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.

Integrated distribution and quantile functions or simple transformations of them play an important role in probability theory, mathematical statistics, and their applications such as insurance, finance, economics etc. They frequently appear in the literature, often under different names. Moreover, in many occasions they are defined under additional assumption of integrability of a random variable or at least integrability of the positive or the negative part of a variable. Let us point out only few references. For a random variable

The function

The integrated survival function

The potential function

The examples of integrated quantile functions or their simple transformations are:

The absolute Lorenz curve

The Hardy–Littlewood maximal function

The Conditional Value at Risk

The main goal of this paper is a systematic exposition of basic properties of integrated distribution and quantile functions. In particular, we define the integrated distribution and quantile functions for any random variable

One of the key points of our approach is that we define integrated distribution and quantile functions as Fenchel conjugates of each other. This is due to the fact that their derivatives, distribution functions and quantile functions, are generalized inverses (see, e. g., [

Let us note that we consider only univariate distributions in this paper. However, it is reasonable to mention a possible generalization to the multidimensional case based on ideas from optimal transport. The integrated quantile function of a random variable

It is more convenient for us to speak about random variables rather than distributions. However, if a probability space is not specified, the symbols

For the reader’s convenience, we recall some terminology and elementary facts concerning convex functions of one real variable. A convex function

The distribution function

Throughout the paper, if

The

It is clear from (vi), that the integrated distribution function uniquely determines the distribution.

It is evident that (i) holds and

Next, by Fubini’s theorem, for

Let us prove (iv). The function

Finally, (v) and (vii) follow from (

A typical graph of an integrated distribution function if the expectations

Since

We call every function

It follows directly from the definitions that, for any

The Fenchel transform of the integrated distribution function of a random variable

This definition is motivated by the fact mentioned in the introduction, that a function whose derivative is a quantile function must coincide with the Fenchel transform of

It is clear from (ii) and the similar remark after Theorem

A typical graph of an integrated quantile function

Since

Putting

Statement (v) follows from the definition of

Finally, (vii) follows from the definition of

Put

Under our assumptions

An alternative way to prove Theorem

It is convenient to introduce

Now we can express the functions defined in (

Let us recall the definitions of convex orders in the univariate case.

For an arbitrary function

Let

It is trivial that

The following theorem is well known. We provide its proof which reduces to the duality between integrated distribution and quantile functions.

First, let us prove (ii). It is well known (see, e. g., [

Now, (iii) follows from (ii) and the first part of the remark before Theorem

Next, the equalities in (b) and (c) are both equivalent to

Further properties of convex orders see, e.g., in [

In this subsection we demonstrate how the developed techniques can be used to derive two elementary well-known inequalities, see [

Let

We solve a converse problem. Namely, let

Graphs of shifted integrated quantile functions

The above class of distributions has a minimal element with respect to the convex order. Indeed, let

Let

We will proceed in the similar way as in the previous example. Namely, let

The above class of distributions has a minimal element with respect to the convex order. Indeed, let

Graphs of shifted integrated quantile functions

In this subsection we study conditions for tightness and uniform integrability of a family of random variables in terms of integrated quantile function. It is a natural question because both tightness and uniform integrability are characterized in terms of one-dimensional distributions of these variables.

(i) ⇒ (iv) Let

(ii) ⇒ (iii) Let

(ii) ⇒ (i) Let

Since implications (iv) ⇒ (ii) and (iii) ⇒ (ii) are obvious, the claim follows. □

Let us consider the probability space

Let us recall that a family

We shall check that the uniform boundedness and the equicontinuity of

By the properties of integrated quantile functions,

For a fixed

Conversely, fix

The following criterion of uniform integrability is proved in [

Without loss of generality, we may assume that all

The sufficiency is evident. Indeed, if

Let us define

In this subsection

If

In contrast to Remark

First, let us suppose that

Assume for the moment that

If no assumptions on

To complete the proof of Theorem

It is enough to prove Theorem

The theory of statistical experiments deals with the problem of comparing the information in different experiments. The foundation of the theory of experiments was laid by Blackwell [

In this paper we consider only binary statistical experiments, or dichotomies,

Let us introduce some notation.

For an experiment

Denote

The shaded area represents the set

Let

Let

It is clear that

A usual way to prove that the set

Let us also introduce the

Conversely, using Definition

Finally, let us introduce one more characteristic of binary models, namely the distribution of the ‘likelihood ratio’

Now let us present some basic notions and results from the theory of comparison of dichotomies. All these facts are well known, see e. g. [

Let

(i)

(i) Let

(iii) First, it is evident that

(ii) First, it follows from the definition of the error function that

Let us note that the proofs of (i) and (iii) give more than it is stated. Starting with a function

Let

It is easy to check that

(i)

To simplify the notation, put

Since

Conversely, let (

As a consequence, we obtain the following expressions for

The subset of concave functions

Let

The equivalences (i)

Assume (i). By (

Now the converse implication (ii)

The Skorokhod embedding problem was posed and solved by Skorokhod [

Cox [

It is easy to observe that the Chacon–Walsh construction has a graphical interpretation in terms of integrated quantile functions as well; moreover, in our opinion, the picture is more simple. We give alternative proofs of the result in [

Let us recall the definition of the balayage. For a probability measure

Let

Graphs of shifted integrated quantile functions

If

The next lemma is a key tool in our future construction.

Without loss of generality, we may assume that

Using Corollary

From now on, we assume that there is a probability space with filtration

By the strong Markov property, in view of boundedness of the random variables under the conditional expectations below,

Let us also recall that

Put

Moreover, the construction in Lemma

It remains to prove that

Let

It has been already mentioned in the proof of Theorem

Let

We thank three anonymous referees for constructive comments and remarks that helped improving the exposition.