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 | References | Similar Articles | 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 -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