# Monge-Kantorovich superquantiles and expected shortfalls with applications
to multivariate risk measurements

Bernard Bercu, Jérémie Bigot and Gauthier Thurin  Université de Bordeaux
Institut de Mathématiques de Bordeaux et CNRS UMR 5251,
351 Cours de la libération, 33400 Talence cedex, France

###### Abstract.

We propose center-outward superquantile and expected shortfall functions, with
applications to multivariate risk measurements, extending the standard notion
of value at risk and conditional value at risk from the real line to
${\mathbb{R}}^{d}$. Our new concepts are built upon the recent definition of
Monge-Kantorovich quantiles based on the theory of optimal transport, and they
provide a natural way to characterize multivariate tail probabilities and
central areas of point clouds. They preserve the univariate interpretation of
a typical observation that lies beyond or ahead a quantile, but in a
meaningful multivariate way. We show that they characterize random vectors and
their convergence in distribution, which underlines their importance. Our new
concepts are illustrated on both simulated and real datasets.

The authors gratefully acknowledge financial support from the Agence Nationale
de la Recherche (MaSDOL grant ANR-19-CE23-0017).

_Keywords:_ Monge-Kantorovich quantiles, center-outward quantiles, tails of a
multivariate distribution, conditional value at risk, expected shortfall.

_AMS classifications:_ 62H05, 62P99, 60F05.

##  1\. Introduction

###  1.1. Superquantile, expected shortfall

Modeling the dependency between the components of a random vector is at the
core of multivariate statistics. To that end, one way to proceed is to
characterize the multivariate probability tails. For distributions supported
on the real line, this is often tackled with the use of superquantiles or
expected shortfalls, that complement the information given by the quantiles.
Let $X$ be an integrable absolutely continuous random variable with cumulative
distribution function $F$. For all $\alpha\in]0,1[$, the quantile $Q(\alpha)$
of level $\alpha$ is given by

$Q(\alpha)=\inf\\{x:F(x)\geq\alpha\\},$

whereas the superquantile $S(\alpha)$ and expected shortfall $E(\alpha)$ are
defined by

(1.1) $S(\alpha)=\mathbb{E}[X\big{|}X\geq
Q(\alpha)]=\frac{\mathbb{E}[X\mathds{1}_{X\geq Q(\alpha)}]}{\mathbb{P}(X\geq
Q(\alpha))}=\frac{1}{1-\alpha}\mathbb{E}[X\mathds{1}_{X\geq Q(\alpha)}],$

and

(1.2) $E(\alpha)=\mathbb{E}[X\big{|}X\leq
Q(\alpha)]=\frac{\mathbb{E}[X\mathds{1}_{X\leq Q(\alpha)}]}{\mathbb{P}(X\leq
Q(\alpha))}=\frac{1}{\alpha}\mathbb{E}[X\mathds{1}_{X\leq Q(\alpha)}].$

As illustrated in Figure 1, $S(\alpha)$ focuses on the upper-tail while
$E(\alpha)$ targets  the lower-tail. We emphasize that using the terms of
superquantile and expected shortfall is a subjective consideration taken from
[1, 52]. Most of the time, one does not consider the upper and the lower tails
together, so that a single name is required, up to considering the
distribution of $-X$. In this vein, depending on the application, the expected
shortfall may refer to the same as the Conditional-Value-at-Risk, Conditional-
Tail-Expectation, see $e.g.$ [2], or even the superquantile, that aims to be a
neutral alternative name in statistics [52].

Figure 1.  Illustration of the notions of superquantile $S(\alpha)$ and
expected shortfall $E(\alpha)$ for an univariate Gaussian distribution.

The main contribution of the present paper is to extend (1.1) and (1.2)
towards a notion of multivariate superquantile and expected shortfall. As part
of the difficulty, both the mathematical meanings of “ahead”, “beyond” and
“typical” do not adapt canonically in ${\mathbb{R}}^{d}$. We argue that
sufficient notions are provided by the Monge-Kantorovich (MK) quantiles, ranks
and signs, introduced in [13, 37]. In particular, the traditional left-to-
right ordering is replaced in our approach by a center-outward one that is
more intuitive for a point cloud [13]. Hence, the two subsets of observations
that we are interested in are located at the outward or near the mean value,
which requires to adapt the concepts of (1.1) and (1.2) in ${\mathbb{R}}^{d}$.
It is well-known from a simple change of variables in ${\mathbb{R}}$ that $S$
and $E$ average observations beyond and ahead the quantile of level $\alpha$,
in the sense that

(1.3) $S(\alpha)=\frac{1}{1-\alpha}\int_{\alpha}^{1}Q(t)dt\hskip
28.45274pt\text{and}\hskip
28.45274ptE(\alpha)=\frac{1}{\alpha}\int_{0}^{\alpha}Q(t)dt.$

In Section 2, our definitions generalize the formulation (1.3). If $Q$ stands
for the multivariate MK quantile function instead of the classical univariate
one, center-outward superquantile and expected shortfall functions are
defined, for any $u$ in  the unit ball $\mathbb{B}(0,1)\backslash\\{0\\}$, by

