Fractal properties of random variables with independent -symbols

Authors:
R. O. Nikiforov and G. M. Torbin

Translated by:
S. V. Kvasko

Original publication:
Teoriya Imovirnostei ta Matematichna Statistika, tom **86** (2012).

Journal:
Theor. Probability and Math. Statist. **86** (2013), 169-182

MSC (2010):
Primary 60G30, 11K55, 28A80

DOI:
https://doi.org/10.1090/S0094-9000-2013-00896-5

Published electronically:
August 20, 2013

MathSciNet review:
2986457

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We study an equivalent definition of the Hausdorff-Besicovitch dimension in terms of a system of cylinders of the -expansion. Sufficient conditions for the system to be faithful for the evaluation of the Hausdorff-Besicovitch dimension in the unit interval are found; fine fractal properties of probability measures with independent -digits are investigated (we do not assume that the -digits are identically distributed).

**1.**Sergio Albeverio and Grygoriy Torbin,*Fractal properties of singular probability distributions with independent 𝑄*-digits*, Bull. Sci. Math.**129**(2005), no. 4, 356–367 (English, with English and French summaries). MR**2134126**, https://doi.org/10.1016/j.bulsci.2004.12.001**2.**S. Albeverio, Yu. Kondratiev, R. Nikiforov, and G. Torbin,*On fractal phenomena connected with infinite linear IFS and related singular probability measures*, J. London Math. Soc. (submitted)**3.**S. Albeverio, V. Koshmanenko, M. Pratsiovytyi, and G. Torbin,*On fine structure of singularly continuous probability measures and random variables with independent 𝑄-symbols*, Methods Funct. Anal. Topology**17**(2011), no. 2, 97–111. MR**2849470****4.**Sergio Albeverio, Vyacheslav Koval, Mykola Pratsiovytyi, and G. Torbin,*On classification of singular measures and fractal properties of quasi-self-affine measures in 𝑅²*, Random Oper. Stoch. Equ.**16**(2008), no. 2, 181–211. MR**2446437**, https://doi.org/10.1515/ROSE.2008.010**5.**A. S. Besicovitch,*On existence of subsets of finite measure of sets of infinite measure*, Nederl. Akad. Wetensch. Proc. Ser. A. 55 = Indagationes Math.**14**(1952), 339–344. MR**0048540****6.**M. P. Bernardi and C. Bondioli,*On some dimension problems for self-affine fractals*, Z. Anal. Anwendungen**18**(1999), no. 3, 733–751. MR**1718162**, https://doi.org/10.4171/ZAA/909**7.**Patrick Billingsley,*Ergodic theory and information*, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR**0192027****8.**Patrick Billingsley,*Hausdorff dimension in probability theory. II*, Illinois J. Math.**5**(1961), 291–298. MR**0120339****9.**B. H. Bissinger,*A generalization of continued fractions*, Bull. Amer. Math. Soc.**50**(1944), 868–876. MR**11338**, https://doi.org/10.1090/S0002-9904-1944-08254-2**10.**S. D. Chatterji,*Certain induced measures and the fractional dimensions of their “supports”*, Z. Wahrscheinlichkeitstheorie und Verw. Gebiete**3**(1964), 184–192 (1964). MR**174691**, https://doi.org/10.1007/BF00534907**11.**C. D. Cutler,*A note on equivalent interval covering systems for Hausdorff dimension on 𝑅*, Internat. J. Math. Math. Sci.**11**(1988), no. 4, 643–649. MR**959443**, https://doi.org/10.1155/S016117128800078X**12.**C. J. Everett,*Representations for real numbers*, Bull. Amer. Math. Soc.**52**(1946), 861–869. MR**18221**, https://doi.org/10.1090/S0002-9904-1946-08659-0**13.**Kenneth Falconer,*Fractal geometry*, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR**1102677****14.**John E. Hutchinson,*Fractals and self-similarity*, Indiana Univ. Math. J.