{
  "id": "1409.7975",
  "version": "v1",
  "published": "2014-09-29T00:53:01.000Z",
  "updated": "2014-09-29T00:53:01.000Z",
  "title": "The smallest singular value of random rectangular matrices with no moment assumptions on entries",
  "authors": [
    "Konstantin E. Tikhomirov"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\delta>1$ and $\\beta>0$ be some real numbers. We prove that there are positive $u,v,N_0$ depending only on $\\beta$ and $\\delta$ with the following property: for any $N,n$ such that $N\\ge \\max(N_0,\\delta n)$, any $N\\times n$ random matrix $A=(a_{ij})$ with i.i.d. entries satisfying $\\sup\\limits_{\\lambda\\in {\\mathbb R}}{\\mathbb P}\\bigl\\{|a_{11}-\\lambda|\\le 1\\bigr\\}\\le 1-\\beta$ and any non-random $N\\times n$ matrix $B$, the smallest singular value $s_n$ of $A+B$ satisfies ${\\mathbb P}\\bigl\\{s_n(A+B)\\le u\\sqrt{N}\\bigr\\}\\le \\exp(-vN)$. The result holds without any moment assumptions on distribution of the entries of $A$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-09-29T00:53:01.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "smallest singular value",
      "random rectangular matrices",
      "moment assumptions",
      "result holds",
      "real numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1409.7975T"
    }
  }
}