On the Uniform Convergence of Relative Frequencies of Events to Their Probabilities

  • V. N. Vapnik
  • A. Ya. Chervonenkis


This chapter reproduces the English translation by B. Seckler of the paper by Vapnik and Chervonenkis in which they gave proofs for the innovative results they had obtained in a draft form in July 1966 and announced in 1968 in their note in Soviet Mathematics Doklady. The paper was first published in Russian as Вапник В. Н. and Червоненкис А. Я. О равномерноЙ сходимости частот появления событиЙ к их вероятностям. Теория вероятностеЙ и ее применения 16(2), 264–279 (1971).


Relative Frequency Uniform Convergence Growth Function Distribution Property Entire Class 
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  1. 1.
    Vapnik, V.N., Chervonenkis, A.Ya.: On the uniform convergence of relative frequencies of events to their probabilities. Dokl. Akad. Nauk SSSR 181(4), 781–783 (1968) (In Russian)Google Scholar
  2. 2.
    Gnedenko, B.V.: Theory of Probability, 4th edn. Chelsea Publishing Co., New York (1968)Google Scholar
  3. 3.
    Khinchin, A.Ya.: On basic theorems of information theory. Uspekhi Mat. Nauk XI(1), 17–75 (1956) (In Russian)Google Scholar
  4. 4.
    Feller, W.: Introduction to Probability Theory and Its Applications, vol. 1, 2nd edn. Wiley, New York (1957)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • V. N. Vapnik
    • 1
  • A. Ya. Chervonenkis
    • 1
  1. 1.Institute of Automation and Remote ControlMoscowRussia

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