Stationarityan

Stationarityan is a theoretical construct in probability theory and time series analysis that extends the notion of stationarity to settings where the timing of observations may be altered by a structured transformation, yet certain statistical properties remain invariant. Unlike classical strict or wide-sense stationarity, which require invariance under fixed time shifts, stationarityan allows invariance under a prescribed family of time reparameterizations or state-space transformations.

Formally, let {X_t} be a stochastic process indexed by t in a time set T, and let G be a group of transformations acting on T or on the state space. The process is said to be stationaryan of order k if, for every s in G, the k-dimensional joint distribution of (X_{t1}, ..., X_{tk}) is the same as the joint distribution of (H_s(X_{t1+s}), ..., H_s(X_{tk+s})), where H_s is a measurable transformation associated with s. In a weak sense, moments of order up to p are invariant under the same transformations.

In practice, stationarityan is a theoretical generalization used to model processes that display recurring nonstationarities that can be absorbed into a known transformation. Common examples include processes obtained by applying a fixed time change to a stationary process or processes with periodic structure after a non-linear time warp.

Relation to other concepts: it subsumes strict and weak stationarity as special cases when G contains only the identity and H_s is the identity. It is distinct from ergodicity and from nonstationarity in that invariance is retained under the transformation. Applications are largely in theoretical studies and in the development of generalized inference methods. See also: stationarity, nonstationary processes, time series analysis, ergodicity.