Let $X$ and $Y$ be random variables. I am trying to understand if we can conclude that $X$ and $Y$ are independent given that their joint MGF factorises into a product of the individual MGFs of $X$ and $Y$. I found the following discussion, where the top answer states a relevant theorem and its proof:
Theorem (Kac's theorem) Let $X,Y$ be $\mathbb{R}^d$-valued random variables. Then the following statements are equivalent.
- $X,Y$ are independent
- $\forall \eta,\xi \in \mathbb{R}^d: \mathbb{E}e^{\imath \, (X,Y) \cdot (\xi,\eta)} = \mathbb{E}e^{\imath \, X \cdot \xi} \cdot \mathbb{E}e^{\imath \, Y \cdot \eta}$
So if the joint characteristic function factorises as above, then $X$ and $Y$ are independent. However can the same be said for moment generating functions? I feel like it might not be true, as some of the comments on the linked question were implying, but I would like to understand why a similar proof to the linked answer would not work for the following statement (supposing the terms are well defined for all $a,b$): $$\forall a,b \in \mathbb{R}^d : \mathbb{E}e^{(X,Y) \cdot (a,b)} = \mathbb{E}e^{X \cdot a} \cdot \mathbb{E}e^{Y \cdot b} \implies X,Y \text{ are independent.}$$
Also if this statement is false, what would be a counterexample?
