Fractional characteristic functions, and a fractional calculus approach for moments of random variables

In this paper we introduce a fractional variant of the characteristic function of a random variable. It exists on the whole real line, and is uniformly continuous. We show that fractional moments can be expressed in terms of Riemann–Liouville integrals and derivatives of the fractional characteristic function. The fractional moments are of interest in particular for distributions whose integer moments do not exist. Some illustrative examples for particular distributions are also presented.