{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nFor each $n \\ge 1$, let $X_n$ and $Y_n$ be real random variables on one probability space.\nLet $X$ be a real random variable and $c \\in \\R$ a constant.\nAssume that $X_n \\to X$ [[in distribution|0SRYJGXKR4]] and $Y_n \\to c$ [[in probability|0QA1HBD9Z4]].\nThen, in distribution:\n\n1. $X_n + Y_n \\to X + c$;\n2. $Y_n X_n \\to c X$;\n3. $X_n / Y_n \\to X / c$, provided $c \\ne 0$, where $X_n / Y_n$ is given any fixed value on the event $\\set{Y_n = 0}$.\n\n## Remarks\n\nProof idea: since the limit of $Y_n$ is a constant, the pair $(X_n, Y_n)$ converges in distribution to $(X, c)$.\nEach of the three maps $(x, y) \\mapsto x + y$, $xy$ and $x/y$ is continuous at the points $(x, c)$, so the [[continuous mapping theorem|CFRN0G7GDB]] applies.\nExample: let $X_1, X_2, \\dots$ be i.i.d. with mean $\\mu$ and variance $\\sigma^2 \\in (0, \\infty)$, with [[sample mean|03Y6AQ2S0J]] $\\bar{X}_n$ and sample standard deviation $S_n$. The [[central limit theorem|0EPZC4Z1FC]] gives $\\sqrt{n}(\\bar{X}_n - \\mu)/\\sigma \\to N(0, 1)$, and the law of large numbers gives $S_n \\to \\sigma$ in probability; so $\\sqrt{n}(\\bar{X}_n - \\mu)/S_n \\to N(0, 1)$ in distribution.\nThe limit $c$ must be a constant.\nLet $Z$ be standard normal and $X_n = Z$; the variables $Z$ and $-Z$ have the same law.\nWith $Y_n = -Z$ one gets $X_n + Y_n = 0$, and with $Y_n = Z$ one gets $2Z$.\nSo convergence in law of both terms does not determine the law of the limit of the sum.\nThe theorem holds in the same way for random vectors, and for random matrices $Y_n$ that converge in probability to a constant matrix $C$.\nFor the quotient $Y_n^{-1} X_n$ one needs $C$ to be invertible.\n\n## Explanations\n\n### L1 human\n\nSuppose one sequence settles into a random limit law and another settles down to a fixed number.\nThen their sum, product and quotient behave as if the second sequence were already that number.\nThis is what allows statisticians to replace an unknown parameter by an estimate of it in an approximate normal law.\n\n### L2 human\n\nSlutsky's theorem is used constantly in asymptotic statistics, together with the [[central limit theorem|0EPZC4Z1FC]] and the delta method.\nIt justifies the use of estimated standard errors in confidence intervals and test statistics such as the Student $t$ statistic for large samples.\nIt follows from the [[continuous mapping theorem|CFRN0G7GDB]] applied to the pair $(X_n, Y_n)$.\nIts key point is that joint convergence comes for free when one limit is constant.\n",
  "commit": "5fc425f97f1f6b11da5892c2948ef8c5d8805665",
  "front": {
    "about": [
      "0SRYJGXKR4"
    ],
    "kind": "statement",
    "msc": [
      "60F05",
      "62E20"
    ],
    "name": "Slutsky's theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q643826",
      "wikipedia": "Slutsky's theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Slutsky%27s_theorem"
      }
    ],
    "tag": "GCV10B5HES",
    "uses": [
      "0SRYJGXKR4",
      "0QA1HBD9Z4",
      "CFRN0G7GDB"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/GCV10B5HES/",
  "links": [
    {
      "field": "about",
      "target": "0SRYJGXKR4"
    },
    {
      "field": "uses",
      "target": "0SRYJGXKR4"
    },
    {
      "field": "uses",
      "target": "0QA1HBD9Z4"
    },
    {
      "field": "uses",
      "target": "CFRN0G7GDB"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/GCV10B5HES.md",
  "msc": [
    "60F05",
    "62E20"
  ],
  "name": "Slutsky's theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GCV/GCV10B5HES.md",
  "status": null,
  "tag": "GCV10B5HES",
  "uri": "m2://GCV10B5HES",
  "url": "https://math2.tamnd.com/wiki/slutskys-theorem/",
  "writing": {
    "tag": "GCV10B5HES",
    "kind": "statement",
    "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
      }
    ]
  }
}