Theory of Probability and Mathematical Statistics

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

 

 

Singularity of the distribution of a random variable represented by an $ A_2$-continued fraction with independent elements


Authors: M. V. Prats’ovytyĭ and D. V. Kyurchev
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 81 (2010).
Journal: Theor. Probability and Math. Statist. 81 (2010), 159-175
MSC (2010): Primary 11K55; Secondary 11K50, 60E05, 26A30, 28A80
Published electronically: January 20, 2011
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Abstract | References | Similar Articles | Additional Information

Abstract: We study the properties of the distribution of the random variable

$\displaystyle \xi=\frac{1}{\eta_1+\frac{1}{\eta_2+\cdots}}, $

where $ \eta_k$ are independent random variables such that $ \mathsf{P}\{\eta_k=\alpha_1\}=p_{\alpha_1k}\geq 0$, $ \mathsf{P}\{\eta_k=\alpha_2\}=p_{\alpha_2k}\geq 0$, $ 0<\alpha_1<\alpha_2$, $ \alpha_1\alpha_2\geq\frac{1}{2}$, $ p_{\alpha_1k}+p_{\alpha_2k}=1$. It is proved that the distribution of $ \xi$ cannot be absolutely continuous. We find the criteria for the distribution of $ \xi$ to belong to one of the two types of singular distributions, Cantor and Salem types, depending on topological and metric properties of the topological support of the distribution.


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  • 1. Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0192027
  • 2. Ya. F. Vinnishin, Random continued fractions defined by independent elements and their distribution functions, Proceedings of the National Dragomanov Pedagogical University, Physics and Mathematics, 2 (2001), 319-326. (Ukrainian)
  • 3. S. O. Dmitrenko, D. V. Kyurchev, and M. V. Prats′ovitiĭ, 𝐴₂-continued fraction representation of real numbers and its geometry, Ukraïn. Mat. Zh. 61 (2009), no. 4, 452–463 (Ukrainian, with English and Russian summaries); English transl., Ukrainian Math. J. 61 (2009), no. 4, 541–555. MR 2588672, 10.1007/s11253-009-0236-7
  • 4. D. V. Kyurchev, On the Hausdorff-Besicovitch dimension of some sets of continued fractions, Proceedings of the National Dragomanov Pedagogical University. Ser. 1, Physics and Mathematics 4 (2004), 285-291. (Ukrainian)
  • 5. O. L. Leschinskiĭ and M. V. Prats'ovytyĭ, A certain class of singular distributions of random variables represented by elementary continued fractions with independent elements, Current Researches of Young Scientists of Universities of Ukraine in Physics and Mathematics, National Taras Shevchenko University, Kyiv, 1995, pp. 20-30. (Ukrainian)
  • 6. M. V. Prats′ovitiĭ, Singularity of distributions of random variables defined by distributions of elements of the corresponding continued fraction, Ukraïn. Mat. Zh. 48 (1996), no. 8, 1086–1095 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J. 48 (1996), no. 8, 1229–1240 (1997). MR 1429595, 10.1007/BF02383869
  • 7. M. V. Prats'ovytyĭ, Fractal Approach to the Studies of Singular Distributions, National Dragomanov Pedagogical University, Kyiv, 1998. (Ukrainian)
  • 8. A. Ya. Khinchin, Continued fractions, The University of Chicago Press, Chicago, Ill.-London, 1964. MR 0161833
  • 9. Alok Goswami, Random continued fractions: a Markov chain approach, Econom. Theory 23 (2004), no. 1, 85–105. Symposium on Dynamical Systems Subject to Random Shock. MR 2032898, 10.1007/s00199-002-0333-4
  • 10. Russell Lyons, Singularity of some random continued fractions, J. Theoret. Probab. 13 (2000), no. 2, 535–545. MR 1778585, 10.1023/A:1007837306508
  • 11. M. Pratsiovytyi and D. Kyurchev, Properties of the distribution of the random variable defined by 𝐴₂-continued fraction with independent elements, Random Oper. Stoch. Equ. 17 (2009), no. 1, 91–101. MR 2519460, 10.1515/ROSE.2009.006

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Additional Information

M. V. Prats’ovytyĭ
Affiliation: Department of Higher Mathematics, Institute for Physics and Mathematics, National Dragomanov Pedagogical University, Pirogova Street 9, Kyiv 01030, Ukraine
Email: prats4@yandex.ru

D. V. Kyurchev
Affiliation: Department of Fractal Analysis, Institute for Physics and Mathematics, National Dragomanov Pedagogical University, Pirogova Street 9, Kyiv 01030, Ukraine
Email: d_kyurchev@ukr.net

DOI: http://dx.doi.org/10.1090/S0094-9000-2011-00817-4
Keywords: Random continued fraction, $A_{2}$-continued fraction, Cantor type singular distribution, Salem type singular distribution
Received by editor(s): July 14, 2009
Published electronically: January 20, 2011
Additional Notes: The first author is supported by DFG 436 UKR Projects #113/80 and #113/97
The second author is supported by DFG 436 UKR Project #113/80
Article copyright: © Copyright 2011 American Mathematical Society