The asymptotic distribution of the scaled remainder for pseudo golden ratio expansions of a continuous random variable
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866916527477882880 |
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| author | Herbst, Ira W. Møller, Jesper Svane, Anne Marie |
| author_facet | Herbst, Ira W. Møller, Jesper Svane, Anne Marie |
| contents | Let $X=\sum_{k=1}^\infty X_k β^{-k}$ be the base-$β$ expansion of a continuous random variable $X$ on the unit interval where $β$ is the positive solution to $β^n = 1 + β+ \cdots + β^{n-1}$ for an integer $n\ge 2$ (i.e., $β$ is a generalization of the golden mean for which $n=2$). We study the asymptotic distribution and convergence rate of the scaled remainder $\sum_{k=1}^\infty X_{m+k} β^{-k}$ when $m$ tends to infinity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_08387 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The asymptotic distribution of the scaled remainder for pseudo golden ratio expansions of a continuous random variable Herbst, Ira W. Møller, Jesper Svane, Anne Marie Probability 60F25 (Primary) 62E17, 37A50 (Secondary) Let $X=\sum_{k=1}^\infty X_k β^{-k}$ be the base-$β$ expansion of a continuous random variable $X$ on the unit interval where $β$ is the positive solution to $β^n = 1 + β+ \cdots + β^{n-1}$ for an integer $n\ge 2$ (i.e., $β$ is a generalization of the golden mean for which $n=2$). We study the asymptotic distribution and convergence rate of the scaled remainder $\sum_{k=1}^\infty X_{m+k} β^{-k}$ when $m$ tends to infinity. |
| title | The asymptotic distribution of the scaled remainder for pseudo golden ratio expansions of a continuous random variable |
| topic | Probability 60F25 (Primary) 62E17, 37A50 (Secondary) |
| url | https://arxiv.org/abs/2404.08387 |