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

• 成果类型: Article
• 署名作者: El Karoui, Noureddine
• 署名单位: University of California System; University of California Berkeley
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/009117906000000502
• 发表日期: 2006
• 页码: 2077-2117
• 关键词: sample covariance matrices; spacing distributions; ensembles; edge

摘要

It has been recently shown that if X is an n x N matrix whose entries are i.i.d. standard complex Gaussian and l(1) is the largest eigenvalue of X*X, there exist sequences m(n,N) and s(n,N) such that (l(1) - m(n,N))/s(n,N) converges in distribution to W-2, 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 F-2. In this paper we show that, under the assumption that n/N -> gamma is an element of (0, infinity), we can find a function M, continuous and nonincreasing, and sequences mu(n), N and sigma(n,N) such that, for all real s(0), there exists an integer N (s(0), gamma) for which, if (n boolean AND N) >= N (s(0), gamma), we have, with l(n,N) = (l(1) - mu(n,N))/sigma(n,N), for all s >= S-0 (n boolean AND N)(2/3) vertical bar P(l(n,N) <= s) F-2(s)vertical bar <= M(s(0))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 l(n,N) in simulations, an important fact from the point of view of (e.g., statistical) applications.