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The Tracy-Widom limit for the largest eigenvalues of singular complex Wishart matrices. (English) Zbl 1141.60009

N. El Karoui [Ann. Probab. 35, No. 2, 663–714 (2007; Zbl 1117.60020)] proved that the asymptotic distribution of the properly scaled and centered largest eigenvalue of a nonsingular complex Wishart matrix, \(W_{\mathcal{C}}(\sum_p/n,n),\) is the Tracy-Widom distribution of type 2, which refers to a distribution with the distribution function (\(\sum_p-\)some covariance matrix) \[ F(x)=\exp\left(-\int_x^{\infty}(x-s)q^2(s)ds\right), \] \(q(s)\) is the solution of an ordinary differential equation \(q''(x)=sq(s)+2q^3(s).\) The main goal of this paper is to extend his result and to show that the asymptotic distribution of the scaled and centered \(m\) largest eigenvalues \((m<\infty)\) of a possibly singular \(p-\)dimensional complex Wishart matrix, \(W_{\mathcal{C}}(\sum_p/n,n)\) (as both \(n\) and \(p\) tend to infinity in such a way that \(n/p\) remains in a compact subset of \((0,\infty)\)) is the same as the joint asymptotic distribution of \(\tilde{d}_1,\dots,\tilde{d}_m,\) where \(\tilde{d}_i=N^{2/3}(d_i-2),\;d_1\geq \cdots\geq d_N\) are eigenvalues of a matrix from the so-called Gaussian Unitary Ensemble which is the collection of all \(N\times N\) Hermitian matrices with iid complex Gaussian \(N_{\mathcal{C}}(0,1/N)\) lower triangular entries and (independent of them) iid real Gaussian \(N(0,1/N)\) diagonal entries, and \(m\) is any fixed positive number. As a by-product all results of J. Baik, G. Ben Arous and S. Péch’e [Ann. Probab. 33, No. 5, 1643–1697 (2005; Zbl 1086.15022)] are extended to the singular Wishart matrix case and then all author’s findings are applied to obtain a 95% confidence set for the number of common risk factors in excess stock returns. Besides are described in introduction 1) G. Chamberlain and M. Rothschild’s approximate factor model [Econometrica 51, 1281–1304 (1983; Zbl 0523.90017)], which researchers in macroeconomics and finance use to handle high-dimensional data sets and in which in contrast to the classical factor model the error-variables are allowed to be correlated over the \(i\)-th dimension, 2) a powerful method of analysis of the joint asymptotic distribution of a few of the largest eigenvalues of various random matrices as the dimensionality of the matrices tends to infinity.

MSC:

60F05 Central limit and other weak theorems
62E20 Asymptotic distribution theory in statistics
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