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 | References | Similar Articles | Additional Information

Abstract: Several metric relations for representations of real numbers by the Ostrogradski type 1 series are obtained. These relations are used to prove that a random variable with independent differences of consecutive elements of the Ostrogradski 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.

**1.**E. Ya. Remez,*On series with alternating sign which may be connected with two algorithms of M. V. Ostrogradski 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.**F. Schweiger,*Ergodic Theory of Fibred Systems and Metric Number Theory*, Oxford Univ. Press, New York, 1995. MR**1419320 (97h:11083)****5.**A. Ya. Khinchin,*Continued Fractions*, 3rd ed., Fizmatgiz, Moscow, 1961; English transl., Dover Publications Inc., Mineola, NY, 1997. MR**1451873 (98c:11008)****6.**P. I. Bodnarchuk, V. Ya. Skorobogat'ko,*Branching Continued Fractions and Their Applications*, ``Naukova Dumka'', Kyïv, 1974. (Ukrainian) MR**0353626 (50:6109)****7.**Yu. V. Mel'nichuk,*-adic continued fractions constructed using the Euclid algorithm and the Ostrogradski algorithm*, Proc. Conf. ``Computative Mathematics in Modern Scientific and Technical Progress'', Kanev, 1974, pp. 259-265. (Russian)**8.**Yu. V. Mel'nichuk,*On representation of real numbers by quickly convergent series*, Continued Fractions and Their Applications, Inst. Mat., Akad. Nauk Ukrain. SSR, Kiev, 1976, pp. 77-78. (Russian) MR**0505986 (58:21908)****9.**K. G. Valeev and E. D. Zlebov,*The metric theory of the Ostrogradski algorithm*, Ukrain. Mat. Zh.**27**(1975), no. 1, 64-69; English transl. in Ukrainian Math. J.**27**(1975), no. 1, 47-51. MR**0366855 (51:3101)****10.**S. Kalpazidou, A. Knopfmacher, and J. Knopfmacher,*Lüroth-type alternating series representations for real numbers*, Acta Arith.**55**(1990), 311-322. MR**1069185 (91i:11011)****11.**M. V. Prats'oviti,*A Fractional Approach to Studying Singular Distributions*, M. P. Dragomanov National Pedagog. Univ., Kyïv, 1998. (Ukrainian)**12.**M. V. Prats'oviti, O. L. Leshchins'ki,*Properties of random variables defined by the distributions of the elements of their -representation*, Teor. Imovr. Mat. Stat.**57**(1997), 134-140; English transl. in Theory Probab. Math. Statist.**57**(1998), 143-148. MR**1806893 (2003b:60024)****13.**O. M. Baranovs'kii,*Ostrogradski 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 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,*Theory of Functions of a Real Variable*, ``Nauka'', Moscow, 1972; English transl., Frederick Ungar Publishing Co., New York, 1955. MR**0039790 (12:598d)**; 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:
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