On the speed of convergence in the ergodic theorem |
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Authors: | Prof Dr Ulrich Krengel |
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Institution: | 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 $$\lim \sup \alpha _{\rm N}^{ - 1} \left| {N^{ - 1} \sum\limits_{k = 0}^{N - 1} {f \circ \tau ^k - \smallint f} } \right| = \infty a. e.,$$ 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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