Fluctuation of ergodic averages and other stochastic processes

Abstract

For an ergodic map $T$ and a non-constant, real-valued $f \in L^1$, the ergodic averages $\mathbb{A}_N f(x) = \frac{1} {N} \sum_{n=1}^N f(T^n x)$ converge a.e., but the convergence is never monotone. Depending on particular properties of the function $f$, the averages $\mathbb{A}_N f(x)$ may or may not actually fluctuate around the mean value infinitely often a.e. We will prove that a.e. fluctuation around the mean is the generic behavior. That is, for a fixed ergodic $T$, the generic non-constant $f\in L^1$ has the averages $\mathbb{A}_N f(x)$ fluctuating around the mean infinitely often for almost every $x$. We also consider fluctuation 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 fluctuation around the limit infinitely often persists as the generic behavior.