Fractal random variables defined by probability distributions of digits of their $G_2$-representation having two bases with different signs

Abstract

In this paper, we study distributions of two random variables \begin{gather*} τ= τ_1 g_{1-τ_1} + \sum_{k=2}^\infty τ_k g_{1-τ_k} \prod_{i=1}^{k-1} g_{τ_i} \equiv Δ^{G_2}_{τ_1τ_2...τ_n...}, ξ= ξ_1 g_{1-ξ_1} + \sum_{k=2}^\infty ξ_k g_{1-ξ_k} \prod_{i=1}^{k-1} g_{ξ_i} \equiv Δ^{G_2}_{ξ_1ξ_2...ξ_n...}, \end{gather*} where $g_0$ is a given number belonging to interval $[\frac{1}{2};1)$, $g_1\equiv g_0-1$, $(\ddot{I}{„}_n)$ and $(\hat{I}{¾}_n)$ are sequences of random variables taking the values $0$ and $1$, and $(\ddot{I}{„}_n)$ is a sequence of random variables that form a Markov chain with positive initial probabilities $p_0$, $p_1$ and matrix of transition probabilities $\left(\begin{array}{ccc}{p}_{00}& p_{01}{p}_{10}& p_{11}\end{array}\right),$ $(\hat{I}{¾}_n)$ is a sequence of independent random variables taking the specified values with probabilities $p_{0n}$ and $p_{1n}$, respectively ($p_{0n}+p_{1n}=1$). We study structural, spectral, and fractal properties of distributions of $\ddot{I}„$ and $\hat{I}¾$. For random variable $\ddot{I}„$, point spectrum (the set of atoms) and continuous spectrum (minimal closed support) of its distribution are studied exhaustively. We prove a theorem on the Lebesgue purity of distribution of random variable $\hat{I}¾$ (an analog of the Jessen--Wintner theorem), i.e., conditions for the distribution to belong to one of the types: pure discrete, pure absolutely continuous, and pure singular.