Singular distributions of random variables with independent digits of representation in numeral system with natural base and redundant alphabet
Mykola Pratsiovytyi, Sofiia Ratushniak
TL;DR
We study the distribution of $ξ=\sum_{k=1}^{\infty} s^{-k}ξ_k$ with independent digits in a redundant base-$s$ representation, focusing on the Jessen–Wintner dichotomy between absolute continuity and singularity. For the case $s=3=r$, the paper links $ξ$ to infinite Bernoulli convolutions and analyzes the geometry of representations in the base-3 expansion, including cylinder overlaps and the fractal nature of the spectrum. It provides a near-complete classification: typical parameter choices yield singular distributions with Cantor-type spectra, whereas the special case $p_1=p_2=1/3$ yields an absolutely continuous distribution as a convolution of a Uniform$[0,1]$ component and a Cantor-type component. The work also demonstrates when $ξ$ can be expressed as convolutions of Cantor-type distributions and outlines several open problems, notably the detailed description of the density and the essential support, due to non-uniqueness in Δ-representations and cylinder overlaps.
Abstract
Given natural parameters s and r, where $2\leq s\leq r$, we consider the distribution of a random variable $ξ=\sum\limits_{k=1}^{\infty}s^{-k}ξ_k\equivΔ^{r_s}_{ξ_1ξ_2...ξ_k...},$ where $(ξ_k)$ 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.