Asymptotic expansions for normal deviations of random walks conditioned to stay positive

TL;DR

This paper develops new asymptotic expansions for the probabilities of a one-dimensional random walk remaining positive, focusing on deviations of order \,\sqrt{n}, extending previous results to larger deviations.

Contribution

It introduces an alternative asymptotic expansion for local probabilities of positive random walks, valid for deviations of order \,\sqrt{n}, improving upon earlier expansions limited to smaller zones.

Findings

Derived an asymptotic expansion for deviations of order \,\sqrt{n}

Extended the validity zone of local probability approximations

Provided a more accurate description of the walk's behavior in the large deviation regime

Abstract

We consider a one-dimensional random walk $S_n$ having i.i.d. increments with zero mean and finite variance. We continue our study of asymptotic expansions for local probabilities $\mathbf P(S_n=x,\tau_0>n)$, which has been started in \cite{DTW23}. Obtained there expansions make sense in the zone $x=o(\frac{\sqrt{n}}{\log^{1/2} n})$ only. In the present paper we derive an alternative expansion, which deals with $x$ of order $\sqrt{n}$.