Abstract
In the paper, we study the strong law of large numbers for general weighted sums of asymptotically almost negatively associated random variables (AANA, in short) with nonidentical distribution. As an application, the Marcinkiewicz strong law of large numbers for AANA random variables is obtained. In addition, we present some sufficient conditions to prove the strong law of large numbers for weighted sums of AANA random variable. Our results generalize the corresponding ones (sufficient conditions) for independent random variables.
MSC: 60F15.
Keywords:
asymptotically almost negatively associated random variables; Marcinkiewicz strong law of large numbers; weighted sums; the three series theorem1 Introduction
Throughout the paper, let be a sequence of random variables defined on a fixed probability space . C denotes a positive constant not depending on n, which may be different in various places. Let denote the integer part of x and be the indicator function of the set A.
Recently, Jajte [1] studied the strong law of large numbers for general weighted sums of independent and identically distributed random variables. The main result of Jajte [1] is as follows.
Theorem 1.1Letbe a sequence of independent and identically distributed random variables. Letbe a positive, increasing function andbe a positive function such thatsatisfies the following conditions.
(i) For some, is strictly increasing onwith range.
(ii) There existCand a positive integersuch thatfor all.
(iii) There exist constantsaandbsuch that for all,
Then the following two conditions are equivalent:
whereandis the inverse of a functionϕ.
Inspired by Jajte [1], Jing and Liang [2] extended the result of Jajte [1] for independent and identically distributed random variables to the case of negatively associated random variables with identical distribution. Meng and Lin [3] and Wang [4] extended Theorem 1.1 to the case of mixing random variables and nonidentically distributed negatively associated random variables, respectively. Sung [5] gave some sufficient conditions to prove the strong law of large numbers for weighted sums of random variables. The main purpose of the paper is to generalize the result of Theorem 1.1 to the case of asymptotically almost negatively associated random variables, which contains independent random variables and negatively associated (NA, in short) random variables as special cases. In addition, we present some sufficient conditions to prove the strong law of large numbers for weighted sums of asymptotically almost negatively associated random variables.
The concept of asymptotically almost negatively associated random variables is as follows.
Definition 1.1 A sequence of random variables is called asymptotically almost negatively associated (AANA, in short) if there exists a nonnegative sequence as such that
for all and for all coordinatewise nondecreasing continuous functions f and g whenever the variances exist.
The family of AANA sequence contains NA (in particular, independent) sequences (with , ) and some more sequences of random variables which are not much deviated from being negatively associated. For more details about NA random variables, one can refer to JoagDev and Proschan [6], and so forth. An example of an AANA sequence which is not NA was constructed by Chandra and Ghosal [7].
Since the concept of AANA sequence was introduced by Chandra and Ghosal [7], many applications have been found. For example, Chandra and Ghosal [7] derived the Kolmogorov type inequality and the strong law of large numbers of MarcinkiewiczZygmund; Chandra and Ghosal [8] obtained the almost sure convergence of weighted averages; Wang et al.[9] established the law of the iterated logarithm for product sums; Ko et al.[10] studied the HájekRényi type inequality; Yuan and An [11] established some Rosenthal type inequalities for maximum partial sums of AANA sequence; Yuan and Wu [12] studied the limiting behavior of the maximum of the partial sum for asymptotically negatively associated random variables under residual Cesàro alphaintegrability assumption; Wang et al.[13] obtained some strong growth rate and the integrability of supremum for the partial sums of AANA random variables; Yuan and An [14] established some laws of large numbers for Cesàro alphaintegrable random variables under dependence condition AANA or AQSI; Wang et al.[15] studied the complete convergence for weighted sums of arrays of rowwise AANA random variables; and Yang et al.[16] investigated the complete convergence of moving average process for AANA sequence, and so forth.
The structure of this paper is as follows. Some important lemmas are presented in Section 2. Main results and their proofs are provided in Section 3 and Section 4, respectively.
2 Preliminaries
In this section, we will present some important lemmas which will be used to prove the main results of the paper. The first one is a basic property of AANA random variables, which was given by Yuan and An [11].
Lemma 2.1 (cf. Yuan and An [11])
Letbe a sequence of AANA random variables with mixing coefficients, , , … be all nondecreasing (or nonincreasing) continuous functions, thenis still a sequence of AANA random variables with mixing coefficients.
The next one, the KhintchineKolmogorov type convergence theorem for AANA random variables proved by Wang et al.[17], will play an essential role in proving the main results of the paper.
Lemma 2.2 (cf. Wang et al. [17])
Letbe a sequence of AANA random variables with mixing coefficientsand. Assume that, thenconverges a.s.
By Lemma 2.2 and the standard method, we can easily get the following three series theorem for AANA random variables. The proof is easy, so we omit the details.
Lemma 2.3Letbe a sequence of AANA random variables with mixing coefficientsand. Denote, wherecis a positive constant. If the following three conditions are satisfied:
The following concept of stochastic domination will be used in this paper.
Definition 2.1 A sequence of random variables is said to be stochastically dominated by a random variable X if there exists a positive constant C such that
By the definition of stochastic domination and integration by parts, we can get the following basic inequalities. The proof is standard, so we omit it.
