Strongly consistent autoregressive predictors in abstract Banach spaces

TL;DR

This paper establishes strong consistency results for autoregressive predictors in Banach spaces, extending previous work by defining a Rigged Hilbert space structure and analyzing the autocorrelation operator.

Contribution

It introduces new strong consistency results for autoregressive processes in Banach spaces using a Rigged Hilbert space framework, extending prior research.

Findings

Strong consistency of the autocorrelation operator estimator

Consistency of the plug-in predictor in the Banach space norm

Extension of previous autoregressive process results to Banach spaces

Abstract

This work derives new results on strong consistent estimation and prediction for autoregressive processes of order 1 in a separable Banach space B. The consistency results are obtained for the componentwise estimator of the autocorrelation operator in the norm of the space $\mathcal{L}(B)$ of bounded linear operators on B. The strong consistency of the associated plug-in predictor then follows in the $B$-norm. A Gelfand triple is defined through the Hilbert space constructed in Kuelbs' Lemma \cite{Kuelbs70}. A Hilbert--Schmidt embedding introduces the Reproducing Kernel Hilbert space (RKHS), generated by the autocovariance operator, into the Hilbert space conforming the Rigged Hilbert space structure. This paper extends the work of \cite{Bosq00} and \cite{LabbasMourid02}.