Topological metric and fractal properties of the convolution of two singular distributions with independent binary digits. (Ukrainian, English) Zbl 1051.60013

Goncharenko, Ya. V.; Prats’ovytyj, M. V.; Torbin, G. M.

Teor. Jmovirn. Mat. Stat. 67, 9-19 (2002); translation in Theory Probab. Math. Stat. 67, 11-22 (2003).

The paper starts with a brief survey of the metric-topological and fractal properties of random variables \(\eta\) with independent binary digits, i.e. \(\eta=\sum_{k=1}^{\infty} 2^k \xi_k\), where \(\{\xi_k\}\) are i.i.d. Bernoulli random variables. Then the fractal and topological-metric properties of the convolution distribution of the sum \(\xi^*=\eta_1+\eta_2\) of two independent binary digits are investigated. The obtained results are extended to the case of the random variables \(\xi=\sum_{k=1}^{\infty} 2^k \tau_k\), where \(\{\tau_k\}\) are i.i.d. random variables which can take three values \(0,1,2\) with probabilities \(p_{01},p_{02},p_{03}\), respectively. In particular, conditions of absolute continuity and singularity of the distribution of \(\xi\) are derived.

Reviewer: N. M. Zinchenko (Kyïv)

MSC:

60E05	Probability distributions: general theory
28A80	Fractals

Keywords:

random variables with binary digits; metric-topological and fractal properties; convolution; Hausdorff-Basicovich dimension; singularity of distribution