Asymptotic expansions for normal deviations of random walks conditioned to stay positive
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910742833266688 |
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| author | Denisov, Denis Tarasov, Alexander Wachtel, Vitali |
| author_facet | Denisov, Denis Tarasov, Alexander Wachtel, Vitali |
| contents | 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 $\sqrt{n}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_09145 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotic expansions for normal deviations of random walks conditioned to stay positive Denisov, Denis Tarasov, Alexander Wachtel, Vitali Probability 60G50, 60G40, 60F17 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 $\sqrt{n}$. |
| title | Asymptotic expansions for normal deviations of random walks conditioned to stay positive |
| topic | Probability 60G50, 60G40, 60F17 |
| url | https://arxiv.org/abs/2412.09145 |