Why can we split product averages of stochastic processes? $\langle x(t_1)x(t_2)\dots x(t_N) \rangle = \langle x(t_1)x(t_2)\rangle \dots$

Question

For white Gaussian or dichotomous stochastic processes it is often stated that averages can be split into a sum of all possible products of two point correlation functions: $$ \langle \prod_n x(t_{n})\rangle = \sum_{(s_1,s_2,\dots)}\langle x(t_{s_1})x(t_{s_2})\rangle \langle x(t_{s_3})x(t_{s_4}) \rangle \dots. $$ Why is this so, and for what reason is this identity limited to only Gaussian and dichotomous noises? It has been tough for me to find a clear answer on this from the books I have access to.