Pointwise estimates for first passage times of perpetuity sequences

TL;DR

This paper derives precise asymptotics for the probability that the first passage time of a perpetuity sequence exceeds a large threshold, improving upon previous limit theorems and analyzing its relation to random walks.

Contribution

It provides the first detailed asymptotic estimates for the distribution of first passage times of perpetuity sequences, extending known limit theorems.

Findings

Established asymptotics for $ ext{P}[ au_u = rac{ ext{log } u}{ ho}]$ as $u o \infty$

Identified probabilities of $ au_u$ within large intervals around $k_u$

Discussed similarities and differences with random walk behavior

Abstract

We consider first passage times $\tau_u = \inf\{n:\; Y_n>u\}$ for the perpetuity sequence $$ Y_n = B_1 + A_1 B_2 + \cdots + (A_1\ldots A_{n-1})B_n, $$ where $(A_n,B_n)$ are i.i.d. random variables with values in ${\mathbb R} ^+\times {\mathbb R}$. Recently, a number of limit theorems related to $\tau_u$ were proved including the law of large numbers, the central limit theorem and large deviations theorems. We obtain a precise asymptotics of the sequence ${\mathbb P}[\tau_u = \log u/\rho ]$, $\rho >0$, $u\to \infty $ which considerably improves the previous results. There, probabilities ${\mathbb P}[\tau_u \in I_u]$ were identified, for some large intervals $I_u$ around $k_u$, with lengths growing at least as $\log\log u$. Remarkable analogies and differences to random walks are discussed.