Prediction of long memory processes on same-realisation

TL;DR

This paper analyzes the properties of least-squares predictors for stationary Gaussian long memory processes, providing bounds, asymptotic error expressions, and a CLT for predictions based on the same realisation.

Contribution

It offers new theoretical results on the convergence and distribution of least-squares predictors for long memory processes, including moment bounds and a CLT.

Findings

Moment bounds for inverse empirical covariance matrix

Asymptotic mean-squared error expression for predictors

Central limit theorem for predictor convergence

Abstract

For the class of stationary Gaussian long memory processes, we study some properties of the least-squares predictor of X_{n+1} based on (X_n, ..., X_1). The predictor is obtained by projecting X_{n+1} onto the finite past and the coefficients of the predictor are estimated on the same realisation. First we prove moment bounds for the inverse of the empirical covariance matrix. Then we deduce an asymptotic expression of the mean-squared error. In particular we give a relation between the number of terms used to estimate the coefficients and the number of past terms used for prediction, which ensures the L^2-sense convergence of the predictor. Finally we prove a central limit theorem when our predictor converges to the best linear predictor based on all the past.