4.2 Stationarity

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This module introduces stationarity, such as strict sense stationarity (SSS) and wide sense stationarity (WSS).

Stationarity in a Random Process implies that its statistical characteristics do not change with time . Put another way, if one were to observe a stationary random process at some time $t$ it would be impossible to distinguish the statistical characteristics at that time from those at some other time ${t}^{\prime }$ .

Strict sense stationarity (sss)

Choose a Random Vector of length $N$ from a Random Process:

$X=\begin{pmatrix}X({t}_{1}) & X({t}_{2}) & \dots & X({t}_{N})\\ \end{pmatrix}^T$
Its $N$ th order cdf is
$X(t)$ is defined to be Strict Sense Stationary iff:
for all time shifts $c$ , all finite $N$ and all sets of time points $\{{t}_{1}, \dots , {t}_{N}\}$ .

Wide sense (weak) stationarity (wss)

If we are only interested in the properties of moments up to 2nd order (mean, autocorrelation, covariance, ...), which isthe case for many practical applications, a weaker form of stationarity can be useful:

$X(t)$ is defined to be Wide Sense Stationary (or Weakly Stationary) iff:

• The mean value is independent of $t$ , for all $t$
$(X(t))=\mu$
• Autocorrelation depends only upon $\tau ={t}_{2}-{t}_{1}$ , for all ${t}_{1}$
$(X({t}_{1})X({t}_{2}))=(X({t}_{1})X({t}_{1}+\tau ))={r}_{XX}(\tau )$
Note that, since 2nd-order moments are defined in terms of2nd-order probability distributions, strict sense stationary processes are always wide-sense stationary, but not necessarily vice versa .

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