Corrected diffusion approximation for random walks conditioned to stay positive
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866912915630587904 |
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| author | Denisov, Denis Tarasov, Alexander Wachtel, Vitali |
| author_facet | Denisov, Denis Tarasov, Alexander Wachtel, Vitali |
| contents | Let $S_n$ be a random walk with i.i.d. increments which have zero mean and finite variance. For every $x\ge0$ we define the stopping time $τ_x:=\inf\{n\ge1:x+S_n\le0\}$ and consider the probabilities $\mathbb{P}(x+S_n\ge y,τ_x>n)$. We study the quality of the normal approximation for these probabilities and derive a Berry-Esseen-type inequality for $\mathbb{P}(x+S_n\ge y|τ_x>n)$. Our Theorem 1 is an extension of the results in our previous paper (arXiv:2412.08502) where we have considered the special case $x=0$. It is also worth mentioning that Theorem 1 complements the results of Siegmund and Yuh (1982) on the corrected diffusion approximation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_18120 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Corrected diffusion approximation for random walks conditioned to stay positive Denisov, Denis Tarasov, Alexander Wachtel, Vitali Probability 60G50 Let $S_n$ be a random walk with i.i.d. increments which have zero mean and finite variance. For every $x\ge0$ we define the stopping time $τ_x:=\inf\{n\ge1:x+S_n\le0\}$ and consider the probabilities $\mathbb{P}(x+S_n\ge y,τ_x>n)$. We study the quality of the normal approximation for these probabilities and derive a Berry-Esseen-type inequality for $\mathbb{P}(x+S_n\ge y|τ_x>n)$. Our Theorem 1 is an extension of the results in our previous paper (arXiv:2412.08502) where we have considered the special case $x=0$. It is also worth mentioning that Theorem 1 complements the results of Siegmund and Yuh (1982) on the corrected diffusion approximation. |
| title | Corrected diffusion approximation for random walks conditioned to stay positive |
| topic | Probability 60G50 |
| url | https://arxiv.org/abs/2602.18120 |