Finite-dimensional distribution

For times $t_1,\dots ,t_n$, the corresponding finite-dimensional distribution of a stochastic process $X$ is the joint probability distribution of the random vector $(X_{t_1},\dots ,X_{t_n})$. 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.

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 MIT, The basics of stochastic processes, section 1.1.

Consistent probability distributions on finite collections of coordinates in standard Borel spaces determine a unique probability measure on the product sigma-algebra. Consistency means that discarding coordinates by marginalization agrees with the law already assigned to the smaller collection, and relabeling coordinates relabels the law. The theorem turns finite recipes into a complete stochastic process. In particular, products of one fixed probability distribution construct an independent and identically distributed sequence; putting one common random variable in every coordinate constructs the perfectly shared process above. The theorem supplies a law on the coordinate space, but does not by itself ensure continuous sample paths.