Second order ergodic theorems for ergodic transformations of infinite measure spaces

Abstract

For certain pointwise dual ergodic transformations T T we prove almost sure convergence of the log-averages \[ 1 log ⁡ N ∑ n = 1 N 1 n a ( n ) ∑ k = 1 n f ∘ T k ( f ∈ L 1 ) \frac {1}{{\log N}}\sum \limits _{n = 1}^N {\frac {1}{{na\left ( n \right )}}\sum \limits _{k = 1}^n {f \circ {T^k}\left ( {f \in {L_1}} \right )} } \] and the Chung-Erdös averages \[ 1 log ⁡ a ( N ) ∑ k = 1 N 1 a ( k ) f ∘ T k ( f ∈ L 1 + ) \frac {1}{{\log a\left ( N \right )}}\sum \limits _{k = 1}^N {\frac {1}{{a\left ( k \right )}}f \circ {T^k}} \left ( {f \in L_1^ + } \right ) \] towards ∫ f \smallint f , where a ( n ) a\left ( n \right ) denotes the return sequence of T T .