Tail behavior of stationary solutions of random difference equations: the case of regular matrices

TL;DR

This paper investigates the tail behavior of stationary solutions to multidimensional random difference equations with regular matrices, using regeneration methods and extending Goldie's renewal theory to provide a concise proof of key properties.

Contribution

It introduces a shorter, regeneration-based proof for the tail asymptotics of solutions to RDEs with regular matrices, extending Goldie's theory to multiple dimensions.

Findings

Established tail asymptotics for stationary solutions of RDEs.

Provided a simplified proof for the positivity of the tail limit function.

Extended Goldie's implicit renewal theory to higher dimensions.

Abstract

Given a sequence $(M_{n},Q_{n})_{n\ge 1}$ of i.i.d. random variables with generic copy $(M,Q)$ such that $M$ is a regular $d\times d$ matrix and $Q$ takes values in $\mathbb{R}^{d}$, we consider the random difference equation (RDE) $R_{n}=M_{n}R_{n-1}+Q_{n}$, $n\ge 1$. Under suitable assumptions, this equation has a unique stationary solution $R$ such that, for some $\kappa>0$ and some finite positive and continuous function $K$ on $S^{d-1}:=\{x \in \mathbb{R}^{d}:|x|=1\}$, $ \lim_{t \to \infty} t^{\kappa} P(xR>t)=K(x)$ for all $x \in S^{d-1} $ holds true. This result is originally due to Kesten and Le Page. The purpose of this article is to show how regeneration methods can be used to provide a much shorter argument (in particular for the positivity of K). It is based on a multidimensional extension of Goldie's implicit renewal theory.