Strictly stationary solutions of ARMA equations in Banach spaces

TL;DR

This paper establishes necessary and sufficient conditions for the existence of strictly stationary solutions to ARMA equations in Banach spaces, extending classical results to infinite-dimensional settings with spectral analysis.

Contribution

It provides a comprehensive spectral condition framework for the existence of stationary solutions of ARMA equations in Banach spaces, including ARMA(1,q) and ARMA(p,q).

Findings

Conditions for ARMA(1,q) exclude zero and the unit circle from the spectrum.

Extended spectral conditions to ARMA(p,q) equations.

Discussed various examples illustrating the theoretical results.

Abstract

We obtain necessary and sufficient conditions for the existence of strictly stationary solutions of ARMA equations in Banach spaces with independent and identically distributed noise under certain assumptions. First, we obtain conditions for ARMA(1,$q$) equations by excluding zero and the unit circle from the spectrum of the operator of the AR part. We then extend this to ARMA($p$,$q$) equations. Finally, we discuss various examples.