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  "body": "\n## Definition\n\nLet $(\\Omega, \\mathcal{F}, P)$ be a [[probability space|0NFPJVJJHX]], and let $(S, \\mathcal{S})$ be a measurable space.\nLet $X = (X_t)_{t \\in T}$ be a [[stochastic process|0QH5ENKZM1]] with values in $S$, indexed by a set $T$.\nLet $\\mathcal{S}^{\\otimes k}$ be the product [[sigma-algebra|0SV8K0TDGY]] on $S^k$.\nFor $k \\ge 1$ and distinct $t_1, \\dots, t_k \\in T$, the *finite-dimensional distribution* of $X$ at $(t_1, \\dots, t_k)$ is the measure $\\mu_{t_1, \\dots, t_k}$ on $(S^k, \\mathcal{S}^{\\otimes k})$ given by\n$$\\mu_{t_1, \\dots, t_k}(B) = P\\bigl((X_{t_1}, \\dots, X_{t_k}) \\in B\\bigr) \\quad \\text{for } B \\in \\mathcal{S}^{\\otimes k}.$$\nSo $\\mu_{t_1, \\dots, t_k}$ is the [[law|01EQPZ6RYZ]] of the random vector $(X_{t_1}, \\dots, X_{t_k})$.\nThe *finite-dimensional distributions* of $X$ are the family of all these measures.\n\n## Remarks\n\nThe family is consistent.\nPermuting the times permutes the coordinates of the measure, and $\\mu_{t_1, \\dots, t_k}(B \\times S) = \\mu_{t_1, \\dots, t_{k-1}}(B)$.\nConversely, for $S$ a Polish space, the [[Kolmogorov extension theorem|WR4X7MMTH2]] gives a process with any prescribed consistent family.\n\nExample: for the standard [[Wiener process|22735FGJV6]] $W$ on $T = [0, \\infty)$, the law of $(W_{t_1}, \\dots, W_{t_k})$ is the [[multivariate normal distribution|0XN3K7RSZY]] with mean $0$ and covariance matrix $(\\min(t_i, t_j))_{i, j}$.\n\nTwo processes with the same finite-dimensional distributions can have different paths.\nLet $U$ be uniform on $[0, 1]$, let $X_t = 0$ and let $Y_t = \\mathbf{1}_{\\set{t = U}}$ for $t \\in [0, 1]$.\nFor any fixed times, $P(Y_{t_i} = 0 \\text{ for all } i) = 1$, so $X$ and $Y$ have the same finite-dimensional distributions.\nBut every path of $X$ is continuous, and no path of $Y$ is continuous.\n",
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