Fractional calculus approach to the statistical characterization of random variables and vectors

TL;DR

This paper introduces a fractional calculus-based method using complex moments for the statistical characterization of random variables, especially those with divergent integer moments, demonstrating its practical effectiveness.

Contribution

It presents a novel approach employing complex moments and fractional calculus to analyze random variables with divergent moments, enhancing data representation and analysis.

Findings

Effective representation of variables with inverse power-law tails

Good numerical convergence for practical data analysis

Applicable to both univariate and multivariate data

Abstract

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.