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Strongly ergodic sequences of integers and the individual ergodic theorem

Authors: J. R. Blum and J. I. Reich
Journal: Proc. Amer. Math. Soc. 86 (1982), 591-595
MSC: Primary 28D05; Secondary 47A35, 60F15
MathSciNet review: 674086
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Abstract: Let $ S = \{ {k_1},{k_2}, \ldots \} $ be an increasing sequence of positive integers. We call $ S$ strongly ergodic if for every measure preserving transformation $ T$ on a probability space $ (\Omega ,\mathcal{F},P)$ and every $ f \in {L_1}(\Omega )$ we have $ {\lim _{n \to \infty }}(1/n)\sum\nolimits_{j = 1}^n {f({T^{kj}}\omega ) = Pf(\omega )} $ a.e. where $ Pf$ is the appropriate limit guaranteed by the individual ergodic theorem. We give sufficient conditions for a sequence $ S$ to be strongly ergodic and provide examples.

References [Enhancements On Off] (What's this?)

  • [1] J. R. Blum and B. Eisenberg, Generalized summing sequences and the mean ergodic theorem, Proc. Amer. Math. Soc. 42 (1974), 423-429. MR 0330412 (48:8749)
  • [2] J. R. Blum and R. Cogburn, On ergodic sequences of measures, Proc. Amer. Math. Soc. 51 (1975), 359-365. MR 0372529 (51:8736)
  • [3] W. R. Emerson, The pointwise ergodic theorem for amenable groups, Amer. J. Math. 96 (1974), 472-487. MR 0354926 (50:7403)
  • [4] A. M. Garsia, Topics in almost everywhere convergence, Markham, Chicago, Ill., 1970. MR 0261253 (41:5869)
  • [5] J. I. Reich, On the individual ergodic theorem for subsequences, Ann. Probability 5 (1977), 1039-1046. MR 0444906 (56:3252)

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Keywords: Individual ergodic theorems, subsequences, strongly ergodic sequences
Article copyright: © Copyright 1982 American Mathematical Society

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