Lemma 2.4Letbe a sequence of random variables which is stochastically dominated by a random variableX. For anyand, the following two statements hold:
3 Main results
Hypothesis ALetandbe real positive functions defined on the same domainand (). is strictly increasing on, , and its range is.
Hypothesis BThere exist constantssuch that for every,
Throughout the paper, let be a sequence of AANA random variables with mixing coefficients and , which is stochastically dominated by a random variable X. Denote for each . Based on Hypothesis A and Hypothesis B, we will establish the strong law of large numbers for the weighted sums of AANA random variables. Our main results are as follows.
Theorem 3.1Let, andbe functions satisfying the conditions of Hypothesis A and Hypothesis B. If, then
If we further assume thatis increasing on its domain and, thena.s. as.
Theorem 3.2Let, andbe functions satisfying the conditions of Hypothesis A and Hypothesis B. Assume further thatsatisfies the following conditions:
(i) Ifis finite, then, whereandare constants.
(ii) Ifdoes not exist or is infinite, thenis nondecreasing and, where, andare constants.
Suppose thatandwhen (ii) holds, then
If we further assume thatis increasing on its domain and, thena.s. as.
By Theorem 3.1 and Theorem 3.2, we can get the Marcinkiewicz strong law of large numbers for AANA random variables as follows.
Corollary 3.1Letbe a sequence of identically distributed AANA random variables with mixing coefficientsand. Iffor some, then for some finite constanta,
If, thenacan take an arbitrary real number. If, then.
The next two results are based on different conditions from Hypothesis B. The main idea is inspired by Sung [5]. Here we consider stochastic domination, not identical distribution.
Theorem 3.3Let, andbe functions satisfying the conditions of Hypothesis A. Assume further that the following conditions hold:
Then (3.2) holds true. If we further assume thatis increasing on its domain and, thena.s. as.
Theorem 3.4Let, andbe functions satisfying the conditions of Hypothesis A. Assume further that the following conditions hold:
Then
If we further assume thatis increasing on its domain and, thena.s. as.
Remark 3.1 The methods used in this paper are the KhintchineKolmogorov type convergence theorem and the three series theorem for AANA random variables, which are partially the same as that in Jajte [1] and Jing and Liang [2]. But here we consider some different conditions.
4 Proofs of the main results
Proof of Theorem 3.1 For every , denote
It is easily seen that
which together with the BorelCantelli lemma yields that
Now, we consider the series . By ’s inequality, Lemma 2.4 and (4.1), we can get that
Since , it follows that a.s. Hence, we have by (3.1) that
which implies that
Combining (4.3) and (4.4), we can see that
It follows by Lemma 2.1 that is also a sequence of AANA random variables. Hence, we have by (4.5) and Lemma 2.2 that
which together with (4.2) implies that
To complete the proof of (3.2), it suffices to show that
But this follows from (4.1) immediately. This completes the proof of (3.2). Finally, a.s. as follows from (4.1) and Kronecker’s lemma immediately. The proof is complete. □
Proof of Theorem 3.2 We use the same notations as those in Theorem 3.1. In the proof of Theorem 3.1, we have proved that
and
Note that is also a sequence of AANA random variables. By Lemma 2.3, we can see that in order to prove (3.3), we only need to show
Suppose that (i) holds, we have by Lemma 2.4 that
which implies (4.6).
Suppose that (ii) holds. Note that . We have by Lemma 2.4 again that
which implies (4.6).
Thus, (4.6) holds both in case (i) and in case (ii). This completes the proof of the theorem. □
Proof of Corollary 3.1 If , then for an arbitrary real number a,
Thus, in order to prove (3.4), we only need to show that
If , , without loss of generality, we assume that . Hence, for and , we only need to show (4.7) holds true.
Taking
in Theorem 3.2, we can get the desired result (3.4) immediately.
For , , and satisfy the conditions of Theorem 3.1, hence,
where . By the dominated convergence theorem, we can get that , which implies that
Therefore, the desired result (3.4) follows from (4.8) and (4.9) immediately. This completes the proof of the corollary. □
Proof of Theorem 3.3 We use the same notations as those in Theorem 3.1. By (i), we can see that
which together with the BorelCantelli lemma yields that
Note that is still a sequence of AANA random variables, and by (4.3)
We have by Lemma 2.2 that
The fact and the monotone convergence theorem yield that
Combining (4.11) and (4.12), we can get that
The desired result (3.2) follows from (4.10) and (4.13) immediately. This completes the proof of the theorem. □
Proof of Theorem 3.4 By (i), we can easily get
Applying Theorem 3.3, we can get (3.2) immediately. In order to prove (3.5), it suffices to show that
By (i) again and Lemma 2.4, we have
which implies (4.15). The desired result (3.5) follows from (3.2) and (4.15) immediately. The proof is complete. □
Competing interests
The author declares that they have no competing interests.
Author’s contributions
The author read and approved the final manuscript.
Acknowledgements
The author are most grateful to the editor Andrei Volodin and an anonymous referee for careful reading of the manuscript and valuable suggestions which helped in improving an earlier version of this paper. This work was supported by the Project of the Feature Specialty of China (TS11496) and the Scientific Research Projects of Fuyang Teachers College (2009FSKJ09).
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