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.)



Fourier inversion formula for complex-valued random variables?
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,


$P(X\_1,X\_2)=(2\pi)^{-2}\int\_{-\infty}^{\infty}dt\_1 \int\_{-\infty}^{\infty}dt\_2 exp(-it\_1 X\_1-i t\_2 X\_2) \phi(t\_1,t\_2)$


