA sequence. Wang et al. [10] generalized and improved the results of Ko et al. [6] and studied the large deviation and Marcinkiewicz-type strong law of large numbers for AANA sequences. Yang et al. [11] investigated the complete convergence of moving average process for AANA sequence. Hu et al. [12] and Shen and Wu [13, 14] studied strong convergence property for weighted sums of AANA sequence. Wang et al. [15, 16] and Shen et al. [17] obtained some results on complete convergence for AANA sequence, and so forth.

In this paper, we mainly study convergence results for AANA random variables, and the strong stability for weighted sums of AANA random variables, which extend the corresponding ones for independent sequences and NA sequences without necessarily adding extra conditions. The techniques used in the paper are the truncated method, the Khintchine-Kolmogorov-type convergence theorem and three-series theorem for AANA random variables.

Throughout this paper, let𝐼(𝐴)be the indicator function of the set𝐴, and𝑋_𝑛^(𝑐)=−𝑐𝐼(𝑋_𝑛<−𝑐)+𝑋_𝑛𝐼(|𝑋_𝑛|≤𝑐)+𝑐𝐼(𝑋_𝑛>𝑐)for some𝑐>0.𝑎_𝑛=𝑂(𝑏_𝑛)denotes that there exists a positive constant𝐶such that|𝑎_𝑛/𝑏_𝑛|≤𝐶. The symbol𝐶represents a positive constant which may be different in various places. The main results of this paper depend on the following lemmas.

Lemma 7 (cf. Yuan and An [8]). Let{𝑋_𝑛,𝑛≥1}be a sequence of AANA random variables with mixing coefficients{𝑞(𝑛),𝑛≥1}, and let𝑓₁,𝑓₂,…be all nondecreasing (or nonincreasing) continuous functions; then{𝑓_𝑛(𝑋_𝑛),𝑛≥1}is still a sequence of AANA random variables with mixing coefficients{𝑞(𝑛),𝑛≥1}.

Lemma 8 (cf. Wang et al. [9]). Let1<𝑝≤2and{𝑋_𝑛,𝑛≥1}be a sequence of AANA random variables with mixing coefficients{𝑞(𝑛),𝑛≥1}. Assume that𝐸𝑋_𝑛=0for all𝑛≥1and∑^∞_𝑛=1𝑞²(𝑛)<∞; then there exists a positive constant𝐶_𝑝depending only on𝑝such that 𝐸max_{1≤𝑗≤𝑛}^𝑗_𝑖=1𝑋_𝑖^𝑝≤𝐶_𝑝^𝑛_𝑖=1𝐸𝑋_𝑖^𝑝 (6) for all𝑛≥1, where𝐶_𝑝=2^𝑝[2^2−𝑝𝑝+(6𝑝)^𝑝(∑^∞_𝑛=1𝑞²(𝑛))^(𝑝−1)].

By Lemmas 7 and 8, we can get the following Khintchine-Kolmogorov-type convergence theorem and three series theorem for AANA sequences, which can be applied to prove the main results of the paper. The proofs are standard, so we omit them.

Corollary 9 (Khintchine-Kolmogorov-type convergence theorem). Let{𝑋_𝑛,𝑛≥1}be a sequence of AANA random variables with mixing coefficients{𝑞(𝑛),𝑛≥1}and∑^∞_𝑛=1𝑞²(𝑛)<∞. If ^∞_𝑛=1Var𝑋_𝑛<∞, (7) then∑^∞_𝑛=1(𝑋_𝑛−𝐸𝑋_𝑛)converges almost surely.

Corollary 10 (three-series theorem for AANA random variables). Let{𝑋_𝑛,𝑛≥1}be a sequence of AANA random variables with mixing coefficients{𝑞(𝑛),𝑛≥1}and∑^∞_𝑛=1𝑞²(𝑛)<∞. Assume that for some𝑐>0, ^∞_𝑛=1𝑃𝑋_𝑛>𝑐<∞,^∞_𝑛=1𝐸𝑋_𝑛^(𝑐)converges,_∞𝑛=1Var𝑋_𝑛^(𝑐)<∞. (8) Then,∑^∞_𝑛=1𝑋_𝑛converges almost surely.

Remark 11. Since NA implies AANA, Corollaries 9 and 10 extend corresponding results for NA random variables (see Matula [18]) to AANA random variables without adding any extra condition.

Lemma 12 (cf. Wu [19] or Shen [20]). Let{𝑋_𝑛,𝑛≥1}be a sequence of random variables which is stochastically dominated by a random variable𝑋. For any𝛼>0and𝑏>0, the following two statements hold: 𝐸𝑋_𝑛^𝛼𝐼𝑋_𝑛≤𝑏≤𝐶₁𝐸|𝑋|^𝛼𝐼(|𝑋|≤𝑏)+𝑏^𝛼𝑃𝑋|,𝐸𝑋(|>𝑏)_𝑛𝑎𝐼𝑋_𝑛>𝑏≤𝐶₂𝐸|𝑋|^𝛼𝐼(|𝑋|>𝑏), (9) where𝐶₁and𝐶₂are positive constants.

Lemma 13 (cf. Wu [1]). Letℎ(𝑥)>0be a slowly varying function; then for any𝛿>0,𝑥^𝛿ℎ(𝑥)is a quasimonotonically increasing function and𝑥^−𝛿ℎ(𝑥)is a quasimonotonically decreasing function.

2. Strong Convergence Properties of Weighted Sums for AANA Sequence

Theorem 14. 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 defined on𝑅. For each𝑛≥1,𝑔_𝑛(𝑥)is a positive and nondecreasing function in(0,∞)and satisfies one of the following conditions:(i)for some0<𝑟≤1,𝑥^𝑟/𝑔_𝑛(𝑥)is a nondecreasing function in(0,∞);(ii)for some1<𝑟≤2,𝑥/𝑔_𝑛(𝑥)and𝑔_𝑛(𝑥)/𝑥^𝑟are nonincreasing functions in(0,∞); furthermore, assume that𝐸𝑋_𝑛=0for each𝑛≥1.
For any positive number sequence{𝑎_𝑛,𝑛≥1}with𝑎_𝑛↑∞ such that ^∞_𝑛=1𝐸𝑔_𝑛𝑋_𝑛𝑔_𝑛𝑎_𝑛<∞, (10) then ∑^∞_𝑛=1𝑋_𝑛/𝑎_𝑛converges a.s., and 𝑎_𝑛^𝑛−1_𝑖=1𝑋_𝑖⟶0𝑎.𝑠.,𝑎𝑠𝑛⟶∞. (11)

