Geodesic
Effective convergence in probability and an ergodic theorem for individual random sequences
Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 1, pp. 35-50

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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. 
@article{TVP_1997_42_1_a2,
     author = {V. V. V'yugin},
     title = {Effective convergence in probability and an ergodic theorem for individual random sequences},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
     pages = {35--50},
     publisher = {mathdoc},
     volume = {42},
     number = {1},
     year = {1997},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TVP_1997_42_1_a2/}
}
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V. V. V'yugin. Effective convergence in probability and an ergodic theorem for individual random sequences. Teoriâ veroâtnostej i ee primeneniâ, Tome 42 (1997) no. 1, pp. 35-50. http://geodesic.mathdoc.fr/item/TVP_1997_42_1_a2/