## 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.

### MSC:

 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

### Citations:

Zbl 0942.60099; Zbl 0789.35152

DLMF
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### References:

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