The Annals of Probability

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

Jeffrey F. Collamore and Sebastian Mentemeier

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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 Cramér-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.

Article information

Source
Ann. Probab., Volume 46, Number 4 (2018), 2064-2120.

Dates
Received: August 2016
Revised: July 2017
First available in Project Euclid: 13 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.aop/1528876822

Digital Object Identifier
doi:10.1214/17-AOP1221

Mathematical Reviews number (MathSciNet)
MR3813986

Zentralblatt MATH identifier
06919019

Subjects
Primary: 60K15: Markov renewal processes, semi-Markov processes 60F10: Large deviations
Secondary: 60J05: Discrete-time Markov processes on general state spaces 60G70: Extreme value theory; extremal processes 60G17: Sample path properties

Keywords
Random recurrence equations stochastic fixed-point equations products of random matrices Markov chain theory in general state space nonlinear Markov renewal theory large deviations first passage times conditional limit theorems extreme value theory

Citation

Collamore, Jeffrey F.; Mentemeier, Sebastian. Large excursions and conditioned laws for recursive sequences generated by random matrices. Ann. Probab. 46 (2018), no. 4, 2064--2120. doi:10.1214/17-AOP1221. https://projecteuclid.org/euclid.aop/1528876822


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