A note on strong-consistency of componentwise ARH(1) predictors

TL;DR

This paper establishes new strong-consistency results in trace norm for componentwise ARH(1) predictors, even when eigenvectors are unknown and the autocorrelation operator lacks a diagonal spectral form.

Contribution

It introduces a novel strong-consistency result for a diagonal componentwise estimator of the autocorrelation operator in ARH(1) processes under general conditions.

Findings

Proves strong-consistency of the estimator in trace norm

Allows for unknown eigenvectors of the autocovariance operator

Supports strong-consistency of the plug-in predictor

Abstract

This paper presents a new result on strong-consistency, in the trace norm, of a diagonal componentwise parameter estimator of the autocorrelation operator of an autoregressive process of order one (ARH(1) process), allowing strong-consistency of the associated plug-in predictor. These results are derived, when the eigenvectors of the autocovariance operator are unknown, and the autocorrelation operator does not admit a diagonal spectral representation with respect to the eigenvectors of the autocovariance operator.