Theorem

factorization criterion for independence

A family of random variables $\{ X_t : t \in T \}$ indexed by a set $T$ is independent if and only if for all finite $J \subset T$ and all $x_t \in \mathbb{R}$ we have $F_{J}(x_t, t \in J) \overset{\Delta}{=} \mathbb{P}[X_t \leq x_t, t \in J] = \prod_{t \in J} \mathbb{P}[X_t \leq x_t]$.