Probability theory and stochastic processes Theorem

Wold's decomposition theorem

Builds on Stationary process Hilbert space Orthogonal projection Moving-average model

Statement

Let $(X_t)_{t \in \Z}$ be a real weakly stationary process with mean $0$ . Let $L^2$ be the Hilbert space of square-integrable random variables. For $s \in \Z$ , let $H_s$ be the closed linear span of $\set{X_r : r \le s}$ in $L^2$ . Let $P_s$ be the orthogonal projection onto $H_s$ . Assume that the one-step prediction error variance $\sigma^2 = E[(X_t - P_{t-1} X_t)^2]$ is positive. Then $X_t = \sum_{j=0}^{\infty} \psi_j \varepsilon_{t-j} + V_t \qquad \text{for all } t \in \Z,$ where:

$\varepsilon_t = X_t - P_{t-1} X_t$ , and $(\varepsilon_t)$ is a white noise with variance $\sigma^2$ : mean $0$ , variance $\sigma^2$ , pairwise uncorrelated.
$\psi_0 = 1$ , $\sum_{j \ge 0} \psi_j^2 < \infty$ , and $\psi_j = E[X_t \varepsilon_{t-j}] / \sigma^2$ .
$E[\varepsilon_s V_t] = 0$ for all $s, t \in \Z$ .
$V_t$ 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.

The sequences $(\psi_j)$ , $(\varepsilon_t)$ and $(V_t)$ are uniquely determined by these conditions.

Remarks

The 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 $\bigcap_s H_s$ . The first sum is the projection of $X_t$ onto the closed span of the innovations, with coefficients given by inner products. The remainder $V_t$ is the projection onto the remote past. The process is called purely nondeterministic when $V_t = 0$ ; then $X_t$ is a moving-average model of infinite order. Example: a causal AR( $1$ ) process $X_t = \phi X_{t-1} + \varepsilon_t$ with $\abs{\phi} < 1$ has $\psi_j = \phi^j$ and $V_t = 0$ . Example of a deterministic process: let $A$ and $B$ be uncorrelated with mean $0$ and variance $1$ , let $0 < \lambda < \pi$ , and let $X_t = A \cos(\lambda t) + B \sin(\lambda t)$ . This 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. The coefficients $\psi_j$ need not be absolutely summable, and the shocks $\varepsilon_t$ are only uncorrelated, not independent in general.

Going deeper

For a student

The 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. The innovations $\varepsilon_t$ are the one-step prediction errors, which is why residual checks in model fitting test them for whiteness. The proof is pure Hilbert space geometry: only means and covariances enter, so the result says nothing about nonlinear dependence. It was proved by Herman Wold in his 1938 thesis.

Reading path

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Tag	`2ZK4CE9MF8`
URI	`m2://2ZK4CE9MF8`
Kind	statement, theorem
Writing	W0 stub
MSC	60G10, 62M10, 62M20
Status	This copy of the site has no build data. Run `m2 status 2ZK4CE9MF8`.
About	Stationary process
Uses	Stationary process, Hilbert space, Orthogonal projection, Moving-average model
Same as	Wikidata Q2588432 Wikipedia Wold's theorem