some autoregressive and moving average processes which are both subsets of the autoregressive moving average model. Models with a non-trivial autoregressive component may be either stationary or non-stationary, depending on the parameter values, and important non-stationary special cases are where unit roots exist in the model. Example 1 Let be any scalar random variable, and define a time-series , by Then is a stationary time series, for which realisations consist of a series of constant values, with a different constant value for each realisation. A law of large numbers does not apply on this case, as the limiting value of an average from a single realisation takes the random value determined by , rather than taking the expected value of . The time average of does not converge since the process is not ergodic. Example 2 As a further example of a stationary process for which any single realisation has an apparently noise-free structure, let have a uniform distribution on and define the time series by Then is strictly stationary since ( modulo ) follows the same uniform distribution as for any . Example 3 Keep in mind that a white noise is not necessarily strictly stationary. Let be a random variable uniformly distributed in the interval and define the time series Then So is a white noise, however it is not strictly stationary. Nth-order stationarity In , the distribution of samples of the stochastic process must be equal to the distribution of the samples shifted in time for all . N-th-order stationarity is a weaker form of stationarity where this is only requested for all up to a certain order . A random process is said to be N-th-order stationary if: Weak or wide-sense stationarity Definition A weaker form of stationarity commonly employed in signal processing is known as weak-sense stationarity, wide-sense stationarity (WSS), or covariance stationarity. WSS random processes only require that 1st moment (i.e. the mean) and autocovariance do not vary with respect to time and that the 2nd moment is finite for all times. Any strictly stationary process which has a finite mean and a covariance is also WSS. So, a continuous time random process which is WSS has the following restrictions on its mean function and autocovariance function : The first property implies that the mean function must be constant. The second property implies that the autocovariance function depends only on the difference between and