Tail probabilities of random linear functions of regularly varying random vectors

We provide a new extension of Breiman’s Theorem on computing tail probabilities of a product of random variables to a multivariate setting. In particular, we give a characterization of regular variation on cones in [0,∞)d\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$[0,\infty )^d$$\end{document} under random linear transformations. This allows us to compute probabilities of a variety of tail events, which classical multivariate regularly varying models would report to be asymptotically negligible. We illustrate our findings with applications to risk assessment in financial systems and reinsurance markets under a bipartite network structure.