Extremal eigenvalues of sample covariance matrices with general population
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We analyze the behavior of the largest eigenvalues of sample covariance matrices of the form Q=(Σ^1/2X)(Σ^1/2X)^*. The sample X is an M× N rectangular random matrix with real independent entries and the population covariance matrix Σ is a positive definite diagonal matrix independent of X. In the limit M, N →∞ with N/M→ d∈[1,∞), we prove the relation between the largest eigenvalues of Q and Σ that holds when d is above a certain threshold. When the entries of Σ are i.i.d., the limiting distribution of the largest eigenvalue of Q is given by a Weibull distribution.
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