{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0DFWHQ839T"
    },
    {
      "field": "uses",
      "target": "0FQ968YKA1"
    },
    {
      "field": "uses",
      "target": "0SVN6X695K"
    }
  ],
  "body": "\n## Definition\n\nLet $X, X_1, X_2, \\dots$ be real random variables on one probability space $(\\Omega, \\mathcal{F}, P)$.\nThe sequence $(X_n)$ converges almost surely to $X$ if\n$$P\\left(\\left\\{\\omega \\in \\Omega : \\lim_{n \\to \\infty} X_n(\\omega) = X(\\omega)\\right\\}\\right) = 1.$$\nOne writes $X_n \\to X$ a.s. or $X_n \\xrightarrow{\\text{a.s.}} X$.\n\n## Remarks\n\nThe set in the definition is an event, because the $X_n$ and $X$ are measurable.\nAn equivalent condition: for each $\\varepsilon \u003e 0$, $P(|X_n - X| \u003e \\varepsilon \\text{ for infinitely many } n) = 0$.\nThe limit is unique up to equality [[almost surely|0R46A1XY60]].\nThe strong law of large numbers is a result about almost sure convergence.\n\n## Explanations\n\n### L1 human\n\nAlmost sure convergence means that, apart from a set of outcomes of probability $0$, each run of the experiment gives a convergent sequence of numbers.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0AAXWGT5TD",
    "kind": "definition",
    "msc": [
      "60F15"
    ],
    "name": "Almost sure convergence",
    "sources": [
      {
        "locator": "Almost sure convergence",
        "url": "https://en.wikipedia.org/wiki/Convergence_of_random_variables"
      }
    ],
    "tag": "04P7D63W8P",
    "uses": [
      "0R46A1XY60",
      "0DKRR22AEQ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/04P7D63W8P/",
  "links": [
    {
      "field": "defines",
      "target": "0AAXWGT5TD"
    },
    {
      "field": "uses",
      "target": "0R46A1XY60"
    },
    {
      "field": "uses",
      "target": "0DKRR22AEQ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/04P7D63W8P.md",
  "msc": [
    "60F15"
  ],
  "name": "Almost sure convergence",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/04P/04P7D63W8P.md",
  "status": null,
  "tag": "04P7D63W8P",
  "uri": "m2://04P7D63W8P",
  "url": "https://math2.tamnd.com/wiki/almost-sure-convergence-definition/",
  "writing": {
    "tag": "04P7D63W8P",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}