AI 中文总结
本文研究两类由$G_2$-表示定义的分形随机变量的分布性质，详尽分析了马尔可夫链数字序列对应变量的谱性质，并证明了独立数字序列对应变量分布的勒贝格纯度定理。
AI 中文摘要
本文研究两个随机变量的分布：\n\begin{gather*}\nτ= τ_1 g_{1-τ_1}\n+ \sum_{k=2}^\infty τ_k g_{1-τ_k} \prod_{i=1}^{k-1} g_{τ_i}\n\equiv Δ^{G_2}_{τ_1τ_2...τ_n...},\n\nξ= ξ_1 g_{1-ξ_1}\n+ \sum_{k=2}^\infty ξ_k g_{1-ξ_k} \prod_{i=1}^{k-1} g_{ξ_i}\n\equiv Δ^{G_2}_{ξ_1ξ_2...ξ_n...},\n\end{gather*}\n其中$g_0$是属于区间$[\frac{1}{2}; 1)$的给定数，$g_1\equiv g_0 - 1$，$(τ_n)$和$(ξ_n)$是取值为0和1的随机变量序列。$(τ_n)$是构成马尔可夫链的随机变量序列，具有正初始概率$p_{0}$、$p_{1}$以及转移概率矩阵$\begin{pmatrix} p_{00} & p_{01} \\\\ p_{10} & p_{11} \end{pmatrix}$；$(ξ_n)$是独立随机变量序列，分别以概率$p_{0n}$和$p_{1n}$取上述值（$p_{0n}+p_{1n}=1$）。\n我们研究了$τ$和$ξ$分布的结构、谱和分形性质。对于随机变量$τ$，我们详尽研究了其分布的点谱（原子集）和连续谱（最小闭支撑）。我们证明了关于随机变量$ξ$分布的勒贝格纯度定理（Jessen--Wintner定理的类似结果），即该分布属于纯离散、纯绝对连续和纯奇异三类之一的条件。
英文摘要
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 $[\frac{1}{2}; 1)$, $g_1\equiv g_0 - 1$, $(τ_n)$ and $(ξ_n)$ are sequences of random variables taking the values $0$ and $1$, and $(τ_n)$ 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},$ $(ξ_n)$ 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 $τ$ and $ξ$. For random variable $τ$, 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 $ξ$ (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.
