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November 2006 A rate of convergence result for the largest eigenvalue of complex white Wishart matrices
Noureddine El Karoui
Ann. Probab. 34(6): 2077-2117 (November 2006). DOI: 10.1214/009117906000000502


It has been recently shown 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 (l1mn,N)/sn,N converges in distribution to W2, the Tracy–Widom law appearing in the study of the Gaussian unitary ensemble. This probability law has a density which is known and computable. The cumulative distribution function of W2 is denoted F2.

In this paper we show that, under the assumption that n/Nγ∈(0, ∞), we can find a function M, continuous and nonincreasing, and sequences μ̃n,N and σ̃n,N such that, for all real s0, there exists an integer N(s0, γ) for which, if (nN)≥N(s0, γ), we have, with ln,N=(l1μ̃n,N)/σ̃n,N,

∀ ss0 (nN)2/3|P(ln,Ns)−F2(s)|≤M(s0)exp(−s).

The surprisingly good 2/3 rate and qualitative properties of the bounding function help explain the fact that the limiting distribution W2 is a good approximation to the empirical distribution of ln,N in simulations, an important fact from the point of view of (e.g., statistical) applications.


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Noureddine El Karoui. "A rate of convergence result for the largest eigenvalue of complex white Wishart matrices." Ann. Probab. 34 (6) 2077 - 2117, November 2006.


Published: November 2006
First available in Project Euclid: 13 February 2007

zbMATH: 1108.62014
MathSciNet: MR2294977
Digital Object Identifier: 10.1214/009117906000000502

Primary: 62E20
Secondary: 60F05

Keywords: Fredholm determinant , Liouville–Green approximation , Random matrix theory , trace class operators , Tracy–Widom distribution , Wishart matrices

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 6 • November 2006
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