# On Fourier-based inequality indices

Giuseppe Toscani

Department of Mathematics of the University of Pavia, and IMATI CNR, Pavia,
Italy.
e.mail<EMAIL_ADDRESS>
Abstract. Originally developed for measuring the heterogeneity of wealth
measures, inequality indices are quantitative scores that take values in the
unit interval, with the zero score characterizing perfect equality. In this
paper, we draw attention to a new inequality index, based on the Fourier
transform, which exhibits a number of interesting properties that make it very
promising in applications. As a by-product, it is shown that other inequality
measures, including Gini and Pietra indices can be fruitfully expressed in
terms of the Fourier transform, which allows to enlighten properties in a new
and simple way.

Keywords. Inequality measures; Pietra and Gini indices; Fourier transform;
Sub-additivity for convolutions.

##  1\. Introduction

Denote by $P_{s}(\mathbb{R})$, $s\geq 1$, the class of all probability
measures $F$ on the Borel subsets of $\mathbb{R}$ such that

$m_{s}(F)=\int_{\mathbb{R}}|x|^{s}dF(x)<+\infty.$

Further, denote by $\tilde{P}_{s}(\mathbb{R})$ the class of probability
measures $F\in P_{s}(\mathbb{R})$ which possess a positive mean value

$m(F)=\int_{\mathbb{R}}x\,dF(x)>0,$

and with $P_{s}^{+}(\mathbb{R})$ the subset of probability measures $F\in
P_{s}(\mathbb{R})$ such that $F(x)=0$ for $x\leq 0$. Let $\mathcal{F}_{s}$ be
the set of Fourier transforms

$\widehat{f}(\xi)=\int_{\mathbb{R}}e^{-i\xi x}\,dF(x).$

of probability measures $F$ in $\tilde{P}_{s}(\mathbb{R})$. On
$\mathcal{F}_{s}$ we introduce an inequality index, named $T(F)$, given by the
formula

(1.1)
$T(F)=\frac{1}{2}\sup_{\xi\in\mathbb{R}}\left|\widehat{f}(\xi)-\frac{\widehat{f}^{\prime}(\xi)}{\widehat{f}^{\prime}(0)}\right|,$

In definition (1.1),
$\widehat{f}^{\prime}(\xi)=\widehat{f}^{\prime}_{\xi}(\xi)$ denotes the
derivative of the Fourier transform $\widehat{f}(\xi)$ with respect to its
argument $\xi$. Indeed, $F\in P_{s}(\mathbb{R})$ implies that
$\widehat{f}(\xi)$ is continuously differentiable on the entire real line.

It is immediate to show that the functional $T(F)$ is invariant with respect
to the scaling (dilation)

$F(x)\to F(cx),\quad c>0.$

Let $F\in P_{s}^{+}(\mathbb{R})$. Since
$|\widehat{f}(\xi)|\leq\widehat{f}(0)=1$, and

$\widehat{f}^{\prime}(\xi)=-i\int_{\mathbb{R}_{+}}xe^{-i\xi x}\,dF(x),$

so that $\widehat{f}^{\prime}(0)=-i\,m(F)=-i\,m$, and
$|\widehat{f}^{\prime}(\xi)|\leq|\widehat{f}^{\prime}(0)|=m$, it is immediate
to conclude, by the triangular inequality, that $T(F)$ satisfies the bounds

(1.2) $0\leq T(F)\leq 1,$

and $T(F)=0$ if and only if $\widehat{f}(\xi)$ satisfies the differential
equation

$\widehat{f}^{\prime}(\xi)={\widehat{f}^{\prime}(0)}\,\widehat{f}(\xi),$

with $\widehat{f}(0)=1$, so that the unique solution is given by
$\widehat{f}(\xi)=e^{-im\xi}$, namely by the Fourier transform of a Dirac
delta function located in the mean value $x=m(F)>0$. Note however that, even
if the functional $F$ is defined in the whole class
$\tilde{P}_{s}(\mathbb{R})$, the upper bound is lost if the probability
measure $F\notin P_{s}^{+}(\mathbb{R})$, since in this case the inequality
$|\widehat{f}^{\prime}(\xi)/\widehat{f}^{\prime}(0)|\leq 1$ does not hold.

In the following, we will show that the functional $T(F)$ is a measure of
inequality which satisfies most of the properties required to be a good
measure of sparsity and/or heterogeneity [22]. The interest in having a
measure of inequality based on the Fourier transform, such as $T(F)$, is
twofold. On the one hand, it is very simple to calculate the value taken by
this measure at probability distributions for which the characteristic
function is explicitly available. This is the case, among others, of the
Poisson distribution and, for probability measures defined on the whole real
line $\mathbb{R}$, of the stable laws. On the other hand, in the case of
dealing with a discrete probability measure, the use of the Fourier transform
makes it possible to develop very fast computational procedures [3, 4].

As recently discussed in [5, 15, 16], the challenge of measuring the
statistical heterogeneity of measures arises in most fields of science and
engineering. In economics and social sciences size measures of interest are
wealth measures, and in the context of wealth measures many inequality indices
have been introduced [7, 11, 12, 21] . Specifically, inequality indices
quantify the socio-economic divergence of a given wealth measures from the
state of perfect equality. In economics, the most used measure of inequality
is the Gini index [18, 19]. However, although it has had an economic origin,
the use of the Gini index has not been limited to wealth alone [22].

A second important index of inequality, still introduced in economics, is the
Pietra index [28]. As discussed in [14], the Pietra index is an elemental
measure of statistical heterogeneity which has a number of properties that
render it not only an alternative to the popular Gini index, but rather, a far
more natural and meaningful quantitative tool for the measurement of
egalitarianism, and, consequently, for the measurement of statistical
heterogeneity at large.

In addition, other indices have been introduced so far. An alternative to Gini
index was introduced by Bonferroni in 1930 in a textbook for students at
Bocconi University in Milan [8]. The main properties and representations of
Bonferroni index and its connections with the index of Gini and other measures
were studied in [16]. Also, it is important to mention the Kolkata index,
first introduced in [20] as a measure of inequality, whose connections with
the Gini and Pietra indices have been studied in [5, 6].

An indispensable tool for measuring income and wealth inequality is the Lorenz
function and its graphical representation, the Lorenz curve [24]. The Lorenz
curve plots the percentage of total income earned by the various sectors of
the population, ordered by the increasing size of their incomes. The Lorenz
curve is typically represented as a curve in the unit square of opposite
vertices in the origin of the axes and the point $(1,1)$, starting from the
origin and ending at the point $(1,1)$.

The diagonal of the square exiting the origin is the line of perfect equality,
representing a situation in which all individuals have the same income. Since
the diagonal is the line of perfect equality, we can say that that the closer
the Lorenz curve is to the diagonal, the more equal is the distribution of
income.

This idea of _closeness_ between the line of perfect equality and the Lorenz
curve can be expressed in many ways, each of which gives rise to a possible
measure of inequality. Thus, starting from the Lorenz curve, several indices
of inequality were defined, including the Gini index. Various indices were
obtained by looking at the maximal distance between the line of perfect
equality and the Lorenz curve, either horizontally or vertically, or
alternatively parallel to the other diagonal of the unit square [15].

From a certain point of view, the measure of inequality defined by (1.1) has
many points of contact with the inequality measures obtained from the Lorenz
curve through the concept of _maximum distance_.

In fact, the index (1.1) expresses the maximum value of the modulus of the
difference between the Fourier transforms of a probability measure $F$ of
positive mean $m$ and its derivative normalized by the mean. In the economic
context, the closer the Fourier transform of the probability measure is to its
derivative normalized by dividing it by the mean, the more equal is the
distribution of income. In other words, the line of perfect equality in the
Lorenz square is here substituted by the Fourier transform of a Dirac delta
function located in a point different from zero.

It is interesting to note that, as will become clear from the examples, the
maximum value is usually taken in the finite interval $(-2\pi,2\pi)$.

Despite the enormous amount of research illustrating the fields of application
of inequality indices, the use of arguments based on Fourier transforms
appears rather limited. In particular, although the Gini index can be easily
expressed in terms of the Fourier transform, at least to our knowledge, its
expression in Fourier has never been considered in applications. The same
conclusion can be drawn for the Pietra index, whose expression in Fourier
transform is very useful to understand its nature, and to introduce from that
other Fourier-based measures of inequality, including the one considered in
this paper.

