Absolute moments in terms of characteristic functions

TL;DR

This paper revisits and simplifies the formulas relating absolute moments of probability distributions to their characteristic functions, introduces new results, and explores the determination of distributions of nonnegative variables via fractional moments.

Contribution

It provides a simpler derivation of formulas connecting absolute moments to characteristic functions and extends results on distribution determination from fractional moments for nonnegative variables.

Findings

Revised formulas for absolute moments in terms of characteristic functions.

New results on the characterization of distributions using fractional moments.

Improved conditions for determining distributions of nonnegative variables.

Abstract

The absolute moments of probability distributions are much more complicated than conventional ones. By using a direct and simpler approach, we retreat P. L. Hsu's (1951, J. Chinese Math. Soc., Vol. 1, pp. 257-280) formulas in terms of the characteristic function (which have been ignored in the literature) and provide some new results as well. The case of nonnegative random variables is also investigated through both characteristic function and Laplace-Stieltjes transform. Besides, we prove that the distribution of a nonnegative random variable with a finite fractional moment can be completely determined by a proper subset of the translated fractional moments. This improves significantly P. Hall's (1983, Z. W., Vol. 62, 355-359) result for distributions on the right-half line.