stationarity

Stationarity is a property of a stochastic process describing the stability of its statistical characteristics over time. In practice, a stationary process exhibits consistent behavior when observed at different times, meaning its distribution does not change with time in a fundamental way.

There are two common notions of stationarity. Strict (or strong) stationarity requires that the joint distribution of any collection of observations is unchanged by a shift in time: for all t1, ..., tn and all lags h, the vector (X_t1, ..., X_tn) has the same distribution as (X_{t1+h}, ..., X_{tn+h}). Weak (or second-order) stationarity requires only that the mean is constant over time, the variance is finite and constant, and the autocovariance depends only on the lag between observations, not on the actual time. Weak stationarity is the most commonly used notion in time series analysis.

Stationarity is central to modeling and inference because many methods, including ARMA models and forecasting procedures, assume some form of stationarity. For example, a covariance-stationary process has autocovariances that depend solely on lag, which simplifies estimation and prediction. A process that is not stationary may be difference-stationary or trend-stationary, often requiring transformation, detrending, or differencing to achieve stationarity.

Non-stationary behavior arises from evolving means, changing variances, structural breaks, or unit roots. Techniques to induce stationarity include differencing, log or power transformations to stabilize variance, and removing deterministic trends. Diagnostic tools such as ACF/PACF plots and unit-root tests (e.g., ADF, KPSS) help assess stationarity in practice.