Makarchuk O. The problem of deeping of the Jessen-Wintner theorem for infinite Bernoulli convolutions and their generalizations.

The thesis is devoted to study of probability measures in dimensional Euclidean space with complicated local structure, namely random variables represented by s-adic fraction with redundant set of digits. Lebesgue structure and asymptotic properties of the characteristic function are studied in detail. The main object of investigation is the random variable , which are independent random variables. In the thesis, we obtain the conditions on stochastic matrix when distribution of, random variable having binary representation with redundant set of digits 2 and 3, is pure absolutely continuous or pure singular. We describe properties of the distribution of respectively random variable , including spectral, diferential and other. The problem on Lebesgue type of distribution of the respectively random variable for identically distributed digits is completely solved. The problem on Lebesgue type of distribution of the random variable wich having binary representation with redundant digits 2,3 and 4 for identically distributed digits is completely solved.