On an $L^2$ norm for stationary ARMA processes

TL;DR

This paper introduces an $L^2$ norm for stationary ARMA processes, based on the Wold decomposition, and uses it to derive and empirically verify bounds on mean square prediction errors.

Contribution

It proposes a novel $L^2$ norm for ARMA models and applies it to establish and validate bounds on prediction errors.

Findings

Derived bounds on mean square prediction error for AR(1) models of MA(1) processes.

Verified bounds empirically using sample data.

Abstract

We propose an $L^2$ norm for stationary Autoregressive Moving Average (ARMA) models. We look at ARMA models within the Hilbert space of the past with present of a true purely linearly non-deterministic stationary process $X_t$, and compute the $L^2$ norm based on its Wold decomposition. As an application of this $L^2$ norm, we derive bounds on the mean square prediction error for AR(1) models of MA(1) processes, and verify these bounds empirically for sample data.