Proof. For each𝑛≥1, denote 𝑋^(𝑎𝑛)_𝑛=−𝑎_𝑛𝐼𝑋_𝑛<−𝑎_𝑛+𝑋_𝑛𝐼𝑋_𝑛≤𝑎_𝑛+𝑎_𝑛𝐼𝑋_𝑛>𝑎_𝑛. (12) By Lemma 7, we can see that, for fixed𝑛≥1,{𝑋^(𝑎𝑛)_𝑛}is still a sequence of AANA random variables. So by Corollary 10 in order to prove (11), we need only to prove the convergence of three series of (8), where𝑐=1.
Firstly, we prove that ∑^∞_𝑛=1𝑃(|𝑋_𝑛/𝑎_𝑛|>1)<∞under condition (i) or (ii).
For each𝑛≥1, if𝑔_𝑛(𝑥)satisfies condition (i), noting that{𝑔_𝑛(𝑥),𝑛≥1}is a sequence of positive and nondecreasing even function in(0,+∞). Combining Markov’s inequality with (10), it follows that ^∞_𝑛=1𝑃𝑋_𝑛𝑎_𝑛>1≤^∞_𝑛=1𝑃𝑔_𝑛𝑋_𝑛>𝑔_𝑛𝑎_𝑛≤^∞_𝑛=1𝐸𝑔_𝑛𝑋_𝑛𝑔_𝑛𝑎_𝑛<∞. (13) If𝑔_𝑛(𝑥)satisfies condition (ii), it is easy to prove that (13) also holds.
Secondly, we will show∑^∞_𝑛=1Var(𝑋^(𝑎𝑛)_𝑛/𝑎_𝑛)<∞.
If𝑔_𝑛(𝑥)satisfies (i), when|𝑥|≤𝑎_𝑛, we have(|𝑥|^𝑟/𝑔_𝑛(𝑥))≤(𝑎^𝑟_𝑛/𝑔_𝑛(𝑎_𝑛)), which implies that |𝑥|^𝑟𝑎^𝑟_𝑛≤𝑔_𝑛(𝑥)𝑔_𝑛𝑎_𝑛,𝑥²𝑎²_𝑛≤𝑔_𝑛(𝑥)^2/𝑟𝑔_𝑛𝑎_𝑛^2/𝑟. (14) Note that{𝑔_𝑛(𝑥),𝑛≥1}is a sequence of positive and nondecreasing functions in(0,+∞), so0≤(𝑔_𝑛(𝑥)/𝑔_𝑛(𝑎_𝑛))≤1when|𝑥|≤𝑎_𝑛. Consequently, 𝑥²𝑎²_𝑛≤𝑔_𝑛(𝑥)𝑔_𝑛𝑎_𝑛^2/𝑟≤𝑔_𝑛(𝑥)𝑔_𝑛𝑎_𝑛,for0<𝑟≤1. (15) On the other hand, if𝑔_𝑛(𝑥)satisfies condition (ii), then we can also get that 𝑥²𝑎²_𝑛≤𝑔_𝑛(𝑥)𝑔_𝑛𝑎_𝑛^2/𝑟≤𝑔_𝑛(𝑥)𝑔_𝑛𝑎_𝑛,for1<𝑟≤2. (16) Therefore, whether even function𝑔_𝑛(𝑥)satisfies condition (i) or (ii), we can obtain 𝑋Var^{(𝑎𝑛)𝑛}𝑎_𝑛𝑋≤𝐸^{(𝑎𝑛)𝑛}𝑎_𝑛²𝑋=𝐸^2𝑛𝑎²_𝑛𝐼𝑋_𝑛≤𝑎_𝑛+𝐸𝐼𝑋_𝑛>𝑎_𝑛𝑔≤𝐸^𝑛𝑋_𝑛𝑔_𝑛𝑎_𝑛𝐼𝑋_𝑛≤𝑎_𝑛𝑔+𝐸^𝑛𝑋_𝑛𝑔_𝑛𝑎_𝑛𝐼𝑋_𝑛>𝑎_𝑛=𝐸𝑔_𝑛𝑋_𝑛𝑔_𝑛𝑎_𝑛. (17) Therefore, it follows from (10) that ^∞_𝑛=1𝑋Var^{(𝑎𝑛)𝑛}𝑎_𝑛≤^∞_𝑛=1𝐸𝑔_𝑛𝑋_𝑛𝑔_𝑛𝑎_𝑛<∞. (18)
Finally, we prove that∑^∞_𝑛=1𝐸|𝑋^(𝑎𝑛)_𝑛/𝑎_𝑛|<∞.
If𝑔_𝑛(𝑥)satisfies condition (i), when|𝑥|≤𝑎_𝑛, we have(|𝑥|/𝑎_𝑛)≤(|𝑥|^𝑟/𝑎^𝑟_𝑛), for0<𝑟≤1. It follows that 𝐸𝑋^(𝑎𝑛)_𝑛𝑎_𝑛𝑋≤𝐸^𝑛𝑎_𝑛𝐼𝑋_𝑛≤𝑎_𝑛+𝐸𝐼𝑋_𝑛>𝑎_𝑛𝑋≤𝐸^𝑛𝑟𝑎^𝑟_𝑛𝐼𝑋_𝑛≤𝑎_𝑛𝑔+𝐸^𝑛𝑋_𝑛𝑔_𝑛𝑎_𝑛𝐼𝑋_𝑛>𝑎_𝑛𝑔≤𝐸^𝑛𝑋_𝑛𝑔_𝑛𝑎_𝑛𝐼𝑋_𝑛≤𝑎_𝑛𝑔+𝐸^𝑛𝑋_𝑛𝑔_𝑛𝑎_𝑛𝐼𝑋_𝑛>𝑎_𝑛=𝐸𝑔_𝑛𝑋_𝑛𝑔_𝑛𝑎_𝑛. (19)
If𝑔_𝑛(𝑥)satisfies condition (ii), then by the fact that𝐸𝑋_𝑛=0and𝑥/𝑔_𝑛(𝑥)is a nonincreasing function in(0,∞), we get 𝐸𝑋^(𝑎𝑛)_𝑛𝑎_𝑛≤𝐸𝑋_𝑛𝑎_𝑛𝐼𝑋_𝑛≤𝑎_𝑛+𝐸𝐼𝑋_𝑛>𝑎_𝑛=𝐸𝑋_𝑛𝑎_𝑛𝐼𝑋_𝑛>𝑎_𝑛+𝐸𝐼𝑋_𝑛>𝑎_𝑛𝑋≤𝐸^𝑛𝑎_𝑛𝐼𝑋_𝑛>𝑎_𝑛+𝐸𝐼𝑋_𝑛>𝑎_𝑛𝑔≤𝐸^𝑛𝑋_𝑛𝑔_𝑛𝑎_𝑛𝐼𝑋_𝑛>𝑎_𝑛𝑔+𝐸^𝑛𝑋_𝑛𝑔_𝑛𝑎_𝑛𝐼𝑋_𝑛>𝑎_𝑛≤2𝐸𝑔_𝑛𝑋_𝑛𝑔_𝑛𝑎_𝑛. (20) Therefore, whether𝑔_𝑛(𝑥)satisfies condition (i) or (ii), it also follows from (10) that ^∞_𝑛=1𝐸𝑋^(𝑎𝑛)_𝑛𝑎_𝑛<∞. (21) The proof of Theorem 14 is completed by (13), (18), and (21).

Corollary 15. Let{𝑋_𝑛,𝑛≥1}be a sequence of AANA random variables with∑^∞_𝑛=1𝑞²(𝑛)<∞, and let{𝑎_𝑛,𝑛≥1}be a sequence of positive numbers with𝑎_𝑛↑∞. There exists some0<𝑝≤2such that ^∞_𝑛=1𝐸𝑋_𝑛^𝑝𝑎^𝑝_𝑛<∞. (22) If1<𝑝≤2, we further assume that𝐸𝑋_𝑛=0. Then, (11) holds.

Proof. We take𝑔_𝑛(𝑥)=|𝑥|^𝑟,𝑥∈(−∞,∞). If0<𝑟≤1, one can find that𝑥^𝑟/𝑔_𝑛(𝑥)≡1for𝑥≥0. So condition (i) of Theorem 14 is satisfied. If1<𝑟≤2, we have that𝑥/𝑔_𝑛(𝑥)=𝑥^1−𝑟and𝑔_𝑛(𝑥)/𝑥^𝑟≡1for𝑥≥0. Thus, condition (ii) of Theorem 14 is satisfied. Consequently, the desired result (11) follows from Theorem 14 immediately.

