---
_id: '14775'
abstract:
- lang: eng
text: We establish a quantitative version of the Tracy–Widom law for the largest
eigenvalue of high-dimensional sample covariance matrices. To be precise, we show
that the fluctuations of the largest eigenvalue of a sample covariance matrix
X∗X converge to its Tracy–Widom limit at a rate nearly N−1/3, where X is an M×N
random matrix whose entries are independent real or complex random variables,
assuming that both M and N tend to infinity at a constant rate. This result improves
the previous estimate N−2/9 obtained by Wang (2019). Our proof relies on a Green
function comparison method (Adv. Math. 229 (2012) 1435–1515) using iterative cumulant
expansions, the local laws for the Green function and asymptotic properties of
the correlation kernel of the white Wishart ensemble.
acknowledgement: K. Schnelli was supported by the Swedish Research Council Grants
VR-2017-05195, and the Knut and Alice Wallenberg Foundation. Y. Xu was supported
by the Swedish Research Council Grant VR-2017-05195 and the ERC Advanced Grant “RMTBeyond”
No. 101020331.
article_processing_charge: No
article_type: original
author:
- first_name: Kevin
full_name: Schnelli, Kevin
id: 434AD0AE-F248-11E8-B48F-1D18A9856A87
last_name: Schnelli
orcid: 0000-0003-0954-3231
- first_name: Yuanyuan
full_name: Xu, Yuanyuan
id: 7902bdb1-a2a4-11eb-a164-c9216f71aea3
last_name: Xu
orcid: 0000-0003-1559-1205
citation:
ama: Schnelli K, Xu Y. Convergence rate to the Tracy–Widom laws for the largest
eigenvalue of sample covariance matrices. The Annals of Applied Probability.
2023;33(1):677-725. doi:10.1214/22-aap1826
apa: Schnelli, K., & Xu, Y. (2023). Convergence rate to the Tracy–Widom laws
for the largest eigenvalue of sample covariance matrices. The Annals of Applied
Probability. Institute of Mathematical Statistics. https://doi.org/10.1214/22-aap1826
chicago: Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom
Laws for the Largest Eigenvalue of Sample Covariance Matrices.” The Annals
of Applied Probability. Institute of Mathematical Statistics, 2023. https://doi.org/10.1214/22-aap1826.
ieee: K. Schnelli and Y. Xu, “Convergence rate to the Tracy–Widom laws for the largest
eigenvalue of sample covariance matrices,” The Annals of Applied Probability,
vol. 33, no. 1. Institute of Mathematical Statistics, pp. 677–725, 2023.
ista: Schnelli K, Xu Y. 2023. Convergence rate to the Tracy–Widom laws for the largest
eigenvalue of sample covariance matrices. The Annals of Applied Probability. 33(1),
677–725.
mla: Schnelli, Kevin, and Yuanyuan Xu. “Convergence Rate to the Tracy–Widom Laws
for the Largest Eigenvalue of Sample Covariance Matrices.” The Annals of Applied
Probability, vol. 33, no. 1, Institute of Mathematical Statistics, 2023, pp.
677–725, doi:10.1214/22-aap1826.
short: K. Schnelli, Y. Xu, The Annals of Applied Probability 33 (2023) 677–725.
date_created: 2024-01-10T09:23:31Z
date_published: 2023-02-01T00:00:00Z
date_updated: 2024-01-10T13:31:46Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/22-aap1826
ec_funded: 1
external_id:
arxiv:
- '2108.02728'
isi:
- '000946432400021'
intvolume: ' 33'
isi: 1
issue: '1'
keyword:
- Statistics
- Probability and Uncertainty
- Statistics and Probability
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2108.02728
month: '02'
oa: 1
oa_version: Preprint
page: 677-725
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
call_identifier: H2020
grant_number: '101020331'
name: Random matrices beyond Wigner-Dyson-Mehta
publication: The Annals of Applied Probability
publication_identifier:
issn:
- 1050-5164
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
scopus_import: '1'
status: public
title: Convergence rate to the Tracy–Widom laws for the largest eigenvalue of sample
covariance matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 33
year: '2023'
...
