  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.