Remark 16. If taking𝑟=1in (i) and𝑟=2in (ii), Theorem 14 and Corollary 15 extend the corresponding ones for NA random variables (see Gan [21]) to AANA random variables.

Theorem 17. Let1<𝛼<2and{𝑋_𝑛,𝑛≥1} be a sequence of AANA random variables with∑^∞_𝑛=1𝑞²(𝑛)<∞and identical distribution 𝑃𝑋₁=𝐿(𝑥)𝑥>𝑥^−𝛼,𝑥≥11,𝑥<1, (23) where𝐿(𝑥)is a slowly varying function. Let{𝑎_𝑛,𝑛≥1}and{𝑏_𝑛,𝑛≥1}be sequences of positive constants satisfying0<𝑏_𝑛↑∞. Denote𝑐_𝑛=𝑏_𝑛/𝑎_𝑛for each𝑛≥1. Assume that ^∞_𝑛=1𝑃𝑋_𝑛>𝑐_𝑛<∞; (24) then 𝑏_𝑛^𝑛−1_𝑘=1𝑎_𝑘𝑋_𝑘⟶0𝑎.𝑠.,𝑎𝑠𝑛⟶∞. (25)

Proof. Since (23) and (24) imply that𝑐_𝑘≥1for all sufficiently large𝑘. Without loss of generality, we assume𝑐_𝑘≥1for all𝑘≥1.
By Borel-Cantelli Lemma, it is easily seen that (24) implies that ^𝑛_𝑘=1𝑎_𝑘𝑋_𝑘𝐼𝑋_𝑘>𝑐_𝑘=𝑜𝑏_𝑛a.s. (26) Denote 𝑌_𝑘=−𝑐_𝑘𝐼𝑋_𝑘<−𝑐_𝑘+𝑋_𝑘𝐼𝑋_𝑘≤𝑐_𝑘+𝑐_𝑘𝐼𝑋_𝑘>𝑐_𝑘,𝑘≥1; (27) thus,{𝑌_𝑘,𝑘≥1}is still AANA from Lemma 7. It is easy to check that ^𝑛_𝑘=1𝑎_𝑘𝑋_𝑘=^𝑛_𝑘=1𝑎_𝑘𝑌_𝑘−𝐸𝑌_𝑘+^𝑛_𝑘=1𝑎_𝑘𝐸𝑌_𝑘+^𝑛_𝑘=1𝑎_𝑘𝑐_𝑘𝐼𝑋_𝑘<−𝑐_𝑘−𝐼𝑋_𝑘>𝑐_𝑘+^𝑛_𝑘=1𝑎_𝑘𝑋_𝑘𝐼𝑋_𝑘>𝑐_𝑘. (28) In order to show that 𝑏_𝑛⁻¹∑^𝑛_𝑘=1𝑎_𝑘𝑋_𝑘→0a.s., we only need to show that the first three terms above are𝑜(𝑏_𝑛)or𝑜(𝑏_𝑛)a.s.
By𝐶_𝑟inequality, Theorem  1b in [22, page 281] (or see Adler [23]) and (24), we can get ^∞_𝑘=1𝑌Var^𝑘𝑐_𝑘≤^∞_𝑘=1𝑐_𝑘⁻²𝐸𝑌_𝑘²≤3^∞_𝑘=1𝑐_𝑘⁻²𝐸𝑐_𝑘²𝐼𝑋_𝑘>𝑐_𝑘+𝑋²_𝑘𝐼𝑋_𝑘≤𝑐_𝑘=3^∞_𝑘=1𝑃𝑋_𝑘>𝑐_𝑘+3^∞_𝑘=1𝑐_𝑘⁻²𝐸𝑋²_𝑘𝐼𝑋_𝑘≤𝑐_𝑘≤𝐶+6^∞_𝑘=1𝑐_𝑘^−2𝑐𝑘₀𝑡𝑃𝑋_𝑘>𝑡𝑑𝑡≤𝐶+𝐶^∞_𝑘=1𝐿𝑐_𝑘𝑐_𝑘^−𝛼≤𝐶+𝐶^∞_𝑘=1𝑃𝑋_𝑘>𝑐_𝑘<∞. (29) It follows from Corollary 9 and Kronecker’s lemma that ^𝑛_𝑘=1𝑎_𝑘𝑌_𝑘−𝐸𝑌_𝑘=𝑜𝑏_𝑛a.s. (30) By (24) again, ^∞_𝑘=1𝐸𝑎_𝑘𝑐_𝑘𝐼𝑋_𝑘<−𝑐_𝑘−𝐼𝑋_𝑘>𝑐_𝑘𝑏_𝑘≤^∞_𝑘=1𝐸𝐼𝑋_𝑘<−𝑐_𝑘+𝐼𝑋_𝑘>𝑐_𝑘=^∞_𝑘=1𝑃𝑋_𝑘>𝑐_𝑘<∞, (31) which implies that ^∞_𝑘=1𝑎_𝑘𝑐_𝑘𝐼𝑋_𝑘<−𝑐_𝑘−𝐼𝑋_𝑘>𝑐_𝑘𝑏_𝑘convergesa.s. (32) By Kronecker’s lemma, it follows that ^𝑛_𝑘=1𝑎_𝑘𝑐_𝑘𝐼𝑋_𝑘<−𝑐_𝑘−𝐼𝑋_𝑘>𝑐_𝑘=𝑜𝑏_𝑛a.s. (33) By Theorem  1b in [22, page 281] (or see Adler [23]) and (24) again, we have ^∞_𝑘=1𝑎_𝑘𝐸𝑌_𝑘𝑏_𝑘≤^∞_𝑘=1𝑐_𝑘⁻¹𝑐_𝑘𝑃𝑋_𝑘>𝑐_𝑘+𝐸𝑋_𝑘𝐼𝑋_𝑘>𝑐_𝑘=^∞_𝑘=1𝑃𝑋_𝑘>𝑐_𝑘+^∞_𝑘=1𝑐_𝑘⁻¹𝐸𝑋_𝑘𝐼𝑋_𝑘>𝑐_𝑘=2^∞_𝑘=1𝑃𝑋_𝑘>𝑐_𝑘+^∞_𝑘=1𝑐_𝑘^−1∞_𝑐𝑘𝑃𝑋_𝑘>𝑡𝑑𝑡≤𝐶+𝐶^∞_𝑘=1𝑐_𝑘^−1∞_𝑐𝑘𝐿(𝑡)𝑡^−𝛼𝑑𝑡≤𝐶+𝐶^∞_𝑘=1𝐿𝑐_𝑘𝑐_𝑘^−𝛼≤𝐶+𝐶^∞_𝑘=1𝑃𝑋_𝑘>𝑐_𝑘<∞, (34) which implies that ^∞_𝑘=1𝑎_𝑘𝐸𝑌_𝑘𝑏_𝑘converges. (35) By Kronecker’s Lemma, it follows that ^𝑛_𝑘=1𝑎_𝑘𝐸𝑌_𝑘=𝑜𝑏_𝑛. (36) Hence, the desired result (25) follows from (26)–(36) immediately.

Remark 18. Theorem 17 generalizes and extends the corresponding one for NA random variables (see Wang et al. [24]) to AANA random variables.

