Wold's theorem
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 can be written as the sum of two time series, one deterministic and one stochastic.
Formally
where:
Note that the moving average coefficients have these properties:
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 could define an acausal and non-minimum delay model. Nevertheless the theorem assures the existence of a causal minimum delay moving average that exactly represents this process. How this all works for the case of causality and the minimum delay property is discussed in Scargle (1981), where an extension of the Wold Decomposition is discussed.
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