Singularity of the distribution of a random variable represented by an 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), 159175
MSC (2010):
Primary 11K55; Secondary 11K50, 60E05, 26A30, 28A80
Published electronically:
January 20, 2011
Fulltext PDF
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Additional Information
Abstract: We study the properties of the distribution of the random variable where are independent random variables such that , , , , . It is proved that the distribution of cannot be absolutely continuous. We find the criteria for the distribution of 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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 P. Billingsley, Ergodic Theory and Information, John Wiley and Sons, Inc., New York, 1965. MR 0192027 (33:254)
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 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), 319326. (Ukrainian)
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 M. Pratsiovytyi and D. Kyurchev, Properties of the distribution of the random variable defined by continued fraction with independent elements, Random Oper. Stoch. Equ. 17 (2009), no. 1, 91101. MR 2519460 (2010d:60032)
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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/S009490002011008174
PII:
S 00949000(2011)008174
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
