Joint exceedances of random products

TL;DR

This paper investigates the extremal joint behavior of products of independent regularly varying random variables, revealing that their exceedance probabilities are governed by solutions to a related linear program.

Contribution

It introduces a novel approach combining linear optimization and generalized regular variation to analyze joint exceedances of random products.

Findings

Joint exceedance probabilities are characterized by linear program solutions.

The method applies to products of independent regularly varying variables.

Results are relevant for analyzing extremal events in time series with gamma-like innovations.

Abstract

We analyze the joint extremal behavior of $n$ random products of the form $\prod_{j=1}^m X_j^{a_{ij}}, 1 \leq i \leq n,$ for non-negative, independent regularly varying random variables $X_1, \ldots, X_m$ 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 $\mathbf{A}=(a_{ij})$.