Expansions for random walks conditioned to stay positive
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909076618739712 |
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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 $S_n$ with i.i.d. increments with zero mean and finite variance. We study the asymptotic expansion for the tail distribution $\mathbf P(τ_x>n)$ of the first passage times $τ_x:=\inf\{n\ge1:x+S_n\le0\}$ for $\ x\ge0.$ We also derive asymptotic expansion for local probabilities $\mathbf P(S_n=x,τ_0>n)$. Studying the asymptotic expansions we obtain a sequence of discrete polyharmonic functions and obtain analogues of renewal theorem for them. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09929 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Expansions for random walks conditioned to stay positive Denisov, Denis Tarasov, Alexander Wachtel, Vitali Probability Primary 60G50, Secondary 60G40, 60F17 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 $\mathbf P(τ_x>n)$ of the first passage times $τ_x:=\inf\{n\ge1:x+S_n\le0\}$ for $\ x\ge0.$ We also derive asymptotic expansion for local probabilities $\mathbf P(S_n=x,τ_0>n)$. Studying the asymptotic expansions we obtain a sequence of discrete polyharmonic functions and obtain analogues of renewal theorem for them. |
| title | Expansions for random walks conditioned to stay positive |
| topic | Probability Primary 60G50, Secondary 60G40, 60F17 |
| url | https://arxiv.org/abs/2401.09929 |