Random Oper. Stoch. Equ. 2018; 26(4): 193ś200 Research Article Mykola Pratsiovytyi*, Iryna Lysenko and Oksana Voitovska Distribution of values of classic singular Cantor function of random argument https://doi.org/10.1515/rose-2018-0016 Received April 3, 2018; accepted June 18, 2018 Abstract: Let X be a random variable with independent ternary digits and let y = F(x) be a classic singular Cantor function. For the distribution of the random variable Y = F(X), the Lebesgue structure (i.e., the content of discrete, absolutely continuous and singular components), the structure of its point and the continuous spectra are exhaustively studied. Keywords: s-adic representation of numbers, singular Cantor function, random variable with independent ternary digits, Lebesgue structure of probability distribution, discrete probability distribution, mixture of discrete and continuous probability distributions MSC 2010: 11K55, 26A30, 28A80, 60G30, 60G50 || Communicated by: Vyacheslav L. Girko Introduction We introduce a new object: the distribution of the random variable Y = F(X), where X and y = F(x) are relatively well-known objects; the distribution of the random variable X is induced by the distributions of its ternary digits and y = F(x) is a classic singular Cantor function [6]. We are interested in the Lebesgue structure (i.e., the content of discrete, absolutely continuous and singularly continuous components) of the distribution of Y . Let us recall the main notions and specify a problem. Suppose that 1 < s is a őxed positive integer, A s ={0, 1,..., s − 1} is an alphabet of the s-adic numeral system, and L s ≡ A s × A s ×⋅⋅⋅× A s ×⋅⋅⋅ is the space of sequences of elements from the alphabet. Then an expansion of number x in series x = α 1 s ⋇ α 2 s 2 ⋇⋅⋅⋅⋇ α n s n ⋇⋅⋅⋅= ∆ s α 1 α 2 ... α n ... , α n = α n (x)∈ A s , is called its s-adic expansion, and its symbolic notation ∆ s α 1 α 2 ... α n ... is called s-adic representation of this number. The number α n ∈ A s is called s-adic digit of this representation. The set of all numbers x ∈[0; 1] having s-adic representation such that the őrst digit is equal to c 1 , the second digit is equal to c 2 , etc., the kth digit is equal to c k is called cylinder of rank k and base c 1 c 2 ... c k and *Corresponding author: Mykola Pratsiovytyi, Faculty of Physics and Mathematics, National Pedagogical Dragomanov University, Pyrogova Str. 9, 01030 Kyiv; and Institute of Mathematics of NAS of Ukraine, Tereshchenkivs’ka Str. 3, 01601 Kyiv, Ukraine, e-mail: prats4444@gmail.com Iryna Lysenko, Faculty of Physics and Mathematics, National Pedagogical Dragomanov University, Pyrogova Str. 9, 01030 Kyiv, Ukraine, e-mail: iryna.pratsiovyta@gmail.com Oksana Voitovska, National Pedagogical Dragomanov University, Pyrogova Str. 9, 01030 Kyiv, Ukraine, e-mail: ovoitovskaya@ukr.net Brought to you by | Göteborg University - University of Gothenburg Authenticated Download Date | 11/29/18 7:30 PM