### PROBABILITY DISTRIBUTIONS WITH INDEPENDENT Q-SYMBOLS AND TRANSFORMATIONS PRESERVING THE HAUSDORFF DIMENSION

**GRYGORIY TORBIN**

*Theory of Stochastic Processes*

*Vol.13 (29), no.1-2, 2007, pp.281-293*

The paper is devoted to the study of connections between fractal properties of one-dimensional singularly continuous probability measures and the preservation of the Hausdorff dimension of any subset of the unit interval under the corresponding distribution function. Conditions for the distribution function of a random variable with independent $Q$-digits to be a transformation preserving the Hausdorff dimension (DP-transformation) are studied in details. It is shown that for a large class of probability measures the distribution function is a DP-transformation if and only if the corresponding probability measure is of full Hausdorff dimension.

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