Berry-Esseen bounds for step-reinforced random walks
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866908299262164992 |
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| author | Hu, Zhishui |
| author_facet | Hu, Zhishui |
| contents | We study both the positively and negatively step-reinforced random walks with parameter $p$. For a step distribution $μ$ with finite second moment, the positively step-reinforced random walk with $p\in [1/2,1)$ and the negatively step-reinforced random walk with $p\in (0,1)$ converge to a normal distribution under suitable normalization. In this work, we obtain the rates of convergence to normality for both cases under the assumption that $μ$ has a finite third moment. In the proofs, we establish a Berry-Esseen bound for general functionals of independent random variables, utilize the randomly weighted sum representations of step-reinforced random walks, and apply special comparison arguments to quantify the Kolmogorov distance between a mixed normal distribution and its corresponding normal distribution. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02502 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Berry-Esseen bounds for step-reinforced random walks Hu, Zhishui Probability We study both the positively and negatively step-reinforced random walks with parameter $p$. For a step distribution $μ$ with finite second moment, the positively step-reinforced random walk with $p\in [1/2,1)$ and the negatively step-reinforced random walk with $p\in (0,1)$ converge to a normal distribution under suitable normalization. In this work, we obtain the rates of convergence to normality for both cases under the assumption that $μ$ has a finite third moment. In the proofs, we establish a Berry-Esseen bound for general functionals of independent random variables, utilize the randomly weighted sum representations of step-reinforced random walks, and apply special comparison arguments to quantify the Kolmogorov distance between a mixed normal distribution and its corresponding normal distribution. |
| title | Berry-Esseen bounds for step-reinforced random walks |
| topic | Probability |
| url | https://arxiv.org/abs/2504.02502 |