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),
Strong laws of large numbers (SLLN) for weighted averages are proved under various dependence assumptions when the variables are not necessarily independent or identically distributed. The results considerably extend the existing results. Weighted versions of the Marcinkiewicz-Zygmund SLLN are also formulated and proved under a similar set up. It seems that such results are not known even for independent and identically distributed random variables.
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