Limit theorems for sums of heavy-tailed variables with random dependent weights

Let U-j, j is an element of N be independent and identically distributed random variables with heavy- tailed distributions. Consider a sequence of random weights {W-j} j is an element of N, independent of {U-j} (j is an element of N) and focus on the weighted sums Sigma([nt])(j=1) W-j( U-j - mu), where mu involves a suitable centering. We establish sufficient conditions for these weighted sums to converge to non- trivial limit processes, as n -> infinity, when appropriately normalized. The convergence holds, for example, if {Wj} j. N is strictly stationary, dependent, and W-1 has lighter tails than U-1. In particular, the weights W-js can be strongly dependent. The limit processes are scale mixtures of stable Levy motions. We establish weak convergence in the Skorohod J(1)-topology. We also consider multivariate weights and show that they converge weakly in the strong Skorohod M-1- topology. The M-1-topology, while weaker than the J(1)-topology, is strong enough for the supremum and infimum functionals to be continuous.