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  "body": "\n## Summary\n\nThe finite-dimensional distributions of a [[stochastic process|0QH5ENKZM1]] $(X_t)_{t \\in T}$ are the joint laws of the random vectors $(X_{t_1}, \\dots, X_{t_k})$ for finitely many times $t_1, \\dots, t_k$.\nThey determine the law of the process on the product $\\sigma$-algebra.\nBy the [[Kolmogorov extension theorem|WR4X7MMTH2]], a consistent family of such laws comes from some process, which is the usual way to construct processes.\nThey do not determine path properties such as continuity.\n\n## Explanations\n\n### L1 human\n\nA random process is a random quantity that evolves over time.\nTo describe it, one can list, for each finite set of times, the joint distribution of its values at those times.\nThese are the finite-dimensional distributions.\nFor example, take a sequence of independent fair coin tosses.\nThe values at any $k$ times are independent and each is fair.\nSo each such joint law gives probability $2^{-k}$ to each of the $2^k$ outcomes.\n",
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