---
_id: '1157'
abstract:
- lang: eng
  text: 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.
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-\r\nful to the
  anonymous referee for carefully reading our manuscript and suggesting several improvements."
article_processing_charge: No
arxiv: 1
author:
- first_name: Ji
  full_name: Lee, Ji
  last_name: Lee
- first_name: Kevin
  full_name: Schnelli, Kevin
  id: 434AD0AE-F248-11E8-B48F-1D18A9856A87
  last_name: Schnelli
  orcid: 0000-0003-0954-3231
citation:
  ama: 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>
  apa: Lee, J., &#38; 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>
  chicago: 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>.
  ieee: 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.
  ista: 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.
  mla: 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>.
  short: J. Lee, K. Schnelli, Annals of Applied Probability 26 (2016) 3786–3839.
date_created: 2018-12-11T11:50:27Z
date_published: 2016-12-15T00:00:00Z
date_updated: 2025-09-22T09:55:43Z
day: '15'
department:
- _id: LaEr
doi: 10.1214/16-AAP1193
ec_funded: 1
external_id:
  arxiv:
  - '1409.4979'
  isi:
  - '000391240100016'
fulldoi: https://doi.org/10.1214/16-AAP1193
intvolume: '        26'
isi: 1
issue: '6'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/abs/1409.4979
month: '12'
oa: 1
oa_version: Preprint
page: 3786 - 3839
project:
- _id: 258DCDE6-B435-11E9-9278-68D0E5697425
  call_identifier: FP7
  grant_number: '338804'
  name: Random matrices, universality and disordered quantum systems
publication: Annals of Applied Probability
publication_status: published
publisher: Institute of Mathematical Statistics
publist_id: '6201'
quality_controlled: '1'
scopus_import: '1'
status: public
title: Tracy-widom distribution for the largest eigenvalue of real sample covariance
  matrices with general population
type: journal_article
user_id: 317138e5-6ab7-11ef-aa6d-ffef3953e345
volume: 26
year: '2016'
...