Is the product of two different characteristic functions also a characteristic function?

Question

Suppose that $\phi_{X}(t)$ and $\phi_{Y}(t)$ are characteristic functions of $X, Y$, respectively. Moreover, $X$ and $Y$ are NOT independent random variables. I want to know if $\phi_{X}(t)\cdot\phi_{Y}(t)$ also a characteristic function?

Answer 1

Let $X',Y'$ be independent random variables such that $X$ and $X'$ have the same distribution and $Y$ and $Y'$ have the same distribution. (Such random variables always exist). Then the characteristic function of $X'+Y'$ is $\phi_X \phi_Y$. So the answer is YES.

Answer 2

$\phi_{X+Y}(t) = \mathbb{E}(e^{it(X+Y)}) = \mathbb{E}(e^{itX} \cdot e^{itY}) = \{ \text{independence} \} = \mathbb{E}(e^{itX}) \cdot \mathbb{E}(e^{itY}) = \phi_{X}(t) \cdot \phi_{Y}(t)$.

So if $X \perp Y \Rightarrow \phi_{X+Y}(t) = \phi_{X}(t) \cdot \phi_{Y}(t)$