{
  "aliases": [
    "converse-borel-cantelli-lemma"
  ],
  "backlinks": null,
  "body": "\n## Statement\n\nLet $A_1, A_2, \\dots$ be [[mutually independent|0GCVEV6BJA]] events in a probability space $(\\Omega, \\mathcal{F}, P)$ with $\\sum_{n=1}^{\\infty} P(A_n) = \\infty$.\nThen\n$$P\\left(\\limsup_{n \\to \\infty} A_n\\right) = 1.$$\nSo almost surely, infinitely many of the events $A_n$ occur.\n\n## Remarks\n\nProof: the complement of $\\limsup_n A_n$ is $\\bigcup_n \\bigcap_{m \\ge n} A_m^c$. By independence and $1 - x \\le e^{-x}$, $P\\left(\\bigcap_{m = n}^{N} A_m^c\\right) = \\prod_{m=n}^{N} (1 - P(A_m)) \\le \\exp\\left(-\\sum_{m=n}^{N} P(A_m)\\right)$.\nThe right side tends to $0$ as $N \\to \\infty$. So each $\\bigcap_{m \\ge n} A_m^c$ has probability $0$, and so does their union.\nIndependence is necessary. For $A_n = A$ with $0 \u003c P(A) \u003c 1$ for all $n$, the sum diverges but $P(\\limsup_n A_n) = P(A) \u003c 1$.\nWith the [[first lemma|0T6WS7TT77]], it follows that for independent events, $P(\\limsup_n A_n)$ is $0$ or $1$.\n\n## Explanations\n\n### L1 human\n\nIf independent events have chances that add up to infinity, then infinitely many of them happen, with probability $1$.\nFor example, in an infinite sequence of tosses of a fair coin, the pattern \"ten heads in a row\" appears infinitely often.\n",
  "commit": "f591989ed15704e9d8755a7a5a6426d73de7a3ae",
  "front": {
    "aliases": [
      "converse-borel-cantelli-lemma"
    ],
    "kind": "statement",
    "msc": [
      "60F20"
    ],
    "name": "Second Borel–Cantelli lemma",
    "role": "lemma",
    "sources": [
      {
        "locator": "Converse result",
        "url": "https://en.wikipedia.org/wiki/Borel–Cantelli_lemma"
      }
    ],
    "tag": "0NWXRAMEE7",
    "uses": [
      "0T6WS7TT77",
      "0GCVEV6BJA",
      "0PQEBTYDZM"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0NWXRAMEE7/",
  "links": [
    {
      "field": "uses",
      "target": "0T6WS7TT77"
    },
    {
      "field": "uses",
      "target": "0GCVEV6BJA"
    },
    {
      "field": "uses",
      "target": "0PQEBTYDZM"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0NWXRAMEE7.md",
  "msc": [
    "60F20"
  ],
  "name": "Second Borel–Cantelli lemma",
  "prov": null,
  "role": "lemma",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0NW/0NWXRAMEE7.md",
  "status": null,
  "tag": "0NWXRAMEE7",
  "uri": "m2://0NWXRAMEE7",
  "url": "https://math2.tamnd.com/wiki/second-borel-cantelli-lemma/",
  "writing": {
    "tag": "0NWXRAMEE7",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}