It is worth mentioning that, unlike the classical Gini and Pietra indices,
neither the Bonferroni index nor the Kolkata index seem to be expressible in
closed form in terms of the Fourier transform.

Before studying the new index $T(\cdot)$ defined in (1.1) and listing its
properties, we will begin with a brief introduction to the use of the Fourier
transform to express the classical Gini and Pietra indices. This will be done
in Section 2. As we shall see, the use of Fourier transform allows to clarify
the functional setting where these indices live.

Next, Section 3 will be devoted to the study of the main properties of the new
inequality measure. Various examples will be collected in Sections 4 and 5.

##  2\. A Fourier approach to Gini and Pietra indices

###  2.1. A Fourier-based expression of Gini index

In the rest of the paper, for any fixed constant $a>0$, we will denote by
$F_{a}(x)$ the Heaviside step function defined by

(2.1) $F_{a}(x):=\left\\{\begin{array}[]{cc}0&x<a\\\ 1&x\geq
a\end{array}\right.$

Clearly, $F_{a}(x)$ is the cumulative measure function of a random variable
which is almost surely equal to $a$. It belongs to $P_{s}(\mathbb{R})$ for any
$s\geq 1$, and $m(F_{a})=a$.

To obtain an explicit expression in Fourier transform for the Gini index,
which admits many equivalent formulations [32], we will resort to its well-
known form in terms of a continuous probability measure. For a probability
measure $F\in P_{s}^{+}(\mathbb{R})$ with mean $m$, the Gini index is defined
by the formula

(2.2) $G(F)=1-\frac{1}{m}\int_{\mathbb{R}_{+}}(1-F(x))^{2}\,dx$

Since $F\in P_{s}^{+}(\mathbb{R})$, $F(x)=0$ for $x\leq 0$. Hence, resorting
to the definition of the Heaviside step function $F_{0}(x)$, we have the
identity

$\int_{\mathbb{R}_{+}}(1-F(x))^{2}\,dx=\int_{\mathbb{R}}|F_{0}(x)-F(x)|^{2}\,dx.$

For any given pair of probability measures $F,G\in P_{s}(\mathbb{R})$,
Parseval formula implies

(2.3)
$\int_{\mathbb{R}}|F(x)-G(x)|^{2}\,dx=\frac{1}{2\pi}\int_{\mathbb{R}}|\widehat{F}(\xi)-\widehat{G}(\xi)|^{2}\,d\xi,$

where $\widehat{F}$ and $\widehat{G}$ are the Fourier transforms of the
probability measures $F,G$. It holds

(2.4)
$\widehat{F}(\xi)-\widehat{G}(\xi)=\frac{\widehat{f}(\xi)-\widehat{g}(\xi)}{i\xi}.$

Indeed, considering that $F(-\infty)-G(-\infty)=F(+\infty)-G(+\infty)=0$,
integration by parts gives

$\displaystyle\int_{\mathbb{R}}(F(x)-G(x))e^{-i\xi
x}\,dx=\int_{\mathbb{R}}(F(x)-G(x))\frac{d}{d\xi}\left(\frac{e^{-i\xi
x}}{-i\xi}\right)\,dx=$ $\displaystyle\left[(F(x)-G(x))\frac{e^{-i\xi
x}}{-i\xi}\right]_{-\infty}^{+\infty}+\int_{\mathbb{R}}\frac{e^{-i\xi
x}}{i\xi}\,d(F(x)-G(x))=\frac{\widehat{f}(\xi)-\widehat{g}(\xi)}{i\xi}.$

Consequently, we have the identity

(2.5)
$\int_{\mathbb{R}}|F(x)-G(x)|^{2}\,dx=\frac{1}{2\pi}\int_{\mathbb{R}}\frac{|\widehat{f}(\xi)-\widehat{g}(\xi)|^{2}}{\xi^{2}}\,d\xi.$

Therefore, for any probability measure $F\in P_{s}^{+}(\mathbb{R})$, Gini
index has a simple expression in Fourier transform, given by

(2.6) $G(F)=1-\frac{1}{2\pi
m}\int_{\mathbb{R}}\frac{|1-\widehat{f}(\xi)|^{2}}{\xi^{2}}\,d\xi.$

######  Remark 2.1.

For a given constant $q>0$, let $\dot{H}_{-q}$ denote the homogeneous Sobolev
space of fractional order with negative index ${-q}$, endowed with the norm

(2.7)
$\|h\|_{\dot{H}_{-q}}=\int_{\mathbb{R}}|\xi|^{-2q}|\widehat{h}(\xi)|^{2}\,d\xi.$

Then, the variable part of the Gini index coincides with the scaling invariant
distance between the probability measure $F$ and the the heaviside step
function $F_{0}$ in the homogeneous Sobolev space $\dot{H}_{-1}$.

######  Remark 2.2.

Considering that the value zero in (2.6) is obtained when
$\widehat{f}(\xi)=e^{-im\xi}$, namely when $F=F_{m}$, we can rewrite Gini
index as

(2.8) $G(F)=\frac{1}{2\pi
m}\left[\int_{\mathbb{R}}\frac{|1-e^{-im\xi}|^{2}}{\xi^{2}}\,d\xi-\int_{\mathbb{R}}\frac{|1-\widehat{f}(\xi)|^{2}}{\xi^{2}}\,d\xi\right].$

###  2.2. Another Fourier-based inequality measure

Expression (2.8) suggests considering a related expression in which the
dispersion of the probability measure $F$ of mean value $m>0$ coincides with
its scale invariant $\dot{H}_{-1}$–distance from the Heaviside step function
$F_{m}$ with the same mean value $m$. We define

(2.9) $H(F)=\frac{1}{2\pi
m}\int_{\mathbb{R}}\frac{|\widehat{f}(\xi)-e^{-im\xi}|^{2}}{\xi^{2}}\,d\xi.$

At difference with Gini index, which requires $F\in P_{s}^{+}(\mathbb{R})$,
the inequality measure $H(F)$ is well-defined for any measure
$F\in\tilde{P}_{s}(\mathbb{R})$.

It is interesting to remark that, similarly to Gini index, the inequality
measure $H(F)$, for $F\in P_{s}^{+}(\mathbb{R})$, is bounded above by $1$.
This property is shown in the following

######  Lemma 2.3.

Let $F\in P_{s}^{+}(\mathbb{R})$ be a probability measure of mean value $m$.
Then

(2.10) $0\leq H(F)<1.$

###### Proof.

Let $F\in P_{s}^{+}(\mathbb{R})$ be a probability measure of mean value $m$.
Thanks to Parseval formula, the value of th expression (2.9) in the Fourier
space coincides with the value

(2.11) $H(F)=\frac{1}{m}\int_{\mathbb{R}_{+}}|F(x)-F_{m}(x)|^{2}\,dx.$

The simplest case in which we can explicitly evaluate $H(F)$ is when $F(x)\in
P_{s}^{+}(\mathbb{R})$ is the measure function of a random variable $X$ taking
only two non-negative values, that is, for $0<p<1$

$P(X=m-a)=p;\quad P(X=m+b)=1-p,\quad a,b>0;\quad a\leq m.$

Since $X$ has mean value $m$, $a$ and $b$ are related to $p$ by the relation

(2.12) $pa=(1-p)b.$

In this case, it is a simple exercise to verify that

$\frac{1}{m}\int_{\mathbb{R}_{+}}|F(x)-F_{m}(x)|^{2}\,dx=\frac{1}{m}\left[p^{2}a+(1-p)^{2}b\right],$

so that, thanks to (2.12)

$H(F)=\frac{pa}{m}.$

Therefore, since $a\leq m$ and $p<1$, we conclude with $H(F)<1$.

An interesting application of the previous expression is obtained by assuming
$a=m$, and $p=1-\epsilon$, with $0<\epsilon\ll 1$. In this case the random
variable $X$ of mean value $m$ is such that

$P(X=0)=1-\epsilon;\quad P\left(X=\frac{m}{\epsilon}\right)=\epsilon.$

In economics, this situation describes a population in which most of agents
have zero wealth, while one small part possesses an extremely high wealth,
while maintaining the mean wealth fixed. In this case $H(F)=1-\epsilon$.

