A rate of convergence result for the largest eigenvalue of complex white Wishart matrices
| dc.creator | Karoui, Noureddine El | |
| dc.date | 2004-09-30 | |
| dc.date | 2007-02-28 | |
| dc.date.accessioned | 2026-07-07T08:06:29Z | |
| dc.date.available | 2026-07-07T08:06:29Z | |
| dc.description | 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 $n/N\to γ\in(0,\infty)$, we can find a function $M$, continuous and nonincreasing, and sequences $\tildeμ_{n,N}$ and $\tildeσ_{n,N}$ such that, for all real $s_0$, there exists an integer $N(s_0,γ)$ for which, if $(n\wedge N)\geq N(s_0,γ)$, we have, with $l_{n,N}=(l_1-\tildeμ_{n,N})/\tildeσ_{n,N}$, \[\forall s\geq s_0\qquad (n\wedge N)^{2/3}|P(l_{n,N}\leq s)-F_2(s)|\leq M(s_0)\exp(-s).\] 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. | |
| dc.description | 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) | |
| dc.identifier | https://arxiv.org/abs/math/0409610 | |
| dc.identifier | http://arxiv.org/abs/math/0409610 | |
| dc.identifier | Annals of Probability 2006, Vol. 34, No. 6, 2077-2117 | |
| dc.identifier | doi:10.1214/009117906000000502 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/130621 | |
| dc.subject | Probability | |
| dc.subject | Statistics Theory | |
| dc.subject | 62E20 (Primary) 60F05 (Secondary) | |
| dc.title | A rate of convergence result for the largest eigenvalue of complex white Wishart matrices | |
| dc.type | text |