Stationary Process Invertibility and the Unilateral Shift Operator

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.

Theorems & Definitions (13)

• Definition 1: Unilateral and Bilateral Shift Operators
• Definition 2: Wiener Algebra
• Definition 3: Stationary Process
• Theorem 1
• Definition 4: Stationary Process Invertibility
• Theorem 2: $\ell^1$ Invertibility Condition
• Theorem 3: Nagy dilation
• Proposition 1
• proof
• Proposition 2