The talk is devoted to the study of fractal properties of spectra and minimal dimensional supports of probability distributions of random variables with independent digits of geometrical Cantor series expansions.

In particular, we prove faithfulness of the corresponding family of cylinders and show that this family is extremely non-comparable. Based on these results we study fine fractal properties of the related probability measures and prove the formulae for the calculation of the Hausdorff dimension of the probability distributions of random variables with independent digits of geometrical Cantor series expansions.

Fractal properties of the spectra of such measures are also studied in detail.