= Finite-dimensional distribution
{wiki}
For times $t_1,\ldots,t_n$, the corresponding finite-dimensional distribution of a <stochastic process> $X$ is the <joint probability distribution> of the <random vector> $(X_{t_1},\ldots,X_{t_n})$. The collection of these distributions records every finite set of coordinates of the process.
Knowing only the one-time <marginal distributions> is insufficient. Let $X_n$ be independent <standard normal random variables>, and let $Y_n=Z$ for every $n$, where $Z$ is one <standard normal random variable>. Every $X_n$ and $Y_n$ has the same <standard normal distribution>. Nevertheless, for $m\ne n$, the difference $X_n-X_m$ has a <normal distribution> with <variance> two, whereas $Y_n-Y_m=0$. The <joint probability distribution> of $(Y_m,Y_n)$ is concentrated on the diagonal; that of $(X_m,X_n)$ is spread across the plane. A model for tomorrow's value therefore needs the relation between observations as well as their separate distributions.
\Image[/covariance-process-comparison.png]
{title=Identical standard normal marginals can give independent histories or perfectly shared histories}
For a real-valued <stochastic process>, its <finite-dimensional distributions> determine its <probability law> on the <product sigma-algebra>. With an uncountable <index set>, a path property such as <continuity> need not be measurable in that <sigma-algebra>; specifying a suitable space of <sample paths> and a regular <modification of a stochastic process> is a separate step. This distinction is explained in https://ocw.mit.edu/courses/6-436j-fundamentals-of-probability-fall-2018/b9342e718ec4c58d98ec97dcfcb0d56e_MIT6_436JF18_lec20.pdf[MIT, The basics of stochastic processes, section 1.1].
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