Let us now consider a random variable $X$ of mean value $m$ that takes three
non negative values $x_{1}<x_{2}<m<x_{3}$, where

$P(X=x_{k})=p_{k},\quad k=1,2,3;\quad p_{1}+p_{2}+p_{3}=1.$

In this case

$\int_{\mathbb{R}_{+}}|F(x)-F_{m}(x)|^{2}\,dx=p_{1}^{2}(x_{2}-x_{1})+(p_{1}+p_{2})^{2}(m-x_{2})+p_{3}^{2}(x_{3}-m).$

Let

(2.13) $x=\frac{p_{1}}{p_{1}+p_{2}}\,x_{1}+\frac{p_{2}}{p_{1}+p_{2}}\,x_{2}.$

Then, $x\in(x_{1},x_{2})$, and

$(p_{1}+p_{2})x=p_{1}x_{1}+p_{2}x_{2}.$

Let $Y$ be the two valued random variable defined by

$P(Y=x)=p_{1}+p_{2};\quad P(Y=x_{3})=p_{3}.$

Then, thanks to (2.13), $Y$ has mean value $m$. Moreover, if we denote by $G$
the measure function of $Y$,

$\int_{\mathbb{R}_{+}}|G(x)-F_{m}(x)|^{2}\,dx=(p_{1}+p_{2})^{2}(m-x)+p_{3}^{2}(x_{3}-m).$

Owing to (2.13) we obtain

$\displaystyle(p_{1}+p_{2})^{2}(x_{2}-x)=(p_{1}+p_{2})^{2}\left(x_{2}-\frac{p_{1}}{p_{1}+p_{2}}x_{1}+\frac{p_{2}}{p_{1}+p_{2}}x_{2}\right)=$
$\displaystyle p_{1}(p_{1}+p_{2})(x_{2}-x_{1})\geq p_{1}^{2}(x_{2}-x_{1}).$

Consequently

$\displaystyle
p_{1}^{2}(x_{2}-x_{1})+(p_{1}+p_{2})^{2}\left(m-x_{2}\right)\leq(p_{1}+p_{2})^{2}(x_{2}-x)+(p_{1}+p_{2})^{2}\left(m-x_{2}\right)=$
$\displaystyle(p_{1}+p_{2})^{2}(m-x),$

that implies

(2.14) $H(F)\leq H(G)<1.$

The same conclusion holds if we consider a random variable $X$ of mean value
$m$ that takes the three non negative values $x_{1}<m<x_{2}<x_{3}$, and we
choose the value $x\in(x_{2},x_{3})$ like in (2.13). The previous computations
show that, by suitably choosing the point, we can built, starting from a
random variable with three values, a random variable with two values, with the
same mean and with a bigger value of the functional $H$, which by the previous
computations is less than $1$. At this point, we can iterate the procedure and
conclude that the upper bound in (2.10) holds for the measure function $F\in
P_{s}^{+}(\mathbb{R})$ of any discrete random variable $X$, and finally for
any $F\in P_{s}^{+}(\mathbb{R})$. ∎

The interest in having an inequality index that quantifies the statistical
heterogeneity of probability measures defined on the whole real line
$\mathbb{R}$ in terms of the Fourier transform is evident. As an example, let
us compute the value of the functional $H$ for a Gaussian probability measure
$F$ of mean $m>0$ and variance $\sigma^{2}$. Since the Fourier transform of
the Gaussian density is given by

(2.15)
$\widehat{f}(\xi)=\exp\left\\{-i\,m\,\xi-\frac{\sigma^{2}}{2}\xi^{2}\right\\},$

we easily obtain

$H(F)=\frac{1}{2\pi
m}\int_{\mathbb{R}}\frac{\left(1-\exp\left\\{-\frac{\sigma^{2}}{2}\xi^{2}\right\\}\right)^{2}}{\xi^{2}}\,d\xi.$

Integration by parts yields

$\displaystyle\int_{0}^{\infty}\frac{\left(1-\exp\left\\{-\frac{\sigma^{2}}{2}\xi^{2}\right\\}\right)^{2}}{\xi^{2}}\,d\xi=\left[-{\left(1-\exp\left\\{-\frac{\sigma^{2}}{2}\xi^{2}\right\\}\right)^{2}}\frac{1}{\xi}\right]_{0}^{\infty}+$
$\displaystyle+\int_{0}^{\infty}\frac{1}{\xi}\left(1-\exp\left\\{-\frac{\sigma^{2}}{2}\xi^{2}\right\\}\right)\sigma^{2}\xi\exp\left\\{-\frac{\sigma^{2}}{2}\xi^{2}\right\\}\,d\xi=$
$\displaystyle\sigma\int_{0}^{\infty}\left(e^{-x^{2}/2}-e^{-x^{2}}\right)\,dx=\sigma\frac{(\sqrt{2}-1)\sqrt{\pi}}{2}.$

Thus, for a Gaussian probability measure $F$ of mean $m>0$ and variance
$\sigma^{2}$ we have the value

(2.16) $H(F)=\frac{(\sqrt{2}-1)}{2\sqrt{2\pi}}\frac{\sigma}{m},$

namely a value proportional to the coefficient of variation $\sigma/m$, with
an explicit constant strictly less than one.

###  2.3. A Fourier-based expression of Pietra index

For a probability measure $F\in P_{s}^{+}(\mathbb{R})$ with mean $m$, the
Pietra index $P(F)$ [14, 28] is defined by the formula

(2.17) $P(F)=\int_{m}^{+\infty}(1-F(x))\,dx.$

As remarked in [14], the definition (2.17) seems to disregard the part of the
measure below the mean. This, however, is not true, and (2.17) makes use of
the full information encapsulated in the probability law of the random
variable $X$ of measure $F$.

There is a simple way to verify the previous assertion. Indeed, since

$m=\int_{\mathbb{R}_{+}}(1-F(x))\,dx,$

it holds

$\displaystyle
H(F)=\frac{1}{m}\int_{\mathbb{R}_{+}}[F(x)-F_{m}(x)]^{2}\,dx=\frac{1}{m}\int_{\mathbb{R}_{+}}[1-F(x)-(1-F_{m}(x))]^{2}\,dx=$
$\displaystyle\frac{1}{m}\int_{\mathbb{R}_{+}}[1-F(x)]^{2}\,dx+\frac{1}{m}\int_{\mathbb{R}_{+}}[1-F_{m}(x)]^{2}\,dx-\frac{2}{m}\int_{\mathbb{R}_{+}}[1-F(x)][1-F_{m}(x)]\,dx=$
$\displaystyle\frac{1}{m}\int_{\mathbb{R}_{+}}[1-F(x)]^{2}\,dx+1-\frac{2}{m}\int_{0}^{m}[1-F(x)]\,dx=$
$\displaystyle\frac{1}{m}\int_{\mathbb{R}_{+}}[1-F(x)]^{2}\,dx-1+\frac{2}{m}\int_{m}^{+\infty}[1-F(x)]\,dx=-G(F)+2P(F).$

Hence, we have the identity

(2.18) $P(F)=\frac{1}{2}\left[G(F)+H(F)\right].$

In other words, the Pietra index of a probability measure
$F\in\tilde{P}_{s}^{+}$ is represented by the mean value of the two indices
$G(F)$ and $H(F)$, where $H$ is defined in 2.9, identically weighted.

Resorting to the Fourier expressions of Gini and $H$ indices we then obtain
for Pietra index the expression

(2.19) $P(F)=\frac{1}{2}\left[1-\frac{1}{2\pi
m}\int_{\mathbb{R}}\frac{|1-\widehat{f}(\xi)|^{2}}{\xi^{2}}\,d\xi+\frac{1}{2\pi
m}\int_{\mathbb{R}}\frac{|\widehat{f}(\xi)-e^{-im\xi}|^{2}}{\xi^{2}}\,d\xi\right].$

######  Remark 2.4.

The Fourier expression (2.19) clarifies that the Pietra index is obtained by
taking into account at the same time the distances in $\dot{H}_{-1}$ of a
probability measure in $P_{s}^{+}(\mathbb{R})$ from the Dirac delta functions
located in zero, and, respectively in mean value $m$. From this point of view,
the Pietra index appears as a well-balanced inequality index. This feature is
hidden in the classical definition.

######  Remark 2.5.

Since

$H(F)=2P(F)-G(F),$

the values of the inequality index $H(F)$ for a large number of probability
measures can be easily computed resorting to the tables of values assumed by
Gini and Pietra indices.

######  Remark 2.6.

If one considers only one-dimensional discrete measures, the inequality index
$H(F)$ defined by (2.9) coincides with a particular case of the discrepancy
function recently introduced in [4], where the discrepancy measures the
distance in $L^{2}$ distance between the characteristic functions of two given
discrete measures weighted by the function $k^{2}$, with $k=1,2,\dots,N$. In
this case, one of the two discrete measure is a Dirac delta function located
in the mean value.

