A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices
| dc.creator | Soshnikov, Alexander | |
| dc.date | 2001-04-10 | |
| dc.date | 2001-06-25 | |
| dc.date.accessioned | 2026-07-07T04:41:15Z | |
| dc.date.available | 2026-07-07T04:41:15Z | |
| dc.description | Recently Johansson and Johnstone proved that the distribution of the (properly rescaled) largest principal component of the complex (real) Wishart matrix $ X^* \* X (X^t \*X) $ converges to the Tracy-Widom law as $ n, p $ (the dimensions of $ X $) tend to $ \infty $ in some ratio $ n/p \to γ>0. $ We extend these results in two directions. First of all, we prove that the joint distribution of the first, second, third, etc. eigenvalues of a Wishart matrix converges (after a proper rescaling) to the Tracy-Widom distribution. Second of all, we explain how the combinatorial machinery developed for Wigner matrices allows to extend the results by Johansson and Johnstone to the case of $ X $ with non-Gaussian entries, provided $ n-p =O(p^{1/3}) . $ We also prove that $ λ_{max} \leq (n^{1/2}+p^{1/2})^2 +O(p^{1/2}\*\log(p)) $ (a.e.) for general $ γ>0.$ | |
| dc.description | This is a preliminary version. Minor misprints are corrected | |
| dc.identifier | https://arxiv.org/abs/math/0104113 | |
| dc.identifier | http://arxiv.org/abs/math/0104113 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/61278 | |
| dc.subject | Probability | |
| dc.subject | Mathematical Physics | |
| dc.title | A note on universality of the distribution of the largest eigenvalues in certain sample covariance matrices | |
| dc.type | text |