Probability, Random Variables, and Random Processes: Theory and Signal Processing Applications

6.5 STATIONARITY

There are several different types of stationarity for a random process. Generally, stationarity refers to the degree to which the probabilistic model of the time-indexed random variables is “constant” with time.

Definition: First-Order Stationary Random process X(t) is first-order stationary if the pdf does not depend on time:

(6.19)

As a result, all moments (raw and central), cumulants, and so on, are constant. For this type of stationarity, we cannot say anything about moments across two or more time instants, such as correlation and covariance.

Definition: Nth-Order Stationary Random process X(t) is Nth-order stationary if the joint probability distribution of the process at N time instants does not change by any time shift τ of all :

(6.20)

This type of stationarity is obviously stronger than first-order stationarity; it implies that moments across time instants (up to N−1 time differences) are unchanged due to a time shift. We can view Nth-order stationarity as though there is a “sliding window” (of width N) within which the joint pdf does not change.

Definition: Strictly Stationary Random process X(t) is strictly stationary if it is Nth-order stationary for ...