Author(s):

Lee, Ji O; Schnelli, Kevin

Article Title: 
Tracywidom 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 positivedefinite 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 subcritical regime, we show that the distribution of the largest rescaled eigenvalue of Q is given by the type1 TracyWidom 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 subexponential decay.

Keywords: 
TracyWidom distribution; Sample covariance matrix; edge universality

Journal Title:

Annals of Applied Probability

Volume: 
26

Issue 
6

ISSN:

10505164

Publisher:

Institute of Mathematical Statistics

Date Published:

20161201

Start Page: 
3786

End Page:

3839

URL: 

DOI: 
10.1214/16AAP1193

Notes: 
We thank HorngTzer 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) 