Stationary process

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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. As a result, 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 the price level. Processes are described as trend stationary if they are a linear combination of a stationary process and one or more processes exhibiting a trend. Transforming these data to leave a stationary data set for analysis is referred to as de-trending.

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[edit] Definition

Formally, let Xt be a stochastic process and let F_{X_{t_1},\ldots, X_{t_k}}(x_{t_1}, \ldots, x_{t_k}) represent the cumulative distribution function of the joint distribution of Xt at times t_1, \ldots, t_k. Then, Xt is said to be stationary if, for all k, for all τ, and for all t_1, \ldots, t_k,

F_{X_{t_1} ,\ldots, X_{t_k}}(x_{t_1}, \ldots, x_{t_k}) = F_{X_{t_1+\tau} ,\ldots, X_{t_k+\tau}}(x_{t_1}, \ldots, x_{t_k}).

[edit] Law of large numbers

A stationary sequence of random variables can be written as Xt + 1 = TXt, where T is a measure-preserving operator on some probability space,[1] thus a law of large numbers in the form of Birkhoff-Khinchin theorem applies.

[edit] Examples

As an example, white noise is stationary. However, the sound of a cymbal crashing is not stationary because the acoustic power of the crash (and hence its variance) diminishes with time.

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 autoregressive and moving average processes which are both subsets of the autoregressive moving average model.

If one also assumes divisibility (independent increments) and continuity, one obtains a Lévy process.

[edit] Weaker forms of stationarity

[edit] 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 continuous-time random process x(t) which is WSS has the following restrictions on its mean function

\mathbb{E}\{x(t)\} = m_x(t) = m_x(t + \tau) \,\, \forall \, \tau \in \mathbb{R}

and autocorrelation function

\mathbb{E}\{x(t_1)x(t_2)\} = R_x(t_1, t_2) = R_x(t_1 + \tau, t_2 + \tau) = R_x(t_1 - t_2, 0) \,\, \forall \, \tau \in \mathbb{R}.

The first property implies that the mean function mx(t) must be constant. The second property implies that the correlation function depends only on the difference between t1 and t2 and only needs to be indexed by one variable rather than two variables. Thus, instead of writing,

\,\!R_x(t_1 - t_2, 0)\,

we usually abbreviate the notation and write

R_x(\tau) \,\! \mbox{ where } \tau = t_1 - t_2.

This also implies that the autocovariance depends only on τ = t1 − t2, since

\,\! C_x(t_1,t_2) = C_x(t_1-t_2,0) = C_x(\tau).

When processing WSS random signals with linear, time-invariant (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.

[edit] Second-order stationarity

The case of second-order stationarity arises when the requirements of strict stationarity are only applied to pairs of random variables from the time-series. The definition of second order stationarity can be generalised 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

~f_X(x_1 : t_1 ) = f_X(x_1 : t_1 + \Delta),
~f_X(x_1 ,x_2 : t_1, t_2 ) = f_X(x_1 ,x_2 : t_1 + \Delta, t_2 +\Delta )~ ,

for all t1, t2, 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.

[edit] Other terminology

The terminology used for types of stationarity other than strict stationarity can be rather mixed. Some examples follow.

  • Priestley[2][3] 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 second-order stationarity given here.

[edit] See also

[edit] References

  1. ^ A.N. Shiryaev, Probability, 2nd ed., Springer 1996, pages 405 and 409, ISBN 0-387-94549-0.
  2. ^ Priestley, M.B. (1981) Spectral Analysis and Time Series, Academic Press. ISBN 0-12-564922-3
  3. ^ Priestley, M.B. (1988) Non-linear and Non-stationary Time Series Analyis, Academic Press. ISBN 0-12-564911-8
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