Gaussian Fluctuations in Complex Sample Covariance Matrices

Abstract

Let $X=(X_{i,j})_{m\times n}, m\ge n$, be a complex Gaussian random matrix with mean zero and variance $\frac 1n$, let $S=X^*X$ be a sample covariance matrix. In this paper we are mainly interested in the limiting behavior of eigenvalues when $\frac mn\rightarrow \gamma\ge 1$ as $n\rightarrow\infty$. Under certain conditions on $k$, we prove the central limit theorem holds true for the $k$-th largest eigenvalues $\lambda_{(k)}$ as $k$ tends to infinity as $n\rightarrow\infty$. The proof is largely based on the Costin-Lebowitz-Soshnikov argument and the asymptotic estimates for the expectation and variance of the number of eigenvalues in an interval. The standard technique for the RH problem is used to compute the exact formula and asymptotic properties for the mean density of eigenvalues. As a by-product, we obtain a convergence speed of the mean density of eigenvalues to the Marchenko-Pastur distribution density under the condition $|\frac mn-\gamma|=O(\frac 1n)$.