{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Definition\n\nFor each $n \\ge 1$, let $(\\Omega_n, \\mathcal{F}_n)$ be a measurable space, with $\\mathcal{F}_n$ a [[sigma-algebra|04WSW8T3SY]], and let $P_n$ and $Q_n$ be [[probability measures|059TJN0NZ1]] on it.\nThe sequence $(Q_n)$ is *contiguous with respect to* $(P_n)$, written $Q_n \\triangleleft P_n$, if for every sequence of events $A_n \\in \\mathcal{F}_n$,\n$$P_n(A_n) \\to 0 \\implies Q_n(A_n) \\to 0.$$\nThe sequences are *mutually contiguous*, written $P_n \\triangleleft \\triangleright Q_n$, if $Q_n \\triangleleft P_n$ and $P_n \\triangleleft Q_n$.\n\n## Remarks\n\nContiguity is the asymptotic analogue of [[absolute continuity|2G173YJC54]]; for constant sequences $P_n = P$ and $Q_n = Q$ it is the same as $Q \\ll P$.\nIf $Q_n \\triangleleft P_n$, then every sequence of statistics that tends to $0$ in $P_n$-probability also tends to $0$ in $Q_n$-probability.\n\nExample: let $P_n = \\mathcal{N}(0, 1)^{\\otimes n}$ and $Q_n = \\mathcal{N}(h/\\sqrt{n}, 1)^{\\otimes n}$, products of [[normal distributions|07RXBVTVZB]], for a fixed $h \\in \\R$.\nUnder $P_n$, the log-likelihood ratio $\\log(dQ_n/dP_n) = h n^{-1/2}\\sum_i X_i - h^2/2$ has the law $\\mathcal{N}(-h^2/2, h^2)$ for every $n$.\nBy Le Cam's first lemma, a limit law $L$ of $\\log(dQ_n/dP_n)$ under $P_n$ with $E e^{L} = 1$ implies $Q_n \\triangleleft P_n$, and here $E e^{L} = e^{-h^2/2 + h^2/2} = 1$; by symmetry the sequences are mutually contiguous.\n\nNon-example: let $P_n = \\mathcal{N}(0, 1)^{\\otimes n}$ and $Q_n = \\mathcal{N}(1, 1)^{\\otimes n}$, and let $A_n = \\set{\\bar{x}_n \u003e 1/2}$, with $\\bar{x}_n$ the mean of the coordinates.\nThen $P_n(A_n) = P(Z \u003e \\sqrt{n}/2) \\to 0$ but $Q_n(A_n) = P(Z \u003e -\\sqrt{n}/2) \\to 1$ for $Z$ standard normal, so $Q_n$ is not contiguous with respect to $P_n$.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "defines": "DQGJMP9WHC",
    "kind": "definition",
    "msc": [
      "62E20",
      "60B10"
    ],
    "name": "Contiguity of measures",
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Contiguity_(probability_theory)"
      }
    ],
    "tag": "6WTHKQG40F",
    "uses": [
      "059TJN0NZ1",
      "2G173YJC54",
      "04WSW8T3SY",
      "07RXBVTVZB"
    ]
  },
  "kind": "definition",
  "link": "https://tamnd.github.io/math2/node/6WTHKQG40F/",
  "links": [
    {
      "field": "defines",
      "target": "DQGJMP9WHC"
    },
    {
      "field": "uses",
      "target": "059TJN0NZ1"
    },
    {
      "field": "uses",
      "target": "2G173YJC54"
    },
    {
      "field": "uses",
      "target": "04WSW8T3SY"
    },
    {
      "field": "uses",
      "target": "07RXBVTVZB"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/6WTHKQG40F.md",
  "msc": [
    "62E20",
    "60B10"
  ],
  "name": "Contiguity of measures",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/6WT/6WTHKQG40F.md",
  "status": null,
  "tag": "6WTHKQG40F",
  "uri": "m2://6WTHKQG40F",
  "url": "https://tamnd.github.io/math2/wiki/contiguity-of-measures-definition/",
  "writing": {
    "tag": "6WTHKQG40F",
    "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
      }
    ]
  }
}