# Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population

Lee J, Schnelli K. 2016. Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. Annals of Applied Probability. 26(6), 3786–3839.

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https://arxiv.org/abs/1409.4979
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*Journal Article*|

*Published*|

*English*

**Scopus indexed**

Author

Lee, Ji;
Schnelli, Kevin

^{ISTA}^{}Department

Abstract

We consider sample covariance matrices of the form Q = ( σ1/2X)(σ1/2X)∗, where the sample X is an M ×N random matrix whose entries are real independent random variables with variance 1/N and whereσ is an M × M positive-definite deterministic matrix. We analyze the asymptotic fluctuations of the largest rescaled eigenvalue of Q when both M and N tend to infinity with N/M →d ϵ (0,∞). For a large class of populations σ in the sub-critical regime, we show that the distribution of the largest rescaled eigenvalue of 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 σ is diagonal and that the entries of X have a sub-exponential decay.

Publishing Year

Date Published

2016-12-15

Journal Title

Annals of Applied Probability

Acknowledgement

We thank Horng-Tzer Yau for numerous discussions and remarks. We are grateful to Ben Adlam, Jinho Baik, Zhigang Bao, Paul Bourgade, László Erd ̋os, Iain Johnstone and Antti Knowles for comments. We are also grate-
ful to the anonymous referee for carefully reading our manuscript and suggesting several improvements.

Volume

26

Issue

6

Page

3786 - 3839

IST-REx-ID

### Cite this

Lee J, Schnelli K. Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population.

*Annals of Applied Probability*. 2016;26(6):3786-3839. doi:10.1214/16-AAP1193Lee, J., & Schnelli, K. (2016). Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population.

*Annals of Applied Probability*. Institute of Mathematical Statistics. https://doi.org/10.1214/16-AAP1193Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.”

*Annals of Applied Probability*. Institute of Mathematical Statistics, 2016. https://doi.org/10.1214/16-AAP1193.J. Lee and K. Schnelli, “Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population,”

*Annals of Applied Probability*, vol. 26, no. 6. Institute of Mathematical Statistics, pp. 3786–3839, 2016.Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.”

*Annals of Applied Probability*, vol. 26, no. 6, Institute of Mathematical Statistics, 2016, pp. 3786–839, doi:10.1214/16-AAP1193.**All files available under the following license(s):**

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