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Properties and distributions of values of fractal functions related to \(Q_2\)-representations of real numbers. (English. Ukrainian original) Zbl 1453.28014

Theory Probab. Math. Stat. 99, 211-228 (2019); translation from Teor. Jmovirn. Mat. Stat. 99, 187-202 (2018).
Summary: A \(Q_2\)-representation of numbers \(x\in [0;1]\) is determined by a parameter \(q_0\in (0;1)\) and provides an expansion of a number \(x\in [0;1]\) in the form of the following series: \[x=\alpha_1 q_{1-\alpha_1}+\sum_{k=2}^{\infty}\left( \alpha_k q_{1-\alpha_k} \prod_{j=1}^{k-1} q_{\alpha_j}(x)\right)\equiv \Delta^{Q_2}_{\alpha_1\alpha_2\ldots \alpha_n\ldots},\] where \(\alpha_k\in \{0,1\}\equiv A\) and \(q_1\equiv 1-q_0\). The structural and local as well as global topological/metric and fractal properties are studied for the function \(f_{\varphi }\) defined by \[\begin{aligned} f_{\varphi}(x)&=f_{\varphi}\left(\Delta^{Q_2}_{\alpha_1\alpha_2\alpha_3\ldots \alpha_{n-1}\alpha_n\alpha_{n+1}\ldots}\right)\\ & =\Delta^{Q_2}_{\varphi(\alpha_1,\alpha_2)\varphi(\alpha_2,\alpha_3)\ldots\varphi(\alpha_{n-1},\alpha_n)\varphi(\alpha_n,\alpha_{n+1})\ldots},\end{aligned}\] where \(\varphi\) is a given function, \( \varphi : A^2\rightarrow A\).
For the random variable \(Y=f_{\varphi }(X)\) where \(X\) is a random variable with a known distribution, we study its Lebesgue structure (the discrete, absolutely continuous, and singular components) and spectral properties of the set of points of growth of the distribution function of \(Y\).

MSC:

28A80 Fractals
26A30 Singular functions, Cantor functions, functions with other special properties
60E05 Probability distributions: general theory
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