Characteristic functions

Oct 2009
If the so-called inversion theorem does not hold for some characteristic function \(\displaystyle \psi_X(t)\), ie the function is not integrable on R, how can we tell whether or not there exists some probability distribution with this particular characteristic function? Two examples: \(\displaystyle \cos t\) and \(\displaystyle cos^2 t\). For the first it is obvious that the ChF is just the real part of the complex exponential, so I split the definition of the ChF up into real and imaginary parts and tried to see why equality could not hold (I don't believe there should be any distribution with that ChF, but I could be wrong). However, there I am stuck. I would be very appreciative if anyone has any clues for me!