Prediction of long memory processes on same-realisation

Abstract

For the class of stationary Gaussian long memory processes, we study some properties of the leastsquares 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.

Estimation of inverse autocovariance matrices for long memory processes

“…Remark 3.2. It would be interesting to compare Theorem 3.1 the moment bounds for Γ−1 Kn,n given by Godet [8]. If {u t } is a Gaussian process satisfying (1.2)-(1.5) and (1.8), then Theorem 2.1 of Godet [8] yields that for…”