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 m_n,N and s_n,N such that (l₁−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 μ̃_n,N and σ̃_n,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 l_n,N=(l₁−μ̃_n,N)/σ̃_n,N,

∀ s≥s₀ (n∧N)^2/3|P(l_n,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 l_n,N in simulations, an important fact from the point of view of (e.g., statistical) applications.