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

2008/03/28 by Alexei Onatski
Mathematics · #math.PR #msc:60F05 #msc:62E20

paper · pdf · doi:10.1214/07-aap454

published as Annals of Applied Probability 2008, Vol. 18, No. 2, 470-490 · 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)

arxiv created 2008/03/28 · arxiv updated 2009/12/01

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_ℂ(Ω_p,n) to the case of several of the largest eigenvalues of the possibly singular (n<p) matrix W_ℂ(Ω_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.

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