The Spectral-Domain $\mathcal{W}_2$ Wasserstein Distance for Elliptical Processes and the Spectral-Domain Gelbrich Bound

TL;DR

This paper introduces a spectral-domain Wasserstein distance for elliptical stochastic processes based on their power spectra, along with a spectral-domain Gelbrich bound applicable to more general processes, enhancing spectral analysis tools.

Contribution

The paper presents a novel spectral-domain Wasserstein distance and Gelbrich bound, expanding the analytical framework for stochastic processes in spectral space.

Findings

Defined spectral-domain $ ext{W}_2$ Wasserstein distance for elliptical processes.

Introduced spectral-domain Gelbrich bound for non-elliptical processes.

Provides new spectral analysis tools for stochastic process comparison.

Abstract

In this short note, we introduce the spectral-domain $\mathcal{W}_2$ Wasserstein distance for elliptical stochastic processes in terms of their power spectra. We also introduce the spectral-domain Gelbrich bound for processes that are not necessarily elliptical.