Theorem 19. Let1<𝑟<2and{𝑋_𝑛,𝑛≥1}a sequence of mean zero AANA random variables with∑^∞_𝑛=1𝑞²(𝑛)<∞, which is stochastically dominated by a random variable𝑋. Let{𝑎_𝑛,𝑛≥1}be a sequence of positive constants satisfying𝐴_𝑛≐∑^𝑛_𝑘=1𝑎_𝑘↑∞. Denote𝑐_𝑛=𝐴_𝑛/𝑎_𝑛for each𝑛≥1. Assume that 𝐸|𝑋|^𝑟𝑁(𝑛)<∞,≐Card𝑖∶𝑐_𝑖≤𝑛=𝑂𝑛^𝑟,𝑛≥1; (37) then 𝐴_𝑛^𝑛−1_𝑘=1𝑎_𝑘𝑋_𝑘⟶0𝑎.𝑠.,𝑎𝑠𝑛⟶∞. (38)

Proof. Let𝑁(0)=0and denote 𝑋^(𝑐𝑛)_𝑛=−𝑐_𝑛𝐼𝑋_𝑛<−𝑐_𝑛+𝑋_𝑛𝐼𝑋_𝑛≤𝑐_𝑛+𝑐_𝑛𝐼𝑋_𝑛>𝑐_𝑛,𝑛≥1. (39) It follows by (37) that ^∞_𝑖=1𝑃𝑋_𝑖≠𝑋^(𝑐𝑖)_𝑖=^∞_𝑖=1𝑃𝑋_𝑖>𝑐_𝑖=^∞_{𝑗=1𝑐𝑖≤𝑗<𝑐𝑖+1}𝑃𝑋_𝑖>𝑐_𝑖≤𝐶^∞_{𝑗=1𝑗−1<𝑐𝑖≤𝑗}𝑃|𝑋|>𝑗−1=𝐶^∞_𝑗=1𝑁𝑗−𝑁𝑗−1𝑃|𝑋|>𝑗−1=𝐶^∞_𝑗=1𝑁𝑗−𝑁𝑗−1^∞_𝑛=𝑗𝑃(𝑛−1<|𝑋|≤𝑛)=𝐶^∞𝑛_𝑛=1𝑗=1𝑁𝑗−𝑁𝑗−1𝑃(𝑛−1<|𝑋|≤𝑛)≤𝐶^∞_𝑛=1𝑛^𝑟𝑃(𝑛−1<|𝑋|≤𝑛)≤𝐶𝐸|𝑋|^𝑟<∞. (40) By the equality above and Borel-Cantelli lemma, we can get𝑃(𝑋_𝑖≠𝑋^(𝑐𝑖)_𝑖,i.o.)=0. Therefore, in order to prove (38), we only need to prove that 𝐴_𝑛^𝑛−1_𝑖=1𝑎_𝑖𝑋^(𝑐𝑖)_𝑖⟶0a.s.,𝑛⟶∞. (41)
By𝐶_𝑟inequality, Lemma 12, and (37) again, ^∞_𝑘=1𝑎Var^𝑘𝑋^(𝑐𝑘)_𝑘𝐴_𝑘≤^∞_𝑘=1𝑐_𝑘⁻²𝐸𝑋^(𝑐𝑘)_𝑘²≤3^∞_𝑘=1𝑐_𝑘⁻²𝐸𝑐_𝑘²𝐼𝑋_𝑘>𝑐_𝑘+𝑋_𝑘²𝐼𝑋_𝑘≤𝑐_𝑘≤𝐶^∞_𝑘=1𝑃|𝑋|>𝑐_𝑘+𝐶^∞_𝑘=1𝑐_𝑘⁻²𝐸𝑋²𝐼|𝑋|≤𝑐_𝑘≤𝐶+𝐶^∞_{𝑗=1𝑗−1<𝑐𝑘≤𝑗}𝑐_𝑘⁻²𝐸𝑋²𝐼|𝑋|≤𝑐_𝑘≤𝐶+𝐶^∞_{𝑗=1𝑗−1<𝑐𝑘≤𝑗}𝑐_𝑘⁻²𝐸𝑋²𝐼|𝑋|≤𝑗≤𝐶+𝐶^∞_𝑗=2𝑁𝑗−𝑁𝑗−1𝑗−1⁻²×^𝑗_𝑘=1𝐸𝑋²𝐼(𝑘−1<|𝑋|≤𝑘)≤𝐶+𝐶^∞∞_{𝑘=2𝑗=𝑘}𝑁𝑗×−𝑁𝑗−1𝑗−1⁻²𝐸𝑋²𝐼(𝑘−1<|𝑋|≤𝑘)≤𝐶+𝐶^∞∞_{𝑘=2𝑗=𝑘}𝑁𝑗𝑗−1⁻²−𝑗⁻²×𝐸𝑋²𝐼(𝑘−1<|𝑋|≤𝑘)≤𝐶+𝐶^∞∞_{𝑘=2𝑗=𝑘}𝑗^𝑟−3𝐸𝑋²𝐼(𝑘−1<|𝑋|≤𝑘)≤𝐶+𝐶^∞_𝑘=2𝑘^𝑟−2𝐸|𝑋|^𝑟𝑘^2−𝑟𝐼(𝑘−1<|𝑋|≤𝑘)=𝐶+𝐶^∞_𝑘=2𝐸|𝑋|^𝑟𝐼(|𝑋|)|𝑋|𝑘−1<≤𝑘≤𝐶+𝐶𝐸^𝑟<∞. (42) Hence, by the inequality above, Corollary 9 and Kronecker’s lemma, we have 𝐴_𝑛^𝑛−1_𝑖=1𝑎_𝑖𝑋^(𝑐𝑖)_𝑖−𝐸𝑋^(𝑐𝑖)_𝑖⟶0a.s.,𝑛⟶∞. (43) In order to prove (41), it suffices to prove that 𝐴_𝑛^𝑛−1_𝑖=1𝑎_𝑖𝐸𝑋^(𝑐𝑖)_𝑖⟶0𝑛⟶∞. (44) Notice that𝐸𝑋_𝑛=0for each𝑛≥1, we have 𝐸𝑋_𝑛𝐼𝑋_𝑛≤𝑐_𝑛=𝐸𝑋_𝑛𝐼𝑋_𝑛>𝑐_𝑛. (45) It follows from Lemma 12 and (37) that, ^∞_𝑘=1𝑎_𝑘𝐸𝑋^(𝑐𝑘)_𝑘𝐴_𝑘≤^∞_𝑘=1𝑃𝑋_𝑘>𝑐_𝑘+𝑐_𝑘⁻¹𝐸𝑋_𝑘𝐼𝑋_𝑘≤𝑐_𝑘=^∞_𝑘=1𝑃𝑋_𝑘>𝑐_𝑘+𝑐_𝑘⁻¹𝐸𝑋_𝑘𝐼𝑋_𝑘>𝑐_𝑘≤^∞_𝑘=1𝑃𝑋_𝑘>𝑐_𝑘+^∞_𝑘=1𝑐_𝑘⁻¹𝐸𝑋_𝑘𝐼𝑋_𝑘>𝑐_𝑘≤𝐶^∞_𝑗=1𝑃|𝑋|>𝑐_𝑘+𝐶^∞_𝑘=1𝑐_𝑘⁻¹𝐸|𝑋|𝐼|𝑋|>𝑐_𝑘≤𝐶+𝐶^∞_{𝑗=1𝑐𝑘≤𝑗<𝑐𝑘+1}𝑐_𝑘⁻¹𝐸|𝑋|𝐼|𝑋|>𝑐_𝑘≤𝐶+𝐶^∞_{𝑗=1𝑗−1<𝑐𝑘≤𝑗}𝑐_𝑘⁻¹𝐸|𝑋|𝐼|𝑋|>𝑗−1≤𝐶+𝐶^∞_𝑗=2𝑁𝑗−𝑁𝑗−1𝑗−1⁻¹×^∞_{𝑘=𝑗−1}𝐸|𝑋|𝐼(𝑘<|𝑋|≤𝑘+1)≤𝐶+𝐶^∞_𝑘=1^𝑘+1_𝑗=2𝑁𝑗𝑗−1⁻¹−𝑗⁻¹×𝐸|𝑋|𝐼(𝑘<|𝑋|≤𝑘+1)≤𝐶+𝐶^∞_𝑘=1^𝑘+1_𝑗=2𝑗^𝑟−2𝐸|𝑋|𝐼(𝑘<|𝑋|≤𝑘+1)≤𝐶+𝐶^∞_𝑘=1𝑘^𝑟−1𝐸|𝑋|𝐼(𝑘<|𝑋|≤𝑘+1)≤𝐶+𝐶^∞_𝑘=1𝐸|𝑋|^𝑟𝐼(|𝑋|)|𝑋|𝑘<≤𝑘+1≤𝐶+𝐶𝐸^𝑟<∞. (46) By Kronecker’s lemma, we can get (44) immediately. The proof is complete.

