The Annals of Probability

Large deviation estimates for exceedance times of perpetuity sequences and their dual processes

Dariusz Buraczewski, Jeffrey F. Collamore, Ewa Damek, and Jacek Zienkiewicz

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Abstract

In a variety of problems in pure and applied probability, it is relevant to study the large exceedance probabilities of the perpetuity sequence $Y_{n}:=B_{1}+A_{1}B_{2}+\cdots+(A_{1}\cdots A_{n-1})B_{n}$, where $(A_{i},B_{i})\subset(0,\infty)\times\mathbb{R}$. Estimates for the stationary tail distribution of $\{Y_{n}\}$ have been developed in the seminal papers of Kesten [Acta Math. 131 (1973) 207–248] and Goldie [Ann. Appl. Probab. 1 (1991) 126–166]. Specifically, it is well known that if $M:=\sup_{n}Y_{n}$, then $\mathbb{P}\{M>u\}\sim\mathcal{C}_{M}u^{-\xi}$ as $u\to\infty$. While much attention has been focused on extending such estimates to more general settings, little work has been devoted to understanding the path behavior of these processes. In this paper, we derive sharp asymptotic estimates for the normalized first passage time $T_{u}:=(\log u)^{-1}\inf\{n:Y_{n}>u\}$. We begin by showing that, conditional on $\{T_{u}<\infty\}$, $T_{u}\to\rho$ as $u\to\infty$ for a certain positive constant $\rho$. We then provide a conditional central limit theorem for $\{T_{u}\}$, and study $\mathbb{P}\{T_{u}\in G\}$ as $u\to\infty$ for sets $G\subset[0,\infty)$. If $G\subset[0,\rho)$, then we show that $\mathbb{P}\{T_{u}\in G\}u^{I(G)}\to C(G)$ as $u\to\infty$ for a certain large deviation rate function $I$ and constant $C(G)$. On the other hand, if $G\subset(\rho,\infty)$, then we show that the tail behavior is actually quite complex and different asymptotic regimes are possible. We conclude by extending our results to the corresponding forward process, understood in the sense of Letac [In Random Matrices and Their Applications (Brunswick, Maine, 1984) (1986) 263–273 Amer. Math. Soc.], namely to the reflected process $M_{n}^{\ast}:=\max\{A_{n}M_{n-1}^{\ast}+B_{n},0\}$, $n\in\mathbb{Z}_{+}$. Using Siegmund duality, we relate the first passage times of $\{Y_{n}\}$ to the finite-time exceedance probabilities of $\{M_{n}^{\ast}\}$, yielding a new result concerning the convergence of $\{M_{n}^{\ast}\}$ to its stationary distribution.

Article information

Source
Ann. Probab., Volume 44, Number 6 (2016), 3688-3739.

Dates
Received: November 2014
Revised: August 2015
First available in Project Euclid: 14 November 2016

Permanent link to this document
https://projecteuclid.org/euclid.aop/1479114261

Digital Object Identifier
doi:10.1214/15-AOP1059

Mathematical Reviews number (MathSciNet)
MR3572322

Zentralblatt MATH identifier
1362.60023

Subjects
Primary: 60H25: Random operators and equations [See also 47B80]
Secondary: 60K05: Renewal theory 60F10: Large deviations 60J10: Markov chains (discrete-time Markov processes on discrete state spaces) 60G70: Extreme value theory; extremal processes 60K25: Queueing theory [See also 68M20, 90B22] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Random recurrence equations stochastic fixed point equations first passage times Siegmund duality asymptotic behavior ruin probabilities

Citation

Buraczewski, Dariusz; Collamore, Jeffrey F.; Damek, Ewa; Zienkiewicz, Jacek. Large deviation estimates for exceedance times of perpetuity sequences and their dual processes. Ann. Probab. 44 (2016), no. 6, 3688--3739. doi:10.1214/15-AOP1059. https://projecteuclid.org/euclid.aop/1479114261


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