February 2024 Tracy-Widom law for the extreme eigenvalues of large signal-plus-noise matrices
Zhixiang Zhang, Yiming Liu, Guangming Pan
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Bernoulli 30(1): 448-474 (February 2024). DOI: 10.3150/23-BEJ1604

Abstract

Let S=R+X be an M×N matrix where R is the signal matrix and X is the noise matrix consisting of i.i.d. standardized entries. The signal matrix R is allowed to be full rank, which is rarely studied in literature compared with the low rank cases. Under a regularity condition of R that assures the square root behaviour of the spectral density near the edge, we prove that the largest eigenvalue of SS has the Tracy-Widom distribution under a tail condition on the entries of X. Moreover, such a condition is proved to be necessary and sufficient to assure the Tracy-Widom law.

Funding Statement

Yiming Liu (corresponding author) was supported in part by the China Postdoctoral Science Foundations under Grant 2022M711342. Guangming Pan was supported in part by the Ministry of Education under Grant MOE2018-T2-2-112, in part by the Ministry of Education (MOE) Academic Research Grant (AcRF) Tier 1 under Grant RG76/21.

Acknowledgements

The authors thank the Editor and the reviewers for their valuable comments that improved the paper.

Citation

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Zhixiang Zhang. Yiming Liu. Guangming Pan. "Tracy-Widom law for the extreme eigenvalues of large signal-plus-noise matrices." Bernoulli 30 (1) 448 - 474, February 2024. https://doi.org/10.3150/23-BEJ1604

Information

Received: 1 May 2022; Published: February 2024
First available in Project Euclid: 8 November 2023

MathSciNet: MR4665585
zbMATH: 07788891
Digital Object Identifier: 10.3150/23-BEJ1604

Keywords: edge universality , Extreme eigenvalues , signal-plus-noise matrix , Tracy-Widom law

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Vol.30 • No. 1 • February 2024
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