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

TL;DR

This paper introduces a fractional calculus approach to probabilistically characterize random variables, especially those with non-existent moments, providing a new dual representation of PDFs and CFs that improves tail accuracy.

Contribution

It demonstrates that fractional derivatives of the characteristic function relate to fractional moments and develops a generalized Taylor expansion using fractional moments for better PDF and CF representation.

Findings

Fractional derivatives of CF at zero equal fractional moments.

Generalized Taylor series using fractional moments for CF and PDF.

Improved tail accuracy in PDF representation for engineering applications.

Abstract

In this paper, the classical problem of the probabilistic characterization of a random variable is re-examined. 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 $\alpha$--stable random variables. Here, the problem of representing the CF or the PDF of random variables (r.vs) is examined by introducing fractional calculus. Two very remarkable results are obtained. Firstly, it is shown that the fractional derivatives of the CF in zero coincide with fractional moments. This is true also in case of CF not derivable in zero (like the CF of $\alpha$--stable r.vs). Moreover, it is shown that the CF may be represented by a generalized Taylor expansion involving fractional moments. The generalized Taylor series proposed is also able to represent the PDF in a perfect dual representation to that in terms of CF. The PDF representation in terms of fractional moments is especially accurate in the tails and this is very important in engineering problems, like estimating structural safety.