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For any
we can define
, whose autocorrelation
function
is related to that of
(using the time-average form (2.30)) by
![\begin{displaymath}
C(\tau) \; = \; -\frac{d^2}{d\tau^2} C_{\mathcal{G}}(\tau)
...
...
\frac{d}{dt}{\mathcal{G}}(t+\tau) \right]_{t=-T/2}^{t=T/2} .
\end{displaymath}](img1661.gif) |
(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