---
_id: '14780'
abstract:
- lang: eng
text: In this paper, we study the eigenvalues and eigenvectors of the spiked invariant
multiplicative models when the randomness is from Haar matrices. We establish
the limits of the outlier eigenvalues λˆi and the generalized components (⟨v,uˆi⟩
for any deterministic vector v) of the outlier eigenvectors uˆi with optimal convergence
rates. Moreover, we prove that the non-outlier eigenvalues stick with those of
the unspiked matrices and the non-outlier eigenvectors are delocalized. The results
also hold near the so-called BBP transition and for degenerate spikes. On one
hand, our results can be regarded as a refinement of the counterparts of [12]
under additional regularity conditions. On the other hand, they can be viewed
as an analog of [34] by replacing the random matrix with i.i.d. entries with Haar
random matrix.
acknowledgement: The authors would like to thank the editor, the associated editor
and two anonymous referees for their many critical suggestions which have significantly
improved the paper. The authors are also grateful to Zhigang Bao and Ji Oon Lee
for many helpful discussions. The first author also wants to thank Hari Bercovici
for many useful comments. The first author is partially supported by National Science
Foundation DMS-2113489 and the second author is supported by ERC Advanced Grant
“RMTBeyond” No. 101020331.
article_processing_charge: Yes (in subscription journal)
article_type: original
author:
- first_name: Xiucai
full_name: Ding, Xiucai
last_name: Ding
- first_name: Hong Chang
full_name: Ji, Hong Chang
id: dd216c0a-c1f9-11eb-beaf-e9ea9d2de76d
last_name: Ji
citation:
ama: Ding X, Ji HC. Spiked multiplicative random matrices and principal components.
Stochastic Processes and their Applications. 2023;163:25-60. doi:10.1016/j.spa.2023.05.009
apa: Ding, X., & Ji, H. C. (2023). Spiked multiplicative random matrices and
principal components. Stochastic Processes and Their Applications. Elsevier.
https://doi.org/10.1016/j.spa.2023.05.009
chicago: Ding, Xiucai, and Hong Chang Ji. “Spiked Multiplicative Random Matrices
and Principal Components.” Stochastic Processes and Their Applications.
Elsevier, 2023. https://doi.org/10.1016/j.spa.2023.05.009.
ieee: X. Ding and H. C. Ji, “Spiked multiplicative random matrices and principal
components,” Stochastic Processes and their Applications, vol. 163. Elsevier,
pp. 25–60, 2023.
ista: Ding X, Ji HC. 2023. Spiked multiplicative random matrices and principal components.
Stochastic Processes and their Applications. 163, 25–60.
mla: Ding, Xiucai, and Hong Chang Ji. “Spiked Multiplicative Random Matrices and
Principal Components.” Stochastic Processes and Their Applications, vol.
163, Elsevier, 2023, pp. 25–60, doi:10.1016/j.spa.2023.05.009.
short: X. Ding, H.C. Ji, Stochastic Processes and Their Applications 163 (2023)
25–60.
