stationary process

Let $\{X(t)\mid t\in T\}$ be a stochastic process where $T\subseteq\mathbb{R}$ 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 $n<\infty$ , any $t_{1},\ldots,t_{n}$ and $s\in T$ , the random vectors

$(X(t_{1}),\ldots,X(t_{n}))$ and $(X(t_{1}+s),\ldots,X(t_{n}+s))$ 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 $n<\infty$ , any $t_{1},\ldots,t_{n}$ and $s\in T$ , the joint distributions of the random vectors

$(X(t_{1}),\ldots,X(t_{n}))$ and $(X(t_{1}+s),\ldots,X(t_{n}+s))$ 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