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Second order ergodic theorems for ergodic transformations of infinite measure spaces. (English) Zbl 0738.28011

Summary: For certain pointwise dual ergodic transformations \(T\) we prove almost sure convergence of the log-averages \[ {1 \over \log N}\sum_{n=1}^ N {1 \over na(n)} \sum_{k=1}^ n f\circ T^ k \qquad (f\in L_ 1) \] and the Chung-Erdős averages \[ {1 \over \log a(N)} \sum_{k=1}^ N {1 \over a(k)} f\circ T^ k \qquad (f\in L_ 1^ +) \] towards \(\int f\), where \(a(n)\) denotes the return sequence of \(T\).

MSC:

28D05 Measure-preserving transformations
60F15 Strong limit theorems
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