On the strong law of large numbers for sequences of pairwise negative quadrant dependent random variables
by Deli Li   Andrew Rosalsky   Andrei I. Volodin  

Vol. 1 No. 2 (2006) P.281~P.305

ABSTRACT

For a sequence of pairwise negative quadrant dependent random variables {$X_n, n \ge 1$}, 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 {$\sum^n_{j=1} a_j X_j, n \ge 1$} where {$a_n, n \ge 1$} is a sequence of positive constants and the {$X_n, n \ge 1$} are also identically distributed. A result of Matula [19] is obtained by taking an $a_n \equiv 1$. 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 specied continuous marginal distributions. Illustrative examples are provided, two of which show that the pairwise negative quadrant dependence assumption cannot be dispensed with.

KEYWORDS
Almost certain convergence, sequence of pairwise negative quadrant dependent random variables, strong law of large numbers, weighted sums

MATHEMATICAL SUBJECT CLASSIFICATION 2010
Primary: 60F15

MILESTONES

Received: 2005-01-14
Revised :
Accepted:

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