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Problems of Information Transmission

, Volume 43, Issue 1, pp 48–56 | Cite as

Bound on the cardinality of a covering of an arbitrary randomness test by frequency tests

  • K. Yu. Gorbunov
Large Systems
  • 33 Downloads

Abstract

We improve a well-known asymptotic bound on the number of monotonic selection rules for covering of an arbitrary randomness test by frequency tests. More precisely, we prove that, for any set S (arbitrary test) of binary sequences of sufficiently large length L, where ∨S∨ ≤ 2 L(1−δ), for sufficiently small δ there exists a polynomial (in 1/δ) set of monotonic selection rules (frequency tests) which guarantee that, for each sequence tS, a subsequence can be selected such that the product of its length by the squared deviation of the fraction of zeros in it from 1/2 is of the order of at least 0.5 ln 2 L[δ/ln(1/δ)](1 − 2 ln ln(1/δ)/ln(1/δ)).

Keywords

Selection Rule Information Transmission Binary Sequence Frequency Test Normal Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Muchnik, An.A. and Semenov, A.L., On the Role of the Law of Large Numbers in the Theory of Randomness, Probl. Peredachi Inf., 2003, vol. 39, no. 1, pp. 134–165 [Probl. Inf. Trans. (Engl. Transl.), 2003, vol. 39, no. 1, pp. 119–147].MathSciNetGoogle Scholar
  2. 2.
    Shiryaev, A.N., Veroyatnost’, vol. 1: Elementarnaya teoriya veroyatnostei. Matematicheskie osnovaniya. Predel’nye teoremy (Elementary Probability Theory. Mathematical Foundations. Limit Theorems), Moscow: MCCME, 2004, 3rd ed. Second edition translated under the title Probability, New York: Springer, 1996.Google Scholar
  3. 3.
    Uspensky, V.A., Semenov, A.L., and Shen’, A.Kh., Can an Individual Sequence of Zeros and Ones Be Random?, Uspekhi Mat. Nauk, 1990, vol. 45, no. 1, pp. 105–162 [Russian Math. Surveys (Engl. Transl.), 1990, vol. 45, no. 1, pp. 121–189].MathSciNetGoogle Scholar
  4. 4.
    Muchnik, An.A., Semenov, A.L., and Uspensky, V.A., Mathematical Metaphysics of Randomness, Theor. Comput. Sci., 1998, vol. 207, no. 1–2, pp. 263–317.zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Inc. 2007

Authors and Affiliations

  • K. Yu. Gorbunov
    • 1
  1. 1.Kharkevich Institute for Information Transmission ProblemsRASMoscowRussia

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