We introduce a new fine classification of singularly continuous probability measures on $R^1$ on the basis of spectral properties of such measures (topological and metric properties of the spectrum of the measure as well as local behavior of the measure on subsets of the spectrum). The theorem on the structural representation of any one-dimensional singularly continuous probability measure in the form of a convex combination of three singularly continuous probability measures of pure spectral type is proved.

We introduce into consideration and study a $\widetilde{Q}$-representation of real numbers and a family of probability measures with independent $\widetilde{Q}$-symbols. Topological, metric and fractal properties of the above mentioned probability distributions are studied in details. We also show how the methods of $\widetilde{P}-\widetilde{Q}$-measures can be effectively applied to study properties of generalized infinite Bernoulli convolutions.

Key words: Singularly continuous probability measures, GC-, GP- and GS-types of singular measures; Hausdorff dimension, fractals; $\widetilde{Q}$-representation of real numbers, random variables with independent $\widetilde{Q}$-symbols.

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Title

On fine structure of singularly continuous probability measures and random variables with independent $\widetilde{Q}$-symbols

S. Albeverio, V. Koshmanenko, M. Pratsiovytyi, and G. Torbin, On fine structure of singularly continuous probability measures and random variables with independent $\widetilde{Q}$-symbols, Methods Funct. Anal. Topology 17
(2011), no. 2, 97-111.

BibTex

@article {MFAT570,
AUTHOR = {Albeverio, S. and Koshmanenko, V. and Pratsiovytyi, M. and Torbin, G.},
TITLE = {On fine structure of singularly continuous probability measures and random variables with independent $\widetilde{Q}$-symbols},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {17},
YEAR = {2011},
NUMBER = {2},
PAGES = {97-111},
ISSN = {1029-3531},
URL = {http://mfat.imath.kiev.ua/article/?id=570},
}