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

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$U_{j} ,\;j \in \mathbb{N}$\end{document} be independent and identically distributed random variables with heavy-tailed distributions. Consider a sequence of random weights \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\left\{ {W_{j} } \right\}}_{{j \in \mathbb{N}}}$\end{document}, independent of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\left\{ {U_{j} } \right\}}_{{j \in \mathbb{N}}}$\end{document} and focus on the weighted sums \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\sum\nolimits_{j = 1}^{{\left[ {nt} \right]}} {W_{j} {\left( {U_{j} - \mu } \right)}} }$\end{document}, 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}${\left\{ {W_{j} } \right\}}_{{j \in \mathbb{N}}}$\end{document} is strictly stationary, dependent, and W1 has lighter tails than U1. In particular, the weights Wjs can be strongly dependent. The limit processes are scale mixtures of stable Lévy motions. We establish weak convergence in the Skorohod J1-topology. We also consider multivariate weights and show that they converge weakly in the strong Skorohod M1-topology. The M1-topology, while weaker than the J1-topology, is strong enough for the supremum and infimum functionals to be continuous.