Integrated quantile functions: properties and applications

TL;DR

This paper systematically explores integrated distribution and quantile functions, highlighting their properties, characterizations, and applications in probability theory, including statistical experiment comparison and Skorokhod embedding.

Contribution

It introduces a unified framework for integrated quantile functions, demonstrating their role in characterizing distributions and extending applications in probability theory.

Findings

Integrated quantile functions characterize any probability distribution.

They provide convenient characterizations of uniform integrability, weak convergence, and tightness.

Applications include comparison of statistical experiments and Skorokhod embedding extensions.

Abstract

In this paper we provide a systematic exposition of basic properties of integrated distribution and quantile functions. We define these transforms in such a way that they characterize any probability distribution on the real line and are Fenchel conjugates of each other. We show that uniform integrability, weak convergence and tightness admit a convenient characterization in terms of integrated quantile functions. As an application we demonstrate how some basic results of the theory of comparison of binary statistical experiments can be deduced using integrated quantile functions. Finally, we extend the area of application of the Chacon--Walsh construction in the Skorokhod embedding problem.