Oscillation of ergodic averages and other stochastic processes

For an ergodic map T and a non-constant, real-valued f L¹, the ergodic averages AN f (x) = 1 N ₍=₁N f (Tⁿ x) converge a. e. , but the convergence is never monotone. Depending on particular properties of the function f, the averages AN f (x) may or may not actually oscillate around the mean value infinitely often a. e. We will prove that a. e. oscillation around the mean is the generic behavior. That is, for a fixed ergodic T, the generic non-constant f L¹ has the averages AN f (x) oscillating around the mean infinitely often for almost every x. We also consider oscillation for other stochastic processes like subsequences of the ergodic averages, convolution operators, weighted averages, uniform distribution and martingales. We will show that in general, in these settings oscillation around the limit infinitely often persists as the generic behavior.