Riesz fractional integrals and complex fractional moments for the probabilistic characterization of random variables

The aim of this paper is the probabilistic representation of the probability density function (PDF) or the characteristic function (CF) in terms of fractional moments of complex order. It is shown that such complex moments are related to Riesz and complementary Riesz integrals at the origin. By invoking the inverse Mellin transform theorem, the PDF or the CF is exactly evaluated in integral form in terms of complex fractional moments. Discretization leads to the conclusion that with few fractional moments the whole PDF or CF may be restored.Application to the pathological case of an αα-stable random variable is discussed in detail, showing the impressive capability to characterize random variables in terms of fractional moments.

► Integer moments are not able to characterize various types of random variables.
► We use fractional moments (FMs) or fractional spectral moments (FSMs) to represent probability density functions (PDFs) and characteristic functions (CFs).
► The FMs and FSMs coincide with Mellin transforms of the PDFs and the CFs, respectively.
► Such quantities are related to Riesz and complementary Riesz integrals at the origin.
► Applications of FMs and FSMs of αα-stable distributions are reported.