---
tag: 314WH1WAP1
kind: definition
name: Stationary process
msc: [60G10]
defines: DT0QXJE705
uses: [0QH5ENKZM1, Y0CN7V3B7E, 0WQNRCNY3M, 0KBFAWYRTE]
sources:
  - url: https://en.wikipedia.org/wiki/Stationary_process
---

## Definition

Let $T$ be one of $\Z$, $\N$, $\R$ or $[0, \infty)$, so that $t + h \in T$ whenever $t, h \in T$.
Let $X = (X_t)_{t \in T}$ be a [[stochastic process|0QH5ENKZM1]] with values in a measurable space $S$.
The process $X$ is *stationary*, or *strictly stationary*, if for every $k \ge 1$, all $t_1, \dots, t_k \in T$ and every $h \in T$, the random vectors
$$(X_{t_1 + h}, \dots, X_{t_k + h}) \quad \text{and} \quad (X_{t_1}, \dots, X_{t_k})$$
have the same law.

## Remarks

Equivalently, all [[finite-dimensional distributions|Y0CN7V3B7E]] 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|0KBFAWYRTE]] $\operatorname{Cov}(X_{t+h}, X_t) = r(h)$ depends only on $h$.
A strictly stationary process with finite second moments is weakly stationary.
For [[Gaussian processes|22WEM6RTW0]] 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|04VZE3FZXF]] $S_n$ 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 $\pm 1$.