{
  "id": "0910.0067",
  "version": "v1",
  "published": "2009-10-01T01:53:10.000Z",
  "updated": "2009-10-01T01:53:10.000Z",
  "title": "On the Borel-Cantelli Lemma and its Generalization",
  "authors": [
    "Chunrong Feng",
    "Liangpan Li",
    "Jian Shen"
  ],
  "comment": "5 Pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "Let $\\{A_n\\}_{n=1}^{\\infty}$ be a sequence of events on a probability space $(\\Omega,\\mathcal{F},\\mathbf{P})$. We show that if $\\lim_{m\\to\\infty}\\sum_{n=1}^{m}w_n\\mathbf{P}(A_n)=\\infty$ where each $w_n\\in\\mathbb{R}$, then \\[{\\mathbf{P}}(\\limsup A_n)\\geq\\limsup_{n\\to\\infty} \\frac{\\displaystyle\\big(\\sum_{k=1}^n{w_k\\mathbf{P}}(A_k)\\big)^2}{\\displaystyle\\sum_{i=1}^n\\sum_{j=1}^nw_iw_j{\\mathbf{P}}(A_i\\cap A_j)}.\\]",
  "revisions": [
    {
      "version": "v1",
      "updated": "2009-10-01T01:53:10.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05"
    ],
    "keywords": [
      "borel-cantelli lemma",
      "generalization"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0910.0067F"
    }
  }
}