Abstract
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).
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References
O. V. Shkol’nyi, “On one class of singular complex-valued random variables of the Jessen-Wintner type,” Ukr. Mat. Zh., 49, No. 12. 1653–1660 (1997).
A. F. Turbin and N. V. Pratsevityi, Fractal Sets, Functions, and Distributions[in Russian], Naukova Dumka, Kiev (1992).
N. V. Pratsevityi, “Classification of singular distributions depending on the spectrum properties,” in: Random Evolutions: Theoretical and Applied Problems [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1992), pp. 77–83.
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 50, No. 12, pp. 1715–1720, December, 1998.
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Shkol’nyi, O.V. Random variables determined by the distributions of their digits in a numeration system with complex base. Ukr Math J 50, 1956–1962 (1998). https://doi.org/10.1007/BF02514213
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DOI: https://doi.org/10.1007/BF02514213