Fluctuation of ergodic averages and other stochastic processes

TL;DR

The paper addresses non-monotonicity and fluctuation of convergent stochastic processes in ergodic settings, focusing on classical ergodic averages and extending to subsequences, convolutions, martingales, uniformly distributed sequences, and weighted averages. It shows that for an ergodic map $T$ and a non-constant $f\in L^1$, the averages $\mathbb{A}_{[N]}f(T^n x)$ converge a.e. but fluctuate around the limit for a.e. $x$, and that this fluctuation is generic across a variety of averaging schemes, including subsequences with the complete recurrence property, proper approximate identities, dyadic martingales, UD sequences, and weighted averages. The work introduces the complete recurrence property and ties it to ubiquitous fluctuation phenomena, providing both qualitative and quantitative results and highlighting how recurrence structure governs convergence behavior in ergodic theory. Overall, the results reveal the pervasiveness of a fluctuating regime across diverse stochastic processes linked to ergodic transformations, with implications for understanding rates of convergence and robustness of limits in dynamical systems.

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.

Theorems & Definitions (85)

• Definition 1.1
• Proposition 3.1
• Proposition 3.2
• Theorem 3.3
• Example 3.4
• Definition 3.5
• Theorem 3.6
• Theorem 3.9
• Theorem 3.10
• Definition 3.11