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

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.