On moving averages

TL;DR

This paper explores the mathematical properties and convergence of moving averages, including their connections to fixed point methods, Banach spaces, and convex analysis, providing rigorous proofs and explicit limits.

Contribution

It establishes a rigorous convergence proof for a specific moving average algorithm and extends analysis to Banach spaces and convex function averages.

Findings

Proves convergence of a Gauss-Seidel type fixed point method for moving averages.

Identifies explicit limits of the moving average sequences.

Extends analysis to Banach spaces and convex function averages.

Abstract

We show that the moving arithmetic average is closely connected to a Gauss-Seidel type fixed point method studied by Bauschke, Wang and Wylie, and which was observed to converge only numerically. Our analysis establishes a rigorous proof of convergence of their algorithm in a special case; moreover, limit is explicitly identified. Moving averages in Banach spaces and Kolmogorov means are also studied. Furthermore, we consider moving proximal averages and epi-averages of convex functions.