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A random variable whose digits in the $ \widetilde{L}$-representation have the Markovian dependence

Authors: M. V. Prats’ovytyĭ and Yu. V. Khvorostina
Translated by: S. Kvasko
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 91 (2014).
Journal: Theor. Probability and Math. Statist. 91 (2015), 157-168
MSC (2010): Primary 60E05
Published electronically: February 4, 2016
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Abstract | References | Similar Articles | Additional Information

Abstract: The distribution of the random variable

$\displaystyle \theta =\frac {1}{\theta _1}+\sum ^{\infty }_{n=2}\frac {(-1)^{n-1}} {\theta _1(\theta _{1}+1)\dots \theta _{n-1}(\theta _{n-1}+1)\theta _{n}}$

is studied where $ (\theta _n)$ is a homogeneous Markov chain assuming only positive integer values and having the initial distribution $ (p_1, p_2,\dots , p_n,\dots )$ and transition matrix  $ \Vert p_{ik}\Vert$. The Lebesgue structure of the distribution (discrete, absolutely continuous, and singular components) is studied and topological, metric and fractal properties of the spectrum (the minimal closed support of the distribution) is investigated.

References [Enhancements On Off] (What's this?)

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Additional Information

M. V. Prats’ovytyĭ
Affiliation: Department of Higher Mathematics, Institute for Physics and Mathematics, National Pedagogical Dragomanov University, Pirogov Street, 9, Kyiv, 01601, Ukraine

Yu. V. Khvorostina
Affiliation: Department for Physics and Mathematics #1, Sumy State Pedagogical Makarenko University, Romens’ka Street, 87, Sumy, 40002, Ukraine

Keywords: Alternating L\"uroth series; $\widetilde{L}$-representation; random variable; distribution of the sum of a L\"uroth series whose terms are random variables with the Markov dependence; Lebesgue structure of distributions; singular distribution with an anomalous fractal spectrum
Received by editor(s): September 21, 2014
Published electronically: February 4, 2016
Article copyright: © Copyright 2016 American Mathematical Society