**30**(1981), no. 5, 713–747. MR**625600**, https://doi.org/10.1512/iumj.1981.30.30055**15.**S. Kullback and R. A. Leibler,*On information and sufficiency*, Ann. Math. Statistics**22**(1951), 79–86. MR**39968**, https://doi.org/10.1214/aoms/1177729694**16.**R. O. Nikiforov and G. M. Torbin,*Ergodic properties of -expansions and fractal properties of probability measures with independent -symbols*, Proceedings of the National Dragomanov Pedagogical University, Physics and Mathematics**9**(2008), 80-103. (Ukrainian)**17.**Yu. Peres and G. Torbin,*Continued fractions and dimensional gaps*. (in preparation)**18.**M. V. Pratsevytyĭ,*Fractal Approach in Studies of Singular Distributions*, Dragomanov National Pedagogical University Press, Kyiv, 1998. (Ukrainian)**19.**M. V. Pratsevytyĭ and O. L. Leshchins'kyĭ,*Properties of random variables defined in terms of the distributions of elements of their -representation*, Teor. Imovir. Matem. Statyst.**57**(1997), 134-140; English transl. in Theory Probab. Math. Statist.**57**(1998), 143-148.**20.**M. V. Pratsevytyĭ and G. M. Torbin,*On a classification of one dimensional singularly continuous probability measures with respect to their spectral properties*, Proceedings of the National Dragomanov Pedagogical University, Physics and Mathematics**7**(2006), 140-151. (Ukrainian)**21.**M. V. Pratsevytyĭ, G. M. Torbin,*An analytic (symbol) representation of continuous transformations of that preserve the Hausdorff-Besicovitch dimension*, Proceedings of the National Dragomanov Pedagogical University, Physics and Mathematics**4**(2003), 207-215. (Ukrainian)**22.**A. Rényi,*Representations for real numbers and their ergodic properties*, Acta Math. Acad. Sci. Hungar.**8**(1957), 477–493. MR**97374**, https://doi.org/10.1007/BF02020331**23.**G. M. Torbīn,*Multifractal analysis of singularly continuous probability measures*, Ukraïn. Mat. Zh.**57**(2005), no. 5, 706–721 (Ukrainian, with English and Ukrainian summaries); English transl., Ukrainian Math. J.**57**(2005), no. 5, 837–857. MR**2209816**, https://doi.org/10.1007/s11253-005-0233-4**24.**A. F. Turbin and N. V. Pratsevityĭ,*Fraktal′nye mnozhestva, funktsii, raspredeleniya*, “Naukova Dumka”, Kiev, 1992 (Russian, with Russian and Ukrainian summaries). MR**1353239**

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Additional Information

**R. O. Nikiforov**

Affiliation:
Department of Higher Mathematics, Dragomanov National Pedagogical University, Pirogova Street 9, Kyiv 01130, Ukraine

Email:
rnikiforov@gmail.com

**G. M. Torbin**

Affiliation:
Department of Mathematical Analysis and Differential Equations, Dragomanov National Pedagogical University, Pirogova Street 9, Kyiv 01130, Ukraine – and – Department of Fractal Analysis, Institute for Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street 3, Kyiv 01130, Ukraine

Email:
torbin7@gmail.com, torbin@imath.kiev.ua

DOI:
https://doi.org/10.1090/S0094-9000-2013-00896-5

Keywords:
$Q_{\infty}$-expansions,
faithful systems of coverings,
singularly continuous probability distributions,
Hausdorff--Besicovitch dimension of a set,
Hausdorff dimension of a measure

Received by editor(s):
November 30, 2011

Published electronically:
August 20, 2013

Additional Notes:
The first author was supported by the Project DFG 436113/97

The second author was supported by the Projects DFG 436 UKR 113/97 and DFG KO 1989/6-1 and the Alexander von Humboldt Foundation

Article copyright:
© Copyright 2013
American Mathematical Society