{
  "id": "1004.3848",
  "version": "v2",
  "published": "2010-04-22T07:02:41.000Z",
  "updated": "2011-08-23T11:26:02.000Z",
  "title": "On bilinear forms based on the resolvent of large random matrices",
  "authors": [
    "Walid Hachem",
    "Philippe Loubaton",
    "Jamal Najim",
    "Pascal Vallet"
  ],
  "comment": "35 pp. Extended version of the article accepted for publication in Annales de l'Institut Henri Poincar\\'e: Probabilit\\'e et Statistiques. Additions to the journal version are Section 4.4 and Sections 5.2, 5.3, 5.4, 5.5. These additions provide mathematical details of some aspects of the proofs",
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider a matrix $\\Sigma_n$ with random independent entries, each non-centered with a separable variance profile. In this article, we study the limiting behavior of the random bilinear form $u_n^* Q_n(z) v_n$, where $u_n$ and $v_n$ are deterministic vectors, and Q_n(z) is the resolvent associated to $\\Sigma_n \\Sigma_n^*$ as the dimensions of matrix $\\Sigma_n$ go to infinity at the same pace. Such quantities arise in the study of functionals of $\\Sigma_n \\Sigma_n^*$ which do not only depend on the eigenvalues of $\\Sigma_n \\Sigma_n^*$, and are pivotal in the study of problems related to non-centered Gram matrices such as central limit theorems, individual entries of the resolvent, and eigenvalue separation.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-08-23T11:26:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "large random matrices",
      "central limit theorems",
      "random bilinear form",
      "random independent entries",
      "deterministic vectors"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.3848H"
    }
  }
}