Weighted Birkhoff averages: Deterministic and probabilistic perspectives

TL;DR

This paper surveys weighted Birkhoff averages for deterministic systems, establishing rapid convergence results and probabilistic laws, with applications to Fourier coefficients and ergodic theory.

Contribution

It introduces weighted averaging with compact support, providing universal rapid convergence and new probabilistic results, improving upon classical ergodic theory methods.

Findings

Universal exponential convergence for weighted Fourier coefficients

Quantitative improvements under regularity assumptions

Probabilistic laws like weighted strong law of large numbers

Abstract

In this paper, we survey physically related applications of a class of weighted quasi-Monte Carlo methods from a theoretical, deterministic perspective, and establish quantitative universal rapid convergence results via various regularity assumptions. Specifically, we introduce weighting with compact support to the Birkhoff ergodic averages of quasi-periodic, almost periodic, and periodic systems, thereby achieving universal rapid convergence, including both arbitrary polynomial and exponential types. This is in stark contrast to the typically slow convergence in classical ergodic theory. As new contributions, we not only discuss more general weighting functions but also provide quantitative improvements to existing results; the explicit regularity settings facilitate the application of these methods to specific problems. We also revisit the physically related problems and, for the first time, establish universal exponential convergence results for the weighted computation of Fourier coefficients, in both finite-dimensional and infinite-dimensional cases. In addition to the above, we explore results from a probabilistic perspective, including the weighted strong law of large numbers and the weighted central limit theorem, by building upon the historical results.