On the speed of convergence in the ergodic theorem |
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Authors: | Prof. Dr. Ulrich Krengel |
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Affiliation: | 1. Institut für Mathematische Statistik und Wirtschaftsmathematik, Universit?t G?ttingen, Lotzestra?e 13, D-3400, G?ttingen, Federal Republic of Germany
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Abstract: | Given any ergodic invertible measure preserving transformation τ of [0,1] and any null-sequence (α N ) of positive reals, there exists a continuousf such that $$lim sup alpha _{rm N}^{ - 1} left| {N^{ - 1} sumlimits_{k = 0}^{N - 1} {f circ tau ^k - smallint f} } right| = infty a. e.,$$ i.e. there is no “speed of convergence” in the ergodic theorem for any τ. The analogous result holds also for norm-convergence. |
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