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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.11310 |
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| _version_ | 1866912962350940160 |
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| author | Pratsiovytyi, Mykola Karvatskyi, Dmytro Makarchuk, Oleg |
| author_facet | Pratsiovytyi, Mykola Karvatskyi, Dmytro Makarchuk, Oleg |
| contents | The paper is devoted to infinite Bernoulli convolutions generated by positive multigeometric series and to probability distributions of random variables whose digits in an even integer base-$s$ expansion with two redundant digits form a sequence of independent and identically distributed random variables.
The main objects of the article are random variables:
$ξ=\sum\limits_{n=1}^{\infty}\frac{ξ_n}{s^n}$, where $(ξ_n)$ is a sequence of independent and identically distributed random variables taking values $0, 1, 2, \dots, s-1, s, s+1$ with probabilities $p_0$, $p_1$, $p_2, \dots, p_{s-1}, p_s, p_{s+1}$ respectively $(3<s \in \mathbb{N})$; $$η=\sum\limits_{n=1}^{\infty}\left[\frac{3η_{(n-1)(m+1)+1}}{s^n}+\sum\limits_{j=1}^{m} \frac{2η_{(n-1)(m+1)+1+j}}{s^n}\right],$$ where $(η_n)$ is a sequence of independent and identically distributed random variables that take values $0$ and $1$ with probabilities $q_0>0$ and $q_1=1-q_0>0$. We study conditions under which the above random variables have absolutely continuous or singular distributions as well as topological, metric, and fractal properties of their supports. The main focus is on the case where the spectrum is a Cantorval. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_11310 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Infinite Bernoulli convolutions generated by multigeometric series and their properties Pratsiovytyi, Mykola Karvatskyi, Dmytro Makarchuk, Oleg Probability 60E05, 40A05, 28A80 The paper is devoted to infinite Bernoulli convolutions generated by positive multigeometric series and to probability distributions of random variables whose digits in an even integer base-$s$ expansion with two redundant digits form a sequence of independent and identically distributed random variables. The main objects of the article are random variables: $ξ=\sum\limits_{n=1}^{\infty}\frac{ξ_n}{s^n}$, where $(ξ_n)$ is a sequence of independent and identically distributed random variables taking values $0, 1, 2, \dots, s-1, s, s+1$ with probabilities $p_0$, $p_1$, $p_2, \dots, p_{s-1}, p_s, p_{s+1}$ respectively $(3<s \in \mathbb{N})$; $$η=\sum\limits_{n=1}^{\infty}\left[\frac{3η_{(n-1)(m+1)+1}}{s^n}+\sum\limits_{j=1}^{m} \frac{2η_{(n-1)(m+1)+1+j}}{s^n}\right],$$ where $(η_n)$ is a sequence of independent and identically distributed random variables that take values $0$ and $1$ with probabilities $q_0>0$ and $q_1=1-q_0>0$. We study conditions under which the above random variables have absolutely continuous or singular distributions as well as topological, metric, and fractal properties of their supports. The main focus is on the case where the spectrum is a Cantorval. |
| title | Infinite Bernoulli convolutions generated by multigeometric series and their properties |
| topic | Probability 60E05, 40A05, 28A80 |
| url | https://arxiv.org/abs/2603.11310 |