### Integrated quantile functions: properties and applications

Alexander A. Gushchin, Dmitriy A. Borzykh

#### 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.

#### Keywords

Quantile functions; integrated quantile functions; integrated distribution functions; convex stochastic order; binary experiments; Chacon–Walsh construction

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#### References

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DOI: http://dx.doi.org/10.15559/17-VMSTA88

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