A rate of convergence result for the largest eigenvalue of complex white Wishart matrices. (English) Zbl 1108.62014

Summary: 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 [see C. A. Tracy and H. Widom, Commun. Math. Phys. 159, 151–174 (1994; Zbl 0789.35152); J. Stat. Phys. 92, No. 5–6, 809–835 (1998; Zbl 0942.60099)]. This probability law has a density which is known and computable. The cumulative distribution function of \(W_2\) is denoted \(F_2\).
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 \(\widetilde{\mu}_{n,N}\) and \(\widetilde{\sigma}_{n,N}\) 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 \(l_{n,N}= (l_1- \widetilde{\mu}_{n,N})/ \widetilde{\sigma}_{n,N}\), \[ \forall s\geq s_0, \quad (n\wedge N)^{2/3} |P(l_{n,N}\leq s)- F_2(s)|\leq M(s_0) \exp(-s). \] 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.


62E20 Asymptotic distribution theory in statistics
60F05 Central limit and other weak theorems
15B52 Random matrices (algebraic aspects)
62H10 Multivariate distribution of statistics
15A18 Eigenvalues, singular values, and eigenvectors


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