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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Second order ergodic theorems for ergodic transformations of infinite measure spaces


Authors: Jon Aaronson, Manfred Denker and Albert M. Fisher
Journal: Proc. Amer. Math. Soc. 114 (1992), 115-127
MSC: Primary 28D05; Secondary 60F15
DOI: https://doi.org/10.1090/S0002-9939-1992-1099339-4
MathSciNet review: 1099339
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Abstract: For certain pointwise dual ergodic transformations $T$ we prove almost sure convergence of the log-averages \[ \frac {1}{{\log N}}\sum \limits _{n = 1}^N {\frac {1}{{na\left ( n \right )}}\sum \limits _{k = 1}^n {f \circ {T^k}\left ( {f \in {L_1}} \right )} } \] and the Chung-Erdös averages \[ \frac {1}{{\log a\left ( N \right )}}\sum \limits _{k = 1}^N {\frac {1}{{a\left ( k \right )}}f \circ {T^k}} \left ( {f \in L_1^ + } \right )\] towards $\smallint f$, where $a\left ( n \right )$ denotes the return sequence of $T$.


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Article copyright: © Copyright 1992 American Mathematical Society