##  [Stationarity (Stochastic Process)](/stationarity-stochastic-process-0) 

 Definition

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

Principle

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

 

 

 

 

 





## Demonstration

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

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

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

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

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

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

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.