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

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 $\sqrt{n}$.

40 pages