Asymptotic study of eigenelements of a sequence of random selfadjoint operators

When (Tn) is a sequence of selfadjoint random operators on a separable Hilbert space H converging almost surely to T, and converges in distribution to a random operator U, we give explicitly the asymptotic joint distribution of all the eigenelements of Tn (eigenvalues, eigenvectors and eigenprojectors) as a function of U. The results are obtained for real or complex operators, and for eigenvalues which arc simple or not. They have many applications in Multi-variate Analysis; for example, the asymptotic studies of Principal Component Analysis (real or complex), Canonical Analysis, Discriminant Analysis, Correspondence Analysis, Functional models.

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