1) How is Characteristic function of a complex valued random variable defined? Should it be considered as vector of real random variables or the definition in wiki be used?
2) Also how is the characteristic function defined for vector of complex random variables?
3) If $Z=\begin{bmatrix} X \\ Y \\ \end{bmatrix}$ with $X$ and $Y$ independent real random variables,is the characteristic function of $Z$ equals convolution of C.F of $X$ and C.F of $Y$? (Since $f_Z(z)=f_X(x)f_Y(y)$) and will that be the same as the definition of C.F for vector of random variables?