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arXiv · 2604.02336

The Shift Operator Calculus for Stationary Time Series Analysis

Anand Ganesh·Babhrubahan Bose·Anand Rajagopalan
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Abstract

The article establishes a rigorous shift operator calculus for stationary time series modeling, addressing a certain gap in the literature. It provides proofs of existence and isometry for the transfer function operators $f(B)$ and $f(T)$ where $B$ is the bilateral shift operator and $T$ is the unilateral shift operator for different families of functions $f$. The article establishes convergence of the power series of $f(B)$ and $f(T)$ under the operator norm for the Wiener algebra $\mathbb{W}_+$, and convergence under strong operator topology for $f$ in $H^{\infty}$, based on the use of Abel sums. Based on this calculus, it unifies the notion of stationary process invertibility with the operator invertibility of the transfer function $f(T)$.

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BibTeXRIS

Anand Ganesh, Babhrubahan Bose, Anand Rajagopalan. 2026-01-20. The Shift Operator Calculus for Stationary Time Series Analysis. https://arxiv.org/abs/2604.02336

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