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Homestationary process

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# 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*.

## Mathematics Subject Classification

60G10*no label found*

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