3. Strong Stability for Weighted Sums of AANA Sequence

Theorem 20. Let{𝑋_𝑛,𝑛≥1}be a sequence of AANA random variables with∑^∞_𝑛=1𝑞²(𝑛)<∞, which is stochastically dominated by a random variable𝑋. Let{𝑎_𝑛,𝑛≥1}and{𝑏_𝑛,𝑛≥1}be two sequences of positive numbers with𝑐_𝑛=𝑏_𝑛/𝑎_𝑛and𝑏_𝑛↑∞. Denote𝑁(𝑥)=Card{𝑛∶𝑐_𝑛≤𝑥}, 𝑥>0. If the following conditions are satisfied:(i)𝐸𝑁(|𝑋|)<∞; (ii)∫^∞₀𝑡^𝑝−1𝑃(|𝑋|>𝑡)(∫^∞_𝑡𝑦^−(𝑝+1)𝑁(𝑦)𝑑𝑦)𝑑𝑡<∞, for some𝑝∈[1,2],then there exist𝑑_𝑛∈𝐑,𝑛=1,2,…, such that 𝑏_𝑛^𝑛−1_𝑖=1𝑎_𝑖𝑋_𝑖−𝑑_𝑛⟶0𝑎.𝑠. (47)

Proof. For each𝑖≥1, denote 𝑋^(𝑐𝑖)_𝑖=−𝑐_𝑖𝐼𝑋_𝑖<−𝑐_𝑖+𝑋_𝑖𝐼𝑋_𝑖≤𝑐_𝑖+𝑐_𝑖𝐼𝑋_𝑖>𝑐_𝑖. (48) By Definition 1 and conditions (i), we can obtain ^∞_𝑖=1𝑃𝑋_𝑖≠𝑋^(𝑐𝑖)_𝑖=^∞_𝑖=1𝑃𝑋_𝑖>𝑐_𝑖≤𝐶^∞_𝑖=1𝑃|𝑋|>𝑐_𝑖≤𝐶^∞_𝑖=1^∞₀𝐼𝑐_𝑖≤𝑡𝑑𝑃(|𝑋|≤𝑡)=𝐶^∞₀𝑁(𝑡)𝑋|)𝑋𝑑𝑃(|≤𝑡=𝐶𝐸𝑁(||)<∞. (49) By Borel-Cantelli lemma for any sequence{𝑑_𝑛,𝑛≥1}⊂𝐑, with probability 1, the sequences{𝑏_𝑛⁻¹∑^𝑛_𝑖=1𝑎_𝑖𝑋_𝑖−𝑑_𝑛}and{𝑏_𝑛⁻¹∑^𝑛_𝑖=1𝑎_𝑖𝑋^(𝑐𝑖)_𝑖−𝑑_𝑛}converge on the same set and to the same limit. We will prove that𝑏_𝑛⁻¹∑^𝑛_𝑖=1𝑎_𝑖(𝑋^(𝑐𝑖)_𝑖−𝐸𝑋^(𝑐𝑖)_𝑖)→0a.s., which implies (6) with𝑑_𝑛=𝑏_𝑛⁻¹∑^𝑛_𝑖=1𝑎_𝑖𝐸𝑋^(𝑐𝑖)_𝑖. According to Lemma 7,{𝑎_𝑖(𝑋^(𝑐𝑖)_𝑖−𝐸𝑋^(𝑐𝑖)_𝑖),𝑖≥1}is a sequence of AANA random variables with mean zero. By𝐶_𝑟inequality and Lemma 12, we have ^∞_𝑛=1𝐸𝑎_𝑛𝑋^(𝑐𝑛)_𝑛−𝐸𝑋^(𝑐𝑛)_𝑛^𝑝𝑏^𝑝_𝑛≤𝐶^∞_𝑛=1𝑐_𝑛^−𝑝𝐸𝑋_𝑛^𝑝𝐼𝑋_𝑛≤𝑐_𝑛+𝐶^∞_𝑛=1𝑐_𝑛^−𝑝𝑐^𝑝_𝑛𝑃𝑋_𝑛>𝑐_𝑛≤𝐶^∞_𝑛=1𝑐_𝑛^−𝑝𝑐^𝑝_𝑛𝑃|𝑋|>𝑐_𝑛+𝐸|𝑋|^𝑝𝐼|𝑋|≤𝑐_𝑛+𝐶^∞_𝑛=1𝑃|𝑋|>𝑐_𝑛≤𝐶𝐸𝑁(|𝑋|)+𝐶^∞_𝑛=1𝑐_𝑛^{−𝑝𝑐𝑛}₀𝑡^𝑝−1𝑃(|𝑋|>𝑡)𝑑𝑡. (50) Notice that ^∞_𝑛=1𝑐_𝑛^{−𝑝𝑐𝑛}₀𝑡^𝑝−1𝑃𝑋|)=(|>𝑡𝑑𝑡_∞0𝑡^𝑝−1𝑃(|𝑋|>𝑡)_{𝑛∶𝑐𝑛≥𝑡}𝑐_𝑛^−𝑝𝑑𝑡≤𝐶^∞₀𝑡^𝑝−1𝑃(|𝑋|>𝑡)^∞_𝑡𝑦^−(𝑝+1)𝑁𝑦𝑑𝑦𝑑𝑡, (51) where the last inequality follows from the fact that for𝑡>0_{𝑛∶𝑐𝑛≥𝑡}𝑐_𝑛^−𝑝=lim_{𝑢→∞𝑛∶𝑡≤𝑐𝑛≤𝑢}𝑐_𝑛^−𝑝=lim_𝑢→∞^𝑢_𝑡𝑦^−𝑝𝑑𝑁𝑦=lim_𝑢→∞𝑢^−𝑝𝑁(𝑢)−𝑡^−𝑝𝑁(𝑡)+𝑝^𝑢_𝑡𝑦^−(𝑝+1)𝑁𝑦𝑑𝑦,lim_𝑢→∞𝑢^−𝑝𝑁(𝑢)≤lim_𝑢→∞𝑝^∞_𝑢𝑦^−(𝑝+1)𝑁𝑦𝑑𝑦≤𝑝^∞_𝑡𝑦^−(𝑝+1)𝑁𝑦𝑑𝑦. (52) By (50), (51) and condition (i), (ii), we can get that ^∞_𝑛=1𝐸𝑎_𝑛𝑋^(𝑐𝑛)_𝑛−𝐸𝑋^(𝑐𝑛)_𝑛^𝑝𝑏^𝑝_𝑛<∞. (53) Therefore, it follows from (53) and Corollary 15 that 𝑏_𝑛^𝑛−1_𝑖=1𝑎_𝑖𝑋^(𝑐𝑖)_𝑖−𝐸𝑋^(𝑐𝑖)_𝑖⟶0,𝑎.𝑠. (54) The proof is complete.

