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.
Noureddine El Karoui. "A rate of convergence result for the largest eigenvalue of complex white Wishart matrices." Ann. Probab. 34 (6) 2077 - 2117, November 2006. https://doi.org/10.1214/009117906000000502