Abstract
Kolmogorov discovered in 1933 that the empirical statistics of several independent values of any random variable differs from the true distribution function of this variable in some universal way: the random distribution of the distance of one of these statistics from the other verifies (asymptotically) some stochastic distribution law (called later “Kolmogorov’s distribution”).
The present paper compares the Kolmogorov’s distribution with a similar object, provided by the chain of observations of a nonrandom, deterministic dynamical system, formed by the consecutive members of a geometrical progression.
Say, the Kolmogorov’s distribution is observed for the distribution of the last pairs of digits of the powers of integer 3, that is, for the sequence 01,03,09,27,81,43,29,87,… (which is not random at all and does not verify the Kolmogorov’s theorem conditions).
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References
Kolmogorov AN (1933) Sulla determinazione empirica di una legge di distribuzione. G Ist Ital Attuari 4:83–91
Arnold VI (2006) Adiabatic analysis of the geometrical progressions of residues. In: Dynamics, statistics and projective geometry of Galois fields. Cambridge Univ Press (Moscow, MCCME, 2005, §5)
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Partially supported by RFBR, grant 07-01-00388
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Arnold, V.I. Empirical study of stochasticity for deterministic chaotical dynamics of geometric progressions of residues. Funct. Anal. Other Math. 2, 139–149 (2009). https://doi.org/10.1007/s11853-009-0034-7
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DOI: https://doi.org/10.1007/s11853-009-0034-7
Keywords
- Ergodic theory
- Pseudorandom numbers
- Chaos
- Residues of the division operation
- Random processes
- Independence
- Uniform distribution