 Stationary process

In the mathematical sciences, a stationary process (or strict(ly) stationary process or strong(ly) stationary process) is a stochastic process whose joint probability distribution does not change when shifted in time or space. Consequently, parameters such as the mean and variance, if they exist, also do not change over time or position.
Stationarity is used as a tool in time series analysis, where the raw data are often transformed to become stationary; for example, economic data are often seasonal and/or dependent on a nonstationary price level. An important type of nonstationary process that does not include a trendlike behavior is the cyclostationary process.
Note that a "stationary process" is not the same thing as a "process with a stationary distribution". Indeed there are further possibilities for confusion with the use of "stationary" in the context of stochastic processes; for example a "timehomogeneous" Markov chain (which condition is sometimes called by the name "stationary Markov chain") is sometimes said to have "stationary transition probabilities".
Contents
Definition
Formally, let be a stochastic process and let represent the cumulative distribution function of the joint distribution of at times . Then, is said to be stationary if, for all k, for all τ, and for all ,
Since τ does not affect F_{X}(.), F_{X} is not a function of time.
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 discretetime 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 discretetime 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 nontrivial autoregressive component may be either stationary or nonstationary, depending on the parameter values, and important nonstationary special cases are where unit roots exist in the model.
Let Y be any scalar random variable, and define a timeseries { 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 noisefree structure, let Y have a uniform distribution on (0,2π] and define the time series { X_{t} } by
Then { X_{t} } is strictly stationary.
Weaker forms of stationarity
Weak or wide–sense stationarity
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 and 2nd moments do not vary with respect to time. Any strictly stationary process which has a mean and a covariance is also WSS.
So, a continuoustime random process x(t) which is WSS has the following restrictions on its mean function
and autocorrelation function
The first property implies that the mean function m_{x}(t) must be constant. The second property implies that the correlation function depends only on the difference between t_{1} and t_{2} and only needs to be indexed by one variable rather than two variables. Thus, instead of writing,
the notation is often abbreviated and written as:
This also implies that the autocovariance depends only on τ = t_{1} − t_{2}, since
When processing WSS random signals with linear, timeinvariant (LTI) filters, it is helpful to think of the correlation function as a linear operator. Since it is a circulant operator (depends only on the difference between the two arguments), its eigenfunctions are the Fourier complex exponentials. Additionally, since the eigenfunctions of LTI operators are also complex exponentials, LTI processing of WSS random signals is highly tractable—all computations can be performed in the frequency domain. Thus, the WSS assumption is widely employed in signal processing algorithms.
Secondorder stationarity
The case of secondorder stationarity arises when the requirements of strict stationarity are only applied to pairs of random variables from the timeseries. The definition of second order stationarity can be generalized to Nth order (for finite N) and strict stationary means stationary of all orders.
A process is second order stationary if the first and second order density functions satisfy
for all t_{1}, t_{2}, and Δ. Such a process will be wide sense stationary if the mean and correlation functions are finite. A process can be wide sense stationary without being second order stationary.
Other terminology
The terminology used for types of stationarity other than strict stationarity can be rather mixed. Some examples follow.

 Priestley^{[1]}^{[2]} uses stationary up to order m if conditions similar to those given here for wide sense stationarity apply relating to moments up to order m. Thus wide sense stationarity would be equivalent to "stationary to order 2", which is different from the definition of secondorder stationarity given here.

 Honarkhah^{[3]} also uses the assumption of stationarity in the context of multiplepoint Geostatistics, where higher npoint statistics are assumed to be stationary in the spatial domain.
See also
 Lévy process
 Stationary ergodic process
 WienerKhinchin theorem
References
 ^ Priestley, M.B. (1981) Spectral Analysis and Time Series, Academic Press. ISBN 0125649223
 ^ Priestley, M.B. (1988) Nonlinear and Nonstationary Time Series Analysis, Academic Press. ISBN 0125649118
 ^ Honarkhah, M and Caers, J, 2010, Stochastic Simulation of Patterns Using DistanceBased Pattern Modeling, Mathematical Geosciences, 42: 487  517
External links
Categories: Stochastic processes
 Signal processing
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