{
  "aliases": null,
  "backlinks": [
    {
      "field": "about",
      "target": "BD1CNVV5P0"
    },
    {
      "field": "about",
      "target": "QWEF2WAXJM"
    },
    {
      "field": "about",
      "target": "RZZK51CXMN"
    },
    {
      "field": "defines",
      "target": "MNPWBBNCAD"
    }
  ],
  "body": "\n## Summary\n\nLarge deviations theory studies the exponential decay of the probabilities of rare events.\nIts central notion is the large deviation principle, with a [[rate function|140N2PQSTE]] $I$.\nA family of probability measures $\\mu_n$ satisfies it when $\\mu_n(A)$ behaves like $\\exp(-n \\inf_{x \\in A} I(x))$.\nBasic results are [[Cramér's theorem|RZZK51CXMN]] for sample means, [[Sanov's theorem|QWEF2WAXJM]] for empirical measures and [[Varadhan's lemma|BD1CNVV5P0]] for exponential integrals.\nThe theory is used in statistical mechanics, information theory, queueing and risk analysis.\n\n## Explanations\n\n### L1 human\n\nThe [[law of large numbers|0ZVEAYVRC3]] says that the average of many coin tosses is close to $1/2$.\nLarge deviations theory asks how unlikely it is that the average is far from $1/2$, say above $0.6$.\nThe answer is that such a probability shrinks exponentially in the number of tosses, with a precise exponent.\nFor fair coins, the probability that the average of $n$ tosses is at least $0.6$ is about $e^{-0.020 n}$, up to factors that grow more slowly than any exponential.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "60F10"
    ],
    "name": "Large deviations theory",
    "same_as": {
      "wikidata": "Q3403615",
      "wikipedia": "Large deviations theory"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Large_deviations_theory"
      }
    ],
    "tag": "Q702WCDQQ3"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/Q702WCDQQ3/",
  "links": null,
  "markdown": "https://math2.tamnd.com/node/Q702WCDQQ3.md",
  "msc": [
    "60F10"
  ],
  "name": "Large deviations theory",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/Q70/Q702WCDQQ3.md",
  "status": null,
  "tag": "Q702WCDQQ3",
  "uri": "m2://Q702WCDQQ3",
  "url": "https://math2.tamnd.com/wiki/large-deviations-theory/",
  "writing": {
    "tag": "Q702WCDQQ3",
    "kind": "concept",
    "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
      }
    ]
  }
}