Fractal random variables defined by probability distributions of digits of their $G_2$-representation having two bases with different signs
In this paper, we study distributions of two random variables \begin{gather*} \tau = \tau_1 g_{1-\tau_1} + \sum_{k=2}^\infty \tau_k g_{1-\tau_k} \prod_{i=1}^{k-1} g_{\tau_i} \equiv \Delta^{G_2}_{\tau_1\tau_2...\tau_n...}, \xi = \xi_1 g_{1-\xi_1} + \sum_{k=2}^\infty \xi_k g_{1-\xi_k} \prod_{i=1}^{k-1} g_{\xi_i} \equiv \Delta^{G_2}_{\xi_1\xi_2...\xi_n...}, \end{gather*} where $g_0$ is a given number belonging to interval $[\frac{1}{2}; 1)$, $g_1\equiv g_0 - 1$, $(\tau_n)$ and $(\xi_n)$ are sequences of random variables taking the values $0$ and $1$, and $(\tau_n)$ is a sequence of random variables that form a Markov chain with positive initial probabilities $p_{0}$, $p_{1}$ and matrix of transition probabilities $ \begin{pmatrix} p_{00} & p_{01} p_{10} & p_{11} \end{pmatrix},$ $(\xi_n)$ 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 $\tau$ and $\xi$. For random variable $\tau$, 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 $\xi$ (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.