Local probabilities for random walks conditioned to stay positive
Probability
2007-11-09 v1
Abstract
Let S_0=0,{S_n, n>0} be a random walk generated by a sequence of i.i.d. random variables X_1,X_2,... and let \tau^{-} be the first descending ladder epoch. Assuming that the distribution of X_1 belongs to the domain of attraction of an \alpha-stable law we study the asymptotic behavior of the local probabilities P(\tau ^{-}=n) and the conditional local probabilities P(S_n\in [x,x+y)|\tau^{-}>n) for fixed y and x=x(n)\in (0,\infty).
Cite
@article{arxiv.0711.1302,
title = {Local probabilities for random walks conditioned to stay positive},
author = {Vladimir Vatutin and Vitali Wachtel},
journal= {arXiv preprint arXiv:0711.1302},
year = {2007}
}