Stationarity

Stationary processes
A random process is called stationary if it's FDD (finite-dimensional distribution) does not change with the time origin. Recall that a random process is infinite dimensional, and it's finite dimensional distribution is simply defined as a probability distribution (page 523 ):

$$P_{X[n_1], X[n_2], \dots, X_[n_N]}.$$

Then, it is called stationary if this distribution does not change with time origin, i.e., if

$$P_{X[n_1], X[n_2], \dots, X_[n_N]} = P_{X[n_0+n_1], X[n_0+n_2], \dots, X_[n_0+n_N]}$$

for any time offset $$n_0$$

Wide-sense stationary processes
The condition of stationarity is quite strong. A somewhat more relaxed condition is wide-sense stationarity (WSS). WSS processes are defined in terms of their mean and their covariance (and not their distribution). A random process is called WSS if (page 550 )
 * 1) It has constant mean, \ie $$\mu_X[n] = \mu$$
 * 2) If it's covariance depends only  on the time difference, \ie $$c_X[n_1,n_2] = g(|n_2-n_1|)$$

for some function $$g$$.

Important: For Gaussian random processes, WSS is equivalent to stationarity.