# 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 StationaryProcess 2013-03-22 15:22:42 2013-03-22 15:22:42 CWoo (3771) CWoo (3771) 6 CWoo (3771) Definition msc 60G10 strictly stationary process covariance stationary process evolutionary process