Local behaviour of first passage probabilities
Probability
2010-06-29 v1
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.
Cite
@article{arxiv.1006.5316,
title = {Local behaviour of first passage probabilities},
author = {Ronald Doney},
journal= {arXiv preprint arXiv:1006.5316},
year = {2010}
}