# 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 $$, any ${t}_{1},\mathrm{\dots},{t}_{n}$ and $s\in T$, the
random 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 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 $$, any
${t}_{1},\mathrm{\dots},{t}_{n}$ and $s\in T$, the joint distributions of the random
vectors

$(X({t}_{1}),\mathrm{\dots},X({t}_{n}))$ 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*.

Title | stationary process |
---|---|

Canonical name | StationaryProcess |

Date of creation | 2013-03-22 15:22:42 |

Last modified on | 2013-03-22 15:22:42 |

Owner | CWoo (3771) |

Last modified by | CWoo (3771) |

Numerical id | 6 |

Author | CWoo (3771) |

Entry type | Definition |

Classification | msc 60G10 |

Defines | strictly stationary process |

Defines | covariance stationary process |

Defines | evolutionary process |