{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "0BFDHF2H7N"
    },
    {
      "field": "uses",
      "target": "0EPTDZPJ83"
    },
    {
      "field": "uses",
      "target": "0FQ968YKA1"
    },
    {
      "field": "uses",
      "target": "0SVN6X695K"
    },
    {
      "field": "uses",
      "target": "0XJVF81QF6"
    }
  ],
  "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 in probability to $X$ if for each $\\varepsilon \u003e 0$,\n$$\\lim_{n \\to \\infty} P\\left(|X_n - X| \u003e \\varepsilon\\right) = 0.$$\nOne writes $X_n \\xrightarrow{P} X$.\n\n## Remarks\n\nThe limit is unique up to equality almost surely.\nIf $X_n \\to X$ in probability, then a subsequence of $(X_n)$ converges to $X$ almost surely.\nThe weak law of large numbers is a result about convergence in probability.\nExample: let $X_n$ be $1$ with probability $1/n$ and $0$ otherwise. Then $X_n \\to 0$ in probability.\n\n## Explanations\n\n### L1 human\n\nConvergence in probability means that, for large $n$, $X_n$ is very likely to be close to $X$.\nIt does not say that one fixed run of the experiment gives a convergent sequence.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "defines": "0QA1HBD9Z4",
    "kind": "definition",
    "msc": [
      "60F05"
    ],
    "name": "Convergence in probability",
    "sources": [
      {
        "locator": "Convergence in probability",
        "url": "https://en.wikipedia.org/wiki/Convergence_of_random_variables"
      }
    ],
    "tag": "0EFDDXTVD6",
    "uses": [
      "0DKRR22AEQ"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/0EFDDXTVD6/",
  "links": [
    {
      "field": "defines",
      "target": "0QA1HBD9Z4"
    },
    {
      "field": "uses",
      "target": "0DKRR22AEQ"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0EFDDXTVD6.md",
  "msc": [
    "60F05"
  ],
  "name": "Convergence in probability",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0EF/0EFDDXTVD6.md",
  "status": null,
  "tag": "0EFDDXTVD6",
  "uri": "m2://0EFDDXTVD6",
  "url": "https://math2.tamnd.com/wiki/convergence-in-probability-definition/",
  "writing": {
    "tag": "0EFDDXTVD6",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}