Hidden regular variation for stochastic recursions with diagonal matrices

TL;DR

This paper investigates the hidden regular variation in stochastic recursions with diagonal matrices, revealing how to identify simultaneous extreme events despite apparent asymptotic independence.

Contribution

It introduces a novel analysis of hidden regular variation in stochastic recursions, extending understanding of tail behavior under dependence structures.

Findings

Identifies the proper scaling for observing simultaneous extremes.

Shows the tail measure concentrates on the axes despite asymptotic independence.

Provides a framework for analyzing joint tail behavior in recursive models.

Abstract

We consider random vectors $X$ that satisfy the equation in law $X=AX+B$, where $A$ is a given random diagonal matrix and $B$ a given random vector, both independent of $X$. It is well known by the works of Kesten and Goldie that the marginals of $X$ may exhibit heavy tails, with possibly different tail indices. In recent works (Damek 2025, Mentemeier and Wintenberger 2022) it was observed that asymptotic independence may occur despite strong dependencies in the entries of $A$: The probability that both marginals are simultaneously large decays faster than the marginal probability of an extreme event; the tail measure is concentrated on the axis. In this work, we analyse the hidden regular variation properties of $X$, that is, we find the proper scaling for which one observes simultaneous extremes.