On the use of fractional calculus for the probabilistic characterization of random variables

In this paper, the classical problem of the probabilistic characterization of a random variable is reexamined. A random variable is usually described by the probability density function (PDF) or by its Fourier transform, namely the characteristic function (CF). The CF can be further expressed by a Taylor series involving the moments of the random variable. However, in some circumstances, the moments do not exist and the Taylor expansion of the CF is useless. This happens for example in the case of α$\alpha$-stable random variables.