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), 159-175

MSC (2010):
Primary 11K55; Secondary 11K50, 60E05, 26A30, 28A80

DOI:
https://doi.org/10.1090/S0094-9000-2011-00817-4

Published electronically:
January 20, 2011

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

**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**, https://doi.org/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**, https://doi.org/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**, https://doi.org/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**, https://doi.org/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**, https://doi.org/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:
https://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