Corollary 21. Suppose that the conditions of Theorem 20 are satisfied and𝐸𝑋_𝑛=0for each𝑛≥1. If∫^∞₁𝐸𝑁(|𝑋|/𝑠)𝑑𝑠<∞, then𝑏_𝑛⁻¹∑^𝑛_𝑖=1𝑎_𝑖𝑋_𝑖→0𝑎.𝑠.

Proof. According to the proof of Theorem 20, we need only to prove that 𝑏_𝑛^𝑛−1_𝑖=1𝑎_𝑖𝐸𝑋^(𝑐𝑖)_𝑖⟶0,as𝑛⟶∞. (55) Notice that𝐸𝑋_𝑛=0for each𝑛≥1; then 𝐸𝑋^(𝑐𝑖)_𝑖=−𝐸𝑋_𝑖𝐼𝑋_𝑖>𝑐_𝑖−𝐸𝑐_𝑖𝐼𝑋_𝑖<−𝑐_𝑖+𝐸𝑐_𝑖𝐼𝑋_𝑖>𝑐_𝑖=𝐸𝑋_𝑖−𝑐_𝑖𝐼𝑋_𝑖>𝑐_𝑖+𝐸𝑋_𝑖+𝑐_𝑖𝐼𝑋_𝑖<−𝑐_𝑖≤𝐸𝑋_𝑖+𝑐_𝑖𝐼𝑋_𝑖>𝑐_𝑖,_∞𝑖=1𝑎_𝑖𝐸𝑋^(𝑐𝑖)_𝑖𝑏_𝑖≤^∞_𝑖=1𝑐_𝑖⁻¹𝐸𝑋_𝑖+𝑐_𝑖𝐼𝑋_𝑖>𝑐_𝑖=^∞_𝑖=1𝑃𝑋_𝑖>𝑐_𝑖+^∞_𝑖=1𝑐_𝑖⁻¹𝐸𝑋_𝑖𝐼𝑋_𝑖>𝑐_𝑖≤𝐶𝐸𝑁(|𝑋|)+^∞_𝑖=1𝑐_𝑖^−1∞_𝑐𝑖𝑃𝑋_𝑖𝑋+>𝑡𝑑𝑡=𝐶𝐸𝑁(||)^∞₁^∞_𝑖=1𝑃𝑋_𝑖>𝑠𝑐_𝑖𝑋𝑑𝑠≤𝐶𝐸𝑁(||)+𝐶^∞₁|𝑋|𝐸𝑁𝑠𝑑𝑠<∞. (56)(57) By Kronecker’s lemma, we can get (55) immediately.

Theorem 22. Let{𝑋_𝑛,𝑛≥1}be a sequence of AANA random variables with mean zero and∑^∞_𝑛=1𝑞²(𝑛)<∞, which is stochastically dominated by a random variable𝑋. Let{𝑎_𝑛,𝑛≥1}and{𝑏_𝑛,𝑛≥1}be two sequences of positive numbers with𝑐_𝑛=𝑏_𝑛/𝑎_𝑛and𝑏_𝑛↑∞. Denote𝑁(𝑥)=Card{𝑛∶𝑐_𝑛≤𝑥},𝑥>0. If the following conditions are satisfied:(i)𝐸𝑁(|𝑋|)<∞; (ii)∫^∞₁𝐸𝑁(|𝑋|/𝑠)𝑑𝑠<∞; (iii)max_{1≤𝑗≤𝑛}𝑐^𝑝_𝑗∑^∞_𝑖=𝑛𝑐_𝑖^−𝑝=𝑂(𝑛), for some𝑝∈[1,2],then 𝑏_𝑛^𝑛−1_𝑖=1𝑎_𝑖𝑋_𝑖⟶0𝑎.𝑠. (58)

Proof. By (49), condition (i), and Borel-Cantelli lemma, it suffices to prove𝑏_𝑛⁻¹∑^∞_𝑖=1𝑎_𝑖𝑋^(𝑐𝑖)_𝑖→0,a.s.So we need only to prove 𝑏_𝑛^𝑛−1_𝑖=1𝑎_𝑖𝑋^(𝑐𝑖)_𝑖−𝐸𝑋^(𝑐𝑖)_𝑖𝑏⟶0a.s.,_{𝑛𝑛−1𝑖=1}𝑎_𝑖𝐸𝑋^(𝑐𝑖)_𝑖⟶0,as𝑛⟶∞. (59)(60) We can get (60) from the proof of Corollary 21. In the following, we prove (59). Put𝜀₀=0and𝜀_𝑛=max_{1≤𝑗≤𝑛}𝑐_𝑗for𝑛≥1. According to Lemma 7,{𝑎_𝑖(𝑋^(𝑐𝑖)_𝑖−𝐸𝑋^(𝑐𝑖)_𝑖),𝑖≥1}is a sequence of AANA random variables with mean zero. By𝐶_𝑟inequality and Lemma 12, ^∞_𝑛=1𝐸𝑎_𝑛𝑋^(𝑐𝑛)_𝑛−𝐸𝑋^(𝑐𝑛)_𝑛^𝑝𝑏^𝑝_𝑛≤𝐶^∞_𝑛=1𝑐_𝑛^−𝑝𝐸𝑋_𝑛^𝑝𝐼𝑋_𝑛≤𝑐_𝑛+𝐶^∞_𝑛=1𝑃𝑋_𝑛>𝑐_𝑛≤𝐶^∞_𝑛=1𝑐_𝑛^−𝑝𝑐^𝑝_𝑛𝑃|𝑋|>𝑐_𝑛+𝐸|𝑋|^𝑝𝐼|𝑋|≤𝑐_𝑛+𝐶^∞_𝑛=1𝑃|𝑋|>𝑐_𝑛≤𝐶𝐸𝑁(|𝑋|)+𝐶^∞_𝑛=1𝑐_𝑛^−𝑝𝐸|𝑋|^𝑝𝐼|𝑋|≤𝑐_𝑛. (61) It is easy to see that ^∞_𝑛=1𝑐_𝑛^−𝑝𝐸|𝑋|^𝑝𝐼|𝑋|≤𝑐_𝑛≤^∞_𝑛=1𝑐_𝑛^−𝑝𝐸|𝑋|^𝑝𝐼|𝑋|≤𝜀_𝑛=^∞_𝑛=1𝑐_𝑛^𝑛−𝑝_𝑗=1𝐸|𝑋|^𝑝𝐼𝜀_𝑗−1<|𝑋|≤𝜀_𝑗≤^∞_𝑗=1𝑃𝜀_𝑗−1<|𝑋|≤𝜀_𝑗𝜀^𝑝_𝑗^∞_𝑛=𝑗𝑐_𝑛^−𝑝≤^∞_𝑗=1𝑗𝑃𝜀_𝑗−1<|𝑋|≤𝜀_𝑗≤𝐶^∞_𝑛=1𝑃|𝑋|>𝜀_𝑛−1≤𝐶1+^∞_𝑛=1𝑃|𝑋|>𝑐_𝑛≤𝐶(1+𝐸𝑁(|𝑋|))<∞. (62) Therefore, ^∞_𝑛=1𝐸𝑎_𝑛𝑋^(𝑐𝑛)_𝑛−𝐸𝑋^(𝑐𝑛)_𝑛^𝑝𝑏^𝑝_𝑛<∞ (63) follows from condition (i), (61) and (62). By Corollary 15 and (63), we can obtain (59) immediately. The proof is complete.

