{
  "aliases": null,
  "backlinks": null,
  "body": "\n## Statement\n\nLet $(X_t)_{t \\in \\Z}$ be a real weakly [[stationary process|DT0QXJE705]] with mean $0$.\nLet $L^2$ be the [[Hilbert space|HM2HHE3J51]] of square-integrable random variables.\nFor $s \\in \\Z$, let $H_s$ be the closed linear span of $\\set{X_r : r \\le s}$ in $L^2$.\nLet $P_s$ be the [[orthogonal projection|0B8KZRT28M]] onto $H_s$.\nAssume that the one-step prediction error variance $\\sigma^2 = E[(X_t - P_{t-1} X_t)^2]$ is positive.\nThen\n$$X_t = \\sum_{j=0}^{\\infty} \\psi_j \\varepsilon_{t-j} + V_t \\qquad \\text{for all } t \\in \\Z,$$\nwhere:\n\n1. $\\varepsilon_t = X_t - P_{t-1} X_t$, and $(\\varepsilon_t)$ is a white noise with variance $\\sigma^2$: mean $0$, variance $\\sigma^2$, pairwise uncorrelated.\n2. $\\psi_0 = 1$, $\\sum_{j \\ge 0} \\psi_j^2 \u003c \\infty$, and $\\psi_j = E[X_t \\varepsilon_{t-j}] / \\sigma^2$.\n3. $E[\\varepsilon_s V_t] = 0$ for all $s, t \\in \\Z$.\n4. $V_t$ lies in $\\bigcap_{s \\in \\Z} H_s$, and $(V_t)$ is deterministic: $V_t = P_{s} V_t$ for all $s$, so it is predicted without error from its own remote past.\n\nThe sequences $(\\psi_j)$, $(\\varepsilon_t)$ and $(V_t)$ are uniquely determined by these conditions.\n\n## Remarks\n\nThe idea is to split $H_t$ orthogonally into the innovations $\\varepsilon_t, \\varepsilon_{t-1}, \\dots$, which are orthogonal to each other, and the remote past $\\bigcap_s H_s$.\nThe first sum is the projection of $X_t$ onto the closed span of the innovations, with coefficients given by inner products.\nThe remainder $V_t$ is the projection onto the remote past.\nThe process is called purely nondeterministic when $V_t = 0$; then $X_t$ is a [[moving-average model|PEB9DN0AHA]] of infinite order.\nExample: a causal AR($1$) process $X_t = \\phi X_{t-1} + \\varepsilon_t$ with $\\abs{\\phi} \u003c 1$ has $\\psi_j = \\phi^j$ and $V_t = 0$.\nExample of a deterministic process: let $A$ and $B$ be uncorrelated with mean $0$ and variance $1$, let $0 \u003c \\lambda \u003c \\pi$, and let $X_t = A \\cos(\\lambda t) + B \\sin(\\lambda t)$.\nThis process is weakly stationary, and any two consecutive values determine $A$ and $B$, so $X_t$ lies in every $H_s$ and $\\sigma^2 = 0$; the whole process is its own deterministic part.\nThe coefficients $\\psi_j$ need not be absolutely summable, and the shocks $\\varepsilon_t$ are only uncorrelated, not independent in general.\n\n## Explanations\n\n### L1 human\n\nAny stationary series can be split into two parts that do not interact.\nOne part is fully predictable from the distant past, like a fixed periodic wave.\nThe other part is built from a sequence of unpredictable surprises, each passed on to the future with fading weights.\nSo, as far as linear prediction goes, every stationary series looks like a moving average of white noise plus a predictable part.\n\n### L2 human\n\nThe theorem justifies the use of linear models, such as ARMA models, for stationary series: they are finite-parameter approximations of the infinite moving average in the decomposition.\nThe innovations $\\varepsilon_t$ are the one-step prediction errors, which is why residual checks in model fitting test them for whiteness.\nThe proof is pure Hilbert space geometry: only means and covariances enter, so the result says nothing about nonlinear dependence.\nIt was proved by Herman Wold in his 1938 thesis.\n",
  "commit": "ff8a40749dc16545e3d2fc637f6b0196e81d0bfa",
  "front": {
    "about": [
      "DT0QXJE705"
    ],
    "kind": "statement",
    "msc": [
      "60G10",
      "62M10",
      "62M20"
    ],
    "name": "Wold's decomposition theorem",
    "role": "theorem",
    "same_as": {
      "wikidata": "Q2588432",
      "wikipedia": "Wold's theorem"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Wold%27s_theorem"
      }
    ],
    "tag": "2ZK4CE9MF8",
    "uses": [
      "DT0QXJE705",
      "HM2HHE3J51",
      "0B8KZRT28M",
      "PEB9DN0AHA"
    ]
  },
  "kind": "statement",
  "link": "https://tamnd.github.io/math2/node/2ZK4CE9MF8/",
  "links": [
    {
      "field": "about",
      "target": "DT0QXJE705"
    },
    {
      "field": "uses",
      "target": "DT0QXJE705"
    },
    {
      "field": "uses",
      "target": "HM2HHE3J51"
    },
    {
      "field": "uses",
      "target": "0B8KZRT28M"
    },
    {
      "field": "uses",
      "target": "PEB9DN0AHA"
    }
  ],
  "markdown": "https://tamnd.github.io/math2/node/2ZK4CE9MF8.md",
  "msc": [
    "60G10",
    "62M10",
    "62M20"
  ],
  "name": "Wold's decomposition theorem",
  "prov": null,
  "role": "theorem",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/2ZK/2ZK4CE9MF8.md",
  "status": null,
  "tag": "2ZK4CE9MF8",
  "uri": "m2://2ZK4CE9MF8",
  "url": "https://tamnd.github.io/math2/wiki/wolds-decomposition-theorem/",
  "writing": {
    "tag": "2ZK4CE9MF8",
    "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
      }
    ]
  }
}