date_created: 2024-01-10T09:29:25Z
date_published: 2023-09-01T00:00:00Z
date_updated: 2024-01-16T08:49:51Z
day: '01'
ddc:
- '510'
department:
- _id: LaEr
doi: 10.1016/j.spa.2023.05.009
ec_funded: 1
external_id:
arxiv:
- '2302.13502'
isi:
- '001113615900001'
file:
- access_level: open_access
checksum: 46a708b0cd5569a73d0f3d6c3e0a44dc
content_type: application/pdf
creator: dernst
date_created: 2024-01-16T08:47:31Z
date_updated: 2024-01-16T08:47:31Z
file_id: '14806'
file_name: 2023_StochasticProcAppl_Ding.pdf
file_size: 1870349
relation: main_file
success: 1
file_date_updated: 2024-01-16T08:47:31Z
has_accepted_license: '1'
intvolume: ' 163'
isi: 1
keyword:
- Applied Mathematics
- Modeling and Simulation
- Statistics and Probability
language:
- iso: eng
license: https://creativecommons.org/licenses/by/4.0/
month: '09'
oa: 1
oa_version: Published Version
page: 25-60
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
call_identifier: H2020
grant_number: '101020331'
name: Random matrices beyond Wigner-Dyson-Mehta
publication: Stochastic Processes and their Applications
publication_identifier:
eissn:
- 1879-209X
issn:
- 0304-4149
publication_status: published
publisher: Elsevier
quality_controlled: '1'
status: public
title: Spiked multiplicative random matrices and principal components
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 163
year: '2023'
...
---
_id: '14849'
abstract:
- lang: eng
text: We establish a precise three-term asymptotic expansion, with an optimal estimate
of the error term, for the rightmost eigenvalue of an n×n random matrix with independent
identically distributed complex entries as n tends to infinity. All terms in the
expansion are universal.
acknowledgement: "The second and the fourth author were supported by the ERC Advanced
Grant\r\n“RMTBeyond” No. 101020331. The third author was supported by Dr. Max Rössler,
the\r\nWalter Haefner Foundation and the ETH Zürich Foundation."
article_processing_charge: No
article_type: original
author:
- first_name: Giorgio
full_name: Cipolloni, Giorgio
id: 42198EFA-F248-11E8-B48F-1D18A9856A87
last_name: Cipolloni
orcid: 0000-0002-4901-7992
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Dominik J
full_name: Schröder, Dominik J
id: 408ED176-F248-11E8-B48F-1D18A9856A87
last_name: Schröder
orcid: 0000-0002-2904-1856
- first_name: Yuanyuan
full_name: Xu, Yuanyuan
last_name: Xu
citation:
ama: Cipolloni G, Erdös L, Schröder DJ, Xu Y. On the rightmost eigenvalue of non-Hermitian
random matrices. The Annals of Probability. 2023;51(6):2192-2242. doi:10.1214/23-aop1643
apa: Cipolloni, G., Erdös, L., Schröder, D. J., & Xu, Y. (2023). On the rightmost
eigenvalue of non-Hermitian random matrices. The Annals of Probability.
Institute of Mathematical Statistics. https://doi.org/10.1214/23-aop1643
chicago: Cipolloni, Giorgio, László Erdös, Dominik J Schröder, and Yuanyuan Xu.
“On the Rightmost Eigenvalue of Non-Hermitian Random Matrices.” The Annals
of Probability. Institute of Mathematical Statistics, 2023. https://doi.org/10.1214/23-aop1643.
ieee: G. Cipolloni, L. Erdös, D. J. Schröder, and Y. Xu, “On the rightmost eigenvalue
of non-Hermitian random matrices,” The Annals of Probability, vol. 51,
no. 6. Institute of Mathematical Statistics, pp. 2192–2242, 2023.
ista: Cipolloni G, Erdös L, Schröder DJ, Xu Y. 2023. On the rightmost eigenvalue
of non-Hermitian random matrices. The Annals of Probability. 51(6), 2192–2242.
mla: Cipolloni, Giorgio, et al. “On the Rightmost Eigenvalue of Non-Hermitian Random
Matrices.” The Annals of Probability, vol. 51, no. 6, Institute of Mathematical
Statistics, 2023, pp. 2192–242, doi:10.1214/23-aop1643.
short: G. Cipolloni, L. Erdös, D.J. Schröder, Y. Xu, The Annals of Probability 51
(2023) 2192–2242.
