On the limiting behavior of randomly weighted partial sums

We study the almost sure limiting behavior and convergence in probability of weighted partial sums of the form where {Wnj, 1[less-than-or-equals, slant]j[less-than-or-equals, slant]n, n[less-than-or-equals, slant]1} and {Xnj, 1[less-than-or-equals, slant]j[less-than-or-equals, slant]n, n[greater-or-equal, slanted]1} are triangular arrays of random variables. The results obtain irrespective of the joint distributions of the random variables within each array. Applications concerning the Efron bootstrap and queueing theory are discussed.