Jessen–Wintner type random variables and fractal properties of their distributions

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)