Stationary Process Invertibility and the Unilateral Shift Operator

TL;DR

This paper explores the use of the unilateral shift operator for analyzing stationary process invertibility, establishing a rigorous operator theoretic foundation and connecting it with transfer function invertibility.

Contribution

It unifies stationary process invertibility with algebraic invertibility of transfer functions using the unilateral shift operator and operator theory.

Findings

Proves that for f in Wiener algebra, f(T) is well defined and equals the Toeplitz operator.

Establishes that the norm of f(T) equals the supremum norm of f.

Links process invertibility with algebraic invertibility of transfer functions.

Abstract

The bilateral shift operator $B$ has been the mainstay of stationary process modeling whereas we argue that the unilateral shift operator $T$ may be better suited to analyze invertibility. While doing so, we partially unify the notion of stationary process invertibility (associated with a sufficent but not necessary $\ell^1$ condition) with the algebraic invertibility of the transfer function $f(T)$. We establish a rigorous operator theoretic foundation for these arguments proving that for $f \in \mathbb{W}_+$, the Wiener algebra, $f(T)$ is well defined, that $\| f(T) \| = \| f \|_{\infty}$ and that $f(T) = T_f$, the Toeplitz operator.