Ibrahim M. Fractal properties of random variables with independent Q -symbols and its applications

The thesis is devoted to the development of methods for fractal anal-ysis of singularly continuous probability measures, to the study of ne fractal properties of distributions of random variables with independent Q -symbols and to the application of the results for probabilistic approach to transformations preserving the Hausdor -Besicovitch dimension, and to the fractal analysis of subsets of the set of non-normal numbers. To study ne fractal properties of distributions of random variables with independent Q -symbols and to develop methods for the calculation of the Hausdor -Besicovitch dimension, we introduce new approaches to the proof of faithfulness (for the calculation of the Hausdor -Besicovitch dimmension) of ne coverings generated by polybasic Q -expansion of real numbers. Results on fractal properties of minimal dimensional supports of random variables with independent Q -symbols and exact formulae for the calculation of the Hausdor dimmension of the corresponding prob-ability measures are also obtained. Based on these results in the last chapter of the thesis we developed probabilistic approach to transforma-tions preserving the Hausdor -Besicovitch dimension and found necessary and su cient conditions for the probability distribution functions of ran-dom variables with independent Q -symbols to be DP-functions, as well as to prove the conjecture on superfractality of the set of Q -essentially non-normal numbers