Fractal properties of random variables with independent $Q_{\infty }$-symbols
Authors:
R. O. Nikiforov and G. M. Torbin
Translated by:
S. V. Kvasko
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
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Abstract: We study an equivalent definition of the Hausdorff-Besicovitch dimension in terms of a system $\Phi (Q_\infty )$ of cylinders of the $Q_\infty$-expansion. Sufficient conditions for the system $\Phi (Q_\infty )$ 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 $Q_\infty$-digits are investigated (we do not assume that the $Q_\infty$-digits are identically distributed).
References
- 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, DOI https://doi.org/10.1016/j.bulsci.2004.12.001
- 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)
- 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
- Sergio Albeverio, Vyacheslav Koval, Mykola Pratsiovytyi, and G. Torbin, On classification of singular measures and fractal properties of quasi-self-affine measures in ${\bf R}^2$, Random Oper. Stoch. Equ. 16 (2008), no. 2, 181–211. MR 2446437, DOI https://doi.org/10.1515/ROSE.2008.010
- 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
- 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, DOI https://doi.org/10.4171/ZAA/909
- Patrick Billingsley, Ergodic theory and information, John Wiley & Sons, Inc., New York-London-Sydney, 1965. MR 0192027
- Patrick Billingsley, Hausdorff dimension in probability theory. II, Illinois J. Math. 5 (1961), 291–298. MR 120339
- B. H. Bissinger, A generalization of continued fractions, Bull. Amer. Math. Soc. 50 (1944), 868–876. MR 11338, DOI https://doi.org/10.1090/S0002-9904-1944-08254-2
- 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, DOI https://doi.org/10.1007/BF00534907
- C. D. Cutler, A note on equivalent interval covering systems for Hausdorff dimension on ${\bf R}$, Internat. J. Math. Math. Sci. 11 (1988), no. 4, 643–649. MR 959443, DOI https://doi.org/10.1155/S016117128800078X
- C. J. Everett, Representations for real numbers, Bull. Amer. Math. Soc. 52 (1946), 861–869. MR 18221, DOI https://doi.org/10.1090/S0002-9904-1946-08659-0
- Kenneth Falconer, Fractal geometry, John Wiley & Sons, Ltd., Chichester, 1990. Mathematical foundations and applications. MR 1102677
- John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI https://doi.org/10.1512/iumj.1981.30.30055
- S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statistics 22 (1951), 79–86. MR 39968, DOI https://doi.org/10.1214/aoms/1177729694
- R. O. Nikiforov and G. M. Torbin, Ergodic properties of $Q_{\infty }$-expansions and fractal properties of probability measures with independent $Q_\infty$-symbols, Proceedings of the National Dragomanov Pedagogical University, Physics and Mathematics 9 (2008), 80–103. (Ukrainian)
- Yu. Peres and G. Torbin, Continued fractions and dimensional gaps. (in preparation)
- M. V. Pratsevytyĭ, Fractal Approach in Studies of Singular Distributions, Dragomanov National Pedagogical University Press, Kyiv, 1998. (Ukrainian)
- M. V. Pratsevytyĭ and O. L. Leshchins’kyĭ, Properties of random variables defined in terms of the distributions of elements of their $\tilde Q_{\infty }$-representation, Teor. Imovir. Matem. Statyst. 57 (1997), 134–140; English transl. in Theory Probab. Math. Statist. 57 (1998), 143–148.
