Mathematics – Probability
Scientific paper
2010-09-09
Mathematics
Probability
Scientific paper
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.
Alsmeyer Gerold
Mentemeier Sebastian
No associations
LandOfFree
Tail behavior of stationary solutions of random difference equations: the case of regular matrices does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Tail behavior of stationary solutions of random difference equations: the case of regular matrices, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Tail behavior of stationary solutions of random difference equations: the case of regular matrices will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-558785