Stationary process

Remarks

Equivalently, all finite-dimensional distributions of $X$ are invariant under the time shift by $h$, for every $h \in T$.

A real process with $E[X_t^2] < \infty$ for all $t$ is weakly stationary, or wide-sense stationary, if $E[X_t] = m$ does not depend on $t$ and the covariance $\operatorname{Cov}(X_{t+h}, X_t) = r(h)$ depends only on $h$. A strictly stationary process with finite second moments is weakly stationary. For Gaussian processes the two notions agree.

Example: an i.i.d. sequence $(X_n)_{n \in \Z}$ is stationary, since the law of $(X_{t_1 + h}, \dots, X_{t_k + h})$ is the product of $k$ copies of the common law, for distinct times, whatever $h$ is.

Non-example: a simple symmetric random walk $S_n$ with $S_0 = 0$ is not stationary. Already for $k = 1$, the law of $S_0$ is the point mass at $0$, while $S_1$ takes the values $\pm 1$.