This article is cited in 36 papers

Effective convergence in probability and an ergodic theorem for individual random sequences

V. V. V'yugin

Institute for Information Transmission Problems, Russian Academy of Sciences

Abstract: An algorithmic analysis of the ergodic theorem for a measure-preserving transformation is given. We prove that the classical ergodic theorem is not algorithmically effective. We present a formulation and a proof of the ergodic theorem for individual random sequences based on A. N. Kolmogorov's algorithmic approach to the substantiation of the theory of probability and information theory.

Keywords: ergodic theorem, quasiergodic theorem, stationary measure, convergence in probability, convergence almost everywhere, algorithm, random sequence, algorithmical randomness.

Received: 12.07.1996

DOI: 10.4213/tvp1710

 English version: Theory of Probability and its Applications, 1998, 42:1, 39–50

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