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-ErdĂ¶s 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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*Analogues of the Lebesgue density theorem for fractal subsets of the reals and integers*, Proc. London Math. Soc. (to appear).

*A pathwise central limit theorem for random walks*, Ann. Prob. (to appear). ---,

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Article copyright:
© Copyright 1992
American Mathematical Society