Infinite Bernoulli convolutions generated by multigeometric series and their properties
Mykola Pratsiovytyi, Dmytro Karvatskyi, Oleg Makarchuk
Abstract
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.