The Tracy--Widom limit for the largest eigenvalues of singular complex Wishart matrices

Abstract

This paper extends the work of El Karoui [Ann. Probab. 35 (2007) 663--714] which finds the Tracy--Widom limit for the largest eigenvalue of a nonsingular $p$-dimensional complex Wishart matrix $W_{\mathbb{C}}({\Omega }_p,n)$ to the case of several of the largest eigenvalues of the possibly singular $(n<p)$ matrix $W_{\mathbb{C}}({\Omega }_p,n).$ As a byproduct, we extend all results of Baik, Ben Arous and Peche [Ann. Probab. 33 (2005) 1643--1697] to the singular Wishart matrix case. We apply our findings to obtain a 95% confidence set for the number of common risk factors in excess stock returns.

Published in at http://dx.doi.org/10.1214/07-AAP454 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)

References in corpus (4)

• Level-Spacing Distributions and the Airy Kernel
• Determinantal random point fields
• Correlation Functions, Cluster Functions and Spacing Distributions for Random Matrices
• Tracy--Widom limit for the largest eigenvalue of a large class of complex sample covariance matrices

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