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Theory of Probability and Mathematical Statistics

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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
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 92 (2015).
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
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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.


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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

DOI: https://doi.org/10.1090/tpms/980
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$^{th}$ anniversary of Academician Volodymyr Semenovych Korolyuk
Article copyright: © Copyright 2016 American Mathematical Society

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