In statistics, Wold's decomposition or the Wold representation theorem (not to be confused with the Wold theorem that is the discrete-time analog of the Wiener–Khinchin theorem) named after Herman Wold, says that every covariance-stationary time series
Formally
where:
Note that the moving average coefficients have these properties:
- Stable, that is square summable
∑ j = 1 ∞ | b j | 2 ∞ - Causal (i.e. there are no terms with j < 0)
- Minimum delay
- Constant (
b j - It is conventional to define
b 0 = 1
This theorem can be considered as an existence theorem: any stationary process has this seemingly special representation. Not only is the existence of such a simple linear and exact representation remarkable, but even more so is the special nature of the moving average model. Imagine creating a process that is a moving average but not satisfying these properties 1–4. For example, the coefficients
The usefulness of the Wold Theorem is that it allows the dynamic evolution of a variable
The Wold representation depends on an infinite number of parameters, although in practice they usually decay rapidly. The autoregressive model is an alternative that may have only a few coefficients if the corresponding moving average has many. These two models can be combined into an autoregressive-moving average (ARMA) model, or an autoregressive-integrated-moving average (ARIMA) model if non-stationarity is involved. See Scargle (1981) and references there; in addition this paper gives an extension of the Wold Theorem that allows more generality for the moving average (not necessarily stable, causal, or minimum delay) accompanied by a sharper characterization of the innovation (identically and independently distributed, not just uncorrelated). This extension allows the possibility of models that are more faithful to physical or astrophysical processes, and in particular can sense ″the arrow of time.″