Theorem 23. Let{𝑋_𝑛,𝑛≥1}be a sequence of AANA random variables with∑^∞_𝑛=1𝑞²(𝑛)<∞, which is stochastically dominated by a random variable𝑋. Let{𝑎_𝑛,𝑛≥1}and{𝑏_𝑛,𝑛≥1}be two sequences of positive numbers with𝑐_𝑛=𝑏_𝑛/𝑎_𝑛and𝑏_𝑛↑∞. Define𝑁(𝑥)=Card{𝑛∶𝑐_𝑛≤𝑥},𝑅(𝑥)=∫^∞_𝑥𝑁(𝑦)𝑦⁻³𝑑𝑦,𝑥>0. If(i)𝑁(𝑥)<∞for any𝑥>0;(ii)𝑅(1)=∫^∞₁𝑁(𝑦)𝑦⁻³𝑑𝑦<∞;(iii)𝐸(𝑋²𝑅(|𝑋|))<∞,then there exist𝑑_𝑛∈𝐑,𝑛=1,2,…, such that 𝑏_𝑛^𝑛−1_𝑖=1𝑎_𝑖𝑋_𝑖−𝑑_𝑛⟶0𝑎.𝑠. (64)

Proof. According to Lemma 7,{𝑋^(𝑐𝑖)_𝑖,𝑖≥1}and{𝑋^(𝑐𝑖)_𝑖/𝑐_𝑖,𝑖≥1}are sequences of AANA random variables.
Since𝑁(𝑥)is nondecreasing, then for any𝑥>0𝑅(𝑥)≥𝑁(𝑥)^∞_𝑥𝑦⁻³1𝑑𝑦=2𝑥⁻²𝑁(𝑥), (65) which implies that𝐸𝑁(|𝑋|)≤2𝐸(𝑋²𝑅(|𝑋|))<∞. Therefore, ^∞_𝑖=1𝑃𝑋_𝑖≠𝑋^(𝑐𝑖)_𝑖=^∞_𝑖=1𝑃𝑋_𝑖>𝑐_𝑖≤𝐶^∞_𝑖=1𝑃|𝑋|>𝑐_𝑖≤𝐶𝐸𝑁(|𝑋|)<∞. (66) By Borel-Cantelli lemma for any sequence{𝑑_𝑛,𝑛≥1}⊂𝐑, with probability 1, the sequences{𝑏_𝑛⁻¹∑^𝑛_𝑖=1𝑎_𝑖𝑋_𝑖−𝑑_𝑛}and{𝑏_𝑛⁻¹∑^𝑛_𝑖=1𝑎_𝑖𝑋^(𝑐𝑖)_𝑖−𝑑_𝑛}converge on the same set and to the same limit. We will prove that𝑏_𝑛⁻¹∑^𝑛_𝑖=1𝑎_𝑖(𝑋^(𝑐𝑖)_𝑖−𝐸𝑋^(𝑐𝑖)_𝑖)→0a.s., which implies the theorem with𝑑_𝑛=𝑏_𝑛⁻¹∑^𝑛_𝑖=1𝑎_𝑖𝐸𝑋^(𝑐𝑖)_𝑖. By𝐶_𝑟inequality and Lemma 12, ^∞_𝑛=1𝑎Var^𝑛𝑋^(𝑐𝑛)_𝑛𝑏_𝑛≤^∞_𝑛=1𝑐_𝑛⁻²𝐸𝑋^(𝑐𝑛)_𝑛²≤3^∞_𝑛=1𝑃𝑋_𝑛>𝑐_𝑛+3^∞_𝑛=1𝑐_𝑛⁻²𝐸𝑋²_𝑛𝐼𝑋_𝑛≤𝑐_𝑛≤𝐶^∞_𝑛=1𝑃|𝑋|>𝑐_𝑛+𝐶^∞_𝑛=1𝑐_𝑛⁻²𝑐²_𝑛𝑃|𝑋|>𝑐_𝑛+𝐸𝑋²𝐼|𝑋|≤𝑐_𝑛≤𝐶𝐸𝑁(|𝑋|)+𝐶^∞_𝑛=1𝑐_𝑛⁻²𝐸𝑋²𝐼|𝑋|≤𝑐_𝑛≐𝐶𝐸𝑁(|𝑋|)+CI,𝐼=_{𝑛∶𝑐𝑛≤1}𝑐_𝑛⁻²𝐸𝑋²𝐼|𝑋|≤𝑐_𝑛+_{𝑛∶𝑐𝑛>1}𝑐_𝑛⁻²𝐸𝑋²𝐼|𝑋|≤𝑐_𝑛≐𝐼₁+𝐼₂. (67)(68) Since𝑁(1)=Card{𝑛∶𝑐_𝑛≤1}≤2𝑅(1)<∞, following from (65) and condition (ii), then we have𝐼₁<∞. 𝐼₂=_{𝑛∶𝑐𝑛>1}𝑐_𝑛⁻²𝐸𝑋²𝐼|𝑋|≤𝑐_𝑛=^∞_{𝑘=2()𝑘−1<𝑐𝑛≤𝑘}𝑐_𝑛⁻²𝐸𝑋²𝐼|𝑋|≤𝑐_𝑛≤^∞_𝑘=2(𝑁(𝑘)−𝑁(𝑘−1))(𝑘−1)⁻²𝐸𝑋²𝐼𝑋|≤(|≤𝑘)_∞𝑘=2(𝑁(𝑘)−𝑁(𝑘−1))(𝑘−1)⁻²𝐸𝑋²𝐼𝑋|)+(|≤1_∞𝑘=2(𝑁(𝑘)(×()−𝑁𝑘−1))𝑘−1⁻²𝐸𝑋²𝐼(1<|𝑋|≤𝑘)≐𝐼₂₁+𝐼₂₂. (69) To prove𝐼₂<∞, we need to prove that𝐼₂₁<∞and𝐼₂₂<∞: 𝐼₂₁≤𝐶^∞_𝑘=2(𝑁(𝑘)−𝑁(𝑘−1))^∞_{𝑗=𝑘−1}𝑗⁻³=𝐶^∞_𝑗=1𝑗^−3𝑗+1_𝑘=2(𝑁(𝑘)−𝑁(𝑘−1))≤𝐶^∞_𝑗=1𝑗+1⁻³𝑁𝑗+1≤𝐶^∞₁𝑦⁻³𝑁𝑦𝑑𝑦<∞. (70) Since𝑁(𝑥)is nondecreasing and𝑅(𝑥)is nonincreasing, then 𝐼₂₂=^∞_𝑘=2(𝑁(𝑘)−𝑁(𝑘−1))(𝑘−1)⁻²×^𝑘_𝑚=2𝐸𝑋²𝐼(|𝑋|)=𝑚−1<≤𝑚_∞𝑚=2𝐸𝑋²𝐼(|𝑋|)×𝑚−1<≤𝑚_∞𝑘=𝑚(𝑁(𝑘)−𝑁(𝑘−1))(𝑘−1)⁻²≤^∞_𝑚=2𝐸𝑋²𝐼(|𝑋|)×𝑚−1<≤𝑚_∞𝑘=𝑚𝑁(𝑘)(𝑘−1)⁻²−𝑘⁻²≤𝐶^∞_𝑚=2𝐸𝑋²𝐼(|𝑋|×𝑚−1<≤𝑚)_{∞𝑘=𝑚𝑘𝑘+1}𝑁(𝑥)𝑥⁻³𝑑𝑥=𝐶^∞_𝑚=2𝑅(𝑚)𝐸𝑋²𝐼(𝑚−1<|𝑋|≤𝑚)≤𝐶^∞_𝑚=2𝐸𝑋²𝑅𝑋𝐼(|𝑋|)𝑋(||)𝑚−1<≤𝑚≤𝐶𝐸²𝑅(|𝑋|)<∞. (71) By (67)–(71), we can get that ^∞_𝑛=1𝑎Var^𝑛𝑋^(𝑐𝑛)_𝑛𝑏_𝑛<∞. (72) Therefore, it follows from Corollary 9 and Kronecker’s lemma that 𝑏_𝑛^𝑛−1_𝑖=1𝑎_𝑖𝑋^(𝑐𝑖)_𝑖−𝐸𝑋^(𝑐𝑖)_𝑖⟶0𝑎.𝑠. (73) Taking𝑑_𝑛=𝑏_𝑛⁻¹∑^𝑛_𝑖=1𝑎_𝑖𝐸𝑋^(𝑐𝑖)_𝑖,𝑛≥1, we can get (64). The proof is complete.

