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: is the time series being considered, is an uncorrelated sequence which is the innovation process to the process – that is, a white noise process that is input to the linear filter . is the possibly infinite vector of moving average weights (coefficients or parameters) is a deterministic time series, such as one represented by a sine wave. The moving average coefficients have these properties: Stable, that is square summable < Causal (i.e. there are no terms with j < 0) Minimum delay Constant ( independent of t) It is conventional to define 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 model. Nevertheless the theorem assures the existence of a causal 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. The usefulness of the Wold Theorem is that it allows the dynamic evolution of a variable to be approximated by a linear model. If the innovations are independent, then the linear model is the only possible representation relating the observed value of to its past evolution. However, when 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. In this latter case, it is possible that the linear model may not be very useful, and there would be a nonlinear model relating the observed value of to its past evolution. However, in practical time series analysis, it is often the case that only linear predictors are considered, partly on the grounds of simplicity, in which case the Wold decomposition is directly relevant. The Wold representation depends