Gaussian fluctuations in complex sample covariance matrices
Zhonggen Su, Zhejiang University
Abstract
Let X=(Xi,j)m×n, m ≥ n, be a complex Gaussian
random matrix with mean zero and variance 1/n, let S=X*X
be a sample covariance matrix. In this paper we are mainly
interested in the limiting behavior of eigenvalues when m/n -> γ ≥ 1. Under certain
conditions on k, we prove the central limit theorem holds true
for the k-th largest eigenvalues l(k) as k tends to
infinity as n->∞. 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
| m/n-g|=O(1/n)
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