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 $X^{\ast }X$, 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 $(n\wedge N)\ge N(s_0,\gamma )$, 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.

Published at http://dx.doi.org/10.1214/009117906000000502 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org)