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
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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
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