In mathematics and statistics, a stationary process (also called a strict/strictly stationary process or strong/strongly stationary process) is a stochastic process whose statistical properties, such as mean and variance, do not change over time. More formally, the joint probability distribution of the process remains the same when shifted in time. This implies that the process is statistically consistent across different time periods. Because many statistical procedures in time series analysis assume stationarity, non-stationary data are frequently transformed to achieve stationarity before analysis. A common cause of non-stationarity is a trend in the mean, which can be due to either a unit root or a deterministic trend. In the case of a unit root, stochastic shocks have permanent effects, and the process is not mean-reverting. With a deterministic trend, the process is called trend-stationary, and shocks have only transitory effects, with the variable tending towards a deterministically evolving mean. A trend-stationary process is not strictly stationary but can be made stationary by removing the trend. Similarly, processes with unit roots can be made stationary through differencing. Another type of non-stationary process, distinct from those with trends, is a cyclostationary process, which exhibits cyclical variations over time. Strict stationarity, as defined above, can be too restrictive for many applications. Therefore, other forms of stationarity, such as wide-sense stationarity or N {\displaystyle N} th-order stationarity, are often used. The definitions for different kinds of stationarity are not consistent among different authors (see Other terminology).
Strict-sense stationarity
Definition Formally, let { X t } {\displaystyle \left\{X_{t}\right\}} be a stochastic process and let F X ( x t 1 + τ , … , x t n + τ ) {\displaystyle F_{X}(x_{t_{1}+\tau },\ldots ,x_{t_{n}+\tau })} represent the cumulative distribution function of the unconditional (i.e., with no reference to any particular starting value) joint distribution of { X t } {\displaystyle \left\{X_{t}\right\}} at times t 1 + τ , … , t n + τ {\displaystyle t_{1}+\tau ,\ldots ,t_{n}+\tau } . Then, { X t } {\displaystyle \left\{X_{t}\right\}} is said to be strictly stationary, strongly stationary or strict-sense stationary if
Since τ {\displaystyle \tau } does not affect F X ( ⋅ ) {\displaystyle F_{X}(\cdot )} , F X {\displaystyle F_{X}} is independent of time.
Examples
White noise is the simplest example of 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 {\displaystyle 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 that 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 Y {\displaystyle Y} be any scalar random variable, and define a time-series { X t } {\displaystyle \left\{X_{t}\right\}} by
X t = Y for all t . {\displaystyle X_{t}=Y\qquad {\text{ for all }}t.}
Then { X t } {\displaystyle \left\{X_{t}\right\}} 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 {\displaystyle Y} , rather than taking the expected value of Y {\displaystyle Y} . The time average of X t {\displaystyle X_{t}} does not converge since the process is not ergodic.
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