{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "AE0EZQH4ZM"
    }
  ],
  "body": "\n## Statement\n\nLet $0 \u003c p \u003c \\infty$.\nLet $\\varepsilon_1, \\varepsilon_2, \\ldots$ be independent [[random variables|0PNRKMRC55]] with $\\Pr(\\varepsilon_k = 1) = \\Pr(\\varepsilon_k = -1) = 1/2$.\nThen there are constants $A_p \u003e 0$ and $B_p \u003e 0$, depending only on $p$, with the following property.\nFor every $N \\ge 1$ and all real numbers $a_1, \\ldots, a_N$,\n$$A_p \\Big( \\sum_{k=1}^{N} a_k^2 \\Big)^{1/2} \\le \\Big( \\mathbb{E} \\Big\\lvert \\sum_{k=1}^{N} a_k \\varepsilon_k \\Big\\rvert^p \\Big)^{1/p} \\le B_p \\Big( \\sum_{k=1}^{N} a_k^2 \\Big)^{1/2},$$\nwhere $\\mathbb{E}$ is the [[expected value|0WQNRCNY3M]].\n\n## Remarks\n\nFor $p = 2$ the middle term equals $\\left(\\sum a_k^2\\right)^{1/2}$ exactly, since $\\mathbb{E}[\\varepsilon_j \\varepsilon_k] = 0$ for $j \\ne k$.\nSo one can take $A_p = 1$ for $p \\ge 2$ and $B_p = 1$ for $p \\le 2$, by monotonicity of $L^p$ norms on a probability space.\n\nProof idea for the upper bound: normalize to $\\sum a_k^2 = 1$.\nSince $\\cosh t \\le e^{t^2/2}$, independence gives $\\mathbb{E} \\exp\\left(t \\sum a_k \\varepsilon_k\\right) \\le e^{t^2/2}$ for all real $t$.\nThis gives the tail bound $\\Pr\\left(\\abs{\\sum a_k \\varepsilon_k} \u003e s\\right) \\le 2 e^{-s^2/2}$, and integrating the tail bounds every moment.\nThe lower bound for $p \u003c 2$ follows from the upper bound for some $q \u003e 2$ by Hölder's inequality, which places $L^2$ between $L^p$ and $L^q$.\n\nExample: for $N = 2$, $a_1 = a_2 = 1$ and $p = 1$, the sum $\\varepsilon_1 + \\varepsilon_2$ is $\\pm 2$ with probability $1/2$ and $0$ otherwise, so $\\mathbb{E}\\abs{\\varepsilon_1 + \\varepsilon_2} = 1 = \\frac{1}{\\sqrt 2} \\cdot \\sqrt 2$.\nThis shows that $A_1 \\le 1/\\sqrt 2$; Szarek proved that $A_1 = 1/\\sqrt 2$ is the best constant.\n\n## Explanations\n\n### L1 human\n\nFlip $N$ fair coins and add the numbers $a_k$ with random signs.\nThe typical size of the result is the square root of the sum of the squares of the $a_k$, as in Pythagoras' theorem.\nKhintchine's inequality says this holds for every way of measuring \"typical size\" by a $p$-th moment, up to a constant factor.\nSo random sign sums behave like Gaussian variables with the same variance.\n\n### L2 human\n\nIn harmonic analysis the $\\varepsilon_k$ are realized as the Rademacher functions $r_k(t) = \\operatorname{sgn} \\sin(2^k \\pi t)$ on $[0, 1]$, and the inequality says that all $L^p$ norms are equivalent on their span.\nCombined with Fubini's theorem it turns square-function estimates into estimates for random sums, a basic step in Littlewood–Paley theory.\nIn Banach space theory it is the starting point of the notions of type and cotype.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "0PNRKMRC55"
    ],
    "kind": "statement",
    "msc": [
      "60E15",
      "42A61",
      "46B09"
    ],
    "name": "Khintchine inequality",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q6401711",
      "wikipedia": "Khintchine inequality"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Khintchine_inequality"
      }
    ],
    "tag": "AGWMZ50SSZ",
    "uses": [
      "0PNRKMRC55",
      "0WQNRCNY3M"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/AGWMZ50SSZ/",
  "links": [
    {
      "field": "about",
      "target": "0PNRKMRC55"
    },
    {
      "field": "uses",
      "target": "0PNRKMRC55"
    },
    {
      "field": "uses",
      "target": "0WQNRCNY3M"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/AGWMZ50SSZ.md",
  "msc": [
    "60E15",
    "42A61",
    "46B09"
  ],
  "name": "Khintchine inequality",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AGW/AGWMZ50SSZ.md",
  "status": null,
  "tag": "AGWMZ50SSZ",
  "uri": "m2://AGWMZ50SSZ",
  "url": "https://tamnd.github.io/math2/wiki/khintchine-inequality/",
  "writing": {
    "tag": "AGWMZ50SSZ",
    "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
      }
    ]
  }
}