Geodesic
Probability distributions with
Teoriâ slučajnyh processov, Tome 13 (2007) no. 2, pp. 281-293.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Keywords: Singularly continuous probability distributions, Hausdorff dimension of probability measures, Hausdorff-Billingsley dimension, fractals
Mots-clés : DP-transformations. 
@article{THSP_2007_13_2_a11,
     author = {Grygoriy Torbin},
     title = {Probability distributions with},
     journal = {Teori\^a slu\v{c}ajnyh processov},
     pages = {281--293},
     publisher = {mathdoc},
     volume = {13},
     number = {2},
     year = {2007},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/THSP_2007_13_2_a11/}
}
TY  - JOUR
AU  - Grygoriy Torbin
TI  - Probability distributions with
JO  - Teoriâ slučajnyh processov
PY  - 2007
SP  - 281
EP  - 293
VL  - 13
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/THSP_2007_13_2_a11/
LA  - en
ID  - THSP_2007_13_2_a11
ER  - 
%0 Journal Article
%A Grygoriy Torbin
%T Probability distributions with
%J Teoriâ slučajnyh processov
%D 2007
%P 281-293
%V 13
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/THSP_2007_13_2_a11/
%G en
%F THSP_2007_13_2_a11
Grygoriy Torbin. Probability distributions with. Teoriâ slučajnyh processov, Tome 13 (2007) no. 2, pp. 281-293. http://geodesic.mathdoc.fr/item/THSP_2007_13_2_a11/

[1] S. Albeverio, G. Torbin, “Fractal properties of singular continuous probability distributions with independent $Q^*$-digits”, Bull. Sci. Math., 129:4 (2005), 356–367

[2] S. Albeverio, M. Pratsiovytyi, G. Torbin, “Transformations preserving the fractal dimension and distribution functions”, Journal of the London Math. Soc. (to appear)

[3] S. Albeverio, M. Pratsiovytyi, G. Torbin, “Fractal probability distributions and transformations preserving the Hausdorff-Besicovitch dimension”, Ergodic Theory and Dynamical Systems, 24:1 (2004), 1–16

[4] S. Albeverio, V. Koshmanenko, G. Torbin, “Fine structure of singular continuous spectrum”, Methods Funct. Anal. Topology, 9:2 (2003), 101–119

[5] S. Albeverio, M. Pratsiovytyi, G. Torbin, Fractal probability distributions and transformations preserving the Hausdorff-Besicovitch dimension, SFB 256 Preprint, University of Bonn

[6] Ya. Goncharenko,M. Pratsiovytyi, G. Torbin, “Fractal properties of the set of points of nondifferentiability for absolutely continuous and singular distribution functions”, Theor. Probability and Math. Statist., 65 (2001), 27–34

[7] P. Billingsley, “Hausdorff dimension in probability theory II, III”, J. Math.,, 5 (1961), 291–198

[8] K. J. Falconer, Fractal geometry, John Wiley Sons, 1990

[9] M. Pratsiovytyi, Fractal approach to investigations of singular distributions, National Pedagogical Univ., Kyiv, 1998

[10] M. Pratsiovytyi, “Distributions of random variables with independent $Q$-symbols”, Asymptotic and applied problems in the theory of random evolutions, Akad. Nauk Ukrain. SSR, Inst. Mat., Kiev, 1990

[11] F. Schweiger, Ergodic theory of ?bred systems and metric number theory, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1995

[12] A. F. Turbin, M. V. Pratsiovytyi, Fractal sets, functions, and distributions, Naukova Dumka, Kiev, 1992