A variant of the ergodic maximal function is studied. This maximal function reduces to the usual one for $f \geqq 0$, but the cancellation between the positive and negative parts of $f$ causes interesting behavior. In particular the maximal function can be in $L^1$ even when the function is not in $L \log^+ L$. A relation between this maximal function and the ergodic Hilbert transform is studied.
"The Ergodic Maximal Function with Cancellation." Ann. Probab. 4 (1) 91 - 97, February, 1976. https://doi.org/10.1214/aop/1176996184