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

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.

References in corpus (1)

• The tail of the stationary distribution of a random coefficient AR(q) model

Cited by in corpus (2)

• Heavy tailed solutions of multivariate smoothing transforms
• Two-sided bounds for $L_p$-norms of combinations of products of independent random variables