Probability Theory – Moment Generating Functions and Independence of X,Y

Question

This is solely a reference request. I have heard a few versions of the following theorem:

If the joint moment generating function $\mathbb{E}[e^{uX+vY}] = \mathbb{E}[e^{uX}]\mathbb{E}[e^{vY}]$ whenever the expectations are finite, then $X,Y$ are independent.

And there is a similar version for characteristic functions.
Could anyone provide me a serious reference which proves one or both of these theorems?

Answer 1

Theorem (Kac's theorem) Let $X,Y$ be $\mathbb{R}^d$-valued random variables. Then the following statements are equivalent.

1. $X,Y$ are independent
2. $\forall \eta,\xi \in \mathbb{R}^d: \mathbb{E}e^{\imath \, (X,Y) \cdot (\xi,\eta)} = \mathbb{E}e^{\imath \, X \cdot \xi} \cdot \mathbb{E}e^{\imath \, Y \cdot \eta}$

Proof:

• $(1) \Rightarrow (2)$: Straightforward, use $\mathbb{E}(f(X) \cdot g(Y)) = \mathbb{E}(f(X)) \cdot \mathbb{E}(g(Y))$
• $(2) \Rightarrow (1)$: Let $(\tilde{X},\tilde{Y})$ be such that $\tilde{X}$, $\tilde{Y}$ are independent, $\tilde{X} \sim X$, $\tilde{Y} \sim Y$. Then $$\mathbb{E}e^{\imath \, (X,Y) \cdot (\xi,\eta)} \stackrel{(2)}{=} \mathbb{E}e^{\imath \, X \cdot \xi} \cdot \mathbb{E}e^{\imath \, Y \cdot \eta} = \mathbb{E}e^{\imath \tilde{X} \cdot \xi} \cdot \mathbb{E}e^{\imath \tilde{Y} \cdot \eta} = \mathbb{E}e^{\imath (\tilde{X},\tilde{Y}) \cdot (\xi,\eta)}$$ i.e. the characteristic functions of $(X,Y)$ and $(\tilde{X},\tilde{Y})$ coincide. From the uniqueness of the Fourier transform we conclude $(X,Y) \sim (\tilde{X},\tilde{Y})$. Consequently, $X$ and $Y$ are independent.

Remark: It is not important that $X$ and $Y$ are vectors of the same dimension. The same reasoning works if, say, $X$ is an $\mathbb{R}^k$-valued random variable and $Y$ and $\mathbb{R}^d$-valued random variable.

Reference (not for the given proof, but the result):David Applebaum, B.V. Rajarama Bhat, Johan Kustermans, J. Martin Lindsay, Michael Schuermann, Uwe Franz: Quantum Independent Increment Processes I: From Classical Probability to Quantum Stochastic Calculus (Theorem 2.1).