A central limit theorem for the variation of the sum of digits
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866910364866707456 |
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| author | Hosten, Yohan Janvresse, Élise de la Rue, Thierry |
| author_facet | Hosten, Yohan Janvresse, Élise de la Rue, Thierry |
| contents | We prove a Central Limit Theorem for probability measures defined via the variation of the sum-of-digits function, in base $b\ge 2$. For $r\ge 0$ and $d \in \mathbb{Z}$, we consider $μ^{(r)}(d)$ as the density of integers $n\in \mathbb{N}$ for which the sum of digits increases by $d$ when we add $r$ to $n$. We give a probabilistic interpretation of $μ^{(r)}$ on the probability space given by the group of $b$-adic integers equipped with the normalized Haar measure. We split the base-$b$ expansion of the integer $r$ into so-called "blocks", and we consider the asymptotic behaviour of $μ^{(r)}$ as the number of blocks goes to infinity. We show that, up to renormalization, $μ^{(r)}$ converges to the standard normal law as the number of blocks of $r$ grows to infinity. We provide an estimate of the speed of convergence. The proof relies, in particular, on a $ϕ$-mixing process defined on the $b$-adic integers. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2111_05030 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | A central limit theorem for the variation of the sum of digits Hosten, Yohan Janvresse, Élise de la Rue, Thierry Probability We prove a Central Limit Theorem for probability measures defined via the variation of the sum-of-digits function, in base $b\ge 2$. For $r\ge 0$ and $d \in \mathbb{Z}$, we consider $μ^{(r)}(d)$ as the density of integers $n\in \mathbb{N}$ for which the sum of digits increases by $d$ when we add $r$ to $n$. We give a probabilistic interpretation of $μ^{(r)}$ on the probability space given by the group of $b$-adic integers equipped with the normalized Haar measure. We split the base-$b$ expansion of the integer $r$ into so-called "blocks", and we consider the asymptotic behaviour of $μ^{(r)}$ as the number of blocks goes to infinity. We show that, up to renormalization, $μ^{(r)}$ converges to the standard normal law as the number of blocks of $r$ grows to infinity. We provide an estimate of the speed of convergence. The proof relies, in particular, on a $ϕ$-mixing process defined on the $b$-adic integers. |
| title | A central limit theorem for the variation of the sum of digits |
| topic | Probability |
| url | https://arxiv.org/abs/2111.05030 |