Singular distributions of random variables with independent digits of representation in numeral system with natural base and redundant alphabet

Abstract

Given natural parameters s and r, where $2\leq s\leq r$, we consider the distribution of a random variable $\xi=\sum\limits_{k=1}^{\infty}s^{-k}\xi_k\equiv\Delta^{r_s}_{\xi_1\xi_2...\xi_k...},$ where $(\xi_k)$ is a sequence of independent random variables taking values in $\{0,1,...,r\}$ with probabilities $p_0,p_1,...,p_r$, respectively, and all $ p_i<1$. In the case s=3=r, necessary and sufficient conditions for the singularity and absolute continuity of the distribution of random variable are established. The work also discusses the connection between the distribution of random variable and infinite Bernoulli convolutions governed by the corresponding series as well as representations of numbers in the base-3 numeral system with one redundant digit. Several open problems are formulated.