Joint exceedances of random products

Abstract

We analyze the joint extremal behavior of $n$ random products of the form ${\prod }_{j=1}^m{X}_j^{a_{ij}},1\le i\le n,$ for non-negative, independent regularly varying random variables $X_1,\dots ,X_m$ and general coefficients $a_{ij}\in \mathbb{R}$. Products of this form appear for example if one observes a linear time series with gamma type innovations at $n$ points in time. We combine arguments of linear optimization and a generalized concept of regular variation on cones to show that the asymptotic behavior of joint exceedance probabilities of these products is determined by the solution of a linear program related to the matrix $\mathbf{A}=(a_{ij})$.