Weak convergence of a collection of random functions defined by the eigenvectors of large dimensional random matrices

For each n, let [Formula: see text] be Haar distributed on the group of unitary matrices. Let denote orthogonal nonrandom unit vectors in and text], text]. Define following functions text]: Then it is proven that considered as random processes converge weakly, to independent copies Brownian bridge. The same result holds for real case, where with replaced respectively. This latter will shown hold matrix eigenvectors consisting entries i.i.d. standardized symmetrically distributed, extends [J. W. Silverstein, Ann. Probab. 18 (1990) 1174–1194]. These results are applied detection problem sampling mostly made noise detecting whether sample includes a vector. studied Hermitian or symmetric nonnegative definite either its being Results derived distributional behavior inner product eigenvector associated largest eigenvalue