###  2.4. Towards new inequality indices

A part from the scaling constant, the functional $H(F)$, coincides with the
square of the $L^{2}(\mathbb{R})$–norm of the function

$h(\xi)=\frac{|\widehat{f}(\xi)-e^{-im\xi}|}{|\xi|}.$

It is immediate to verify that a further scaled invariant functional can be
obtained by considering the $L^{\infty}(\mathbb{R})$–norm of $h(\xi)$. This
functional is given by

(2.20)
$H_{\infty}(F)=\frac{1}{2m}\sup_{\xi\in\mathbb{R}}\frac{|\widehat{f}(\xi)-e^{-im\xi}|}{|\xi|}.$

Resorting to the triangular inequality, we can easily conclude that, if
$F\in\tilde{P}_{s}^{+}$, $H_{\infty}$ satisfies the standard bounds

(2.21) $0\leq H_{\infty}(F)<1.$

Indeed, for $F\in P_{s}^{+}$ with mean value $m$

$H_{\infty}(F)=\frac{1}{m}\sup_{\xi\in\mathbb{R}}\frac{|\widehat{f}(\xi)-e^{-im\xi}|}{|\xi|}\leq\frac{1}{m}\sup_{\xi\in\mathbb{R}}\frac{|1-e^{-im\xi}|}{|\xi|}+\frac{1}{m}\sup_{\xi\in\mathbb{R}}\frac{|1-\widehat{f}(\xi)|}{|\xi|}.$

Now

$\frac{1}{m}\sup_{\xi\in\mathbb{R}}\frac{|1-e^{-im\xi}|}{|\xi|}=\frac{1}{m}\sup_{\xi\in\mathbb{R}}\frac{\sqrt{2(1-\cos\xi
m)}}{|\xi|}=\lim_{\xi\to 0}\frac{\sqrt{2(1-\cos\xi m)}}{m|\xi|}=1.$

Moreover, since by (2.4)

$\frac{1-\widehat{f}(\xi)}{i\xi}=\int_{\mathbb{R}}(F_{0}(x)-F(x))e^{-i\xi
x}\,dx,$

we obtain

$\displaystyle\frac{1}{m}\frac{|1-\widehat{f}(\xi)|}{|\xi|}=\frac{1}{m}\left|\int_{\mathbb{R}}(F_{0}(x)-F(x))e^{-i\xi
x}\,dx\right|\leq$
$\displaystyle\frac{1}{m}\int_{\mathbb{R}}\left||(F_{0}(x)-F(x))e^{-i\xi
x}\right|\,dx=\frac{1}{m}\int_{\mathbb{R}_{+}}(1-F(x))\,dx=1.$

The functional $H_{\infty}$ is a particular case of a metric for probability
measures which have been used to study convergence to equilibrium for the
Boltzmann equation. This is an argument that in kinetic theory of rarefied
gases goes back to [17], where convergence to equilibrium for the Boltzmann
equation for Maxwell pseudo-molecules was studied in terms of a metric for
Fourier transforms (cf. also [9, 25, 31] for further applications).

The metric introduced in [17] in connection with the Boltzmann equation for
Maxwell molecules, was subsequently applied in various contexts, which include
kinetic models for wealth measures [27], thus establishing a number of common
points between kinetic modeling and inequality measures.

For a given pair of random variables $X$ and $Y$ distributed according to $F$
and $G$ these metrics read

(2.22)
$d_{r}(X,Y)=d_{r}(F,G)=\sup_{\xi\in\mathbb{R}}\frac{|\widehat{f}(\xi)-\widehat{g}(\xi)|}{|\xi|^{r}},\quad
r>0.$

As shown in [17], the metric $d_{r}(F,G)$ is finite any time the probability
measures $F$ and $G$ have equal moments up to $[r]$, namely the entire part of
$r\in\mathbb{R}_{+}$, or equal moments up to $r-1$ if $r\in\mathbb{N}$, and it
is equivalent to the weak∗ convergence of measures for all $r>0$. Among other
properties, it is easy to see [17, 27] that, for two pairs of random variables
$X,Y$, where $X$ is independent from $Y$, and $Z,\tilde{Z}$ ($Z$ independent
from $\tilde{Z}$), and any constant $c$

(2.23) $\displaystyle d_{r}(X+Y,Z+\tilde{Z})\leq
d_{r}(X,Z)+d_{r}(Y,\tilde{Z})$ $\displaystyle d_{r}(cX,cY)=|c|^{r}d_{r}(X,Y).$

These properties classify $d_{s}$ as an ideal probability metric in the sense
of Zolotarev [33]. Properties of $H_{\infty}(F)$ can be easily extracted from
(LABEL:sca1) considering that, if $X$ is a random variable with probability
measure $F$ of mean value $m$

$H_{\infty}(F)=H_{\infty}(X)=\frac{1}{2m}d_{1}(F,F_{m})$

In particular, the second property in (LABEL:sca1) implies the scaling
invariance of $H_{\infty}$.

Also the first inequality in (LABEL:sca1) implies that, for any pair of
independent variables $X$ and $Y$, with means $m_{X}$ (respectively $m_{Y}$ ),
by choosing $Z$ and $\tilde{Z}$ with probability measures $F_{m_{X}}$
(respectively $F_{m_{Y}}$)

(2.24)
$H_{\infty}(X+Y)\leq\frac{m_{X}}{m_{X}+m_{Y}}H_{\infty}(X)+\frac{m_{Y}}{m_{X}+m_{Y}}H_{\infty}(Y),$

namely a property of sub-additivity for convolutions. Moreover, if $Y$ is
distributed with probability measure $F_{m_{Y}}$, Inequality (2.24) gives

(2.25)
$H_{\infty}(X+Y)\leq\frac{m_{X}}{m_{X}+m_{Y}}H_{\infty}(X)<H_{\infty}(X).$

Inequality (2.25) is a typical feature of sparsity measures, which translates
to the case of a continuous variable the property that adding a constant to
each coefficient decreases sparsity [22].

In view of its properties, the functional $H_{\infty}(\cdot)$ appears to be a
good measure of inequality. Unlikely, the computation of the values of $H$ for
most probability measures is cumbersome. In particular, it seems not possible
to explicitly compute the value of $H_{\infty}(X)$ in the simplest case in
which the variable $X$ takes only two positive values. Consequently, we can
not evaluate if, for a given $\epsilon\ll 1$, there exists a probability
measure with index $1-\epsilon$. Indeed, as we saw in Section 2.2, this basic
property follows from the analysis of the values taken by the inequality index
at two-valued random variables.

The upper bound $1$ can be found also by resorting to Lagrange theorem.
Indeed, since $F\in P_{s}(\mathbb{R})$ the function
$h(\xi)=|\widehat{f}(\xi)-e^{-im\xi}|$ is continuously differentiable on the
entire real line, and satisfies $h(\xi=0)=0$.

Therefore, by Lagrange theorem, for any given $\xi\in\mathbb{R}$, there exists
$\xi_{0}\in\mathbb{R}$ such that

$\frac{h(\xi)}{\xi}=\frac{h(\xi)-h(0)}{\xi-0}=h^{\prime}(\xi_{0}),$

which implies

$\sup_{\xi\in\mathbb{R}}\left|\frac{h(\xi)}{\xi}\right|\leq\sup_{\xi\in\mathbb{R}}|h^{\prime}(\xi)|.$

Since $|e^{im\xi}|=1$, we have the identity

$h(\xi)=|\widehat{f}(\xi)-e^{-im\xi}|=|\widehat{f}(\xi)e^{im\xi}-1|,$

and

$|h^{\prime}(\xi)|\leq\left|\widehat{f}^{\prime}(\xi)e^{im\xi}+im\widehat{f}(\xi)e^{im\xi}\right|=\left|\widehat{f}^{\prime}(\xi)+im\widehat{f}(\xi)\right|.$

We therefore obtain

(2.26)
$\frac{1}{m}\sup_{\xi\in\mathbb{R}}\frac{|\widehat{f}(\xi)-e^{-im\xi}|}{|\xi|}\leq\frac{1}{m}\sup_{\xi\in\mathbb{R}}\left|\widehat{f}^{\prime}(\xi)+im\widehat{f}(\xi)\right|=\sup_{\xi\in\mathbb{R}}\left|\widehat{f}(\xi)-\frac{\widehat{f}^{\prime}(\xi)}{\widehat{f}^{\prime}(0)}\right|.$

Using the argument leading to the upper bound in (1.2) one easily concludes
that $H_{\infty}(F)$ is bounded above by $T(F)$, where $T(F)$ is the
functional defined by (1.1), and that $H_{\infty}(F)\leq 1$.

