Exact asymptotics for the instant of crossing a curve boundary by an asymptotically stable random walk

Abstract

Suppose that $\{S_n,n\ge 0\}$ is an asymptotically stable random walk. Let $g$ be a positive function and $T_g$ be the first time when $S_n$ leaves $[-g(n),\infty )$. In this paper we study asymptotic behaviour of $T_g$. We provide integral tests for function $g$ that guarantee $P(T_g>n)\sim V(g)P(T_0>n)$ where $T_0$ is the first strict descending ladder epoch of $\{S_n\}$

References in corpus (3)

• Random walks in cones
• First passage times of Lévy processes over a one-sided moving boundary
• The first passage time problem over a moving boundary for asymptotically stable Lévy processes

Cited by in corpus (1)

• The first passage time problem over a moving boundary for asymptotically stable Lévy processes