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.
could define an acausal and non-minimum delay[clarification needed] model.
Nevertheless the theorem assures the existence of a causal minimum delay moving average[clarification needed] that exactly represents this process.
is merely an uncorrelated but not independent sequence, then the linear model exists but it is not the only representation of the dynamic dependence of the series.
The Wold representation depends on an infinite number of parameters, although in practice they usually decay rapidly.
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.″