= Stationary process
{wiki}
= Stationarity
{synonym}
A <stationary process>, in the strict sense, has the same <joint probability distribution> after every common shift of its time indices. More precisely, for every positive integer $n$, every choice of times $t_1,\ldots,t_n$, and every integer shift $k$,
$$
(X_{t_1},\ldots,X_{t_n})\overset{d}{=}(X_{t_1+k},\ldots,X_{t_n+k}).
$$
Here equality means equality of <probability distributions>, not equality of realized observations. A <sample path> can fluctuate, stay positive for a while, or show accidental runs while its generating law remains stationary. <Strong white noise> is a basic example: shifting time preserves every joint law of its <independent and identically distributed random variables>.
The distinction between <strict stationarity> and <weak stationarity> is useful. The former preserves all finite joint laws; the latter preserves only the finite <mean> and <covariance> structure. <Strict stationarity> with finite <second moments> implies <weak stationarity>, but strict stationarity itself does not require finite <variance>. A <Gaussian process> with <weak stationarity> is also strictly stationary, because each of its <finite-dimensional distributions>, a joint <normal distribution>, is determined by its <mean> vector and <covariance matrix>.
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