##  3\. A new Fourier-based index of inequality

This Section will be devoted to study in more details the main properties of
the inequality index $T(F)$, as given by (1.1). Depending on convenience,
given a random variable $X$ with probability measure
$F\in\tilde{P}_{s}(\mathbb{R})$, we will write indifferently $T(X)$ or $T(F)$.

As outlined in the introduction, for any constant $c>0$, $T(F)$ is invariant
with respect to the scaling $F(x)\to F(cx)$. The scaling invariance of $T(F)$
can be easily seen by noticing that, if $\widehat{f}(\xi)$ is the Fourier
transform of $F(x)$, $\widehat{f}(\xi/c)$ is the Fourier transform of $F(cx)$,
and

$\frac{\widehat{f}^{\prime}_{\xi}(\xi/c)}{\widehat{f}^{\prime}_{\xi}(0)}=\left.\frac{\widehat{f}^{\prime}_{\eta}(\eta)}{\widehat{f}^{\prime}_{\eta}(0)}\right|_{\eta=\xi/c},$

that implies

(3.1)
$\sup_{\xi\in\mathbb{R}}\left|\widehat{f}(\xi/c)-\frac{\widehat{f}^{\prime}_{\xi}(\xi/c)}{\widehat{f}^{\prime}_{\xi}(0)}\right|=\sup_{\xi\in\mathbb{R}}\left|\widehat{f}(\eta)-\frac{\widehat{f}^{\prime}_{\eta}(\eta)}{\widehat{f}^{\prime}_{\eta}(0)}\right|_{\eta=\xi/c}=T(F).$

Moreover, as shown in the introduction, if $F\in\tilde{P}_{s}(\mathbb{R})$,
$T(F)$ the values of the functional lie between zero and one, where the value
zero (minimal inequality) is assumed in correspondence to a Heaviside
probability measure $F_{m}$, with $m>0$. In addition, the value $T(F)=1$,
corresponding to maximal inequality is approached if we compute the value of
$T(X)$ when, as discussed in Section 2.2, the random variable $X$ of mean
value $m$ is a two-valued random variable where

$P(X=0)=1-\epsilon;\quad
P\left(X=\frac{m}{\epsilon}\right)=\epsilon;\quad\epsilon\ll 1.$

In this case

$\widehat{f}(\xi)=1-\epsilon+\epsilon\exp\left\\{-i\frac{m}{\epsilon}\xi\right\\},$

while

$\widehat{f}^{\prime}(\xi)=-im\exp\left\\{-i\frac{m}{\epsilon}\xi\right\\}.$

Therefore

$\displaystyle
2\,T(X)=\sup_{\xi\in\mathbb{R}}\left|1-\epsilon+\epsilon\exp\left\\{-i\frac{m}{\epsilon}\xi\right\\}-\exp\left\\{-i\frac{m}{\epsilon}\xi\right\\}\right|=$
$\displaystyle(1-\epsilon)\sup_{\xi\in\mathbb{R}}\left|1-\exp\left\\{-i\frac{m}{\epsilon}\xi\right\\}\right|=(1-\epsilon)\sup_{\xi\in\mathbb{R}}\sqrt{2(1-\cos\frac{m}{\epsilon}\xi)}=2(1-\epsilon),$

and $T(X)=1-\epsilon$.

Let $F,G\in\tilde{P}_{s}(\mathbb{R})$ two probability measures with the same
mean value, say $m$. Then, for any given $\tau\in(0,1)$ it holds
$\tau\widehat{f}^{\prime}(0)+(1-\tau)\widehat{g}^{\prime}(0)=\widehat{f}^{\prime}(0)=\widehat{g}^{\prime}(0)$,
so that

$\displaystyle T(\tau
F+(1-\tau)G)=\frac{1}{2}\sup_{\xi\in\mathbb{R}}\left|\tau\widehat{f}(\xi)+(1-\tau)\widehat{g}(\xi)-\frac{\tau\widehat{f}^{\prime}(\xi)+(1-\tau)\widehat{g}^{\prime}(\xi)}{\tau\widehat{f}^{\prime}(0)+(1-\tau)\widehat{g}^{\prime}(0)}\right|=$
$\displaystyle\frac{1}{2}\sup_{\xi\in\mathbb{R}}\left|\tau\widehat{f}(\xi)-\tau\frac{\widehat{f}^{\prime}(\xi)}{\widehat{f}^{\prime}(0)}+(1-\tau)\widehat{g}(\xi)-(1-\tau)\frac{\widehat{g}^{\prime}(\xi)}{\widehat{g}^{\prime}(0)}\right|\leq\tau\,T(F)+(1-\tau)T(G).$

This shows the convexity of the functional $T$ on the set of probability
measures with the same mean.

However, the most important property characterizing the inequality index $T$
is linked to its behavior in presence of convolutions. For any given pair of
Fourier transforms of probability measures in $\tilde{P}_{s}(\mathbb{R})$, let
us set

$\widehat{h}(\xi)=\widehat{f}(\xi)\widehat{g}(\xi).$

Then, since $|\widehat{f}(\xi)|\leq\widehat{f}(0)=1$ and
$|\widehat{g}(\xi)|\leq\widehat{g}(0)=1$

$\displaystyle\sup_{\xi\in\mathbb{R}}\left|\widehat{h}(\xi)-\frac{\widehat{h}^{\prime}(\xi)}{\widehat{h}^{\prime}(0)}\right|=\sup_{\xi\in\mathbb{R}}\left|\widehat{f}(\xi)\widehat{g}(\xi)-\frac{\widehat{f}^{\prime}(\xi)\widehat{g}(\xi)+\widehat{f}(\xi)\widehat{g}^{\prime}(\xi)}{\widehat{f}^{\prime}(0)+\widehat{g}^{\prime}(0)}\right|=$
$\displaystyle\sup_{\xi\in\mathbb{R}}\left|\widehat{f}(\xi)\widehat{g}(\xi)-\frac{\widehat{f}^{\prime}(0)}{\widehat{f}^{\prime}(0)+\widehat{g}^{\prime}(0)}\frac{\widehat{f}^{\prime}(\xi)\widehat{g}(\xi)}{\widehat{f}^{\prime}(0)}-\frac{\widehat{g}^{\prime}(0)}{\widehat{f}^{\prime}(0)+\widehat{g}^{\prime}(0)}\frac{\widehat{f}(\xi)\widehat{g}^{\prime}(\xi)}{\widehat{g}^{\prime}(0)}\right|\leq$
$\displaystyle\frac{\widehat{f}^{\prime}(0)}{\widehat{f}^{\prime}(0)+\widehat{g}^{\prime}(0)}\sup_{\xi\in\mathbb{R}}\left|\widehat{f}(\xi)\widehat{g}(\xi)-\frac{\widehat{f}^{\prime}(\xi)\widehat{g}(\xi)}{\widehat{f}^{\prime}(0)}\right|+$
$\displaystyle\frac{\widehat{g}^{\prime}(0)}{\widehat{f}^{\prime}(0)+\widehat{g}^{\prime}(0)}\sup_{\xi\in\mathbb{R}}\left|\widehat{f}(\xi)\widehat{g}(\xi)-\frac{\widehat{f}(\xi)\widehat{g}^{\prime}(\xi)}{\widehat{g}^{\prime}(0)}\right|\leq$
$\displaystyle\frac{\widehat{f}^{\prime}(0)}{\widehat{f}^{\prime}(0)+\widehat{g}^{\prime}(0)}\sup_{\xi\in\mathbb{R}}\left|\widehat{f}(\xi)-\frac{\widehat{f}^{\prime}(\xi)}{\widehat{f}^{\prime}(0)}\right|+\frac{\widehat{f}^{\prime}(0)}{\widehat{f}^{\prime}(0)+\widehat{g}^{\prime}(0)}\sup_{\xi\in\mathbb{R}}\left|\widehat{g}(\xi)-\frac{\widehat{g}^{\prime}(\xi)}{\widehat{g}^{\prime}(0)}\right|.$

Therefore, if $X$ and $Y$ are independent random variables with probability
measures in $\tilde{P}_{s}(\mathbb{R})$, and mean values $m_{X}$ (respectively
$m_{Y}$) the inequality index $T$ satisfies the inequality

(3.2) $T(X+Y)\leq\frac{m_{X}}{m_{X}+m_{Y}}T(X)+\frac{m_{Y}}{m_{X}+m_{Y}}T(Y).$

In particular, if $Y$ is a random variable that takes the value $m>0$ with
probability $1$ (so that $\widehat{g}(\xi)=e^{-im\xi}$ and $T(Y)=0$),

(3.3) $T(X+Y)\leq\frac{m_{X}}{m_{X}+m}T(X)<T(X).$

Since $X+Y$ corresponds to adding the constant $m$ to $X$, this property
asserts that adding a constant wealth to each agent decreases inequality.

