Extensions of the strong law of large numbers of Marcinkiewicz and Zygmund for dependent variables

Abstract

The classical strong law of large numbers (SLLN) due to Kolmogorov has been extended recently to various weakly dependent random variables (rvs) which are not necessarily identically distributed. It is natural to enquire whether the SLLN of Marcinkiewicz and Zygmund (MZSLLN) (see Theorem 3.2.3 of Stout [11]) holds under similar relaxed conditions. In this paper, we try to fill this gap and show that this weakening is indeed possible; the independence assumption is relaxed to q~-mizing or asymptotically almost negatively associated sequences defined below. Related results are established for asymptotically quadrant sub-independent (AQSI) sequences (see Definition 1). These concepts are interesting as they unify, to some extent, the notion of mixing-type sequences and that of negatively dependent sequences.In contrast to the usual proofs of the MZSLLN (see, e.g., Stout [11] and Chow and Teicher [5]), our proofs use maximal inequalities at a crucial step. We are thus able to relax ,mutual independence to a great extent, and the assumption of identical distribution is also relaxed considerably. This technique is likely to be useful for other dependences also, provided suitable maximal inequalities are available under those dependences. DEFINITION 1. A sequence {Xn} of rvs is called asymptotically quadrant sub-independent (AQSI) if there exists a nonnegative sequence { q(m) } such that for all i ~ j, Note that palrwise negative quadrant dependent, pairwise m-dependent and AQI rvs (introduced by Birkel [2]; see Definition 2 below) are special cases of AQSI rvs. DEFINITION 2. A sequence {Xn) ofrvs is called asymptotically quadrant independent (AQI)if there exists a nonnegative sequence { q(m)} such that for all i ~ j and s,t E R, [P{Xi > s, Xj > t} -P{Xi > s}P{Xj > t}[ <= q(li-jl)aij(s,t),

Complete Convergence of Weighted Sums for Asymptotically Almost Negatively Associated Sequences

“…For example, Chandra and Ghosal [2] [3] obtained the almost sure convergence of weighted average, Kim, Ko and Lee [4] established the Hajek-Renyi type inequalities and Marcinkiewicz-Zygmund type SLLN, Cai [5] investigated the complete convergence of weighted sums, Yuan and An [6] …”

On Complete Convergence and Complete Moment Convergence for a Class of Random Variables

“…Primarily motivated by the definition of NA, Chandra and Ghosal [4] introduced the following dependence. Definition 1.1.…”

“…An example of AANA random variables which are not NA was constructed by Chandra and Ghosal [4]. For various results and applications of AANA random variables, one can refer to Wang et al [18], Yuan and An [21,22], Wang et al [16], Yang et al [20], Hu et al [9], Shen et al [13], Shen [11], Shen and Wu [12], Chen et al [5], and so on.…”

On complete moment convergence for weighted sums of AANA random variables

“…The concept of AANA was introduced by Chandra and Ghosal [3]. It is obviously seen that the family of AANA random variables contains NA random variables (with µ(n) = 0, n ≥ 1) and some more sequences of random variables which are not much deviated from being NA random variables.…”

“…Since the concept of AANA was introduced by Chandra and Ghosal [3], many applications have been established. For more details, we can refer to [3,4,7,9,11,12,14,15,16,17,19,20,21], and so forth.…”