{
  "aliases": [
    "uniqueness-theorem-for-characteristic-functions",
    "levy-inversion-formula"
  ],
  "backlinks": [
    {
      "field": "uses",
      "target": "0FPWF7TTMB"
    },
    {
      "field": "uses",
      "target": "0NXGPJWNFN"
    }
  ],
  "body": "\n## Statement\n\nLet $X$ and $Y$ be real random variables with [[characteristic functions|0MYDNRT7PP]] $\\varphi_X$ and $\\varphi_Y$.\nIf $\\varphi_X(t) = \\varphi_Y(t)$ for all $t \\in \\mathbb{R}$, then $X$ and $Y$ have the same [[law|0TST4MHA3A]].\n\n## Remarks\n\nThe proof uses the Lévy inversion formula. For real $a \u003c b$ that are continuity points of the [[cumulative distribution function|0MXMS3AE13]] $F_X$,\n$$F_X(b) - F_X(a) = \\lim_{T \\to \\infty} \\frac{1}{2\\pi} \\int_{-T}^{T} \\frac{e^{-ita} - e^{-itb}}{it} \\, \\varphi_X(t) \\, dt.$$\nIf $\\int |\\varphi_X(t)| \\, dt \u003c \\infty$, then $X$ has the continuous density $f_X(x) = \\frac{1}{2\\pi} \\int e^{-itx} \\varphi_X(t) \\, dt$.\nThe same result holds for random vectors in $\\mathbb{R}^n$, with $\\varphi_X(t) = E[e^{i \\langle t, X \\rangle}]$.\n\n## Explanations\n\n### L1 human\n\nTwo different distributions cannot have the same characteristic function.\nSo, to identify a distribution, it is enough to compute its characteristic function and recognize it.\n",
  "commit": "6ac761d40018146541484a9e8b9eafd6ecc303dc",
  "front": {
    "aliases": [
      "uniqueness-theorem-for-characteristic-functions",
      "levy-inversion-formula"
    ],
    "kind": "statement",
    "msc": [
      "60E10"
    ],
    "name": "The characteristic function determines the distribution",
    "role": "theorem",
    "sources": [
      {
        "locator": "Inversion formula",
        "url": "https://en.wikipedia.org/wiki/Characteristic_function_(probability_theory)"
      }
    ],
    "tag": "0T8BYAFEXX",
    "uses": [
      "0MYDNRT7PP",
      "0TST4MHA3A",
      "0MXMS3AE13"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/0T8BYAFEXX/",
  "links": [
    {
      "field": "uses",
      "target": "0MYDNRT7PP"
    },
    {
      "field": "uses",
      "target": "0TST4MHA3A"
    },
    {
      "field": "uses",
      "target": "0MXMS3AE13"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/0T8BYAFEXX.md",
  "msc": [
    "60E10"
  ],
  "name": "The characteristic function determines the distribution",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/0T8/0T8BYAFEXX.md",
  "status": null,
  "tag": "0T8BYAFEXX",
  "uri": "m2://0T8BYAFEXX",
  "url": "https://math2.tamnd.com/wiki/the-characteristic-function-determines-the-distribution/",
  "writing": {
    "tag": "0T8BYAFEXX",
    "kind": "statement",
    "level": "W0",
    "next": [
      {
        "what": "an inventory row",
        "note": "the row of the inventory that the node accounts for",
        "have": 0,
        "want": 1
      }
    ]
  }
}