{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $n \\ge 1$, and for $i = 1, \\dots, n$ let $(\\Omega_i, P_i)$ be a finite [[probability space|0NFPJVJJHX]] in which every subset is an event.\nLet $\\Omega = \\Omega_1 \\times \\dots \\times \\Omega_n$ with the [[product measure|77PV0ZAVXT]] $P = P_1 \\otimes \\dots \\otimes P_n$.\nFor a nonempty set $A \\subseteq \\Omega$ and a point $x \\in \\Omega$, define the convex distance\n$$d_T(x, A) = \\sup_{\\alpha} \\ \\inf_{y \\in A} \\sum_{i \\,:\\, x_i \\ne y_i} \\alpha_i,$$\nwhere the supremum is over all $\\alpha \\in [0, \\infty)^n$ with $\\alpha_1^2 + \\dots + \\alpha_n^2 = 1$.\nThen for every nonempty $A \\subseteq \\Omega$ and every $t \\ge 0$,\n$$P(A) \\cdot P(\\set{x \\in \\Omega : d_T(x, A) \\ge t}) \\le e^{-t^2/4}.$$\n\n## Remarks\n\nThe theorem holds, with the same proof, for products of general probability spaces, provided the sets involved are measurable.\nTalagrand proved the stronger bound $\\int_\\Omega e^{d_T(x, A)^2 / 4} \\, P(dx) \\le 1 / P(A)$, by induction on $n$; Markov's inequality applied to $e^{d_T^2/4}$ gives the statement.\nThe convex distance is at least the normalised Hamming distance $\\min_{y \\in A} \\#\\set{i : x_i \\ne y_i} / \\sqrt{n}$, by the choice $\\alpha_i = 1/\\sqrt{n}$, so the theorem implies Hamming concentration.\nA main consequence concerns convex Lipschitz functions.\nLet $X_1, \\dots, X_n$ be independent with values in $[0, 1]$, and let $F \\colon [0, 1]^n \\to \\R$ be [[convex|07Q7CNM5WF]] and $1$-[[Lipschitz|0AW1E6YTJY]] for the Euclidean norm.\nThen $P(\\abs{F(X) - m} \\ge t) \\le C e^{-c t^2}$ for absolute constants $C, c \u003e 0$, where $m$ is a [[median|07Y2E7P9NP]] of $F(X)$.\nThe bound does not depend on $n$, while bounded-difference methods would give a bound that gets worse with the dimension for this class.\nIt is used, for example, to show that the length of the longest increasing subsequence of a random permutation is concentrated around its median.\n\n## Explanations\n\n### L1 human\n\nTake a set $A$ in a product of many independent coordinates that has probability at least one half.\nThen almost every point is close to $A$, in a strong sense: changing a few well-chosen coordinates brings it into $A$, whatever weights one puts on the coordinates.\nThe probability of being at distance $t$ from $A$ drops like $e^{-t^2/4}$, independently of the number of coordinates.\n\n### L2 human\n\nTalagrand's inequality is a central result of the concentration of measure phenomenon in product spaces.\nIt goes beyond [[McDiarmid's inequality|8APFT1RPM4]] because the distance adapts to each point through the weights $\\alpha$, which yields dimension-free bounds for convex Lipschitz functions and for functions with small certificates.\nApplications include the largest eigenvalue of random matrices with bounded entries, longest increasing subsequences, and combinatorial optimisation problems on random inputs.\n",
  "commit": "efe85cee8dbc57d85bedbacfb97b99d1b7bee843",
  "front": {
    "kind": "statement",
    "msc": [
      "60E15",
      "28A35"
    ],
    "name": "Talagrand's concentration inequality",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q7678798",
      "wikipedia": "Talagrand's concentration inequality"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Talagrand%27s_concentration_inequality"
      }
    ],
    "tag": "GGEAGW4RRV",
    "uses": [
      "0NFPJVJJHX",
      "0AW1E6YTJY",
      "07Q7CNM5WF",
      "07Y2E7P9NP"
    ]
  },
  "kind": "statement",
  "link": "https://math2.tamnd.com/node/GGEAGW4RRV/",
  "links": [
    {
      "field": "uses",
      "target": "0NFPJVJJHX"
    },
    {
      "field": "uses",
      "target": "0AW1E6YTJY"
    },
    {
      "field": "uses",
      "target": "07Q7CNM5WF"
    },
    {
      "field": "uses",
      "target": "07Y2E7P9NP"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/GGEAGW4RRV.md",
  "msc": [
    "60E15",
    "28A35"
  ],
  "name": "Talagrand's concentration inequality",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GGE/GGEAGW4RRV.md",
  "status": null,
  "tag": "GGEAGW4RRV",
  "uri": "m2://GGEAGW4RRV",
  "url": "https://math2.tamnd.com/wiki/talagrands-concentration-inequality/",
  "writing": {
    "tag": "GGEAGW4RRV",
    "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
      }
    ]
  }
}