fourier transform of a finite-energy random process
I've got the following problem: I have a random process in time-domain, call it ; assuming that it's a finite-energy process (e.g. it's mean value is non-zero only for ), I can consider the Fourier transform:
Now, if has a joint PDF of N-th order , what's the joint PDF of N-th order of the process ?
I tried to tackle the problem in this way: considering a fixed value for frequency , I can extract a r.v. from the random process :
If I approximate the integral as:
I can consider as a function of the finite set of random variables . This means that the PDF of is a sort of convolution of the PDFs of the r.v. (scaled by the exponentials)... does this make sense? :)
Is there a better solution to my original problem?