Random walks with square-root boundaries: the case of exact boundaries $g(t)=c\sqrt{t+b}-a$
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866912180427816960 |
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| author | Denisov, Denis Sakhanenko, Alexander Terveer, Sara wachtel, Vitali |
| author_facet | Denisov, Denis Sakhanenko, Alexander Terveer, Sara wachtel, Vitali |
| contents | Let $S(n)$ be a real valued random walk with i.i.d. increments which have zero mean and finite variance. We are interested in the asymptotic properties of the stopping time $T(g):=\inf\{n\ge1: S(n)\le g(n)\}$, where $g(t)$ is a boundary function. In the present paper we deal with the parametric family of boundaries $\{g_{a,b}(t)=c\sqrt{t+b}-a, b\ge0, a>c\sqrt{b}\}$. First, assuming that sufficiently many moments of increments of the walk are finite, we construct a positive space-time harmonic function $W(a,b)$. Then we show that there exist $p(c)>0$ and a constant $\varkappa(c)$ such that
$\mathbf{P}(T_{g_{a,b}}>n)\sim \varkappa(c)\frac{W(a,b)}{n^{p(c)/2}}$ as $n\to\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_04554 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Random walks with square-root boundaries: the case of exact boundaries $g(t)=c\sqrt{t+b}-a$ Denisov, Denis Sakhanenko, Alexander Terveer, Sara wachtel, Vitali Probability 60G50 Let $S(n)$ be a real valued random walk with i.i.d. increments which have zero mean and finite variance. We are interested in the asymptotic properties of the stopping time $T(g):=\inf\{n\ge1: S(n)\le g(n)\}$, where $g(t)$ is a boundary function. In the present paper we deal with the parametric family of boundaries $\{g_{a,b}(t)=c\sqrt{t+b}-a, b\ge0, a>c\sqrt{b}\}$. First, assuming that sufficiently many moments of increments of the walk are finite, we construct a positive space-time harmonic function $W(a,b)$. Then we show that there exist $p(c)>0$ and a constant $\varkappa(c)$ such that $\mathbf{P}(T_{g_{a,b}}>n)\sim \varkappa(c)\frac{W(a,b)}{n^{p(c)/2}}$ as $n\to\infty$. |
| title | Random walks with square-root boundaries: the case of exact boundaries $g(t)=c\sqrt{t+b}-a$ |
| topic | Probability 60G50 |
| url | https://arxiv.org/abs/2501.04554 |