Pointwise estimates for first passage times of perpetuity sequences

Abstract

We consider first passage times ${\tau }_u=\inf \{n:Y_n>u\}$ for the perpetuity sequence $$Y_n=B_1+A_1{B}_2+\cdots +(A_1\dots A_{n-1})B_n,$$ 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.