Also, if the random variables $X_{1}$ and $X_{2}$ are distributed with the
same law of $X$, thanks to the scale property

(3.4) $T\left(\frac{X_{1}+X_{2}}{2}\right)=T\left(X_{1}+X_{2}\right)\leq
T(X),$

while the mean of $(X_{1}+X_{2})/2$ is equal to the mean of $X$.

######  Remark 3.1.

Inequality (3.4) is fully operational in the case where the two variables
$X_{1}$ and $X_{2}$ are characterized either by a continuous probability
measure or take on an infinite number of values. Only in this case, in fact,
do the probability measure remain of the same type under the operation of
convolution.

Suppose in fact that the variables $X_{i}$, $i=1,2$, are Bernoulli variables,
such that

$P(X_{i}=0)=P(X_{i}=1)=\frac{1}{2},\quad i=1,2,$

The probability measure of $X_{i}$, $i=1,2$, has Fourier transform

$\widehat{f}(\xi)=\frac{1}{2}\left(1+e^{-i\,\xi}\right),$

and the probability measure of the convolution corresponds to the Fourier
transform

$\widehat{f}(\xi)^{2}=\frac{1}{4}\left(1+2e^{-i\,\xi}+e^{-2i\,\xi}\right).$

Hence, the random variable $Y=X_{1}+X_{2}$ takes the three values $0,1,2$ with
probabilities

$P(Y=0)=P(Y=2)=\frac{1}{4},\quad P(Y=1)=\frac{1}{2}.$

Clearly, it makes little sense to relate the heterogeneity of a two-valued
random variable to a three-valued random variable.

Another important consequence of inequality (3.2) is related to the situation
in which the random variable $Y$ represents a noise (of mean value $m>0$) that
is present when measuring the inequality index of $X$. The classical choice is
that the additive noise is represented by a Gaussian variable of mean $m$ and
variance $\sigma^{2}$.

If this is the case, the Fourier transform of the Gaussian density is given by
(2.15), which is such that

$\widehat{f}^{\prime}(\xi)=(-im-\sigma^{2}\xi)\widehat{f}(\xi);\quad\widehat{f}^{\prime}(0)=-im.$

Hence, since $|\widehat{f}(\xi)|\leq\widehat{f}(0)=1$,

$\left|\widehat{f}(\xi)-\frac{\widehat{f}^{\prime}(\xi)}{\widehat{f}^{\prime}(0)}\right|=\left|\widehat{f}(\xi)-\frac{-im-\sigma^{2}\xi}{-im}\widehat{f}(\xi)\right|=\frac{\sigma}{m}\left|\sigma\xi\widehat{f}(\xi)\right|.$

Finally, if $Y$ denotes the Gaussian random variable of mean $m>0$ and
variance $\sigma^{2}$ we obtain

$T(Y)=\frac{\sigma}{2m}\sup_{\xi\in\mathbb{R}}\left|\xi
e^{-\xi^{2}/2}\right|=\frac{\sigma}{2m}\frac{1}{\sqrt{e}}.$

As we showed in Section 2.2 for the index $H$ defined by (2.9), for a Gaussian
variable, the inequality index $T(Y)$ is proportional to the coefficient of
variation of $Y$. We have in this case

(3.5)
$T(X+Y)\leq\frac{m_{X}}{m_{X}+m}T(X)+\frac{\sigma}{m_{X}+m}\frac{1}{\sqrt{e}},$

namely an explicit upper bound for the inequality index in terms of the mean
value and the variance of the Gaussian noise.

##  4\. Examples

In this section we will recover the values of the inequality index $T$ for
some well-known probability measures. With few exceptions, any time the
explicit expression of the Fourier transform of the probability measure is
available, the computation of the value of the inequality index $T(\cdot)$ is
straightforward. The list of probability measures that can be treated via
Fourier transform is consistent, and includes both discrete and continuous
distributions. For an in-depth look at this topic, the interested reader can
consult the book [26].

We do not consider in this paper the possibility to make use of the fast
Fourier transform to compute the values of the functional $T$ in the case of a
random variable taking only a finite number of values, a situation that we
intend to treat in a companion paper.

###  4.1. Two-valued random variables

Let $X$ be a Bernoulli random variable, characterized by the probability
measure with Fourier transform

$\widehat{f}(\xi)=1-p+pe^{-i\xi},\quad 0<p<1.$

Then, since $\widehat{f}^{\prime}(\xi)=-ipe^{-i\xi}$, and it is immediate to
conclude that

(4.1)
$T(X)=\frac{1}{2}(1-p)\sup_{\xi\in\mathbb{R}}\left|1-e^{-i\xi}\right|=1-p.$

For given positive constants $a,b$, let $Y=aX+b$: Then $Y$ is characterized by
the Fourier transform

$\widehat{h}(\xi)=\widehat{f}(a\xi)e^{-ib\,\xi}.$

We have

$\displaystyle\left|\widehat{h}(\xi)-\frac{\widehat{h}^{\prime}(\xi)}{\widehat{h}^{\prime}(0)}\right|=\left|\frac{ap\widehat{f}(a\xi)-\widehat{f}^{\prime}(a\xi)}{ap+b}\right|=\frac{ap}{ap+b}(1-p)\left|1-e^{-i\xi}\right|,$

so that

$T(Y)=\frac{ap(1-p)}{ap+b}.$

Choosing $\alpha=b$ and $\beta=a+b$, where $\beta>\alpha$, we then conclude
that a two valued random variable $Y$ such that

$P(Y=\alpha)=1-p,\quad P(Y=\beta)=p$

has an inequality index

(4.2) $T(Y)=\frac{(\beta-\alpha)p(1-p)}{\alpha(1-p)+\beta p}.$

The same value is assumed by the Gini and Pietra indices of $Y$.

###  4.2. Poisson distribution

Poisson distribution is characterized by the Fourier transform

$\widehat{f}(\xi)=\exp\left\\{\lambda\left(e^{-i\xi}-1\right)\right\\}.$

In this case

$\displaystyle\left|\widehat{f}(\xi)-\frac{\widehat{f}^{\prime}(\xi)}{\widehat{f}^{\prime}(0)}\right|=\left|\left(e^{-i\xi}-1\right)\widehat{f}(\xi)\right|=\sqrt{2(1-\cos\xi)}\exp\left\\{-\lambda(1-\cos\xi)\right\\}.$

Let us set $0\leq 1-\cos\xi=x^{2}\leq 2$. Then

$T(F)=\frac{\sqrt{2}}{2}\sup_{0\leq x\leq\sqrt{2}}x\,e^{-\lambda x^{2}}.$

If $\lambda\leq 1/4$, the maximum is taken in $\bar{x}=\sqrt{2}$, and
$T(F)=e^{-2\lambda}$. If $\lambda>1/4$, the maximum is taken at the point
$\bar{x}=1/\sqrt{2\lambda}$, and in this case

$T(F)=\frac{1}{2\sqrt{\lambda}}e^{-1/2}.$

Hence, if $F$ is a Poisson probability measure of mean $\lambda$ we have

(4.3)
$T(F)=\left\\{\begin{array}[]{cc}&e^{-2\lambda}\quad\rm{if}\,\,\,\lambda\leq\frac{1}{4}\\\
&\frac{1}{2\sqrt{\lambda}}e^{-1/2}\quad\rm{if}\,\,\,\lambda>\frac{1}{4}.\end{array}\right.$

Note that, as a function of $\lambda$, the functional $T(F)$ is differentiable
at the point $\lambda=1/4$, and it decreases as $\lambda$ increases. Hence,
small values of $\lambda$ corresponds to large heterogeneity.

######  Remark 4.1.

It is interesting to remark that the value of the Gini index of a Poisson
distribution, say $F$, can not be computed explicitly by resorting to its
expression in Fourier transform, as given by formula (2.6). The same
conclusion holds if we try to compute the values of $H(F)$, as given by (2.9),
and $H_{\infty}(F)$, defined in (2.20).

