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

Abstract

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

Locations

• Random Matrices Theory and Application - View
• arXiv (Cornell University) - View - PDF