Time-average approximation of the principal components analysis of a stochastic process

The principal components analysis (PCA) of a L-2-continuous stochastic process (X-t)(t)epsilon[o,T] is often realized by approximating the principal factors in a finite dimensional space of functions. In this paper we study the time-average approximations obtained after discretizing [0,T] in p subintervals. For these approximations we establish bounds for the errors as well as a criterion and a necessary condition for the optimal choice of the discretizations. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.