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_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.