A wide-sense stationary Gaussian process
is one in which the second-order statistics,
that is the expected value of the
product of two samples, depends only upon the time difference between the
samples and not upon the absolute times.
The ACF coefficient of lag , denoted by , is an estimate of the mean or
expected value of the product
which for stationary time-series, is independent of .
The ACF is the fundamental feature extraction behind many
stationary time-series analysis models including autoregressive (AR),
moving average (MA), and autoregressive moving average (ARMA) processes
are popular models.
Such a process is said to be wide-sense stationary and is fully
defined by its power spectrum
The power spectrum is defined by the Fourier transform pair
with the ACF (Wiener-Kinchine theorem)
A stationary Gaussian process is thus defined by either the power spectrum
or the ACF.
Since the power spectrum is the Fourier transform
of the autocorrelation function (ACF) with integration taken over all time,
the existence of the power spectrum means that the ACF must die down to zero.
This, of course, assumes there are no sinusiodal components which result in
Dirac delta functions in the power spectrum.