Theory of Probability and Mathematical Statistics
The distributions of random incomplete sums of a series with positive terms satisfying the property of non-linear homogeneity
M. V. Prats’ovytyĭ, I. O. Savchenko
Download PDF
Abstract: The Lebesgue type as well as topological, metric, and fractal properties of the spectrum of the distribution of the random variable ξ=Σ_{n=1}^{\infty}a_nξ_n are studied, where \sum_{n=1}^{\infty}a_n=a_1+a_2+...+a_n+r_n is a convergent series with positive terms such that r_{n+1}=a_{n+1}a_{n} for any n\in N and (ξ_n) is a sequence of independent random variables taking only two values, 0 and 1, with probabilities p_{0n} and p_{1n}, respectively. We describe the point spectrum in the discrete case, and we prove that the distribution of ξ is of a Cantor singular type with an anomalous fractal spectrum in the continuous case. We also prove that the n-fold convolution of the random variable ξ with itself has an anomalous fractal distribution.
Keywords: Bernoulli convolution, singularly continuous probability distribution, the set of incomplete sums of a series, Hausdorff--Besicovitch dimension of the spectrum of a probability distribution
Bibliography: 1. S. Albeverio, O. Baranovskyi, M. Prats'ovytyĭ, and G. Torbin, The set of incomplete sums of the first Ostrogradsky series and anomalously fractal probability distributions on it, Rev. Roum. Math. Pures. Appl. 54 (2009), no. 2, 85-115.
2. S. Albeverio and G. Torbin, On fine fractal properties of generalized infinite Bernoulli convolutions, Bull. Sci. Math. 132 (2008), no. 8, 711-727.
3. B. Jessen and A. Wintner, Distribution function and the Riemann Zeta-function, Trans. Amer. Math. Soc. 38 (1935), no. 1, 48-88.
4. S. Kakeya, On the partial sums of an infinite series, Tôhoku Sci. Rep. 3 (1915), no. 4, 159-164.
5. P. Lévy, Sur les séries dont les termes sont des variables éventuelles indépendantes, Studia Math. 3 (1931), 119-155.
6. Y. Peres, W. Schlag, and B. Solomyak, Sixty year of Bernoulli convolutions, Fractal Geometry and Stochastics II. Progress in Probability 46 (2000), 39-65.
7. M. V. Prats'ovytyĭ and O. Y. Feshchenko, Topological, metric and fractal properties of probability distributions on the set of incomplete sums of positive series, Theory Stoch. Process. 13(29) (2007), no. 1-2, 205-224.
8. B. Solomyak, On the random series Σ±λ^n (an Erdős problem), Ann. Math. 142 (1995), 611-625.
9. O. M. Baranovs'kyĭ, M. V. Pratsyovytyĭ, and G. M. Torbin, Ostrograds'kyĭ-Sierpiński-Pierce Series and their Applications, Naukove dumka'', Kyiv, 2013. (Ukrainian)
10. Ya. V. Goncharenko, Convolutions of distributions of sums of random series of a special type, Naukovi Zap. Nat. Pedagogical Dragomanov Univ. Ser. Phys. Mat. (2003), no. 4, 216-232. (Ukrainian)
11. Ya. V. Goncharenko, M. V. Prats'ovytyĭ, and G. M. Torbin, Topological, metric and fractal properties of the set of incomplete sums of a positive series and distributions defined on it, Naukovi Zap. Nat. Pedagogical Dragomanov Univ. Ser. Phys. Mat. (2005), no. 6, 210-224. (Ukrainian)
12. Ya. V. Goncharenko, M. V. Prats'ovytyĭ, and G. M. Torbin, Fractal properties of some Bernoulli convolutions, Teor. Imovirnost. Mat. Statist. 79 (2008), 34-49; English transl. in Theor. Probability and Math. Statist. 79 (2009), 39-55.
13. V. M. Zolotarev and V. M. Kruglov, The structure of infinitely divisible distributions on a bicompact Abelian group, Teor. Veroyatnost. Primenen. 20 (1975), no. 4, 712-724; English transl. in Theory Probab. Appl. 20 (1976), no. 4, 698-709.
14. M. V. Prats'ovytyĭ, Fractal approach in studies of singular distributions, National Pedagogical Dragomanov University Publishing House, Kyiv, 1998. (Ukrainian)
15. M. V. Prats'ovytyĭ and G. M. Torbin, A class of random variables of the Jessen-Wintner type, Dop. Nat. Acad. Sci. Ukraine (1998), no. 4, 48-54. (Ukrainian)
16. A. F. Turbin and M. V. Prats'ovytyĭ, Fractal Sets, Functions, Distributions, ''Naukova Dumka'', Kyiv, 1992. (Ukrainian)
