Upper Bounds for Ergodic Sums of Infinite Measure Preserving Transformations

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.

Locations

• Transactions of the American Mathematical Society - View - PDF