######  Remark 4.2.

The previous computations can be extended, at the cost of more complicated
calculations, to evaluate the explicit values of the index $T$ to
distributions which are obtained by summing up independent Poisson variables.
Maybe the most interesting case corresponds to the Skellam distribution [29,
30], that is the discrete probability distribution of the difference of two
independent random variables $X_{1}$ and $X_{2}$, each Poisson-distributed
with expected values $\lambda_{1}$ and, respectively $\lambda_{2}$, with
$\lambda_{1}\not=\lambda_{2}$.

###  4.3. Stable laws

As further example of probability measures defined on the whole real line
$\mathbb{R}$, we will compute the value of $T$ in correspondence to a stable
law [34]. We will restrict here to the case of symmetric alpha-stable
distributions of scale parameter $\sigma>0$ and shift parameter $m>0$,
characterized by the Fourier transform

$\widehat{f}_{\alpha}(\xi)=\exp\left\\{-i\xi\,m-\left|\sigma\xi\right|^{\alpha}\right\\},\qquad\alpha>1.$

Note that the Gaussian distribution of mean $m$ and variance $2\sigma^{2}$
corresponds to the choice $\alpha=2$.

For these distributions

$\displaystyle\left|\widehat{f}_{\alpha}(\xi)-\frac{\widehat{f}_{\alpha}^{\prime}(\xi)}{\widehat{f}_{\alpha}^{\prime}(0)}\right|=\frac{\alpha}{m}\left|\sigma^{\alpha}|\xi|^{\alpha-2}\xi\widehat{f}_{\alpha}(\xi)\right|=\frac{\alpha}{m}\sigma\left|\sigma\xi\right|^{\alpha-1}\exp\left\\{-\left|\sigma\xi\right|^{\alpha}\right\\}.$

Consequently

$T(F_{\alpha})=\frac{\sigma\alpha}{2\,m}\sup_{x\geq
0}x^{(\alpha-1)/\alpha}e^{-x}$

Evaluating the value of the supremum, we obtain

(4.4)
$T(F_{\alpha})=\frac{\sigma\alpha}{2\,m}\left(\frac{\alpha-1}{\alpha}\right)^{(\alpha-1)/\alpha}\exp\left\\{-\frac{\alpha-1}{\alpha}\right\\}$

For $\alpha=1$ the distribution reduces to a Cauchy distribution with scale
parameter $\sigma$ and shift parameter $m$. In this case

$T(F_{1})=\frac{\sigma}{2\,m}\sup_{x\geq 0}e^{-x}=\frac{\sigma}{2\,m}.$

###  4.4. An interesting case: the uniform distribution

The uniform distribution in the interval $(-a,a)$, with $a>0$ is characterized
by the Fourier transform

(4.5) $\widehat{f}(\xi)=\frac{\sin(a\xi)}{a\xi}.$

Hence, if $X$ is a random variable uniformly distributed on $(-a,a)$, for any
constant $b>0$, $X+b$ is uniformly distributed on the interval $(-a+b,a+b)$,
and the Fourier transform of the probability measure of $X+b$, of mean value
$b$ is given by

$\widehat{g}(\xi)=\widehat{f}(\xi)e^{-ib\xi}.$

Then

$\displaystyle\sup_{\xi\in\mathbb{R}}\left|\widehat{g}(\xi)-\frac{\widehat{g}^{\prime}(\xi)}{\widehat{g}^{\prime}(0)}\right|=\sup_{\xi\in\mathbb{R}}\left|\widehat{f}(\xi)e^{-ib\xi}-\frac{\widehat{f}^{\prime}(\xi)e^{-ib\xi}-ib\widehat{f}(\xi)e^{-ib\xi}}{-ib}\right|=\frac{1}{b}\sup_{\xi\in\mathbb{R}}|\widehat{f}^{\prime}(\xi)|.$

Next, since $\widehat{f}$ is expressed by (4.5)

$\widehat{f}^{\prime}(\xi)=\frac{a\xi\cos(a\xi)-\sin(a\xi)}{a\xi^{2}},$

which implies

$\sup_{\xi\in\mathbb{R}}|\widehat{f}^{\prime}(\xi)|=a\sup_{\xi\in\mathbb{R}}\left|\frac{\xi\cos\xi-\sin\xi}{\xi^{2}}\right|=a\,\delta_{u}$

where $\delta_{u}$ is a positive constant. Hence, if $X$ is uniformly
distributed on the interval $(-a,a)$, and $b>0$

$T(X+b)=\frac{a}{2b}\,\delta_{u}.$

In particular, if $b>a$, by setting $\alpha=b-a$ and $\beta=b+a$, we conclude
that, if $Y$ is a random variable uniformly distributed on the interval
$(\alpha,\beta)\in\mathbb{R}_{+}$, it holds

(4.6) $T(Y)=\frac{\delta_{u}}{2}\,\frac{\beta-\alpha}{\beta+\alpha}.$

In this case, at difference with the Gini index, which takes the explicit
value

$G(Y)=\frac{1}{3}\,\frac{\beta-\alpha}{\beta+\alpha},$

the value of the coefficient $\delta_{u}$ can be achieved only numerically. It
is however interesting to remark, in the case of a uniform distribution, the
values of the two indices have deep similarities.

A rough estimation of the constant $\delta_{u}$ follows by studying the
function

$u(x)=\frac{\sin x-x\cos x}{x^{2}},\quad x\geq 0.$

It is immediate to show that any extremal point $\bar{x}$ of the function
$u(x)$ solves the equation

$(x^{2}-2)\sin x-2x\cos x=0,$

that implies

$\sin\bar{x}-\bar{x}\cos\bar{x}=\frac{\bar{x}^{2}}{2}\sin\bar{x}.$

Consequently, if $\bar{x}$ is an extremal point of $u(x)$,

$|u(\bar{x})|=\frac{1}{2}|\sin\bar{x}|\leq\frac{1}{2}.$

Hence $\delta_{u}\leq 1/2$.

To end this Section, we list in Table $4.1$ the values of the inequality index
$T$ for some probability measures in $\mathbb{R}_{+}$ and $\mathbb{R}$
allowing explicit computations. It is remarkable that the Fourier-based index
$T$ is well-adapted to compute the heterogeneity index of discrete probability
measures, like the negative binomial distribution, or the geometric
distribution, which are explicitly expressible in terms of the Fourier
transform. We leave the details of the evaluation to the reader.

Measure | Density | Fourier transform | Index $T(\cdot)$
---|---|---|---
Exponential | $\lambda e^{-\lambda\,x}$ | $\left(1+i\xi/\lambda\right)^{-1}$ | $\frac{1}{4}$
Gamma | $\frac{1}{\Gamma(k)\theta^{k}}x^{k-1}e^{-x/\theta}$ | $(1+i\theta\xi)^{-k}$ |  $\frac{1}{2\sqrt{k}}\left(1+\frac{1}{k}\right)^{-(k+1)/2}$ $k>0$
Chi-squared | $\frac{1}{2^{k/2}\Gamma(k/2)}x^{k/2-1}e^{-x/2}$ | $(1+2i\xi)^{-k/2}$ |  $\frac{1}{\sqrt{2k}}\left(1+2/k\right)^{-(k+2)/4}$ $k\geq 1$
Laplace | $\frac{1}{2\sigma}e^{-|x-m|/\sigma}$ | $e^{-im\,\xi}\left(1+\sigma^{2}\xi^{2}\right)^{-1}$ | $\frac{\sigma}{m}\frac{16}{25}$
Table 4.1. Values of the index $T$ for some probability measures

##  5\. An application to kinetic theory of wealth distribution

Kinetic modelling of agent-based markets are based on few universal
assumptions [27]. First, agents are indistinguishable, so that an agent’s
state at any instant of time $t\geq 0$ is completely characterized by his
current wealth $w\geq 0$. Second, the time variation of the wealth
distribution is entirely due to binary trades between agents. A trade
represents a binary interaction in which part of the money of each agent is
modified according to well-defined rules. When two agents undertake in a
trade, their pre-trade wealths $v$, $w$ change into the post-trade wealths
$v^{*}$, $w^{*}$ according to a linear exchange rule:

(5.1) $v^{*}=p_{1}v+q_{1}w,\quad w^{*}=q_{2}v+p_{2}w.$

The interaction coefficients $p_{i}$ and $q_{i}$, $i=1,2$, are, in general,
non negative random parameters.

