A probabilistic approach to studies of DP-transformations and faithfullness of covering systems to evaluate the Hausdorff–Besicovitch dimension
Authors:
M. H. Ibragim and G. M. Torbin
Translated by:
S. Kvasko
Journal:
Theor. Probability and Math. Statist. 92 (2016), 23-36
MSC (2010):
Primary 60G30, 11K55, 28A80
DOI:
https://doi.org/10.1090/tpms/980
Published electronically:
August 10, 2016
MathSciNet review:
3553424
Full-text PDF Free Access
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Additional Information
Abstract: This paper is devoted to the development of a probabilistic approach to transformations preserving the Hausdorff-Besicovitch dimension. New relations between fractal faithfulness of fine covering systems and DP-properties of related probability distribution functions are found. Necessary and sufficient conditions for the probability distribution functions of random variables with independent $Q^*$-symbols to be DP-functions are obtained.
References
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References
- S. Albeverio, Yu. Kondratiev, R. Nikiforov, and G. Torbin, On fractal properties of non-normal numbers with respect to Rényi $f$-expansions generated by piecewise linear functions, Bull. Sci. Math. 138 (2014), no. 3, 440–455. MR 3206478
- S. Albeverio and G. Torbin, Fractal properties of singular continuous probability distributions with independent $Q^*$-digits, Bull. Sci. Math. 129 (2005), no. 4, 356–367. MR 2134126
- 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, Methods Funct. Anal. Topology 17 (2011), no. 2, 97–111. MR 2849470
- S. Albeverio, M. Pratsiovytyi, and G. Torbin, Fractal probability distributions and transformations preserving the Hausdorff–Besicovitch dimension, Ergodic Theory Dynamic. Systems 24 (2004), no. 1, 1–16. MR 2041258
- S. Albeverio, M. Pratsiovytyi, and G. Torbin, Transformations preserving the Hausddorff–Besicovitch dimension, Central European J. Math. 6 (2008), no. 1, 15–24. MR 2379954
- P. Billingsley, Hausdorff dimension in probability theory. II, Ill. J. Math. 5 (1961), 291–198. MR 0120339
- C. Cutler, A note on equivalent interval covering systems for Hausdorff dimension on $R$, Internat. J. Math. Math. Sci. 2 (1988), no. 4, 643–650. MR 959443
- K. J. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley and Sons, New York, 2003. MR 2118797
- J. R. Kinney and T. S. Pitcher, On dimension of some sets defined in terms of $f$-expansions, Z. Wahrsch. Verw. Geb. 4 (1966), 293–315. MR 0198515
- S. Kullback and R. A. Leibler, On information and sufficiency, Ann. Math. Statist. 22 (1951), 79–86. MR 0039968
- R. O. Nikiforov and G. M. Torbin, Fractal properties of random variables with independent $Q_\infty$-symbols, Teor. Ĭmovir. Mat. Stat. 86 (2012), 150–162; English transl. in Theor. Probability and Math. Statist. 86 (2013), 169–182. MR 2986457
- N. V. Pratsevytyĭ and G. M. Torbin, Analytic (symbol) representation of continuous transformations of $R^1$ that preserve the Hausdorff–Besicovitch dimension, Nauk. Zap. Dragomanov Univ. Fiz. Mat. Nauky 4 (2003), 207–215. (Ukrainian)
- A. N. Shiryaev, Probability, “Nauka”, Moscow, 1989; English transl., Springer-Verlag, New York, 1996. MR 1024077
- G. Torbin, Probability distributions with independent $Q$-symbols and transformations preserving the Hausdorff dimension, Theory Stoch. Process. 13 (2007), no. 29, 281–293. MR 2343830
- G. M. Torbin, Multifractal analysis of singularly continuous probability measures, Ukr. Matem. Zh. 57 (2005), no. 5, 837–857; English transl. in Ukrain. Math. J. 57 (2005), no. 5, 837–857. MR 2209816
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Additional Information
M. H. Ibragim
Affiliation:
Department of Mathematical Analysis and Differential Equations, Naional Pedagogic Dragomanov University, Pyrogov Street, 9, Kyiv 01130, Ukraine
Email:
ibragimmuslem1978@gmail.com
G. M. Torbin
Affiliation:
Department of Mathematical Analysis and Differential Equations, Naional Pedagogic Dragomanov University, Pyrogov Street, 9, Kyiv 01130, Ukraine — and — Department of Fractal Analysis, Institute of Mathematics, National Academy of Science of Ukraine, Tereshchenkivs’ka Street, 3, Kyiv 01130, Ukraine
Email:
torbin7@gmail.com
Keywords:
Singularly continuous probability distributions,
$Q^*$-representations,
DP-transformations,
faithful covering systems,
Hausdorff–Besicovitch dimension of sets,
Hausdorff dimension of probability measures
Received by editor(s):
May 15, 2015
Published electronically:
August 10, 2016
Additional Notes:
The first author was supported by the project “Multilevel analysis of singular probability measures and its applications” (Ministry of Education and Science of Ukraine)
The second author was supported by the projects STREVCOMS and “Multilevel analysis of singular probability measures and its applications” (Ministry of Education and Science of Ukraine) and the Alexander von Humboldt Foundation
Dedicated:
This paper is dedicated to the 90$^{\text {\,th}}$ anniversary of Academician Volodymyr Semenovych Korolyuk
Article copyright:
© Copyright 2016
American Mathematical Society
