2004/09/30 by Noureddine El Karoui · 3 citations
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Bayesian Methods and Mixture Models #Random Matrices and Applications #math.PR #math.ST #msc:60F05 #msc:62E20 #stat.TH
paper · pdf · doi:10.1214/009117906000000502
published as Annals of Probability 2006, Vol. 34, No. 6, 2077-2117 · Published at http://dx.doi.org/10.1214/009117906000000502 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2006/11/01 · arxiv created 2007/02/28 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31
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 (l1−mn,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/N→ γ∈(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 (n∧N)≥N(s0, γ), we have, with ln,N=(l1−μ̃n,N)/σ̃n,N, ∀ s≥s0 (n∧N)2/3|P(ln,N≤s)−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.