Theory of Probability and Mathematical Statistics

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

Author(s): M. V. Prats'ovytyi; O. M. Baranovs'kii
Translated by: V. Zayats
Original publication: Teoriya Imovirnostei ta Matematichna Statistika, vipusk 70 (2004).
Journal: Theor. Probability and Math. Statist. No. 70 (2005), 147-160.
MSC (2000): Primary 60E05, 26A30; Secondary 11A67, 11K55
Posted: August 12, 2005
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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.

1.: E. Ya. Remez, On series with alternating sign which may be connected with two algorithms of M. V. Ostrogradski ${\u{\i}}\kern.15em$ 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 ${\u{\i}}\kern.15em$ 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 ${\u{\i}}\kern.15em$ 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 ${\u{\i}}\kern.15em$ , A Fractional Approach to Studying Singular Distributions, M. P. Dragomanov National Pedagog. Univ., Kyïv, 1998. (Ukrainian)
12.: M. V. Prats'oviti ${\u{\i}}\kern.15em$ , O. L. Leshchins'ki ${\u{\i}}\kern.15em$ , Properties of random variables defined by the distributions of the elements of their $\tilde Q_\infty$ -representation, Teor. Imov ${\={\i}}\kern.15em$ r. 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 ${\u{\i}}\kern.15em$ 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, 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)

Retrieve articles in Theory of Probability and Mathematical Statistics with MSC (2000): 60E05, 26A30, 11A67, 11K55

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: 10.1090/S0094-9000-05-00638-1
PII: S 0094-9000(05)00638-1
Received by editor(s): 11/APR/2003
Posted: August 12, 2005
Copyright of article: Copyright 2005, American Mathematical Society

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