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

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
Published electronically: August 12, 2005
MathSciNet review: 2110871
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Abstract | References | Similar Articles | Additional Information

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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  • 1. E. Ya. Remez, On series with alternating sign which may be connected with two algorithms of M. V. Ostrogradskiĭ for the approximation of irrational numbers, Uspehi Matem. Nauk (N.S.) 6 (1951), no. 5(45), 33–42 (Russian). MR 0044585 (13,444d)
  • 2. W. Sierpinski, O kilku algoritmach dla rozwijania liczb rzeczywistych na szeregi, Sprawozdania z posiedzen Towarzystwa Naukowego Warszawskiego 4 (1911), 56-77.
  • 3. T. A. Pierce, On an algorithm and its use in approximating roots of an algebraic equation, Amer. Math. Monthly 36 (1929), 523-525.
  • 4. Fritz Schweiger, Ergodic theory of fibred systems and metric number theory, Oxford Science Publications, The Clarendon Press Oxford University Press, New York, 1995. MR 1419320 (97h:11083)
  • 5. A. Ya. Khinchin, Continued fractions, Translated from the third (1961) Russian edition, Dover Publications Inc., Mineola, NY, 1997. With a preface by B. V. Gnedenko; Reprint of the 1964 translation. MR 1451873 (98c:11008)
  • 6. P. I. \cyr{B}odnarchuk and V. Ya. \cyr{S}korobogat′ko, Gilljsti lantsyugovi drobi taih zastosuvannya, “Naukova Dumka”, Kiev, 1974 (Ukrainian). MR 0353626 (50 #6109)
  • 7. Yu. V. Mel'nichuk, $p$-adic continued fractions constructed using the Euclid algorithm and the Ostrogradski{\u{\i}}\kern.15em algorithm, Proc. Conf. ``Computative Mathematics in Modern Scientific and Technical Progress'', Kanev, 1974, pp. 259-265. (Russian)
  • 8. V. Ja. Skorobogat′ko (ed.), Tsepnye drobi i ikh primeneniya, Izdanie Inst. Mat., Akad. Nauk Ukrain. SSR, Kiev, 1976 (Russian). MR 0505986 (58 #21908)
  • 9. K. G. Valēēv and E. D. Zlēbov, The metric theory of an algorithm of M. V. Ostrogradskiĭ, Ukrain. Mat. Ž. 27 (1975), 64–69, 142 (Russian). MR 0366855 (51 #3101)
  • 10. Sofia Kalpazidou, Arnold Knopfmacher, and John Knopfmacher, Lüroth-type alternating series representations for real numbers, Acta Arith. 55 (1990), no. 4, 311–322. MR 1069185 (91i:11011)
  • 11. M. V. Prats'oviti{\u{\i}}\kern.15em, A Fractional Approach to Studying Singular Distributions, M. P. Dragomanov National Pedagog. Univ., Kyïv, 1998. (Ukrainian)
  • 12. M. V. Prats′ovitiĭ and O. L. Leshchins′kiĭ, Properties of random variables defined by the distributions of the elements of their 𝑄_{∞}-representation, Teor. Ĭmovīr. Mat. Stat. 57 (1997), 134–139 (Ukrainian, with Ukrainian summary); English transl., Theory Probab. Math. Statist. 57 (1998), 143–148 (1999). MR 1806893 (2003b:60024)
  • 13. O. M. Baranovs'kii, Ostrogradski{\u{\i}}\kern.15em series as a mean of analytic representation of sets and random variables, Fractal Analysis and Related Problems, M. P. Dragomanov National Pedagog. Univ., Kyïv, 1998, pp. 91-102. (Ukrainian)
  • 14. O. M. Baranovs'kii, Some problems in the metric theory of numbers represented by the Ostrogradski{\u{\i}}\kern.15em type 1 series, Scientific Proc., Physical and Math. Sciences, M. P. Dragomanov National Pedagog. Univ., Kyïv, 2002, pp. 391-402. (Ukrainian)
  • 15. I. P. Natanson, Teoriya funkciĭ veščestvennoĭ peremennoĭ, Gosudarstv. Izdat. Tehn.-Teor. Lit.,], Moscow-Leningrad, 1950 (Russian). MR 0039790 (12,598d)
    I. P. Natanson, Theory of functions of a real variable, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron with the collaboration of Edwin Hewitt. MR 0067952 (16,804c)

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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: http://dx.doi.org/10.1090/S0094-9000-05-00638-1
PII: S 0094-9000(05)00638-1
Received by editor(s): April 11, 2003
Published electronically: August 12, 2005
Article copyright: © Copyright 2005 American Mathematical Society