On types of distributions of sums of one class of random power series with independent identically distributed coefficients

Abstract

By using the method of characteristic functions, we obtain sufficient conditions for the singularity of a random variable. $$ξ = \sum_{k=1}^{∞} 2^{−k}ξ_k,$$ where $ξ_k$ are independent identically distributed random variables taking values $x_0, x_1$, and $x_2$ $(x_0 < x_1 < x_2)$ with probabilities $p_0, p_1$ and $p_2$, respectively, such that $p_i ≥ 0,\; p_0 + p_1 + p_2 = 1$ and $2(x_1 − x_0)/(x_2−x_0)$ is a rational number.