{
  "id": "2108.02728",
  "version": "v1",
  "published": "2021-08-05T16:48:17.000Z",
  "updated": "2021-08-05T16:48:17.000Z",
  "title": "Convergence rate to the Tracy--Widom laws for the largest eigenvalue of sample covariance matrices",
  "authors": [
    "Kevin Schnelli",
    "Yuanyuan Xu"
  ],
  "categories": [
    "math.PR",
    "math.ST",
    "stat.TH"
  ],
  "abstract": "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-08-05T16:48:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60B20",
      "62H10"
    ],
    "keywords": [
      "sample covariance matrix",
      "largest eigenvalue",
      "tracy-widom law",
      "convergence rate",
      "high dimensional sample covariance matrices"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}