Large excursions and conditioned laws for recursive sequences generated by random matrices

Abstract

We study the large exceedance probabilities and large exceedance paths of the recursive sequence $V_{n}=M_{n}V_{n-1}+Q_{n}$, where $\{(M_{n},Q_{n})\}$ is an i.i.d. sequence, and $M_{1}$ is a $d\times d$ random matrix and $Q_{1}$ is a random vector, both with nonnegative entries. We impose conditions which guarantee the existence of a unique stationary distribution for $\{V_{n}\}$ and a Cramer-type condition for $\{M_{n}\}$. Under these assumptions, we characterize the distribution of the first passage time $T_{u}^{A}:=\inf\{n:V_{n}\in uA\}$, where $A$ is a general subset of $\mathbb{R}^{d}$, exhibiting that $T_{u}^{A}/u^{\alpha}$ converges to an exponential law for a certain $\alpha>0$. In the process, we revisit and refine classical estimates for $\mathbb{P}(V\in uA)$, where $V$ possesses the stationary law of $\{V_{n}\}$. Namely, for $A\subset\mathbb{R}^{d}$, we show that $\mathbb{P}(V\in uA)\sim C_{A}u^{-\alpha}$ as $u\to\infty$, providing, most importantly, a new characterization of the constant $C_{A}$. As a simple consequence of these estimates, we also obtain an expression for the extremal index of $\{|V_{n}|\}$. Finally, we describe the large exceedance paths via two conditioned limit theorems showing, roughly, that $\{V_{n}\}$ follows an exponentially-shifted Markov random walk, which we identify. We thereby generalize results from the theory of classical random walk to multivariate recursive sequences.