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arXiv · 2610.33612

Asymptotic expansions for random walks conditioned to stay positive with arbitrary endpoints

Alexander Tarasov
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Abstract

We consider a one-dimensional random walk $S_n$ with i.i.d. integer-valued increments having zero mean and finite variance. Under suitable moment assumptions, we derive asymptotic expansions for the local probabilities $\pr(x+S_n=y,τ_x>n)$, where $τ_x=\inf\{n\ge1:x+S_n\le0\}$. These extend previous expansions for walks started at zero to arbitrary starting points. We obtain different expansions according to the sizes of the starting point and endpoint relative to $\sqrt n$.

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Alexander Tarasov. 2026-10-24. Asymptotic expansions for random walks conditioned to stay positive with arbitrary endpoints. https://arxiv.org/abs/2610.33612

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