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  "body": "\n## Definition\n\nLet $T$ be a set and let $X = (X_t)_{t \\in T}$ be a real-valued [[stochastic process|0QH5ENKZM1]].\nThe process $X$ is a *Gaussian process* if, for every $k \\ge 1$ and all $t_1, \\dots, t_k \\in T$, the random vector $(X_{t_1}, \\dots, X_{t_k})$ has a [[multivariate normal distribution|0XN3K7RSZY]], possibly degenerate.\n\n## Remarks\n\nEquivalently, every finite linear combination $\\sum_{i=1}^{k} c_i X_{t_i}$ with real $c_i$ is normal or constant.\n\nThe *mean function* is $m(t) = E[X_t]$ and the *covariance function* is $K(s, t) = \\operatorname{Cov}(X_s, X_t)$.\nThey determine the finite-dimensional distributions, hence the law of $X$.\nThe function $K$ is positive semidefinite: the matrix $(K(t_i, t_j))_{i, j}$ is positive semidefinite for all $t_1, \\dots, t_k$.\nConversely, by the [[Kolmogorov extension theorem|WR4X7MMTH2]], for every function $m$ and every symmetric positive semidefinite $K$ there is a Gaussian process with mean $m$ and covariance $K$.\n\nExample: the [[Wiener process|22735FGJV6]] is a Gaussian process on $[0, \\infty)$ with $m = 0$ and $K(s, t) = \\min(s, t)$.\nCheck: $(W_{t_1}, \\dots, W_{t_k})$ for $t_1 \u003c \\cdots \u003c t_k$ is a linear image of the independent normal increments $W_{t_1}, W_{t_2} - W_{t_1}, \\dots$, hence multivariate normal.\n\nNon-example: let $Z$ be standard normal and let $\\varepsilon$ be independent of $Z$ with $P(\\varepsilon = 1) = P(\\varepsilon = -1) = 1/2$.\nLet $X_1 = Z$ and $X_2 = \\varepsilon Z$ on $T = \\set{1, 2}$.\nEach $X_t$ is standard normal, but $X_1 + X_2 = (1 + \\varepsilon) Z$ equals $0$ with probability $1/2$ and is not constant, so it is neither normal nor constant.\nThus $(X_1, X_2)$ is not multivariate normal, and $X$ is not a Gaussian process.\n",
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