A fixed-point equation approach for the superdiffusive elephant random walk
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866916208955097088 |
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| author | Guérin, Hélène Laulin, Lucile Raschel, Kilian |
| author_facet | Guérin, Hélène Laulin, Lucile Raschel, Kilian |
| contents | We study the elephant random walk in arbitrary dimension $d\geq 1$. Our main focus is the limiting random variable appearing in the superdiffusive regime. Building on a link between the elephant random walk and Pólya-type urn models, we prove a fixed-point equation (or system in dimension two and larger) for the limiting variable. Based on this, we deduce several properties of the limit distribution, such as the existence of a density with support on $\mathbb R^d$ for $d\in\{1,2,3\}$, and we bring evidence for a similar result for $d\geq 4$. We also investigate the moment-generating function of the limit and give, in dimension $1$, a non-linear recurrence relation for the moments. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2308_14630 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A fixed-point equation approach for the superdiffusive elephant random walk Guérin, Hélène Laulin, Lucile Raschel, Kilian Probability 60E05, 60E10, 60J10, 60G50 We study the elephant random walk in arbitrary dimension $d\geq 1$. Our main focus is the limiting random variable appearing in the superdiffusive regime. Building on a link between the elephant random walk and Pólya-type urn models, we prove a fixed-point equation (or system in dimension two and larger) for the limiting variable. Based on this, we deduce several properties of the limit distribution, such as the existence of a density with support on $\mathbb R^d$ for $d\in\{1,2,3\}$, and we bring evidence for a similar result for $d\geq 4$. We also investigate the moment-generating function of the limit and give, in dimension $1$, a non-linear recurrence relation for the moments. |
| title | A fixed-point equation approach for the superdiffusive elephant random walk |
| topic | Probability 60E05, 60E10, 60J10, 60G50 |
| url | https://arxiv.org/abs/2308.14630 |