date_created: 2024-01-22T08:08:41Z
date_published: 2023-11-01T00:00:00Z
date_updated: 2024-01-23T10:56:30Z
day: '01'
department:
- _id: LaEr
doi: 10.1214/23-aop1643
ec_funded: 1
external_id:
arxiv:
- '2206.04448'
intvolume: ' 51'
issue: '6'
keyword:
- Statistics
- Probability and Uncertainty
- Statistics and Probability
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2206.04448
month: '11'
oa: 1
oa_version: Preprint
page: 2192-2242
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
call_identifier: H2020
grant_number: '101020331'
name: Random matrices beyond Wigner-Dyson-Mehta
publication: The Annals of Probability
publication_identifier:
issn:
- 0091-1798
publication_status: published
publisher: Institute of Mathematical Statistics
quality_controlled: '1'
status: public
title: On the rightmost eigenvalue of non-Hermitian random matrices
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 51
year: '2023'
...
---
_id: '15128'
abstract:
- lang: eng
text: "We prove a universal mesoscopic central limit theorem for linear eigenvalue
statistics of a Wigner-type matrix inside the bulk of the spectrum with compactly
supported twice continuously differentiable test functions. The main novel ingredient
is an optimal local law for the two-point function $T(z,\\zeta)$ and a general
class of related quantities involving two resolvents\r\nat nearby spectral parameters. "
acknowledgement: Supported by the ERC Advanced Grant ”RMTBeyond” No. 101020331
article_number: '2301.01712'
article_processing_charge: No
author:
- first_name: Volodymyr
full_name: Riabov, Volodymyr
id: 1949f904-edfb-11eb-afb5-e2dfddabb93b
last_name: Riabov
citation:
ama: Riabov V. Mesoscopic eigenvalue statistics for Wigner-type matrices. arXiv.
doi:10.48550/arXiv.2301.01712
apa: Riabov, V. (n.d.). Mesoscopic eigenvalue statistics for Wigner-type matrices.
arXiv. https://doi.org/10.48550/arXiv.2301.01712
chicago: Riabov, Volodymyr. “Mesoscopic Eigenvalue Statistics for Wigner-Type Matrices.”
ArXiv, n.d. https://doi.org/10.48550/arXiv.2301.01712.
ieee: V. Riabov, “Mesoscopic eigenvalue statistics for Wigner-type matrices,” arXiv.
.
ista: Riabov V. Mesoscopic eigenvalue statistics for Wigner-type matrices. arXiv,
2301.01712.
mla: Riabov, Volodymyr. “Mesoscopic Eigenvalue Statistics for Wigner-Type Matrices.”
ArXiv, 2301.01712, doi:10.48550/arXiv.2301.01712.
short: V. Riabov, ArXiv (n.d.).
date_created: 2024-03-20T09:41:04Z
date_published: 2023-01-04T00:00:00Z
date_updated: 2024-03-25T12:48:20Z
day: '04'
department:
- _id: GradSch
- _id: LaEr
doi: 10.48550/arXiv.2301.01712
ec_funded: 1
external_id:
arxiv:
- '2301.01712'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2301.01712
month: '01'
oa: 1
oa_version: Preprint
project:
- _id: 62796744-2b32-11ec-9570-940b20777f1d
call_identifier: H2020
grant_number: '101020331'
name: Random matrices beyond Wigner-Dyson-Mehta
publication: arXiv
publication_status: submitted
status: public
title: Mesoscopic eigenvalue statistics for Wigner-type matrices
type: preprint
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
year: '2023'
...
---
_id: '12179'
abstract:
- lang: eng
text: We derive an accurate lower tail estimate on the lowest singular value σ1(X−z)
of a real Gaussian (Ginibre) random matrix X shifted by a complex parameter z.
