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# Requirement on time-integral of

For any we can define , whose autocorrelation function is related to that of (using the time-average form (2.30)) by

 (D.3)

is well-defined if is a stationary process [79], i.e. its statistical properties, in particular its average, do not change with time. (We have already assumed , etc. are stationary processes). This condition causes the second term to vanish, since then remains bounded. Integrating the above over all gives an expression for the zeroth moment: if vanishes this implies must be stationary.

Alex Barnett 2001-10-03