The Annals of Probability

A rate of convergence result for the largest eigenvalue of complex white Wishart matrices

Noureddine El Karoui

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Abstract

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.

Article information

Source
Ann. Probab., Volume 34, Number 6 (2006), 2077-2117.

Dates
First available in Project Euclid: 13 February 2007

Permanent link to this document
https://projecteuclid.org/euclid.aop/1171377438

Digital Object Identifier
doi:10.1214/009117906000000502

Mathematical Reviews number (MathSciNet)
MR2294977

Zentralblatt MATH identifier
1108.62014

Subjects
Primary: 62E20: Asymptotic distribution theory
Secondary: 60F05: Central limit and other weak theorems

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

Citation

El Karoui, Noureddine. A rate of convergence result for the largest eigenvalue of complex white Wishart matrices. Ann. Probab. 34 (2006), no. 6, 2077--2117. doi:10.1214/009117906000000502. https://projecteuclid.org/euclid.aop/1171377438


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