The Annals of Applied Probability

Tracy–Widom distribution for the largest eigenvalue of real sample covariance matrices with general population

Ji Oon Lee and Kevin Schnelli

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We consider sample covariance matrices of the form $\mathcal{Q}=(\Sigma^{1/2}X)(\Sigma^{1/2}X)^{*}$, where the sample $X$ is an $M\times N$ random matrix whose entries are real independent random variables with variance $1/N$ and where $\Sigma$ is an $M\times M$ positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of $\mathcal{Q}$ when both $M$ and $N$ tend to infinity with $N/M\to d\in(0,\infty)$. For a large class of populations $\Sigma$ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of $\mathcal{Q}$ is given by the type-1 Tracy–Widom distribution under the additional assumptions that (1) either the entries of $X$ are i.i.d. Gaussians or (2) that $\Sigma$ is diagonal and that the entries of $X$ have a sub-exponential decay.

Article information

Ann. Appl. Probab., Volume 26, Number 6 (2016), 3786-3839.

Received: June 2015
Revised: January 2016
First available in Project Euclid: 15 December 2016

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Zentralblatt MATH identifier

Primary: 60B20: Random matrices (probabilistic aspects; for algebraic aspects see 15B52) 15B52: Random matrices 62H10: Distribution of statistics

Sample covariance matrix Tracy–Widom distribution edge universality


Lee, Ji Oon; Schnelli, Kevin. Tracy–Widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Ann. Appl. Probab. 26 (2016), no. 6, 3786--3839. doi:10.1214/16-AAP1193.

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