Limit Theorems for Sums of Heavy-tailed Variables with Random Dependent Weights

Abstract

Let \(U_{j} ,\;j \in \mathbb{N}\) be independent and identically distributed random variables with heavy-tailed distributions. Consider a sequence of random weights \({\left\{ {W_{j} } \right\}}_{{j \in \mathbb{N}}}\), independent of \({\left\{ {U_{j} } \right\}}_{{j \in \mathbb{N}}}\) and focus on the weighted sums \({\sum\nolimits_{j = 1}^{{\left[ {nt} \right]}} {W_{j} {\left( {U_{j} - \mu } \right)}} }\), where μ involves a suitable centering. We establish sufficient conditions for these weighted sums to converge to non-trivial limit processes, as n→∞, when appropriately normalized. The convergence holds, for example, if \({\left\{ {W_{j} } \right\}}_{{j \in \mathbb{N}}}\) is strictly stationary, dependent, and W 1 has lighter tails than U 1. In particular, the weights W j s 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.