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

TL;DR

This paper generalizes the Tracy--Widom limit to the largest eigenvalues of singular complex Wishart matrices, extending prior results and enabling applications like confidence sets for risk factors in finance.

Contribution

It extends the Tracy--Widom limit to multiple largest eigenvalues of singular Wishart matrices, broadening theoretical understanding and practical applications.

Findings

Extended Tracy--Widom limit to singular matrices

Generalized results of Baik, Ben Arous, and Peche

Provided a 95% confidence set for risk factors

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.