Remote Access Theory of Probability and Mathematical Statistics

Theory of Probability and Mathematical Statistics

ISSN 1547-7363(online) ISSN 0094-9000(print)

Request Permissions   Purchase Content 
 

 

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
DOI: https://doi.org/10.1090/tpms/974
Published electronically: February 4, 2016
Full-text PDF

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?)

  • 1. Sofia Kalpazidou, Arnold Knopfmacher, and John Knopfmacher, Lüroth-type alternating series representations for real numbers, Acta Arith. 55 (1990), no. 4, 311–322. MR 1069185
  • 2. J. Lüroth, Ueber eine eindeutige Entwickelung von Zahlen in eine unendliche Reihe, Math. Ann. 21 (1883), no. 3, 411–423 (German). MR 1510205, https://doi.org/10.1007/BF01443883
  • 3. Mykola Pratsiovytyi and Yuriy Khvorostina, Topological and metric properties of distributions of random variables represented by the alternating Lüroth series with independent elements, Random Oper. Stoch. Equ. 21 (2013), no. 4, 385–401. MR 3139317, https://doi.org/10.1515/rose-2013-0018
  • 4. Yu. I. Zhikhareva and M. V. Prats'ovytyĭ, Properties of the distribution of the random variable whose digits in the alternating Lüroth series form a homogeneous Markov chain, Naukovi Zap. Nat. Pedagogical Dragomanov Univ. Ser. Phys. Mat. (2009), no. 10, 100-107. (Ukrainian)
  • 5. Eugene Lukacs, Characteristic functions, Hafner Publishing Co., New York, 1970. Second edition, revised and enlarged. MR 0346874
  • 6. M. V. Prats'ovytyĭ, Fractal approach in studies of singular distributions, National Pedagogical Dragomanov University Publishing House, Kyiv, 1998. (Ukrainian)
  • 7. M. V. Prats'ovytyĭ and Yu. V. Khvorostina, The essentials of the metric theory of representations of real numbers by alternating Lüroth series and simplest applications, Naukovi Zap. Nat. Pedagogical Dragomanov Univ. Ser. Phys. Mat. (2010), no. 11, 102-118. (Ukrainian)

Similar Articles

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2010): 60E05

Retrieve articles in all journals with MSC (2010): 60E05


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
Email: prats4@yandex.ru

Yu. V. Khvorostina
Affiliation: Department for Physics and Mathematics #1, Sumy State Pedagogical Makarenko University, Romens’ka Street, 87, Sumy, 40002, Ukraine
Email: khvorostina13@mail.ru

DOI: https://doi.org/10.1090/tpms/974
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