On asymptotic expansion and central limit theorem of linear eigenvalue statistics for sample covariance matrices when N/M →0

Abstract

We study the renormalized real sample covariance matrix H = X^T X/ √MN−√M/N with N/M → 0 as N,M →∞. We always assume M = M(N). Here X = [X<inf>jk</inf>]M×N is anM×N real random matrix with independent identically distributed entries, and we assume E|X<inf>11</inf>|^5+δ < ∞ with some small positive δ. The Stieltjes transform mN (z) = N⁻¹Tr(H − z)⁻¹ and the linear eigenvalue statistics of H are considered. We mainly focus on the asymptotic expansion of E{mN (z)} in this paper. Then for some fine test function, a central limit theorem for the linear eigenvalue statistics of H is established. We show that the variance of the limiting normal distribution coincides with the case of a real Wigner matrix with Gaussian entries.