Capacity of the range of random walk: The law of the iterated logarithm
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913252333584384 |
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| author | Dembo, Amir Okada, Izumi |
| author_facet | Dembo, Amir Okada, Izumi |
| contents | We establish both the $\limsup$ and the $\liminf$ law of the iterated logarithm (LIL), for the capacity of the range of a simple random walk in any dimension $d\ge 3$. While for $d \ge 4$, the order of growth in $n$ of such LIL at dimension $d$ matches that for the volume of the random walk range in dimension $d-2$, somewhat surprisingly this correspondence breaks down for the capacity of the range at $d=3$. We further establish such LIL for the Brownian capacity of a $3$-dimensional Brownian sample path and novel, sharp moderate deviations bounds for the capacity of the range of a $4$-dimensional simple random walk. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2208_02184 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Capacity of the range of random walk: The law of the iterated logarithm Dembo, Amir Okada, Izumi Probability We establish both the $\limsup$ and the $\liminf$ law of the iterated logarithm (LIL), for the capacity of the range of a simple random walk in any dimension $d\ge 3$. While for $d \ge 4$, the order of growth in $n$ of such LIL at dimension $d$ matches that for the volume of the random walk range in dimension $d-2$, somewhat surprisingly this correspondence breaks down for the capacity of the range at $d=3$. We further establish such LIL for the Brownian capacity of a $3$-dimensional Brownian sample path and novel, sharp moderate deviations bounds for the capacity of the range of a $4$-dimensional simple random walk. |
| title | Capacity of the range of random walk: The law of the iterated logarithm |
| topic | Probability |
| url | https://arxiv.org/abs/2208.02184 |