Zhykharyeva Y. Singular probability distributions related to representations of numbers by positive Luroth series

It is investigated the Lebesgue structure, topological, metric and fractal properties of spectrum of probability distributions belonging to four families of random variables related to positive Luroth series. Probability distributions from the first family belong to class of infinite Bernoulli convolutions, they are distributions of random subsums of given Luroth series with independent addends. Any representative of the second family is a random subsum of the Luroth series, where addends have a Markovian dependence. Third family consists of random variables with independent elements of representation by the Luroth series. Representatives of the last family are random variables represented by the Luroth series such that their elements form the Markov chain. The asymptotic behaviour at infinity of the absolute value of characteristic function of this infinite Bernoulli convolution is studied. For first, second and third classes of random variables, we proved the purity of distribution and criteria of belonging to each pure type of probability distribution. For fourth family, examples of pure distributions and their mixtures are given, and criterion for probability distribution to be of Cantor-type singular is proved.