{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "BD1CNVV5P0"
    },
    {
      "field": "uses",
      "target": "QWEF2WAXJM"
    },
    {
      "field": "uses",
      "target": "RZZK51CXMN"
    }
  ],
  "body": "\n## Definition\n\nLet $\\mathcal{X}$ be a [[topological space|0CQHFPGJ45]] with its Borel $\\sigma$-algebra.\nLet $(\\mu_n)_{n \\ge 1}$ be a sequence of [[probability measures|059TJN0NZ1]] on $\\mathcal{X}$, and let $I \\colon \\mathcal{X} \\to [0, \\infty]$ be a [[rate function|140N2PQSTE]].\nFor $A \\subseteq \\mathcal{X}$, write $I(A) = \\inf_{x \\in A} I(x)$, with $I(\\emptyset) = \\infty$.\nThe sequence $(\\mu_n)$ satisfies the *large deviation principle* with rate function $I$ if both of the following hold.\n\n1. For every [[closed set|0ZCK8DQYN1]] $F \\subseteq \\mathcal{X}$, $\\displaystyle \\limsup_{n \\to \\infty} \\frac{1}{n} \\log \\mu_n(F) \\le -I(F)$.\n2. For every [[open set|02BH3MCBQB]] $G \\subseteq \\mathcal{X}$, $\\displaystyle \\liminf_{n \\to \\infty} \\frac{1}{n} \\log \\mu_n(G) \\ge -I(G)$.\n\n## Remarks\n\nHere $\\log 0 = -\\infty$.\nA sequence of random elements $Z_n$ of $\\mathcal{X}$ satisfies the principle when their laws do.\nThe factor $1/n$ is the speed; some applications use another speed $1/\\varepsilon$ or $1/a_n$ with $a_n \\to \\infty$.\nIf $A$ is a Borel set whose interior $A^\\circ$ and closure $\\bar A$ satisfy $I(A^\\circ) = I(\\bar A)$, then $\\frac{1}{n} \\log \\mu_n(A) \\to -I(A)$.\n\nExample: let $\\mu_n$ be the normal law with mean $0$ and variance $1/n$ on $\\R$, and let $I(x) = x^2/2$.\nFor an interval $[a, \\infty)$ with $a \u003e 0$, the Gaussian tail estimate gives $\\frac{1}{n} \\log \\mu_n([a, \\infty)) \\to -a^2/2 = -I([a, \\infty))$.\nThe same tail estimate gives both bounds for all closed and open sets, so $(\\mu_n)$ satisfies the principle with this $I$.\n\nNon-example: the same sequence does not satisfy the principle with $I(x) = x^2$.\nThe upper bound fails for the closed set $F = [1, \\infty)$.\nIndeed $\\frac{1}{n} \\log \\mu_n(F) \\to -1/2$, while $-I(F) = -1 \u003c -1/2$.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "defines": "Q702WCDQQ3",
    "kind": "definition",
    "msc": [
      "60F10"
    ],
    "name": "Large deviations theory",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Large_deviations_theory"
      }
    ],
    "tag": "MNPWBBNCAD",
    "uses": [
      "059TJN0NZ1",
      "0CQHFPGJ45",
      "140N2PQSTE",
      "0ZCK8DQYN1",
      "02BH3MCBQB",
      "0QF1MZ9FRG",
      "0T3RZQXDYD",
      "033RZ32T4Q"
    ]
  },
  "kind": "definition",
  "link": "https://math2.tamnd.com/node/MNPWBBNCAD/",
  "links": [
    {
      "field": "defines",
      "target": "Q702WCDQQ3"
    },
    {
      "field": "uses",
      "target": "059TJN0NZ1"
    },
    {
      "field": "uses",
      "target": "0CQHFPGJ45"
    },
    {
      "field": "uses",
      "target": "140N2PQSTE"
    },
    {
      "field": "uses",
      "target": "0ZCK8DQYN1"
    },
    {
      "field": "uses",
      "target": "02BH3MCBQB"
    },
    {
      "field": "uses",
      "target": "0QF1MZ9FRG"
    },
    {
      "field": "uses",
      "target": "0T3RZQXDYD"
    },
    {
      "field": "uses",
      "target": "033RZ32T4Q"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/MNPWBBNCAD.md",
  "msc": [
    "60F10"
  ],
  "name": "Large deviations theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/MNP/MNPWBBNCAD.md",
  "status": null,
  "tag": "MNPWBBNCAD",
  "uri": "m2://MNPWBBNCAD",
  "url": "https://math2.tamnd.com/wiki/large-deviations-theory-definition/",
  "writing": {
    "tag": "MNPWBBNCAD",
    "kind": "definition",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      },
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}