Local behaviour of first passage probabilities

TL;DR

This paper investigates the local behavior of first passage probabilities for asymptotically stable random walks, providing uniform estimates for the probability of first entering a state at a specific time across different regimes of the initial position.

Contribution

It extends previous work by Vatutin and Wachtel to derive three uniform estimates for first passage probabilities in various asymptotic regimes of the initial position.

Findings

Provides estimates for P(T_x=n) in different regimes of x relative to c_n

Extends known asymptotic results to local probabilities in broader regions

Enhances understanding of the behavior of stable random walks at first passage times

Abstract

Suppose that S is an asymptotically stable random walk with norming sequence c_{n} and that T_{x} is the time that S first enters (x,\inf), where x\ge 0. The asymptotic behaviour of P(T_0=n) has been described in a recent paper of Vatutin and Wachtel, \cite{vw}, and here we build on that result to give three estimates for P(T_{x}=n), which hold uniformly as n\to\inf in the regions x=o(c_{n}), x=O(c_{n}), and x/c_{n}\to\inf, respectively.