I am interested on the proof of the following result:
If $X=(X_1,\ldots,X_n)$ is a random vector, then $M_X(t_1,\ldots,t_n)=M_{X_{1}}(t_1)\cdots M_{X_{n}}(t_n)$ for all $(t_1,\ldots,t_n)$ implies that $X_1,\ldots,X_n$ are independent.
I have been searching for a proof on the Internet, and found this one here.
My doubt relies on the proof of P2. More concretely, I do not understand the statement "$M_X$ is equal to the moment generating function of a vector whose components $X_1,\ldots,X_n$ are independent". How do you know that vector exists? If you want to prove that $X_1,\ldots,X_n$ are precisely independent, how can you construct a vector whose components $X_1,\ldots,X_n$ are independent?