Abstract: New lower bound inequalities are obtained for generalized upcrossings of ergodic averages. Results and techniques are presented in such a way that the duality with Bishop's results on upper bounds is emphasized. Finally, the significance of generalized upcrossings as a mean to count spatial oscillations is clarified.

Keywords:Cesaro averages. Upcrossing inequalities. Pointwise a.e. convergence. Asymptotics.