Abstract: | Formulas are given for the Lebesgue measure and the Hausdorff–Besicovitch dimension of the minimal closed set Sξ supporting the distribution of the random variable ξ = 2–k τk, where τk are independent random variables taking the values 0, 1, 2 with probabilities p 0k , p 1k , p 2k , respectively. A classification of the distributions of the r.v. ξ via the metric‐topological properties of Sξ is given. Necessary and sufficient conditions for superfractality and anomalous fractality of Sξ 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 Sξ 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 ξi = , where ηk are independent random variables taking the values 0 and 1. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) |