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Theory of Probability and Mathematical Statistics
Theory of Probability and Mathematical Statistics
ISSN 1547-7363(online) ISSN 0094-9000(print)

Singularity of the second Ostrogradskiĭ random series


Authors: G. M. Torbin and I. M. Pratsyovyta
Translated by: N. Semenov
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 81 (2010).
Journal: Theor. Probability and Math. Statist. 81 (2010), 187-195
MSC (2010): Primary 11K55, 37B10, 60G30
Published electronically: January 24, 2011
MathSciNet review: 2667319
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Abstract: We study properties of the distribution of the second Ostrogradskiĭ series, for which the differences of terms are independent identically distributed random variables. We completely describe the Lebesgue structure of this distribution. In particular, we prove that it cannot be absolutely continuous. We also develop ergodic theory for the second Ostrogradskiĭ expansion. One of the results is that, for almost all (in the sense of Lebesgue measure) real numbers of the unit interval, an arbitrary symbol of an alphabet occurs finitely often in the corresponding Ostrogradskiĭ difference expansion. We also study properties of the dynamical system generated by the one-sided shift transformations $ T$ of the Ostrogradskiĭ difference representation. It is shown that there is no probability measure that is invariant and ergodic with respect to $ T$ and absolutely continuous with respect to Lebesgue measure.


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Additional Information

G. M. Torbin
Affiliation: Department of Higher Mathematics, Institute for Physics and Mathematics, National Dragomanov Pedagogical University, Pirogova Street 9, Kyiv 01030, Ukraine
Email: torbin@imath.kiev.ua

I. M. Pratsyovyta
Affiliation: Department of Higher Mathematics, Institute for Physics and Mathematics, National Dragomanov Pedagogical University, Pirogova Street 9, Kyiv 01030, Ukraine
Email: lightsoul2008@gmail.com

DOI: http://dx.doi.org/10.1090/S0094-9000-2011-00819-8
PII: S 0094-9000(2011)00819-8
Keywords: The second Ostrogradskiĭ series, singular probability measures, symbolic dynamics
Received by editor(s): November 2, 2009
Published electronically: January 24, 2011
Additional Notes: The research of the first author is supported by the projects DFG 436 UKR 113/80 and DFG 436 113/97 and the Alexander von Humboldt Foundation
The research of the second author is supported by the projects DFG 436 UKR 113/80 and DFG 436 113/97
Article copyright: © Copyright 2011 American Mathematical Society