Remark 24. Since NA implies AANA, Theorem 20 extends corresponding result for NA random variable (see Wang et al. [24]) to AANA random variables without adding any extra condition.
Similar to the proof of Corollary 21, we can get the following corollary.

Corollary 25. Let the conditions of Theorem 23 be satisfied and𝐸𝑋_𝑛=0for each𝑛≥1. If∫^∞₁𝐸𝑁(|𝑋|/𝑠)𝑑𝑠<∞, then𝑏_𝑛⁻¹∑^𝑛_𝑖=1𝑎_𝑖𝑋_𝑖→0a.s.

Corollary 26. Let{𝑋_𝑛,𝑛≥1}be a sequence of AANA random variables with∑^∞_𝑛=1𝑞²(𝑛)<∞, which is stochastically dominated by a random variable𝑋. Let{𝑎_𝑛,𝑛≥1}and{𝑏_𝑛,𝑛≥1}be two sequences of positive numbers with𝑐_𝑛=𝑏_𝑛/𝑎_𝑛and𝑏_𝑛↑∞. Let𝑓(𝑥)=𝑥^𝑟ℎ(𝑥), whereℎ(𝑥)>0is a slowly varying function as𝑥→∞,1<𝑟<2. Define𝑁(𝑥)=Card{𝑛∶𝑐_𝑛≤𝑥},𝑥>0. If(i)𝑁(𝑛)=𝑂(𝑓(𝑛))for each𝑛≥1;(ii)𝐸𝑓(|𝑋|)<∞.Then there exist𝑑_𝑛∈𝐑,𝑛=1,2,…, such that 𝑏_𝑛^𝑛−1_𝑖=1𝑎_𝑖𝑋_𝑖−𝑑_𝑛⟶0𝑎.𝑠. (74)

Proof. It is easy to verify that conditions (i)–(iii) of Theorem 23 hold under the conditions of Corollary 25. So Corollary 25 is true by Theorem 23.

Corollary 27. Suppose that the conditions of Corollary 26 are satisfied. If𝐸𝑋_𝑛=0for each𝑛≥1, then𝑏_𝑛⁻¹∑^𝑛_𝑖=1𝑎_𝑖𝑋_𝑖→0𝑎.𝑠.

Proof. According to Corollary 26, we need only to prove (60). By𝐸𝑋_𝑛=0for each𝑛≥1and Lemma 12, we have ^∞_𝑖=1𝑎_𝑖𝐸𝑋^(𝑐𝑖)_𝑖𝑏_𝑖≤^∞_𝑖=1𝑐_𝑖⁻¹𝑐_𝑖𝐸𝐼𝑋_𝑖>𝑐_𝑖+𝐸𝑋_𝑖𝐼𝑋_𝑖≤𝑐_𝑖≤^∞_𝑖=1𝑐_𝑖⁻¹𝐸𝑋_𝑖𝐼𝑋_𝑖>𝑐_𝑖+𝐸𝑋_𝑖𝐼𝑋_𝑖>𝑐_𝑖≤2^∞_𝑖=1𝑐_𝑖⁻¹𝐸𝑋_𝑖𝐼𝑋_𝑖>𝑐_𝑖≤𝐶^∞_𝑖=1𝑐_𝑖⁻¹𝐸|𝑋|𝐼|𝑋|>𝑐_𝑖=𝐶^∞_{𝑘=1()𝑘−1<𝑐𝑖≤𝑘}𝑐_𝑖⁻¹𝐸|𝑋|𝐼|𝑋|>𝑐_𝑖>𝑘−1≤𝐶^∞_𝑘=2(𝑘−1)⁻¹(𝑁(𝑘)(×−𝑁𝑘−1))_∞𝑗=𝑘𝐸|𝑋|𝐼𝑗≤|𝑋|<𝑗+1≤𝐶^∞𝑗_𝑗=2𝑘=2𝑁(𝑘)(𝑘−1)⁻¹−𝑘⁻¹×𝐸|𝑋|𝐼𝑗≤|𝑋|<𝑗+1. (75) Since𝑟>1, we can take𝛿>0such that𝑟−𝛿>1. By Lemma 13 and differential mean value theorem, we can obtain ^𝑗_𝑘=2𝑁(𝑘)(𝑘−1)⁻¹−𝑘⁻¹≤𝐶^𝑗_𝑘=2𝑘^𝑟ℎ(𝑘)(𝑘−1)⁻¹−𝑘⁻¹=𝐶^𝑗_𝑘=2𝑘^𝑟−𝛿𝑘^𝛿ℎ(𝑘)(𝑘−1)⁻¹−𝑘⁻¹≤𝐶𝑗^𝛿ℎ𝑗^𝑗_𝑘=2𝑘^{𝑟−𝛿−2}≤𝐶𝑗^𝛿ℎ𝑗^𝑗₂𝑥^{𝑟−𝛿−2}𝑑𝑥≤𝐶𝑗^𝑟−1ℎ𝑗. (76) It is easily seen that𝑥^𝑟−1ℎ(𝑥)is a quasimonotonically increasing function by Lemma 13. Hence, we have by (75) and (76) that ^∞_𝑖=1𝑎_𝑖𝐸𝑋^(𝑐𝑖)_𝑖𝑏_𝑖≤𝐶^∞_𝑗=2𝑗^𝑟−1ℎ𝑗𝐸|𝑋|𝐼𝑗≤|𝑋|<𝑗+1≤𝐶^∞_𝑗=2𝐸|𝑋|^𝑟−1ℎ(|𝑋|)|𝑋|𝐼𝑗≤|𝑋|<𝑗+1=𝐶^∞_𝑗=2𝐸|𝑋|^𝑟ℎ𝑋𝐼|𝑋|𝑋(||)𝑗≤<𝑗+1≤𝐶𝐸𝑓(||)<∞. (77) By Kronecker’s lemma, we can get (60) immediately. The proof is complete.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are most grateful to the Editor Jaume Giné and anonymous referees for careful reading of the paper and valuable suggestions which helped in improving an earlier version of this paper. This work is supported by the National Natural Science Foundation of China (11171001, 11201001), the Natural Science Foundation of Anhui Province (1208085QA03, 1308085QA03), Doctoral Research Start-Up Funds Projects of Anhui University, Applied Teaching Model Curriculum of Anhui University (XJYYXKC04), the Students Innovative Training Project of Anhui University (201310357004, cxcy2012003), and the Students Science Research Training Program of Anhui University (KYXL2012007, kyxl2013003).