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 Lévy 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.