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

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.

A comment is added in the introduction on the role of positive-part moments

References in corpus (5)

• On Hoeffding's inequalities
• Exact inequalities for sums of asymmetric random variables, with applications
• On Shift Harnack Inequalities for Subordinate Semigroups and Moment Estimates for Lévy Processes
• Exact Rosenthal-type bounds
• Convex cones of generalized multiply monotone functions and the dual cones