Strongly consistent autoregressive predictors in abstract Banach spaces

TL;DR

This paper establishes new 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 a novel framework for strong consistent estimation and prediction of AR(1) processes in Banach spaces using a Rigged Hilbert space approach.

Findings

Proves strong consistency of the component-wise autocorrelation estimator

Demonstrates the plug-in predictor's strong consistency in B-norm

Extends previous AR process results to Banach space setting

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 component-wise estimator of the autocorrelation operator in the norm of the space 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 (1970)' lemma. 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 Bosq (2000) and Labbas and Mourid 2002.