Stationary process

In mathematics and statistics, a stationary process (a.k.a. a strict(ly) stationary process or strong(ly) stationary process) is a whose joint probability distribution does not change when shifted in time. Consequently, parameters such as mean and variance, if they are present, also do not change over time.

Since stationarity is an assumption underlying many statistical procedures used in time series analysis, non-stationary data is often transformed to become stationary. The most common cause of violation of stationarity are trends in mean, which can be due either to the presence of a unit root or of a deterministic trend. In the former case of a unit root, stochastic shocks have permanent effects and the process is not mean-reverting. In the latter case of a deterministic trend, the process is called a trend stationary process, and stochastic shocks have only transitory effects which are mean-reverting (i.e., the mean returns to its long-term average, which changes deterministically over time according to the trend).

A trend stationary process is not strictly stationary, but can easily be transformed into a stationary process by removing the underlying trend, which is solely a function of time. Similarly, processes with one or more unit roots can be made stationary through differencing. An important type of non-stationary process that does not include a trend-like behavior is a cyclostationary process, which is a stochastic process that varies cyclically with time.

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 the term "stationary" in the context of stochastic processes; for example a "time-homogeneous" Markov chain is sometimes said to have "stationary transition probabilities". Besides, all stationary Markov random processes are time-homogeneous.

Formally, let $\left\{X_{t}\right\}$ $\left\{X_{t}\right\}$ be a and let $F_{X}(x_{t_{1}+\tau },\ldots ,x_{t_{k}+\tau })$ $F_{{X}}(x_{{t_{1}+\tau }},\ldots ,x_{{t_{k}+\tau }})$ represent the cumulative distribution function of the joint distribution of $\left\{X_{t}\right\}$ $\left\{X_{t}\right\}$ at times $t_{1}+\tau ,\ldots ,t_{k}+\tau$ $t_{1}+\tau ,\ldots ,t_{k}+\tau$ . Then, $\left\{X_{t}\right\}$ $\left\{X_{t}\right\}$ is said to be strictly (or strongly) stationary if, for all ${k}$ ${k}$ , for all ${\tau }$ ${\tau }$ , and for all $t_{1},\ldots ,t_{k}$ $t_{1},\ldots ,t_{k}$ ,

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