Upper bounds for ergodic sums of infinite measure preserving transformations

Aaronson, Jon; Denker, Manfred

doi:10.1090/S0002-9947-1990-1024766-3

Upper bounds for ergodic sums of infinite measure preserving transformations
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by Jon Aaronson and Manfred Denker

Trans. Amer. Math. Soc. 319 (1990), 101-138

DOI: https://doi.org/10.1090/S0002-9947-1990-1024766-3

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Abstract:

For certain conservative, ergodic, infinite measure preserving transformations $T$ we identify increasing functions $A$, for which \[ \limsup \limits _{n \to \infty } \frac {1} {{A(n)}}\sum \limits _{k = 1}^n {f \circ } {T^k} = \int _X {fd\mu } \quad {\text {a}}{\text {.e}}{\text {.}}\] holds for any nonnegative integrable function $f$. In particular the results apply to some Markov shifts and number-theoretic transformations, and include the other law of the iterated logarithm.

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