An asymptotic Berry-Esseen result for the largest eigenvalue of complex white Wishart matrices

A number of results concerning the convergence in distribution of the largest eigenvalue of a large class of random covariance matrices have recently been obtained. In particular, it was shown in Johansson (2000), Johnstone (2001), and El Karoui (2003) that if X is an n×N matrix whose entries are i.i.d standard complex Gaussian and l1 is the largest eigenvalue of X∗X , there exist sequences mn,N and sn,N such that (l1 −mn,N)/sn,N converges in distribution to the Tracy-Widom law of order 2, denoted W2, a distribution whose density is known and computable. Its cumulative distribution function is denoted F2. In this paper, we show that we can find a function M , and sequences μ̃n,N and σ̃n,N such that when n and N go to infinity, with n/N → γ ∈ (0,∞), we have, with ln,N = (l1 − μ̃n,N )/σ̃n,N , ∀s, (n ∧N)|P (ln,N ≤ s)− F2(s)| ≤M(s) . The surprisingly good 2/3 rate helps explain the fact that the limiting distribution F2 is a good approximation to the empirical distribution of ln,N in simulations, an important fact from the point of view of (for instance, statistical) applications.