Stationary series

In mathematics and statistics, a stationary process (a.k.a. a strict(ly) stationary process or strong(ly) stationary process) is a whose unconditional 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 is a trend in the 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 after which the variable tends toward a deterministically evolving (non-constant) mean.

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.

Formally, let ${\displaystyle \left\{X_{t}\right\}}$ be a and let ${\displaystyle F_{X}(x_{t_{1}+\tau },\ldots ,x_{t_{k}+\tau })}$ represent the cumulative distribution function of the unconditional (i.e., with no reference to any particular starting value) joint distribution of ${\displaystyle \left\{X_{t}\right\}}$ at times ${\displaystyle t_{1}+\tau ,\ldots ,t_{k}+\tau }$. Then, ${\displaystyle \left\{X_{t}\right\}}$ is said to be strictly (or strongly) stationary if, for all ${\displaystyle {k}}$, for all ${\displaystyle {\tau }}$, and for all ${\displaystyle t_{1},\ldots ,t_{k}}$,

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