A Rate of Convergence Result for the Largest Eigenvalue of Complex White Wishart Matrices

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$m_{n,N}$and$s_{n,N}$such that$(l_{1}-m_{n,N})/s_{n,N}$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$\tilde{\mu}_{n,N}$and$\tilde{\sigma}_{n,N}$such that, for all real s₀, there exists an integer N(s₀, γ) for which, if$(n\wedge N)\geq N(s_{0},\gamma)$, we have, with$l_{n,N}=(l_{1}-\tilde{\mu}_{n,N})/\tilde{\sigma}_{n,N}$,$\forall s\geq s_{0}\ (n\wedge N)^{2/3}|P(l_{n,N}\leq s)-F_{2}(s)|\leq M(s_{0}){\rm exp}(-s)$. 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.