It is well known that the probability density function characterizes the independence of random variables in the following sense.
$$X,Y \quad\text{independent}\iff f(x,y)=f_x(x)f_y(y)$$
where $f$ is the pdf of the random variable $(X,Y)$ and $f_x$ and $f_y$ are the pdfs of the marginal distributions of $X$ and $Y$, respectively. My question is, seeing as there is a bijection between characteristic functions and probability distributions, do characteristic functions also characterize independence? In the following sense
$$\varphi_{(X,Y)}(t_1,t_2)=\varphi_X(t_1)\varphi_Y(t_2)$$
where again $\varphi$ is the characteristic function of the random variable in its subscript.
