Fourier inversion formula for complex-valued random variables?

Question

The characteristic function of a complex-valued random variable $X$ with pdf $\mu$ is given by $$ \phi(t) = \int \exp[i \Re(\bar{t} X)] \; d\mu $$ (or, so says Wikipedia). How does one recover the pdf from $\phi$, i.e., what is the Fourier inversion formula for measures on $\mathbb{C}$? The $\mu$ I am working with is as "nice" as one could ask for.

P.S. Where would I find such a result? (Of course, I could try to work out the exact form of Pontryagin Duality for $\mathbb{C}$ from the definitions, but presumably somebody has done this before.)

Answer 1

you have to take the real part of $\bar{t}X$ in the exponent, in order for the integral to make sense; if you then decompose $t=t_1+it_2$ and $X=X_1+iX_2$ into real and imaginary parts, you just have a conventional two-dimensional Fourier transform

$\phi(t_1,t_2)=\int_{-\infty}^{\infty}dX_1 \int_{-\infty}^{\infty}dX_2 exp(it_1 X_1+i t_2 X_2) P(X_1,X_2)$

the inverse is, as usual,

$P(X_1,X_2)=(2\pi)^{-2}\int_{-\infty}^{\infty}dt_1 \int_{-\infty}^{\infty}dt_2 exp(-it_1 X_1-i t_2 X_2) \phi(t_1,t_2)$