Moving Averages

TL;DR

This paper investigates the convergence properties of moving averages in ergodic theory, providing conditions for universal convergence, constructing counterexamples, and exploring the impact of function spaces and growth rates.

Contribution

It characterizes when moving averages converge universally, constructs examples where convergence fails, and extends results to polynomial growth cases in ergodic processes.

Findings

Universal convergence of moving averages is characterized by complete convergence of ergodic averages.

Counterexamples are constructed showing non-convergence for certain functions and maps.

Results are extended to moving averages with polynomial growth.

Abstract

We consider the convergence of moving averages in the general setting of ergodic theory or stationary ergodic processes. We characterize when there is universal convergence of moving averages based on complete convergence to zero of the standard ergodic averages. Using a theorem of Hsu-Robbins (1947) for independent, identically distributed processes, we prove for any bounded measurable function $f$ on a standard probability space $(X,\mathcal{B},\mu)$, there exists a Bernoulli shift $T$, such that all moving averages $M(v_n, L_n)^T f = \frac{1}{L_n} \sum_{i=v_n+1}^{v_n+L_n} f \circ T^i$ with $L_n\geq n$ converge a.e. to $\int_X f d\mu$. We refresh the reader about the cone condition established by Bellow, Jones, Rosenblatt (1990) which guarantees convergence of certain moving averages for all $f \in L^1(\mu)$ and ergodic measure preserving maps $T$. We show given $f \in L^1(\mu)$ and ergodic measure preserving $T$, there exists a moving average $M(v_n,L_n)^T f$ with $L_n$ strictly increasing such that $(v_n,L_n)$ does not satisfy the cone condition, but pointwise convergence holds a.e. We show for any non-zero $f\in L^1(\mu)$, there is a generic class of ergodic maps $T$ such that each map has an associated moving average $M(v_n, L_n)^T f$ which does not converge pointwise. It is known if $f\in L^2(\mu)$ is mean-zero, then there exist solutions $T$ and $g\in L^1(\mu)$ to the coboundary equation: $f = g - g\circ T$. This implies $f$ and $T$ produce universal moving averages. We show this does not generalize to $L^p(\mu)$ for $p<2$ by explicitly defining functions $f\in \cap_{p<2}L^p(\mu)$ such that for each ergodic measure preserving $T$, there exists a moving average $M(v_n, L_n)^T f$ with $L_n\geq n$ such that these moving averages do not converge pointwise. Several of the results are generalized to the case of moving averages with polynomial growth.