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

### Abstract

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.

### Article Information

 Title On fine structure of singularly continuous probability measures and random variables with independent $\widetilde{Q}$-symbols Source Methods Funct. Anal. Topology, Vol. 17 (2011), no. 2, 97-111 MathSciNet MR2849470 Copyright The Author(s) 2011 (CC BY-SA)

### Authors Information

S. Albeverio
Institut fur Angewandte Mathematik, Universitat Bonn, Wegelerstr. 6, D-53115 Bonn, Germany; SFB 611, Bonn, BiBoS, Bielefeld--Bonn; IZKS Bonn; CERFIM, Locarno and Acc. Arch. (USI) Switzerland

V. Koshmanenko
Institute of Mathematics, National Academy of Sciences of Ukraine, 3 Tereshchenkivs'ka, Kyiv, 01601, Ukraine

M. Pratsiovytyi
National Dragomanov Pedagogical University, 9 Pyrogova, Kyiv, 01030, Ukraine

G. Torbin
National Dragomanov Pedagogical University, 9 Pyrogova, Kyiv, 01030, Ukraine

### Citation Example

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},
}