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  "body": "\n## Definition\n\nLet $T$ be one of $\\Z$, $\\N$, $\\R$ or $[0, \\infty)$, so that $t + h \\in T$ whenever $t, h \\in T$.\nLet $X = (X_t)_{t \\in T}$ be a [[stochastic process|0QH5ENKZM1]] with values in a measurable space $S$.\nThe process $X$ is *stationary*, or *strictly stationary*, if for every $k \\ge 1$, all $t_1, \\dots, t_k \\in T$ and every $h \\in T$, the random vectors\n$$(X_{t_1 + h}, \\dots, X_{t_k + h}) \\quad \\text{and} \\quad (X_{t_1}, \\dots, X_{t_k})$$\nhave the same law.\n\n## Remarks\n\nEquivalently, all [[finite-dimensional distributions|Y0CN7V3B7E]] of $X$ are invariant under the time shift by $h$, for every $h \\in T$.\n\nA real process with $E[X_t^2] \u003c \\infty$ for all $t$ is *weakly stationary*, or *wide-sense stationary*, if $E[X_t] = m$ does not depend on $t$ and the [[covariance|0KBFAWYRTE]] $\\operatorname{Cov}(X_{t+h}, X_t) = r(h)$ depends only on $h$.\nA strictly stationary process with finite second moments is weakly stationary.\nFor [[Gaussian processes|22WEM6RTW0]] the two notions agree.\n\nExample: an i.i.d. sequence $(X_n)_{n \\in \\Z}$ is stationary, since the law of $(X_{t_1 + h}, \\dots, X_{t_k + h})$ is the product of $k$ copies of the common law, for distinct times, whatever $h$ is.\n\nNon-example: a simple symmetric [[random walk|04VZE3FZXF]] $S_n$ with $S_0 = 0$ is not stationary.\nAlready for $k = 1$, the law of $S_0$ is the point mass at $0$, while $S_1$ takes the values $\\pm 1$.\n",
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