{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $1 \\le p \u003c \\infty$.\nThere are constants $0 \u003c A_p \\le B_p \u003c \\infty$, depending only on $p$, with the following property.\nLet $N \\ge 1$ and let $X_1, \\dots, X_N$ be [[independent|0GQ938GDE0]] real or complex [[random variables|0PNRKMRC55]] with $\\mathbb{E}[X_i] = 0$ and $\\mathbb{E}[\\abs{X_i}^p] \u003c \\infty$ for each $i$, where $\\mathbb{E}$ is the [[expected value|0WQNRCNY3M]].\nThen\n$$A_p \\, \\mathbb{E}\\left[ \\left( \\sum_{i=1}^N \\abs{X_i}^2 \\right)^{p/2} \\right] \\le \\mathbb{E}\\left[ \\Big\\vert \\sum_{i=1}^N X_i \\Big\\vert^p \\right] \\le B_p \\, \\mathbb{E}\\left[ \\left( \\sum_{i=1}^N \\abs{X_i}^2 \\right)^{p/2} \\right].$$\n\n## Remarks\n\nThe proof uses symmetrization and the [[Khintchine inequality|AGWMZ50SSZ]].\nLet $(\\varepsilon_i)$ be independent random signs, independent of the $X_i$.\nBecause the $X_i$ have mean zero, $\\mathbb{E}\\abs{\\sum X_i}^p$ is comparable, with constants depending only on $p$, to $\\mathbb{E}\\abs{\\sum \\varepsilon_i X_i}^p$.\nConditioning on the $X_i$ and applying Khintchine's inequality to the signs gives the comparison with $\\mathbb{E}(\\sum \\abs{X_i}^2)^{p/2}$.\n\nFor $p = 2$ the inequality is an identity with $A_2 = B_2 = 1$: independence and mean zero give $\\mathbb{E}\\abs{\\sum X_i}^2 = \\sum \\mathbb{E}\\abs{X_i}^2$.\nThe mean zero hypothesis is needed: if every $X_i = 1$, the middle term is $N^p$ while the outer terms are of order $N^{p/2}$, so the upper bound fails for large $N$.\n\n## Explanations\n\n### L1 human\n\nA sum of independent random quantities with average zero has much cancellation.\nThe inequality says that its size, measured by a $p$-th moment, is comparable to the size of the square root of the sum of the squares of the terms.\nSo the sum behaves as if the terms were orthogonal, as in the Pythagorean theorem.\n\n### L2 human\n\nThe inequality is the probabilistic counterpart of the [[Khintchine inequality|AGWMZ50SSZ]], which is the special case of random signs times constants.\nIt is a model for the Littlewood–Paley inequality: the dyadic pieces of a function play the role of independent terms, and the square function plays the role of $\\left( \\sum \\abs{X_i}^2 \\right)^{1/2}$.\nBurkholder extended it to martingale differences for $1 \u003c p \u003c \\infty$, which links it further with Littlewood–Paley theory.\nIt is used to bound moments of sums of independent variables, for example in proofs of laws of large numbers with rates.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "0GQ938GDE0"
    ],
    "kind": "statement",
    "msc": [
      "60E15",
      "60G50",
      "42B25"
    ],
    "name": "Marcinkiewicz–Zygmund inequality",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q6757294",
      "wikipedia": "Marcinkiewicz–Zygmund inequality"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Marcinkiewicz%E2%80%93Zygmund_inequality"
      }
    ],
    "tag": "AE0EZQH4ZM",
    "uses": [
      "0PNRKMRC55",
      "0GQ938GDE0",
      "0WQNRCNY3M",
      "AGWMZ50SSZ"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/AE0EZQH4ZM/",
  "links": [
    {
      "field": "about",
      "target": "0GQ938GDE0"
    },
    {
      "field": "uses",
      "target": "0PNRKMRC55"
    },
    {
      "field": "uses",
      "target": "0GQ938GDE0"
    },
    {
      "field": "uses",
      "target": "0WQNRCNY3M"
    },
    {
      "field": "uses",
      "target": "AGWMZ50SSZ"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/AE0EZQH4ZM.md",
  "msc": [
    "60E15",
    "60G50",
    "42B25"
  ],
  "name": "Marcinkiewicz–Zygmund inequality",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/AE0/AE0EZQH4ZM.md",
  "status": null,
  "tag": "AE0EZQH4ZM",
  "uri": "m2://AE0EZQH4ZM",
  "url": "https://tamnd.github.io/math2/wiki/marcinkiewicz-zygmund-inequality/",
  "writing": {
    "tag": "AE0EZQH4ZM",
    "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
      }
    ]
  }
}