O. V. Shkol’nyi
Random variables determined by the distributions of their digits in a numeration system with complex base
Ukr. Mat. Zh. - 1998. - 50, № 12. - pp. 1715–1720
We study the distributions of complex-valued random variables determined by the distributions of their digits in a numeration system with complex base. We establish sufficient conditions for the singularity of such random variables, in particular, in the cases where their spectrum has Lebesgue measure zero (C-type singular distribution) or is a rectangle (S-type singular distribution).
Ukr. Mat. Zh. - 1997. - 49, № 12. - pp. 1653–1660
We study the structure of the distribution of a complex-valued random variable ξ = Σa k ξ k , where ξ k are independent complex-valued random variables with discrete distribution and a k are terms of an absolutely convergent series. We establish a criterion of discreteness and sufficient conditions for singularity of the distribution of ξ and investigate the fractal properties of the spectrum.