CONVOLUTION OF TWO SINGULAR DISTRIBUTIONS: CLASSIC CANTOR TYPE AND RANDOM VARIABLE WITH INDEPENDENT NINE DIGITS

Abstract

We consider distribution of random variable $\xi=\tau+\eta$, where $\tau$ and $\eta$ independent random variables, moreover $\tau$ has classic Cantor type distribution and $\eta$ is a random variable with independent identically distributed digits of the nine-digit representation. With additional conditions for the distributions of the digits $\eta$, sufficient conditions for the singularity of the Cantor type of the distribution $\xi$ are specified. To substantiate the statements, a topological-metric analysis of the representation of numbers $x\in [0;2]$ in the numerical system with base $9$ and a seventeen-symbol alphabet (a set of numbers) is carried out. The geometry (positional and metric) of this representation is described by the properties of the corresponding cylindrical sets.