stacionaritás

Stacionaritás, or stationarity, is a property of a stochastic process in statistics and time series analysis. A stationary process has statistical characteristics that do not depend on the absolute time at which the process is observed.

There are two common notions of stationarity. Strict (or strong) stationarity means that for any collection of times t1, …, tn and any time shift h, the joint distribution of (X_t1, …, X_tn) is identical to the joint distribution of (X_{t1+h}, …, X_{tn+h}). This implies that all moments and dependencies are time-invariant.

Weak (or second-order) stationarity requires less. A process is weakly stationary if its mean is constant over time, its variance is finite and constant, and its autocovariance Cov(X_t, X_{t+h}) depends only on the lag h, not on the actual time t. Weak stationarity does not require independence, only a time-invariant second moment structure.

Examples help clarify. White noise, where X_t = ε_t with ε_t independent and identically distributed with zero mean, is strictly and weakly stationary. An autoregressive process AR(1): X_t = φ X_{t-1} + ε_t is stationary if |φ| < 1. By contrast, a random walk X_t = X_{t-1} + ε_t is non-stationary in the strict and weak senses.

In practice, stationarity underpins many modelling and forecasting methods. Non-stationary data are often transformed to achieve stationarity, for example by differencing or detrending. Statistical tests such as the Augmented Dickey-Fuller (ADF) test or the KPSS test are used to assess stationarity. It is important to distinguish stationarity from independence, as a stationary process may exhibit temporal dependence; ergodicity is a related concept linking time and ensemble averages.