Weak Convergence of a Collection of Random Functions Defined by the Eigenvectors of Large Dimensional Random Matrices

Abstract

For each n, let Un be Haar distributed on the group of n × n unitary matrices. Let xn,1, . . . ,xn,m denote orthogonal nonrandom unit vectors in C n and let un,k = (uk, . . . , u n k) ∗ = U∗ nxn,k, k = 1, . . . , m. Define the following functions on [0,1]: X k,k n (t) = √ n ∑[nt] i=1(|uk|− 1 n ),X ′ n (t) = √ 2n ∑[nt] i=1 ū i ku i k′ , k < k ′. Then it is proven thatX n ,RXk,k ′ n , IXk,k′ n , considered as random processes in D[0, 1], converge weakly, as n → ∞, to m independent copies of Brownian bridge. The same result holds for the m(m + 1)/2 processes in the real case, where On is real orthogonal Haar distributed and xn,i ∈ R, with √ n in X n and √ 2n in X ′ n replaced with √ n 2 and √ n, respectively. This latter result will be shown to hold for the matrix of eigenvectors of Mn = (1/s)VnV T n where Vn is n × s consisting of the entries of {vij}, i, j = 1, 2, . . . , i.i.d. standardized and symmetrically distributed, with each xn,i = {±1/ √ n, . . . ,±1/√n}, and n/s→ y > 0 as n→ ∞. This result extends the result in J.W. Silverstein Ann. Probab. 18 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 Bn = θvnv ∗ n + Sn is studied where Sn is Hermitian or symmetric and nonnegative definite with either its matrix of eigenvectors being Haar distributed, or Sn = Mn, θ > 0 nonrandom, and vn is a nonrandom unit vector. Results are derived on the distributional behavior of the inner product of vectors orthogonal to vn with the eigenvector associated with the largest eigenvalue of Bn.