Expansions for random walks conditioned to stay positive

TL;DR

This paper develops asymptotic expansions for the tail distribution of first passage times and local probabilities of a one-dimensional random walk conditioned to stay positive, introducing discrete polyharmonic functions and renewal theorem analogues.

Contribution

It introduces a novel asymptotic expansion framework for conditioned random walks, including discrete polyharmonic functions and renewal theorem analogues.

Findings

Derived asymptotic expansion for tail distribution of first passage times.

Established asymptotic expansion for local probabilities of the walk.

Developed discrete polyharmonic functions and renewal theorem analogues.

Abstract

We consider a one-dimensional random walk $S_n$ with i.i.d. increments with zero mean and finite variance. We study the asymptotic expansion for the tail distribution $\mathbf P(\tau_x>n)$ of the first passage times $\tau_x:=\inf\{n\ge1:x+S_n\le0\}$ for $\ x\ge0.$ We also derive asymptotic expansion for local probabilities $\mathbf P(S_n=x,\tau_0>n)$. Studying the asymptotic expansions we obtain a sequence of discrete polyharmonic functions and obtain analogues of renewal theorem for them.