On the strong law of large numbers for sequences of pairwise negative quadrant dependent random variables

Abstract

For a sequence of pairwise negative quadrant dependent random variables {Xn, n ≥ 1}, conditions are given under which normed and centered partial sums converge to 0 almost certainly. As special cases, new results are obtained for weighted sums { ∑n j=1 ajXj, n ≥ 1} where {an, n ≥ 1} is a sequence of positive constants and the {Xn, n ≥ 1} are also identically distributed. A result of Matula [19] is obtained by taking an ≡ 1. Moreover, it is shown that a pairwise negative quadrant dependent sequence (which is not a sequence of independent random variables) can be constructed having any specified continuous marginal distributions. Illustrative examples are provided, two of which show that the pairwise negative quadrant dependence assumption cannot be dispensed with. 1.