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Properties of distributions of random variables with independent differences of consecutive elements of the Ostrogradskii series


Authors: M. V. Prats'ovytyi and O. M. Baranovs'kii
Translated by: V. Zayats
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, tom 70 (2004).
Journal: Theor. Probability and Math. Statist. 70 (2005), 147-160
MSC (2000): Primary 60E05, 26A30; Secondary 11A67, 11K55
DOI: https://doi.org/10.1090/S0094-9000-05-00638-1
Published electronically: August 12, 2005
MathSciNet review: 2110871
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Abstract: Several metric relations for representations of real numbers by the Ostrogradski{\u{\i}}\kern.15em type 1 series are obtained. These relations are used to prove that a random variable with independent differences of consecutive elements of the Ostrogradski{\u{\i}}\kern.15em type 1 series has a pure distribution, that is, its distribution is either purely discrete, or purely singular, or purely absolutely continuous. The form of the distribution function and that of its derivative are found. A criterion for discreteness and sufficient conditions for the distribution spectrum to have zero Lebesgue measure are established.


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Additional Information

M. V. Prats'ovytyi
Affiliation: Dragomanov National Pedagogical University, Pyrogov Street 9, Kyïv 01601, Ukraine
Email: prats@ukrpost.net

O. M. Baranovs'kii
Affiliation: Dragomanov National Pedagogical University, Pyrogov Street 9, Kyïv 01601, Ukraine

DOI: https://doi.org/10.1090/S0094-9000-05-00638-1
Received by editor(s): April 11, 2003
Published electronically: August 12, 2005
Article copyright: © Copyright 2005 American Mathematical Society

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