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On the Role of the Law of Large Numbers in the Theory of Randomness

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Abstract

In the first part of this article, we answer Kolmogorov's question (stated in 1963 in [1]) about exact conditions for the existence of random generators. Kolmogorov theory of complexity permits of a precise definition of the notion of randomness for an individual sequence. For infinite sequences, the property of randomness is a binary property, a sequence can be random or not. For finite sequences, we can solely speak about a continuous property, a measure of randomness. Is it possible to measure randomness of a sequence t by the extent to which the law of large numbers is satisfied in all subsequences of t obtained in an “admissible way”? The case of infinite sequences was studied in [2]. As a measure of randomness (or, more exactly, of nonrandomness) of a finite sequence, we consider the specific deficiency of randomness δ (Definition 5). In the second part of this paper, we prove that the function δ/ln(1/δ) characterizes the connections between randomness of a finite sequence and the extent to which the law of large numbers is satisfied.

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REFERENCES

  1. Kolmogorov, A.N., On Tables of Random Numbers, Sankhyā, Indian J. Statist., Ser. A, 1963, vol. 25, no. 4, pp. 369–376. Reprinted in Theor. Comp. Sci., 1998, vol. 207, pp. 387–395.

    Google Scholar 

  2. Muchnik, An.A., Semenov, A.L., and Uspensky, V.A., Mathematical Metaphysics of Randomness, Theor. Comp. Sci., 1998, vol. 207, no. 1–2, pp. 263–317.

    Google Scholar 

  3. van Lambalgen, M., Von Mises's Definition of Random Sequences Reconsidered, J. Symb. Logic, 1987, vol. 52, pp. 725–755.

    Google Scholar 

  4. Kolmogorov, A.N., Three Approaches to the Quantitative De_nition of Information, Probl. Peredachi Inf., 1965, vol. 1, no. 1, pp. 3–11 [Probl. Inf. Trans. (Engl. Transl.), 1965, vol. 1, no. 1, pp. 1–7].

    Google Scholar 

  5. Uspensky, V.A. and Shen, A.Kh., Relations between Varieties of Kolmogorov Complexities, Math. Syst. Theory, 1996, vol. 29, pp. 271–292.

    Google Scholar 

  6. Kolmogorov, A.N., On Tables of Random Numbers, in Semiotika i informatika, Moscow: VINITI, 1982, vol. 18, pp. 3–13. Reprinted in Kolmogorov, A.N., Teoriya informatsii i teoriya algoritmov (Information Theory and Theory of Algorithms), Moscow: Nauka, 1987, pp. 204–213.

    Google Scholar 

  7. Yaglom, A.M. and Yaglom, I.M., Veroyatnost' i informatsiya, Moscow: Fizmatgiz, 1960, 2nd ed.Translated under the title Probability and Information, Boston: Reidel, 1983.

    Google Scholar 

  8. Durand, B. and Vereshchagin, N., Kolmogorov Loveland Stochasticity for Finite Strings, 2002. Available from http://markov.math.msu.ru/~ver/papers/kolm-love.ps.

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  1. Institute of New Technologies, Nizhnyaya Radishchevskaya 10, Moscow, 109004, Russia

    An. A. Muchnik & A. L. Semenov

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  1. An. A. Muchnik
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  2. A. L. Semenov
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Muchnik, A.A., Semenov, A.L. On the Role of the Law of Large Numbers in the Theory of Randomness. Problems of Information Transmission 39, 119–147 (2003). https://doi.org/10.1023/A:1023638717091

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