Jessen–Wintner type random variables and fractal properties of their distributions
✍ Scribed by Sergio Albeverio; Yana Gontcharenko; Mykola Pratsiovytyi; Grygoriy Torbin
- Publisher
- John Wiley and Sons
- Year
- 2006
- Tongue
- English
- Weight
- 201 KB
- Volume
- 279
- Category
- Article
- ISSN
- 0025-584X
No coin nor oath required. For personal study only.
✦ Synopsis
Abstract
Formulas are given for the Lebesgue measure and the Hausdorff–Besicovitch dimension of the minimal closed set S~ξ~ supporting the distribution of the random variable ξ = $ \sum ^\infty _{k=1} $ 2^–k^ τ~k~, where τ~k~ are independent random variables taking the values 0, 1, 2 with probabilities p ~0__k__~ , p ~1__k__~ , p ~2__k__~ , respectively. A classification of the distributions of the r.v. ξ via the metric‐topological properties of S~ξ~ is given. Necessary and sufficient conditions for superfractality and anomalous fractality of S~ξ~ are found. It is also proven that for any real number a ~0~ ∈ [0, 1] there exists a distribution of the r.v. ξ such that the Hausdorff–Besicovitch dimension of S~ξ~ is equal to a ~0~. The results are applied to the study of the metric‐topological properties of the convolutions of random variables with independent binary digits, i.e., random variables ξ^i^ = $ \sum ^\infty _{k=1} , 2^{-k} \eta^i_k $, where η~k~ are independent random variables taking the values 0 and 1. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)