Weak Convergence of Random Functions Defined by The Eigenvectors of Sample Covariance Matrices

Let ᵢⱼ\, i, j = 1, 2, …, be i.i.d. symmetric random variables with E(ν⁴₁₁) < ∞, and for each n let Mₙ = (1/s)Vₙ VTₙ, where Vₙ = (vᵢⱼ), i = 1, 2, …, n, j = 1, 2, …, s = s(n) and n/s → y > 0 as n → ∞. Denote by Oₙ Λₙ OTₙ the spectral decomposition of Mₙ. Define X ∈ D 0, 1 by Xₙ(t) = √n/2 ∑ⁿᵗi = 1(y²ᵢ - 1/n) where (y₁, y₂, …, yₙ)T = OT(± 1/√n, ± 1/ √n, …, ± 1/√n)T. It is shown that Xₙ →D W⁰ as n → ∞, where W⁰ is a Brownian bridge. This result sheds some light on the problem of describing the behavior of the eigenvectors of Mₙ for n large and for general v₁₁.