Editing Autocorrelation (section)

=== Definition for wide-sense stationary stochastic process ===
If <math>\left\{ X_t \right\}</math> is a [[wide-sense stationary process]] then the mean <math>\mu</math> and the variance <math>\sigma^2</math> are time-independent, and further the autocovariance function depends only on the lag between <math>t_1</math> and <math>t_2</math>: the autocovariance depends only on the time-distance between the pair of values but not on their position in time. This further implies that the autocovariance and autocorrelation can be expressed as a function of the time-lag, and that this would be an [[even function]] of the lag  <math>\tau=t_2-t_1</math>. This gives the more familiar forms for the '''autocorrelation function'''<ref name=Gubner/>{{rp|p.395}}

{{Equation box 1
|indent = :
|title=
|equation = <math>\operatorname{R}_{XX}(\tau) = \operatorname{E}\left[X_{t+\tau} \overline{X}_{t} \right]</math>
|cellpadding= 6
|border colour = #0073CF
|background colour=#F5FFFA}}

and the '''auto-covariance function''':

{{Equation box 1
|indent = :
|title=
|equation = <math>
\begin{align}
\operatorname{K}_{XX}(\tau) &= \operatorname{E}\left[ (X_{t+\tau} - \mu)\overline{(X_{t} - \mu)} \right] \\
&= \operatorname{E} \left[ X_{t+\tau} \overline{X}_{t} \right] - \mu\overline{\mu} \\
&= \operatorname{R}_{XX}(\tau) - \mu\overline{\mu}
\end{align}
</math>
|cellpadding= 6
|border colour = #0073CF
|background colour=#F5FFFA}}

In particular, note that

<math display=block>\operatorname{K}_{XX}(0) = \sigma^2 .</math>