Abstract
In this article, we provide a review of some fundamental theories concerning the local laws of Green's functions of high-dimensional sample covariance matrices, along with their pertinent applications. Employing a cumulant expansion method, we establish an almost sharp local law for the Green's function of a sample covariance matrix ensemble, valid down to the optimal scale of the spectral parameter. We further discuss its applications to the proof of the Tracy-Widom law of the largest eigenvalue and the study of spiked sample covariance matrices. © 2024 Elsevier B.V.
| Original language | English |
|---|---|
| Title of host publication | Handbook of Statistics |
| Editors | Arni S.R. Srinivasa Rao, Zhidong Bai, C.R. Rao |
| Publisher | Elsevier |
| Chapter | 6 |
| Pages | 143-173 |
| Volume | 51 |
| ISBN (Print) | 978-0-443-29328-3 |
| DOIs | |
| Publication status | Published - 2024 |
Funding
ZB is supported in part by Hong Kong RGC grant GRF 16303922. YH is supported in part by the National Key Research and Development Program of China (No. 2023YFA1010400) and NSFC Excellent Young Scientist Scheme (No. 12322121). FY is supported in part by the National Key Research and Development Program of China (No. 2023YFA1010400).
Research Keywords
- Green's function
- Local law
- Sample covariance matrix
- Spiked covariance model
- Tracy-Widom law
RGC Funding Information
- RGC-funded
Fingerprint
Dive into the research topics of 'Random matrix theory: Local laws and applications'. Together they form a unique fingerprint.Cite this
- APA
- Author
- BIBTEX
- Harvard
- Standard
- RIS
- Vancouver