Such shift effectively changes the upper tail behavior of the condition number
κ(X−z) from the slower (κ(X−z)≥t)≲1/t decay typical for real Ginibre matrices
to the faster 1/t2 decay seen for complex Ginibre matrices as long as z is away
from the real axis. This sharpens and resolves a recent conjecture in [J. Banks
et al., https://arxiv.org/abs/2005.08930, 2020] on the regularizing effect of
the real Ginibre ensemble with a genuinely complex shift. As a consequence we
obtain an improved upper bound on the eigenvalue condition numbers (known also
as the eigenvector overlaps) for real Ginibre matrices. The main technical tool
is a rigorous supersymmetric analysis from our earlier work [Probab. Math. Phys.,
1 (2020), pp. 101--146].
article_processing_charge: No
article_type: original
author:
- first_name: Giorgio
full_name: Cipolloni, Giorgio
id: 42198EFA-F248-11E8-B48F-1D18A9856A87
last_name: Cipolloni
orcid: 0000-0002-4901-7992
- first_name: László
full_name: Erdös, László
id: 4DBD5372-F248-11E8-B48F-1D18A9856A87
last_name: Erdös
orcid: 0000-0001-5366-9603
- first_name: Dominik J
full_name: Schröder, Dominik J
id: 408ED176-F248-11E8-B48F-1D18A9856A87
last_name: Schröder
orcid: 0000-0002-2904-1856
citation:
ama: Cipolloni G, Erdös L, Schröder DJ. On the condition number of the shifted real
Ginibre ensemble. SIAM Journal on Matrix Analysis and Applications. 2022;43(3):1469-1487.
doi:10.1137/21m1424408
apa: Cipolloni, G., Erdös, L., & Schröder, D. J. (2022). On the condition number
of the shifted real Ginibre ensemble. SIAM Journal on Matrix Analysis and Applications.
Society for Industrial and Applied Mathematics. https://doi.org/10.1137/21m1424408
chicago: Cipolloni, Giorgio, László Erdös, and Dominik J Schröder. “On the Condition
Number of the Shifted Real Ginibre Ensemble.” SIAM Journal on Matrix Analysis
and Applications. Society for Industrial and Applied Mathematics, 2022. https://doi.org/10.1137/21m1424408.
ieee: G. Cipolloni, L. Erdös, and D. J. Schröder, “On the condition number of the
shifted real Ginibre ensemble,” SIAM Journal on Matrix Analysis and Applications,
vol. 43, no. 3. Society for Industrial and Applied Mathematics, pp. 1469–1487,
2022.
ista: Cipolloni G, Erdös L, Schröder DJ. 2022. On the condition number of the shifted
real Ginibre ensemble. SIAM Journal on Matrix Analysis and Applications. 43(3),
1469–1487.
mla: Cipolloni, Giorgio, et al. “On the Condition Number of the Shifted Real Ginibre
Ensemble.” SIAM Journal on Matrix Analysis and Applications, vol. 43, no.
3, Society for Industrial and Applied Mathematics, 2022, pp. 1469–87, doi:10.1137/21m1424408.
short: G. Cipolloni, L. Erdös, D.J. Schröder, SIAM Journal on Matrix Analysis and
Applications 43 (2022) 1469–1487.
date_created: 2023-01-12T12:12:38Z
date_published: 2022-07-01T00:00:00Z
date_updated: 2023-01-27T06:56:06Z
day: '01'
department:
- _id: LaEr
doi: 10.1137/21m1424408
external_id:
arxiv:
- '2105.13719'
intvolume: ' 43'
issue: '3'
keyword:
- Analysis
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://doi.org/10.48550/arXiv.2105.13719
month: '07'
oa: 1
oa_version: Preprint
page: 1469-1487
publication: SIAM Journal on Matrix Analysis and Applications
publication_identifier:
eissn:
- 1095-7162
issn:
- 0895-4798
publication_status: published
publisher: Society for Industrial and Applied Mathematics
quality_controlled: '1'
scopus_import: '1'
status: public
title: On the condition number of the shifted real Ginibre ensemble
type: journal_article
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 43
year: '2022'
...