We can more generally define the independence between two processes $X=(X_t)_t$ and $Y=(Y_t)_t$ by the following way: if $\mathcal F_X$ and $\mathcal F_Y$ are the smallest $\sigma$-algebra making respectively $X$ and $Y$ measurable, we say that $X$ and $Y$ are independent if and only if so are $\mathcal F_X$ and $\mathcal F_Y$.
We can show the equivalence between
- $X$ and $Y$ are independent;
- The finite-dimensional distributions of $X$ and $Y$ are independent.
If we assume that the finite-dimensional distributions of $X$ and $Y$ are independent, then consider $\pi_1$ and $\pi_2$, the sets which consists of finite intersection of sets of the form $X_t^{-1}(B)$ (respectively $Y_t^{-1}$), where $B$ is a Borel set. These one are $\pi$-systems (i.e. stable by finite intersection), and a monotone class result ensure us that the monotone class generated by $\pi$ and $\pi_2$ are independent.