The first explicit description of a binary wealth-exchange model dates back to
the seminal work of Angle [1], (cf. also [2]), even if the intimate relation
to statistical mechanics was only described about a decade later [13, 23]. In
each binary interaction, winner and loser are randomly chosen, and the loser
pays a random fraction of his wealth to the winner. From here, Chakraborti and
Chakrabarti [10] developed the class of strictly conservative exchange models,
which preserve the total wealth in each individual trade,

(5.2) $v^{*}+w^{*}=v+w.$

In its most basic version, the microscopic interaction is determined by one
single parameter $\lambda\in(0,1)$, which is the global saving propensity. In
the interactions, each agent retains the corresponding fraction of its pre-
trade wealth, while the rest $(1-\lambda)(v+w)$ is equally shared equally
between the two trading partners,

(5.3) $v^{*}=\lambda v+\frac{1}{2}(1-\lambda)(v+w),\quad w^{*}=\lambda
w+\frac{1}{2}(1-\lambda)(v+w).$

The wealth distribution $f(v,t)$ of the system of agents coincides with
agent’s density and satisfies the associated spatially homogeneous Boltzmann
equation,

(5.4) $\partial_{t}f+f=Q_{+}(f,f),$

on the real half-line, $v\geq 0$. The collisional gain operator $Q_{+}$ acts
on test functions $\varphi(v)$ as

$\displaystyle Q_{+}(f,f)[\varphi]=$
$\displaystyle\int_{\mathbb{R}_{+}}\varphi(v)Q_{+}\big{(}f,f\big{)}(v)\,dv$
$\displaystyle=$
$\displaystyle\frac{1}{2}\int_{\mathbb{R}_{+}^{2}}\langle\varphi(v^{*})+\varphi(w^{*})\rangle
f(v)f(w)\,dv\,dw.$

In reason of (5.3), the average wealth of the society is conserved with time,
so that

(5.6) $m(t)=\int_{\mathbb{R}_{+}}wf(w,t)\,dw=m,$

where $m>0$ is finite. A useful way of writing equation (5.4) is to resort to
the Fourier transform [27]. Assuming the initial distribution of wealth in
$P_{s}^{+}$, with $s>1$, the transformed kernel reads

(5.7)
$\widehat{Q}\bigl{(}\widehat{f},\widehat{f}\bigr{)}(\xi)=\widehat{f}\left(\frac{1-\lambda}{2}\xi\right)\widehat{f}\left(\frac{1+\lambda}{2}\xi\right),$

where, since the initial density has a bounded mean,

$\widehat{f_{0}}^{\prime}(0)=-im.$

Hence, the Boltzmann equation (5.4) can be rewritten in terms of the Fourier
transform of $f(v,t)$ as

(5.8) $\frac{\partial\widehat{f}(\xi,t)}{\partial
t}+\widehat{f}(\xi,t)=\widehat{f}\left(\frac{1-\lambda}{2}\xi,t\right)\widehat{f}\left(\frac{1+\lambda}{2}\xi,t\right).$

It is immediate to show that the functions $e^{-i\mu\xi}$, with $\mu>0$,
namely the Fourier transforms of a Dirac delta concentrated at the wealth
$\mu$, are stationary solutions of equation (5.8).

More can be said if we assume that $s\geq 2$. Then, the moment of order two of
the initial distribution is finite, and, applying (5.7) with
$\varphi(v)=(v-m)^{2}$, and recalling that the mean value is preserved during
the evolution, shows that the variance of $f(v,t)$ satisfies

(5.9)
$\frac{d}{dt}\int_{\mathbb{R}_{+}}(v-m)^{2}f(v,t)\,dv=-\frac{1}{2}(1-\lambda^{2})\int_{\mathbb{R}_{+}}(v-m)^{2}f(v,t)\,dv.$

As a result, all agents tend for large times to become equally rich. Indeed,
the steady state $f_{\infty}(v)$ is a Dirac delta concentrated at the mean
wealth, and is approached at the exponential rate $(1-\lambda^{2})/2$.

To remain into the framework of inequality indices, the previous result
implies that, if the initial distribution belongs to $P_{s}^{+}$, with $s\geq
2$, the coefficient of variation is monotonically decreasing towards zero at
the explicit rate $(1-\lambda^{2})/4$.

This result is lost as soon as the value of $s$ is less than $2$. It is
however interesting to remark that the inequality index $T(F(t))$, where
$F(v,t)$ is the probability measure associated to the solution
$\widehat{f}(\xi,t)$ of equation (5.8), is monotonically decreasing in time as
soon as $s\geq 1$. Indeed, if we set

$h(\xi,t)=\widehat{f}(\xi,t)-\frac{\widehat{f}^{\prime}(\xi,t)}{\widehat{f}^{\prime}(0)},$

it is immediate to show that $h(\xi,t)$ satisfies the equation

$\frac{\partial h(\xi,t)}{\partial
t}+h(\xi,t)=\widehat{f}\left(\frac{1-\lambda}{2}\xi,t\right)\widehat{f}\left(\frac{1+\lambda}{2}\xi,t\right)-\frac{\left[\widehat{f}\left(\frac{1-\lambda}{2}\xi,t\right)\widehat{f}\left(\frac{1+\lambda}{2}\xi,t\right)\right]^{\prime}}{\widehat{f}^{\prime}(0,t)},$

which implies

(5.10) $\left|\frac{\partial h(\xi,t)}{\partial
t}+h(\xi,t)\right|\leq\sup_{\xi}\left|\widehat{f}\left(\frac{1-\lambda}{2}\xi,t\right)\widehat{f}\left(\frac{1+\lambda}{2}\xi,t\right)-\frac{\left[\widehat{f}\left(\frac{1-\lambda}{2}\xi,t\right)\widehat{f}\left(\frac{1+\lambda}{2}\xi,t\right)\right]^{\prime}}{\widehat{f}^{\prime}(0,t)}\right|.$

If now $X(t)$ and $Y(t)$ are random variables with probability measures of
Fourier transforms $\widehat{f}\left(\xi\,(1-\lambda)/2,t\right)$
(respectively $\widehat{f}\left(\xi\,(1+\lambda)/2,t\right)$), which have mean
values $m\,(1-\lambda)/2$ (respectively $m\,(1+\lambda)/2$), formula (3.2) for
convolutions gives

(5.11)
$T(X(t)+Y(t))\leq\frac{1-\lambda}{2}T(X(t))+\frac{1-\lambda}{2}T(Y(t)).$

On the other hand, by scaling invariance $T(X(t))=T(Y(t))=T(Z(t)$, where the
probability measure of $Z(t)$ has Fourier transform $\widehat{f}(\xi,t)$.
Hence, equation (5.10) implies

(5.12) $\left|\frac{\partial h(\xi,t)}{\partial
t}+h(\xi,t)\right|\leq\sup_{\xi}\left|h(\xi,t)\right|,$

that, for any given $t_{0}<t$ by Gronwall inequality implies [27]

$T(Z(t))\leq T(Z(t_{0}),$

and, consequently the monotonicity in time of the inequality index $T(F(t)$ of
the probability measure solution of the kinetic equation (5.4). It is
remarkable that this result, which does not require the condition $s>1$, is a
direct consequence of the convolution property of the inequality index $T$.
Hence, the monotonicity result does not hold if we resort to Gini and Pietra
indices.

##  6\. Conclusions

Inequality indices are quantitative scores that take values in the unit
interval, with the zero score characterizing perfect equality. Measuring the
statistical heterogeneity of measures arises in most fields of science and
engineering, which makes it important to know the strengths and possible
weaknesses of heterogeneity measures in applications [5, 7, 11, 12, 15, 21,
22]. In this paper, we draw attention to a new inequality index, based on the
Fourier transform, which exhibits a number of interesting properties that make
it very promising in applications. In comparison with the well-known and
widely used Gini index, which can still be expressed by resorting to Fourier
transform, the new index $T$ allows to compute explicitly the heterogeneity of
various probability measures, like the Poisson distribution, which can not be
measured explicitly resorting to Gini index. Also, this new Fourier-based
index has an interesting property of sub-additivity for convolutions, which in
principle makes it interesting for applications to models of kinetic theory
which contain mass and mean preserving bilinear operators [27].

## Acknowledgement

This work has been written within the activities of GNFM group of INdAM
(National Institute of High Mathematics), and partially supported by IMATI
(Institute of Applied Mathematics and Information Technologies Enrico
Magenes).

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