stationarity (strict and wide-sense)

Some random phenomena have settled into a steady statistical rhythm: the rules of the game no longer depend on what o'clock it is. The hiss of a fan, the daily ups and downs of a long-running queue at equilibrium, background noise — measure them now or an hour from now and the statistics look the same. Stationarity is the precise way of saying 'the probabilistic behaviour does not change as time slides forward'.

There are two grades. Strict (strong) stationarity is the demanding version: the entire joint distribution is invariant under time shifts. That is, for any times t_1, ..., t_n and any shift h, the law of (X_{t_1}, ..., X_{t_n}) is identical to the law of (X_{t_1 + h}, ..., X_{t_n + h}). Sliding the whole window of times left or right leaves every finite-dimensional distribution unchanged. Wide-sense (weak, second-order) stationarity asks much less: only that the mean E[X_t] is constant in t, and the covariance Cov(X_s, X_t) depends only on the time gap t - s, not on where the pair sits on the clock. It controls just the first two moments, not the full distribution.

The distinction matters because the two are not equivalent. Strict stationarity (with finite variance) implies wide-sense, but wide-sense does not imply strict — a process can have a constant mean and shift-invariant covariance while higher-order features still drift. There is one famous exception: for a Gaussian process the first two moments determine everything, so wide-sense and strict stationarity coincide. Stationarity is the backbone assumption of time-series analysis and signal processing, because it is what lets you estimate a process's statistics by averaging over time from a single long record.