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Asymptotic expansions for normal deviations of random walks conditioned to stay positive
Denis Denisov, Alexander Tarasov, Vitali Wachtel · 2024-12-12 · via math.PR updates on arXiv.org

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 $\mathbf P(S_n=x,τ_0>n)$, 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 $\sqrt{n}$.

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