  Fractional moments have been investigated by many authors to represent the
density of univariate and bivariate random variables in different contexts.
Fractional moments are indeed important when the density of the random variable
has inverse power-law tails and, consequently, it lacks integer order moments.
In this paper, starting from the Mellin transform of the characteristic
function and by fractional calculus method we present a new perspective on the
statistics of random variables. Introducing the class of complex moments, that
include both integer and fractional moments, we show that every random variable
can be represented within this approach, even if its integer moments diverge.
Applications to the statistical characterization of raw data and in the
representation of both random variables and vectors are provided, showing that
the good numerical convergence makes the proposed approach a good and reliable
tool also for practical data analysis.
