A property of a stochastic process whose finite-dimensional distributions are invariant under time shifts: for all t and all finite index tuples, the joint law of (X_{t+t1},…,X_{t+tk}) does not depend on t.

Principle

Impose statistical homogeneity in the time domain so that probabilistic descriptions depend only on relative time separations rather than absolute time origins.

Demonstration

Discrete-time white noise with i.i.d. zero-mean increments is strictly stationary since any finite collection of values has the same joint distribution after any integer shift; a Gaussian autoregressive model can be stationary if its parameters lie in the stability region.

Misapplication

Confusing stationarity with independence or with constant mean alone; a process can have constant first moment but time-dependent higher-order structure and thus fail to be stationary.

Consequence

Stationarity permits use of spectral methods and time-invariant correlation functions; when combined with ergodicity, time averages converge to ensemble expectations for almost every sample path.

Reversal

Nonstationary processes exhibit statistics that change with time (trends, periodic modulation of moments, evolving variance) and require time-dependent models or detrending for standard stationary analysis to apply.

Boundary

Defined relative to an index set (discrete or continuous) and concerns invariance of joint distributions; weaker notions (second-order or wide-sense stationarity) require only constancy of mean and shift-invariance of second moments.

Semantic Tension

Contrast between strict stationarity (all finite-dimensional distributions invariant) and wide-sense stationarity (only first two moments are shift-invariant); the latter suffices for many linear-signal methods but omits higher-order structure.

Synthesis

Stationarity is the time-translation invariance of a process's probabilistic law, establishing temporal homogeneity that underpins spectral representations, invariant correlation functions, and many standard inferential techniques.