stationary process

Let $\lbrace X(t)\mid t\in T\rbrace$ be a stochastic process where $T\subseteq\mathbb{R}$ and has the property that $s+t\in T$ whenever $s,t\in T$ . Then $\lbrace X(t)\rbrace$ 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.

$\lbrace X(t) \rbrace$ is said to be a strictly stationary process if it is a strictly stationary process of order $n$ for all positive integers $n$ . Alternatively, $\lbrace X(t)\mid t\in T\rbrace$ is strictly stationary if $\lbrace X(t)\rbrace$ and $\lbrace X(t+s)\rbrace$ 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 $\lbrace X(t)\rbrace$ . Formally, a stochastic process $\lbrace X(t)\mid t\in T\rbrace$ 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.