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

DOI:
https://doi.org/10.1090/S0094-9000-2011-00819-8

Published electronically:
January 24, 2011

MathSciNet review:
2667319

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 of the Ostrogradskiĭ difference representation. It is shown that there is no probability measure that is invariant and ergodic with respect to 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:
https://doi.org/10.1090/S0094-9000-2011-00819-8

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