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Article: Extreme eigenvalues of log-concave ensemble
| Title | Extreme eigenvalues of log-concave ensemble |
|---|---|
| Authors | |
| Keywords | Local law Log-concave distribution Sample covariance matrix Spectral rigidity Spiked model Tracy–Widom law |
| Issue Date | 1-Feb-2025 |
| Publisher | Institute of Mathematical Statistics |
| Citation | Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2025, v. 61, n. 1, p. 155-184 How to Cite? |
| Abstract | In this paper, we consider the log-concave ensemble of random matrices, a class of covariance-type matrices XX∗ with isotropic log-concave X-columns. A main example is the covariance estimator of the uniform measure on isotropic convex body. Non-asymptotic estimates and first order asymptotic limits for the extreme eigenvalues have been obtained in the literature. In this paper, with the recent advancements on log-concave measures (Geom. Funct. Anal. 31 (2021) 34–61; Geom. Funct. Anal. 32 (2022) 1134–1159), we take a step further to locate the eigenvalues with a nearly optimal precision, namely, the spectral rigidity of this ensemble is derived. Based on the spectral rigidity and an additional “unconditional” assumption, we further derive the Tracy–Widom law for the extreme eigenvalues of XX∗, and the Gaussian law for the extreme eigenvalues in case strong spikes are present. |
| Persistent Identifier | http://hdl.handle.net/10722/358649 |
| ISSN | 2023 Impact Factor: 1.2 2023 SCImago Journal Rankings: 1.555 |
| DC Field | Value | Language |
|---|---|---|
| dc.contributor.author | Bao, Zhigang | - |
| dc.contributor.author | Xu, Xiaocong | - |
| dc.date.accessioned | 2025-08-13T07:47:12Z | - |
| dc.date.available | 2025-08-13T07:47:12Z | - |
| dc.date.issued | 2025-02-01 | - |
| dc.identifier.citation | Annales de l'Institut Henri Poincaré, Probabilités et Statistiques, 2025, v. 61, n. 1, p. 155-184 | - |
| dc.identifier.issn | 0246-0203 | - |
| dc.identifier.uri | http://hdl.handle.net/10722/358649 | - |
| dc.description.abstract | <p>In this paper, we consider the log-concave ensemble of random matrices, a class of covariance-type matrices XX<sup>∗</sup> with isotropic log-concave X-columns. A main example is the covariance estimator of the uniform measure on isotropic convex body. Non-asymptotic estimates and first order asymptotic limits for the extreme eigenvalues have been obtained in the literature. In this paper, with the recent advancements on log-concave measures (Geom. Funct. Anal. 31 (2021) 34–61; Geom. Funct. Anal. 32 (2022) 1134–1159), we take a step further to locate the eigenvalues with a nearly optimal precision, namely, the spectral rigidity of this ensemble is derived. Based on the spectral rigidity and an additional “unconditional” assumption, we further derive the Tracy–Widom law for the extreme eigenvalues of XX<sup>∗</sup>, and the Gaussian law for the extreme eigenvalues in case strong spikes are present.</p> | - |
| dc.language | eng | - |
| dc.publisher | Institute of Mathematical Statistics | - |
| dc.relation.ispartof | Annales de l'Institut Henri Poincaré, Probabilités et Statistiques | - |
| dc.subject | Local law | - |
| dc.subject | Log-concave distribution | - |
| dc.subject | Sample covariance matrix | - |
| dc.subject | Spectral rigidity | - |
| dc.subject | Spiked model | - |
| dc.subject | Tracy–Widom law | - |
| dc.title | Extreme eigenvalues of log-concave ensemble | - |
| dc.type | Article | - |
| dc.identifier.doi | 10.1214/23-AIHP1439 | - |
| dc.identifier.scopus | eid_2-s2.0-85218963130 | - |
| dc.identifier.volume | 61 | - |
| dc.identifier.issue | 1 | - |
| dc.identifier.spage | 155 | - |
| dc.identifier.epage | 184 | - |
| dc.identifier.issnl | 0246-0203 | - |