Stationary Process - Examples

Examples

As an example, white noise is stationary. The sound of a cymbal clashing, if hit only once, is not stationary because the acoustic power of the clash (and hence its variance) diminishes with time. However, it would be possible to invent a stochastic process describing when the cymbal is hit, such that the overall response would form a stationary process.

An example of a discrete-time stationary process where the sample space is also discrete (so that the random variable may take one of N possible values) is a Bernoulli scheme. Other examples of a discrete-time stationary process with continuous sample space include 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.

Let Y be any scalar random variable, and define a time-series { X_t }, by

.

Then { X_t } 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 Y, rather than taking the expected value of Y.

As a further example of a stationary process for which any single realisation has an apparently noise-free structure, let Y have a uniform distribution on (0,2π] and define the time series { X_t } by

Then { X_t } is strictly stationary.