On convergence of sample and population Hilbertian functional principal components

In this article we consider the sequences of sample and population covariance operators for a sequence of arrays of Hilbertian random elements. Then under the assumptions that sequences of the covariance operators norm are uniformly bounded and the sequences of the principal component scores are uniformly sumable, we prove that the convergence of the sequences of covariance operators would imply the convergence of the corresponding sequences of the sample andpopulation eigenvalues and eigenvectors, and vice versa. In particular we prove that the principal component scores converge in distribution in certain family of Hilbertian elliptically contoured distributions.