  We consider analytic continuations of Fourier transforms and Stieltjes
transforms. This enables us to define what we call complex moments for some
class of probability measures which do not have moments in the usual sense.
There are two ways to generalize moments accordingly to Fourier and Stieltjes
transforms; however these two turn out to coincide. As applications, we give
short proofs of the convergence of probability measures to Cauchy distributions
with respect to tensor, free, Boolean and monotone convolutions.
