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

(D.3) 
is welldefined 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
20011003