---
tag: DT0QXJE705
kind: concept
name: Stationary process
msc: [60G10]
specializes: [0QH5ENKZM1]
sources:
  - url: https://en.wikipedia.org/wiki/Stationary_process
same_as:
  wikidata: Q1192209
  wikipedia: Stationary process
---

## Summary

A stationary process is a [[stochastic process|0QH5ENKZM1]] whose statistical behaviour does not change under shifts of time.
All its [[finite-dimensional distributions|Y0CN7V3B7E]] are invariant under a common time shift.
A weaker notion, weak or wide-sense stationarity, asks only that the mean be constant and the covariance depend only on the time lag.
Stationary processes are the basic models of time series analysis and of ergodic theory, and their covariance has a spectral representation by the [[Wiener–Khinchin theorem|3ZXN26PW4G]].

## Explanations

### L1 human

A process is stationary if it looks the same whenever you start watching it.
Daily noise in a measuring device is often modelled this way, while a growing population is not.
For example, a sequence of independent rolls of a fair die is stationary: any stretch of five rolls has the same joint law as any other stretch of five rolls.
A random walk is not stationary, since its spread grows with time.