# Consistency of the plug-in functional predictor of the Ornstein-Uhlenbeck process in Hilbert and Banach spaces

## TL;DR

This paper establishes the consistency of the maximum likelihood estimator and the plug-in predictor for the Ornstein-Uhlenbeck process in both Hilbert and Banach space frameworks, advancing functional prediction methods.

## Contribution

It provides new theoretical results on the consistency of estimators and predictors for the Ornstein-Uhlenbeck process in infinite-dimensional spaces.

## Key findings

- Consistency of the maximum likelihood estimator is proven in Hilbert and Banach spaces.
- The plug-in predictor based on the estimator is shown to be consistent.
- The results extend functional prediction theory to broader infinite-dimensional settings.

## Abstract

New results on functional prediction of the Ornstein-Uhlenbeck process in an autoregressive Hilbert-valued and Banach-valued frameworks are derived. Specifically, consistency of the maximum likelihood estimator of the autocorrelation operator, and of the associated plug-in predictor is obtained in both frameworks.