Markitan V. Stochastic and doubly stochastic matrices in the problems of fractal analysis of functions

The dissertation is carried out in the field of constructive theory of functions. It is devoted to the study of functions and distributions of random variables with inhomogeneous local structure and fractal properties using stochastic and double stochastic matrices and different systems of encoding of real numbers, as well as sets which are essential for functions and distributions. Constructed in the work is a family of infinite positive doubly stochastic matrices. Described here are the topological and metric properties of the distribution function spectrum of one continuous random variable given by its Q_{\infty}^{*}-representation defined by an infinite doubly stochastic matrix. Studied in the work are two functions that establish a relationship between the numbers of the segment \left [0; 1 \right]. The first function is given by the direct projection of the digits of the Markov representation determined by the doubly stochastic matrix into the digits of the classical binary representation. The second function is defined in a similar way as a projection of digits of a non-binary representation into the Markov one. Obtained here are the conditions of their continuity, monotony, and singularity. The system of functional equations, whose solution is the second function, is found. Established in the work is a zero-dimensionality (in the sense of Lebesgue measure) of Cantor-type sets defined by prohibitions of usage of certain symbols in the binary Markov representation of a fractional part of a real number determined by a doubly stochastic matrix. The Hausdorf-Bezikovych dimension of the mentioned sets is computed. Investigated here are functions having the form f(x=\Delta_{\alpha_1\alpha_2\ldots\alpha_k\ldots}^{Q_2^*})=\sum\limits_{k=1}^{\infty}\alpha_{k}d_k and the distribution of a random variable \xi=\sum\limits_{k=1}^{\infty}d_k\xi_k defined for two classes of convergent positive series \sum\limits_{k=1}^{\infty}d_k with some homogeneity conditions; the behavior of the module of the characteristic function of the infinite Bernoulli convolution managed by the series \sum\limits_{k=1}^{\infty}d_k from one of the classes. Autoconvolutions of the specified infinite Bernoulli convolutions are considered. The properties of their distribution spectrum and the conditions for singularity of the distribution are studied. It is proved that one of the above classes consists of series whose sets of incomplete sums are cantorvals, and the Lebesgue measures of those cantorvals can be arbitrary close to 1.