Properties of a Moving-Average Process

This section will examine the properties of a moving-average process, MA((7), defined as 

5 Modelling Stochastic Processes with Time Series Analysis 

Theorem 5.1 Autocovariance of a Moving-Average Process. The autocovariance of a moving-average process can be written as 

Proof Substituting the definition of a moving-average series (Eq. 5.41) into the definition of the autocovariance (Eq. 5.3) gives 

Erom the definition of a white noise signal, the autocovariance will be nonzero if and only if the two white noise realisations have the same subscripts. Therefore, solving for the subscripts gives 

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