Asymptotic distributions for some functions of the eigenvalues of a sequence of random selfadjoint operators

We get asymptotic distributions for a class of functions of the eigenvalues of a sequence of random selfadjoint operators having some convergence properties. The results are useful for problems in multivariate analysis involving statistics which are functions of eigenvalues. As an illustration, the case of canonical analysis is consider-ed and a distribution-free method for determining the dimension in sliced inverse regression is discussed. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.