Positive-part moments via the characteristic functions, and more general expressions

TL;DR

This paper introduces a unified approach to representing positive-part and absolute moments of random variables using characteristic functions, providing new insights and computational methods for these probabilistic measures.

Contribution

It offers a general framework that unifies existing and new representations of moments and related functions via characteristic functions, enhancing understanding and computation.

Findings

Derived a single basic representation for various moment expressions

Provided computational techniques for the new representations

Extended existing formulas to more general cases

Abstract

A unifying and generalizing approach to representations of the positive-part and absolute moments $\mathsf{E} X_+^p$ and $\mathsf{E}|X|^p$ of a random variable $X$ for real $p$ in terms of the characteristic function (c.f.) of $X$, as well as to related representations of the c.f.\ of $X_+$, generalized moments $\mathsf{E} X_+^p e^{iuX}$, truncated moments, and the distribution function is provided. Existing and new representations of these kinds are all shown to stem from a single basic representation. Computational aspects of these representations are addressed.