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stationary process
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(Definition)
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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.
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"stationary process" is owned by CWoo.
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strictly stationary process, covariance stationary process, evolutionary process |
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Cross-references: covariance matrices, mean vectors, identically distributed stochastic processes, stationary, strictly, joint distributions, random vectors, integer, positive, order, property, stochastic process
There are 3 references to this entry.
This is version 3 of stationary process, born on 2005-07-05, modified 2008-07-07.
Object id is 7207, canonical name is StationaryProcess.
Accessed 9503 times total.
Classification:
| AMS MSC: | 60G10 (Probability theory and stochastic processes :: Stochastic processes :: Stationary processes) |
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Pending Errata and Addenda
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