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

J. Lee, K. Schnelli, Annals of Applied Probability 26 (2016) 3786–3839.


Journal Article | Published | English
Author
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-AAP1193
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. https://doi.org/10.1214/16-AAP1193
Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” Annals of Applied Probability 26, no. 6 (2016): 3786–3839. 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, pp. 3786–3839, 2016.
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.
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.

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