stationary process

Let $\{X(t)\mid t\in T\}$ be a stochastic process where $T\subseteq ℝ$ and has the property that $s+t\in T$ whenever $s,t\in T$. Then $\{X(t)\}$ is said to be a strictly stationary process of order n if for a given positive integer , any $t_1,\mathrm{\dots },t_n$ and $s\in T$, the random vectors

$(X(t_1),\mathrm{\dots },X(t_n))$ and $(X(t_1+s),\mathrm{\dots },X(t_n+s))$ have identical joint distributions.

$\{X(t)\}$ is said to be a strictly stationary process if it is a strictly stationary process of order $n$ for all positive integers $n$. Alternatively, $\{X(t)\mid t\in T\}$ is strictly stationary if $\{X(t)\}$ and $\{X(t+s)\}$ are identically distributed stochastic processes for all $s\in T$.

A weaker form of the above is the concept of a covariance stationary process, or simply, a stationary process $\{X(t)\}$. Formally, a stochastic process $\{X(t)\mid t\in T\}$ is stationary if, for any positive integer , any $t_1,\mathrm{\dots },t_n$ and $s\in T$, the joint distributions of the random vectors

$(X(t_1),\mathrm{\dots },X(t_n))$ and $(X(t_1+s),\mathrm{\dots },X(t_n+s))$ have identical means (mean vectors) and identical covariance matrices.

So a strictly stationary process is a stationary process. A non-stationary process is sometimes called an evolutionary process.

Title	stationary process
Canonical name	StationaryProcess
Date of creation	2013-03-22 15:22:42
Last modified on	2013-03-22 15:22:42
Owner	CWoo (3771)
Last modified by	CWoo (3771)
Numerical id	6
Author	CWoo (3771)
Entry type	Definition
Classification	msc 60G10
Defines	strictly stationary process
Defines	covariance stationary process
Defines	evolutionary process