Gaussian heat kernel asymptotics for conditioned random walks
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866912153520308224 |
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| author | Grama, Ion Xiao, Hui |
| author_facet | Grama, Ion Xiao, Hui |
| contents | Consider a random walk $S_n=\sum_{i=1}^n X_i$ with independent and identically distributed real-valued increments with zero mean, finite variance and moment of order $2 + δ$ for some $δ>0$. For any starting point $x\in \mathbb R$, let $τ_x = \inf \left\{ k\geq 1: x+S_{k} < 0 \right\}$ denote the first time when the random walk $x+S_n$ exits the half-line $[0,\infty)$. We investigate the uniform asymptotic behavior over $x\in \mathbb R$ of the persistence probability $\mathbb P (τ_x >n)$ and the joint distribution $\mathbb{P} \left( x + S_n \leq u, τ_x > n \right)$, for $u\geq 0$, as $n \to \infty$. New limit theorems for these probabilities are established based on the heat kernel approximations. Additionally, we evaluate the rate of convergence by proving Berry-Esseen type bounds. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_08932 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gaussian heat kernel asymptotics for conditioned random walks Grama, Ion Xiao, Hui Probability Primary 60F05, 60F17, 60G50. Secondary 60G40, 60J05 Consider a random walk $S_n=\sum_{i=1}^n X_i$ with independent and identically distributed real-valued increments with zero mean, finite variance and moment of order $2 + δ$ for some $δ>0$. For any starting point $x\in \mathbb R$, let $τ_x = \inf \left\{ k\geq 1: x+S_{k} < 0 \right\}$ denote the first time when the random walk $x+S_n$ exits the half-line $[0,\infty)$. We investigate the uniform asymptotic behavior over $x\in \mathbb R$ of the persistence probability $\mathbb P (τ_x >n)$ and the joint distribution $\mathbb{P} \left( x + S_n \leq u, τ_x > n \right)$, for $u\geq 0$, as $n \to \infty$. New limit theorems for these probabilities are established based on the heat kernel approximations. Additionally, we evaluate the rate of convergence by proving Berry-Esseen type bounds. |
| title | Gaussian heat kernel asymptotics for conditioned random walks |
| topic | Probability Primary 60F05, 60F17, 60G50. Secondary 60G40, 60J05 |
| url | https://arxiv.org/abs/2412.08932 |