SPHARMA approximations for stationary functional time series on the sphere

TL;DR

This paper introduces SPHARMA processes for modeling stationary isotropic random fields on the sphere-cross-time domain, providing a framework for approximation and spectral analysis in infinite-dimensional settings.

Contribution

The paper extends spherical functional autoregressions to SPHARMA processes and demonstrates their ability to approximate all isotropic stationary sphere-cross-time fields.

Findings

SPHARMA processes can approximate any isotropic stationary sphere-cross-time field.

Spectral representation theorems are established for these processes.

Wold-like decompositions are derived for the proposed models.

Abstract

In this paper, we focus on isotropic and stationary sphere-cross-time random fields. We first introduce the class of spherical functional autoregressive-moving average processes (SPHARMA), which extend in a natural way the spherical functional autoregressions (SPHAR) recently studied in [8, 7]; more importantly, we then show that SPHAR and SPHARMA processes of sufficiently large order can be exploited to approximate every isotropic and stationary sphere-cross-time random field, thus generalizing to this infinite-dimensional framework some classical results on real-valued stationary processes. Further characterizations in terms of functional spectral representation theorems and Wold-like decompositions are also established.