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

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)$forall$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.