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


Author(s): Lee, Ji O; Schnelli, Kevin
Article Title: Tracy-widom distribution for the largest eigenvalue of real sample covariance matrices with general population
Affiliation IST Austria
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.
Keywords: Tracy-Widom distribution; Sample covariance matrix; edge universality
Journal Title: Annals of Applied Probability
Volume: 26
Issue 6
ISSN: 1050-5164
Publisher: Institute of Mathematical Statistics  
Date Published: 2016-12-01
Start Page: 3786
End Page: 3839
URL:
DOI: 10.1214/16-AAP1193
Notes: 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.
Open access: yes (repository)
IST Austria Authors