- 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)
- M. V. Pratsevytyĭ, G. M. Torbin, An analytic (symbol) representation of continuous transformations of $R^1$ that preserve the Hausdorff–Besicovitch dimension, Proceedings of the National Dragomanov Pedagogical University, Physics and Mathematics 4 (2003), 207–215. (Ukrainian)
- A. Rényi, Representations for real numbers and their ergodic properties, Acta Math. Acad. Sci. Hungar. 8 (1957), 477–493. MR 97374, DOI https://doi.org/10.1007/BF02020331
- 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, DOI https://doi.org/10.1007/s11253-005-0233-4
- A. F. Turbin and N. V. Pratsevityĭ, Fraktal′nye mnozhestva, funktsii, raspredeleniya, “Naukova Dumka”, Kiev, 1992 (Russian, with Russian and Ukrainian summaries). MR 1353239
References
- S. Albeverio and G. Torbin, Fractal properties of singularly continuous probability distributions with independent $Q^{*}$-digits, Bull. Sci. Math. 129 (2005), no. 4, 356–367. MR 2134126 (2006b:28013)
- 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)
- S. Albeverio, V. Koshmanenko, M. Pratsiovytyi, and G. Torbin, On fine structure of singularly continuous probability measures and random variables with independent $\widetilde {Q}$-symbols, Meth. Funct. Anal. Topology 17 (2011), no. 2, 97–111. MR 2849470 (2012g:60006)
- S. Albeverio, V. Koval, M. Pratsiovytyi, and G. Torbin, On classification of singular measures and fractal properties of quasi-self-affine measures in $R^2$, Random Oper. Stoch. Equations 16 (2008), no. 2, 181–211. MR 2446437 (2010b:28008)
- A. Besicovitch, On existence of subsets of finite measure of sets of infinite measure, Indag. Math. 14 (1952), 339–344. MR 0048540 (14:28e)
- M. Bernardi and C. Bondioli, On some dimension problems for self-affine fractals, J. Anal. Appl. 18 (1999), no. 3, 733–751. MR 1718162 (2000j:28004)
- P. Billingsley, Ergodic Theory and Information, John Wiley and Sons, New York, 1965. MR 0192027 (33:254)
- P. Billingsley, Hausdorff dimension in probability theory II, Ill. J. Math. 5 (1961), 291–298. MR 0120339 (22:11094)
- B. H. Bissinger, A generalization of continued fractions, Bull. Amer. Math. Soc. 50 (1944), 868–876. MR 0011338 (6:150h)
- S. D. Chatterji, Certain induced measures and the fractional dimensions of their “supports”, Z. Wahrscheinlichkeitstheorie verw. Geb. 3 (1964), 184–192. MR 0174691 (30:4891)
- C. D. Cutler, A note on equivalent interval covering systems for Hausdorff dimension on $R$, Internat. J. Math. Math. Sci. 4 (1988), 643–650. MR 959443 (89h:28008)
- C. I. Everett, Representations for real numbers, Bull. Amer. Math. Soc. 52 (1946), 861–869. MR 0018221 (8:259c)
- K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, Wiley, Chichester, 1990. MR 1102677 (92j:28008)
- J. E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713–747. MR 625600 (82h:49026)
- S. Kullback and R. A Leibler, On information and sufficiency, Ann. Math. Statist. 22 (1951), 79–86. MR 0039968 (12:623a)
- R. O. Nikiforov and G. M. Torbin, Ergodic properties of $Q_{\infty }$-expansions and fractal properties of probability measures with independent $Q_\infty$-symbols, Proceedings of the National Dragomanov Pedagogical University, Physics and Mathematics 9 (2008), 80–103. (Ukrainian)
- Yu. Peres and G. Torbin, Continued fractions and dimensional gaps. (in preparation)
- M. V. Pratsevytyĭ, Fractal Approach in Studies of Singular Distributions, Dragomanov National Pedagogical University Press, Kyiv, 1998. (Ukrainian)
- M. V. Pratsevytyĭ and O. L. Leshchins’kyĭ, Properties of random variables defined in terms of the distributions of elements of their $\tilde Q_{\infty }$-representation, Teor. Imovir. Matem. Statyst. 57 (1997), 134–140; English transl. in Theory Probab. Math. Statist. 57 (1998), 143–148.
- 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)
- M. V. Pratsevytyĭ, G. M. Torbin, An analytic (symbol) representation of continuous transformations of $R^1$ that preserve the Hausdorff–Besicovitch dimension, Proceedings of the National Dragomanov Pedagogical University, Physics and Mathematics 4 (2003), 207–215. (Ukrainian)
- A. Renyi, Representations for real numbers and their ergodic properties, Acta Math. Sci. Hungar. 8 (1957), 477–493. MR 0097374 (20:3843)
- G. M. Torbin, Multifractal analysis of singularly continuous probability measures, Ukr. Matem. Zh. 57 (2005), no. 5, 706–721; English transl. in Ukr. Math. J. 57 (2005), no. 5, 837–857. MR 2209816 (2007f:28010)
- A. F. Turbin and M. V. Pratsevytyĭ, Fractal sets, functions, distributions, “Naukova Dumka”, Kiev, 1992. (Russian) MR 1353239 (96f:28010)
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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
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