Probability Theory – Criterion for Independence Based on Characteristic Function

characteristic-functionsindependenceprobability theory

Let $X$ and $Y$ be real-valued random variables defined on the same space. Let's use $\phi_X$ to denote the characteristic function of $X$. If $\phi_{X+Y}=\phi_X\phi_Y$ then must $X$ and $Y$ be independent?

Best Answer

As user75064 already pointed out, the answer is "no". However, there is the following result:

Let $X,Y$ be $\mathbb{R}^d$-valued random variables. Then the following statements are equivalent.

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

i.e. if the characteristic function of the random vector $(X,Y)$ equals the product of the characteristic function of $X$ and $Y$, then $X$ and $Y$ are independent (proof).

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