On the application of A.N. Kolmogorov’s Theorem
https://doi.org/10.22405/2226-8383-2025-26-5-203-220
Abstract
In the article, on the class K^0 of infinite binary sequences without the runs of ones, a
consistent probability distribution P is constructed which is induced by a time-homogeneous
Markov chain with a one-step transition matrix P𝜑 , and is completely determined by the
golden ratio 𝜑. Using a Markov chain to construct a probability measure P allows us to apply
Kolmogorov’s existence theorem. The asymptotic distribution of the subclass K^0 of infinite
binary sequences without the runs of ones starting with zero coincides with the analogous
asymptotic distribution of the classical equiprobable scheme . And in this case, the asymptotic
distribution of the class K 0 coincides with the probability P(K^0).
Keywords
Kolmogorov’s existence theorem,
Markov chains,
entropy of Markov chains,
Fibonacci numbers,
golden ratio,
binary sequences,
equidistributed sequences,
probability distribution,
uniform distribution,
frequency distribution,
asymptotic distribution,
asymptotic behavior,
limiting distribution,
entropy.
About the Authors
Vitaliy Nikolaevich Sobolev
Lomonosov Moscow State University; RTU MIREA
Russian Federation
Russian Federation
candidate of physical and mathematical sciences
Andrey Alexandrovich Frolov
HSE University
Russian Federation
Russian Federation
senior lecturer
References
1. Kac, M. 1959, Probability and related topics in physical sciences, Lectures in Applied Mathematics, vol. I, Interscience Publishers, New York, 266 p.
2. Borovkov, A.A. 2013, Probability theory, Springer-Verlag, Berlin, 733 p.
3. Goncharov, V.L. 1944, “Du domaine de l’analyse combinatoire”, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 8, no. 1, pp. 3–48.
4. Amelkin, V.A. 2001, “Algorithms for exact solving the problems of enumeration, coding, and generation of serial sequences” [Algoritmy tochnogo resheniya zadach perechisleniya, kodirovaniya i generirovaniya serijnykh posledovatel’nostej], Sibirskii Zhurnal Vychislitel’noi Matematiki, vol. 4, no. 1, pp. 1–12.
5. Mineev, M.P. & Chubarikov, V.N. 2014, Lectures on the arithmetic aspects of cryptography [Lektsii po arifmeticheskim voprosam kriptografii], Izdatel’stvo Luch, Moscow, 224 p.
6. Prokhorov, A.V. (ed.) 2003, Probability and mathematical statistics: Encyclopedia [Veroyatnost’ i matematicheskaya statistika: entsiklopediya], Izdatel’stvo Bol’shaya Rossiiskaya Entsiklopediya, Moscow, 910 p.
7. Shiryaev, A.N. 2004, Probability [Veroyatnost’], Izdatel’stvo MTsNMO, Moscow, 927 p.
8. Feller, W. 1957, An introduction to probability theory and its applications, Vol. 1, John Wiley & Sons, New York, 462 p.
9. Gikhman, I.I. & Skorokhod, A.V. 2004, The theory of stochastic processes, Vol. 1, Springer-Verlag, Berlin, 574 p.
10. Romanovsky, V.I. 1970, Discrete Markov chains, Wolters-Noordhoff, Groningen, 408 p.
11. Vorobyov, N.N. 1964, Fibonacci numbers [Chisla Fibonacci], Izdatel’stvo Nauka, Moscow, 71 p.
12. Arkhipov, G.I., Sadovnichy, V.A. & Chubarikov, V.N. 2008, Lectures on mathematical analysis [Lektsii po matematicheskomu analizu], Izdatel’stvo Drofa, Moscow, 638 p.
13. Klimov, G.P. 1986, Probability theory and mathematical statistics, Izdatel’stvo Moskovskogo Universiteta, Moscow, 336 p.
14. Shannon, C.E. 1963, Works on information theory and cybernetics [Raboty po teorii informatsii i kibernetike], Izdatel’stvo Inostrannoi Literatury, Moscow, 829 p.
15. Khinchin, A.Ya. 1953, “The concept of entropy in probability theory”, Uspekhi Matematicheskikh Nauk, vol. 8, no. 3, pp. 3–20.
16. Yushkevich, A.A. 1953, “On limit theorems connected with the concept of entropy of Markov chains”, Russian Mathematical Surveys, vol. 8, no. 5, pp. 177–180.
17. Vershik, A.M. & Sidorov, N.A. 1994, “Arithmetic expansions associated with the rotation of a circle and continued fractions”, St. Petersburg Mathematical Journal, vol. 5, no. 6, pp. 1121–1136.
18. Kulikova, V.L., Olekhova, E.F. & Oseledets, V.I. 2024, “On absolute continuity of the Erdos measure for the golden ratio, Tribonacci numbers, and second-order Markov chains”, Theory of Probability and Its Applications, vol. 69, no. 2, pp. 265–280.
19. Gelfond, A.O. 1959, “A common property of number systems”, Izvestiya Akademii Nauk SSSR. Seriya Matematicheskaya, vol. 23, no. 6, pp. 809–814.
Review
For citations:
Sobolev V.N., Frolov A.A. On the application of A.N. Kolmogorov’s Theorem. Chebyshevskii Sbornik. 2025;26(5):203-220. (In Russ.) https://doi.org/10.22405/2226-8383-2025-26-5-203-220
Views:
41
JATS XML
Email this article 