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 $n\times N$ 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 $W_2$ is denoted $F_2$. In this paper we show that, under the assumption that $n/N\to \gamma\in(0,\infty)$, 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 $s_0$, there exists an integer $N(s_0,\gamma)$ for which, if $(n\wedge N)\geq N(s_0,\gamma)$, we have, with $l_{n,N}=(l_1-\tilde{\mu}_{n,N})/\tilde{\sigma}_{n,N}$, \[\forall s\geq s_0\qquad (n\wedge N)^{2/3}|P(l_{n,N}\leq s)-F_2(s)|\leq M(s_0)\exp(-s).\] The surprisingly good 2/3 rate and qualitative properties of the bounding function help explain the fact that the limiting distribution $W_2$ 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.