Convergence rate to the Tracy--Widom laws for the largest eigenvalue of sample covariance matrices
| Autor: | Schnelli, Kevin, Xu, Yuanyuan |
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| Rok vydání: | 2021 |
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| Druh dokumentu: | Working Paper |
| Popis: | 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 $M \times N$ 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 [73]. Our proof relies on a Green function comparison method [27] using iterative cumulant expansions, the local laws for the Green function and asymptotic properties of the correlation kernel of the white Wishart ensemble. Comment: arXiv admin note: text overlap with arXiv:2102.04330 |
| Databáze: | arXiv |
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