$\widetilde{Q}$-representation of real numbers and fractal probability distributions

Sergio Albeverio; Volodymyr Koshmanenko; Mykola Pratsiovytyi; Grygoriy Torbin

$\widetilde{Q}$-representation of real numbers and fractal probability distributions

Probability 2007-06-13 v2 Number Theory

Authors: Sergio Albeverio , Volodymyr Koshmanenko , Mykola Pratsiovytyi , Grygoriy Torbin

Abstract

A $\widetilde{Q}-$ representation of real numbers is introduced as a generalization of the $p-$ adic and $Q-$ representations. It is shown that the $\widetilde{Q}-$ representation may be used as a convenient tool for the construction and study of fractals and sets with complicated local structure. Distributions of random variables $\xi$ with independent $\widetilde{Q}-$ symbols are studied in details. Necessary and sufficient conditions for the probability measures $\mu_\xi$ associated with $\xi$ to be either absolutely continuous or singular (resp. pure continuous, or pure point) are found in terms of the $\widetilde{Q}-$ representation. In addition the metric-topological properties for the distribution of $\xi$ are investigated. A number of examples are presented.

Keywords

q-representation real numbers fractals probability distributions random variables metric-topological properties singular measures

Cite

@article{arxiv.math/0308007,
  title  = {$\widetilde{Q}$-representation of real numbers and fractal probability distributions},
  author = {Sergio Albeverio and Volodymyr Koshmanenko and Mykola Pratsiovytyi and Grygoriy Torbin},
  journal= {arXiv preprint arXiv:math/0308007},
  year   = {2007}
}

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13 pages

$\widetilde{Q}$-representation of real numbers and fractal probability distributions

Abstract

Keywords

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