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), 187195
MSC (2010):
Primary 11K55, 37B10, 60G30
Published electronically:
January 24, 2011
MathSciNet review:
2667319
Fulltext PDF
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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 onesided 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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 S. Albeverio, I. Pratsiovyta, M. Pratsiovytyi, and G. Torbin, On Bernoulli Convolutions Generated by Second Ostrogradsky Series and their Fine Fractal Properties, SFB611 Preprint, vol. 459, Bonn University, 2009.
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 T. A. Pierce, On an algorithm and its use in approximating roots of algebraic equations, Amer. Math. Monthly 36 (1929), no. 10, 523525. MR 1521866
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 F. Schweiger, Ergodic Theory of Fibred Systems and Metric Number Theory, Clarendon Press, Oxford, 1995. MR 1419320 (97h:11083)
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 J. O. Shallit, Metric theory of Pierce expansions, Fibonacci Quart. 24 (1986), no. 1, 2240. MR 825872 (87g:11093)
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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/S009490002011008198
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
