Asymptotic normality for the empirical estimator of the autocorrelation operator of an ARH(1) process

We consider the ARH(1) model defined by X-t = rho(Xt-1) + epsilon(t), where rho is a linear operator on a Hilbert space; X-t and epsilon(t) are Hilber t space valued random variables ((epsilon(t))(t is an element of N) is a wh ite noise). We derive a central limit theor-ern for an estimator of the aut ocorrelation operator rho based an the empirical covariance operators of th e process and on the projection of the data on a subspace whose dimension i ncreases with the number of observations. (C) 1999 Academie des sciences/Ed itions scientifiques et medicales Elsevier SAS.