Abstract
By using the method of characteristic functions, we obtain sufficient conditions for the singularity of a random variable.
where ξ k are independent identically distributed random variables taking valuesx 0,x 1, andx 2 (x 0<x 1<x 2) with probabilitiesp 0,p 1 andp 2, respectively, such thatp i≥0,p 0+p 1+p 2 =1 and 2(x 1−x 0)/(x 2−x 0) is a rational number.
References
A. F. Turbin and N. V. Pratsevityi,Fractal Sets, Functions, and Distributions [in Russian], Naukova Dumka, Kiev (1992).
Ya. F. Vinnishin and V. A. Moroka, “On the type of the distribution function of a random power series,” in:Asymptotic Analysis of Random Evolutions [in Ukrainian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1994), pp. 65–73.
N. V. Pratsevityi, “Distribution of sums of random power series,”Dop. Akad. Nauk Ukr., No. 5, 32–37 (1996)
N. V. Pratsevityi, “Fractal, superfractal, and anomalously fractal distribution of random variables with independentn-adic digits an infinite set of which is fixed,” in: A. V. Skorokhod and Yu. V. Borovskikh (editors),Exploring Stochastic Laws, Festschrift in Honor of the 70th Birthday of Academician V. S. Korolyuk, VSP (1995), pp. 409–416.
E. Lukacs,Characteristic Functions, Criffin, London (1970).
W. Feller,An Introduction to Probability Theory and Its Applications [Russian translation], Vol. 2, Mir, Moscow (1984).
Additional information
Pedagogic University, Kiev. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 51, No. 1, pp. 128–132, January, 1999.
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Litvinyuk, A.A. On types of distributions of sums of one class of random power series with independent identically distributed coefficients. Ukr Math J 51, 140–145 (1999). https://doi.org/10.1007/BF02591923
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DOI: https://doi.org/10.1007/BF02591923