Recurrence, transience and anti-concentration of Rademacher random walks
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| Format: | Preprint |
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2025
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| _version_ | 1866908721115824128 |
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| author | Bhattacharya, Satyaki Crane, Edward Johnston, Tom |
| author_facet | Bhattacharya, Satyaki Crane, Edward Johnston, Tom |
| contents | The Rademacher random walk associated with a deterministic sequence $(a_n)_{n \geq 1}$ is the walk which starts at zero and, at step $i$, independently steps either up or down by $a_i$ with equal probability. We continue the study begun by Bhattacharya and Volkov in 2023 of the transience or recurrence of one-dimensional Rademacher random walks. In particular, we show that if the sequence of step sizes is bounded, the walk is weakly recurrent, meaning that it returns infinitely often to a random finite interval, while if the step sizes tend to infinity arbitrarily slowly, the walk may be transient. On the other hand, using a construction with integer step sizes, we show that the step sizes may grow arbitrarily fast and still give a weakly recurrent random walk. We also show, using a construction with non-integer step sizes, that the same conclusion holds even if we restrict to strictly increasing step sizes. However, we prove that if $a_n = n^{α+ o(1)}$ for some $α> 1/2$, then the walk is transient. We show that the bound on $α$ is tight by giving an example where $a_n = Θ(n^{1/2})$ and the walk is weakly recurrent. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_24568 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Recurrence, transience and anti-concentration of Rademacher random walks Bhattacharya, Satyaki Crane, Edward Johnston, Tom Probability The Rademacher random walk associated with a deterministic sequence $(a_n)_{n \geq 1}$ is the walk which starts at zero and, at step $i$, independently steps either up or down by $a_i$ with equal probability. We continue the study begun by Bhattacharya and Volkov in 2023 of the transience or recurrence of one-dimensional Rademacher random walks. In particular, we show that if the sequence of step sizes is bounded, the walk is weakly recurrent, meaning that it returns infinitely often to a random finite interval, while if the step sizes tend to infinity arbitrarily slowly, the walk may be transient. On the other hand, using a construction with integer step sizes, we show that the step sizes may grow arbitrarily fast and still give a weakly recurrent random walk. We also show, using a construction with non-integer step sizes, that the same conclusion holds even if we restrict to strictly increasing step sizes. However, we prove that if $a_n = n^{α+ o(1)}$ for some $α> 1/2$, then the walk is transient. We show that the bound on $α$ is tight by giving an example where $a_n = Θ(n^{1/2})$ and the walk is weakly recurrent. |
| title | Recurrence, transience and anti-concentration of Rademacher random walks |
| topic | Probability |
| url | https://arxiv.org/abs/2510.24568 |