Strictly stationary process

Every finite-dimensional distribution of a strictly stationary process is invariant under a common shift of its time indices. If second moments are finite, strict stationarity implies a weakly stationary process. A constant mean and lag-dependent covariance alone do not imply strict stationarity.

Let ${\mathcal{F}}_{-\infty }^0$ and ${\mathcal{F}}_n^{\infty }$ be the sigma-algebras generated by the past and future observations. The mixing coefficient is $\alpha (n)={\sup }_{A\in {\mathcal{F}}_{-\infty }^0,B\in {\mathcal{F}}_n^{\infty }}|P(A\cap B)-P(A)P(B)|$. Strong mixing means $\alpha (n)\to 0$. Appropriate quantitative mixing and moment conditions can imply a central limit theorem; absolute summability of autocovariance alone does not.

A stationary path law is ergodic when every time-shift-invariant event has probability zero or one. The Birkhoff ergodic theorem then identifies integrable time averages with deterministic ensemble expectations. A shared random scale multiplying iid noise gives a stationary counterexample: it is uncorrelated across distinct times, but the path retains information about the random scale.