article
Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population
published
yes
Ji
Lee
author
Kevin
Schnelli
author 434AD0AE-F248-11E8-B48F-1D18A9856A870000-0003-0954-3231
LaEr
department
Random matrices, universality and disordered quantum systems
project
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.
Institute of Mathematical Statistics2016
eng
Annals of Applied Probability10.1214/16-AAP1193
2663786 - 3839
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.
J. Lee, K. Schnelli, Annals of Applied Probability 26 (2016) 3786–3839.
J. Lee and K. Schnelli, “Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population,” <i>Annals of Applied Probability</i>, 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.” <i>Annals of Applied Probability</i>, vol. 26, no. 6, Institute of Mathematical Statistics, 2016, pp. 3786–839, doi:<a href="https://doi.org/10.1214/16-AAP1193">10.1214/16-AAP1193</a>.
Lee, J., & Schnelli, K. (2016). Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics. <a href="https://doi.org/10.1214/16-AAP1193">https://doi.org/10.1214/16-AAP1193</a>
Lee, Ji, and Kevin Schnelli. “Tracy-Widom Distribution for the Largest Eigenvalue of Real Sample Covariance Matrices with General Population.” <i>Annals of Applied Probability</i>. Institute of Mathematical Statistics, 2016. <a href="https://doi.org/10.1214/16-AAP1193">https://doi.org/10.1214/16-AAP1193</a>.
Lee J, Schnelli K. Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population. <i>Annals of Applied Probability</i>. 2016;26(6):3786-3839. doi:<a href="https://doi.org/10.1214/16-AAP1193">10.1214/16-AAP1193</a>
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