Harmonic analysis meets stationarity: A general framework for series expansions of special Gaussian processes

TL;DR

This paper introduces a harmonic analysis-based framework for series expansions of Gaussian processes, including a new rate-optimal expansion for fractional Brownian motion, with applications to functional quantization.

Contribution

It develops a general, convergent series expansion framework for stationary Gaussian processes and proposes a new, simple, rate-optimal expansion for fractional Brownian motion.

Findings

Proposed a new series expansion for fractional Brownian motion with optimal convergence rate.

Established a general framework for series expansions of stationary Gaussian processes.

Applied the framework to optimal functional quantization.

Abstract

In this paper, we present a new approach to derive series expansions for some Gaussian processes based on harmonic analysis of their covariance function. In particular, we propose a new simple rate-optimal series expansion for fractional Brownian motion. The convergence of the latter series holds in mean square and uniformly almost surely, with a rate-optimal decay of the remainder of the series. We also develop a general framework of convergent series expansions for certain classes of Gaussian processes with stationarity. Finally, an application to optimal functional quantization is described.