The Annals of Applied Probability

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

Alexei Onatski

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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 Wp, n) to the case of several of the largest eigenvalues of the possibly singular (n<p) matrix Wp, 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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Ann. Appl. Probab., Volume 18, Number 2 (2008), 470-490.

First available in Project Euclid: 20 March 2008

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Primary: 60F05: Central limit and other weak theorems 62E20: Asymptotic distribution theory

Singular Wishart matrix Tracy–Widom distribution largest eigenvalues random matrix theory approximate factor model number of factors arbitrage pricing theory


Onatski, Alexei. The Tracy–Widom limit for the largest eigenvalues of singular complex Wishart matrices. Ann. Appl. Probab. 18 (2008), no. 2, 470--490. doi:10.1214/07-AAP454.

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