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 astrictly stationary process of order n if for a givenpositive integer , any $t_1,\mathrm{\dots },t_n$ and $s\in T$, therandom 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 stationaryprocess if it is a strictly stationary process of order $n$ for allpositive integers $n$. Alternatively, $\{X(t)\mid t\in T\}$ is strictly stationary if $\{X(t)\}$ and$\{X(t+s)\}$ are identically distributed stochasticprocesses for all $s\in T$.

A weaker form of the above is the concept of a covariancestationary 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 randomvectors

$(X(t_1),\mathrm{\dots },X(t_n))$ 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.