Second order ergodic theorems for ergodic transformations of infinite measure spaces

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 $T$.