Estimating invertible processes in Hilbert spaces, with applications to functional ARMA processes

TL;DR

This paper develops a general theoretical framework for estimating invertible operators in functional time series within Hilbert spaces, providing consistency and error bounds applicable to a wide range of models.

Contribution

It introduces consistent estimators and explicit asymptotic error bounds for invertible operators in functional time series, extending analysis to general Hilbert spaces and broad classes of models.

Findings

Derived explicit asymptotic error bounds for operator estimation.

Established consistency rates for operators in functional ARMA models.

Applied results to a broad class of functional linear processes.

Abstract

Invertible processes are central to functional time series analysis, making the estimation of their defining operators a key problem. While asymptotic error bounds have been established for specific ARMA models on $L^2[0,1]$, a general theoretical framework has not yet been considered. This paper fills in this gap by deriving consistent estimators for the operators characterizing the invertible representation of a functional time series with white noise innovations in a general separable Hilbert space. Under mild conditions covering a broad class of functional time series, we establish explicit asymptotic error bounds, with rates determined by operator smoothness and eigenvalue decay. These results further provide consistency-rate estimates for operators in Hilbert space-valued causal linear processes, including functional MA, AR, and ARMA models of arbitrary order.