Risk Measures and Multivariate Extensions of Breiman's Theorem

The modeling of insurance risks has received an increasing amount of attention because of solvency capital requirements. The ruin probability has become a standard risk measure to assess regulatory capital. In this paper we focus on discrete-time models for the finite time horizon. Several results are available in the literature to calibrate the ruin probability by means of the sum of the tail probabilities of individual claim amounts. The aim of this work is to obtain asymptotics for such probabilities under multivariate regular variation and, more precisely, to derive them from extensions of Breiman's theorem. We thus present new situations where the ruin probability admits computable equivalents. We also derive asymptotics for the value at risk.