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

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 (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 μn,N and σn,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−μn,N)/σn,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.