Finite-dimensional distributions

Summary

The finite-dimensional distributions of a stochastic process $(X_t)_{t \in T}$ are the joint laws of the random vectors $(X_{t_1}, \dots, X_{t_k})$ for finitely many times $t_1, \dots, t_k$. They determine the law of the process on the product $\sigma$-algebra. By the Kolmogorov extension theorem, a consistent family of such laws comes from some process, which is the usual way to construct processes. They do not determine path properties such as continuity.