Khvorostina Y. Distributions of random variables with fractal properties that are associated with alternating Luroth series

In the thesis we consider the properties of the distributions of the sums of the random alternating Luroth series. We study four classes of random variables. In the first class the random variables are represented by the alternating Luroth series with independent elements. The second class is the random alternating Luroth series which elements are random variables with Markovian dependence. The third class is the random subsums of given series with the independence random coefficients. The fourth class is the random subsums of given series which coefficients form a homogeneous Markov chain. The content of discrete, absolutely continuous and singular continuous components in Lebesgue structure of distributions of these random variables is studied. In addition, the belonging of the singular distribution to Cantor, Salem or quasi-Cantor type is investigated. The topological, metric and fractal properties of the minimal closed support of the distribution of random variables are described. We proved the purity of the distribution of the first three classes of random variables. The conditions of belonging to each pure type of probability distribution are received. We give examples of the pure probability distributions and their mixtures for the fourth class. The problem of the spectral structure of the singular distribution of random variables, that represented by the alternating Luroth series with independent elements, are fully solved. The conditions, under which the distribution of other random variables, belong to the singular distribution of Cantor type are received. The asymptotic behavior of the module of the characteristic function of the random subsums with independent coefficients are studied. The fractal properties of the spectrum for all random variables are studied. In particular, we are looking for conditions under which the spectrum is abnormally fractal set. The conditions, under which some probability distribution functions keep the fractal dimension of sets, are investigated.