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Theory of Probability and Mathematical Statistics
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: 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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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
PII: S 0094-9000(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