$S(u)=\frac{1}{1-\|u\|}\int_{\|u\|}^{1}Q\Big{(}t\frac{u}{\|u\|}\Big{)}dt\hskip
28.45274pt\text{and}\hskip
28.45274ptE(u)=\frac{1}{\|u\|}\int_{0}^{\|u\|}Q\Big{(}t\frac{u}{\|u\|}\Big{)}dt.$

MK quantiles have already led to many applications, among which lie
statistical tests [31, 39, 46, 56, 57], regression [12, 23], risk measurement
[5], or Lorenz maps [27, 38]. We also refer to the recent review [36] on the
concept of MK quantiles. Importantly, the univariate quantile and related
functions are deeply rooted in risk analysis. On the one hand, risk measures
which are both coherent and regular can be characterized by integrated
quantile functions [35]. On the other hand, given a level $\alpha$,
fundamental risk measures are given by $Q(\alpha)$ and $S(\alpha)$, called
Value-at-Risk (VaR) and Conditional-Value-at-Risk (CVaR), respectively. As a
matter of fact, a natural main contribution of the present paper is to provide
meaningful multivariate extensions of VaR and CVaR.

###  1.2. Background on multivariate risk measurement

MK quantiles and the related concepts have already been applied to
multivariate risk measurement in [5], and, in some sense, in the previous
works [26, 29]. The theory in [26] states ideal theoretical properties for
coherent regular risk measures, while the maximal correlation risk measure of
[5] furnishes a real-valued risk measure with these properties. This
constitutes, to the best of our knowledge, the short literature on risk
measurement based on the MK quantile function. In this work, we argue that an
adequate procedure of multivariate risk measurement shall account for all the
information on the tails, both in terms of direction and spreadness. To answer
this issue, vector-valued risk measures are natural candidates.  There also
exist several extensions of VaR or CVaR to the multivariate setting, including
[3, 11, 14, 15, 32, 40, 49, 58, 59], but none of them is based on the theory
of optimal transportation and its associated potential benefits. In
particular, our concepts do not require any assumption on the tail behavior of
the data, nor any statistical model, because MK quantiles adapt naturally to
the shape of a point cloud.  On the real line, the VaR and the CVaR have a
clear interpretation: for a level $\alpha\in[0,1]$, the VaR is the worst
observation encountered with probability $1-\alpha$ whereas CVaR is the
average value beyond this worst observation. Such a meaningful definition is
surely part of the reason for their wide use in practice. Under the name of
Conditional-Tail-Expectation, the idea proposed in [15, 24] preserves this
interpretation, but relies on level sets defined from the theory of copulas.
Specifically, the obtained quantile levels do not adapt automatically to the
shape of the data. Still, this notion averages over a certain quantile level,
and it returns a tail observation of the same dimension as the data. Our work
is inspired by this approach, as we aim to give the same information about
multivariate tails, but we use the MK quantile function, which yields, to our
opinion, concepts with even better interpretability.

###  1.3. Main contributions

Our main contributions are the definitions of a center-outward superquantile
function and related risk measures, both real-valued and vector-valued. Doing
so, we provide an extension to the multivariate case of the fundamental Value-
at-Risk and Conditional-Value-at-Risk. Furthermore, we provide a center-
outward expected shortfall function, that describes the central areas of a
given point cloud. Our center-outward expected shortfalls and superquantiles
are uniquely defined and characterize convergence in distribution, and they
are closely related to the potential whose gradient gives the MK quantile
function.

###  1.4. Outline of the paper

Section 2 details  definitions and properties of our new concepts of center-
outward superquantiles and expected shortfalls. It includes our main results
and the crucial relation between these functions and Kantorovich potentials.
An alternative class of reference measures, which differ from the standard
spherical uniform distribution on the unit ball, is provided in Section 3. The
multivariate definitions of VaR and CVaR are given in Section 4. Finally, we
present in Section 5 a regularized version of our superquantile and expected
shortfall functions, using entropically regularized optimal transport that has
fundamental computional benefits to estimate the center-outward quantile
function. Numerical experiments are also provided to shed some light on the
benefits  of our new concepts of MK superquantile, expected shortfall and
multivariate VaR and CVaR for multivariate data analysis. A conclusion and
some perspectives are given in Section 6.

##  2\. Center-outward superquantiles and expected shortfalls

###  2.1. Main definitions

On the real line, the notion of superquantile and expected shortfall relies
heavily on the one of quantile. It is then natural to make use of the Monge-
Kantorovich (MK) quantile function and its appealing properties in order to
define associated superquantile and expected shortfall functions. We shall
denote by $\mathcal{P}({\mathbb{R}}^{d})$ the set of integrable probability
measures over ${\mathbb{R}}^{d}$. First of all, recall that a probability
measure $\nu\in\mathcal{P}({\mathbb{R}}^{d})$ is the push-forward of
$\mu\in\mathcal{P}({\mathbb{R}}^{d})$ by
$T:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}^{d}$ if $T(U)$ has distribution
$\nu$ as soon as $U$ is distributed according to $\mu$. This is denoted by
$T_{\\#}\mu=\nu$. The following definition has been introduced in [37].

######  Definition 2.1.

The MK quantile function of a multivariate distribution $\nu$, with respect to
a reference distribution $\mu$, is a push-forward map $Q_{\\#}\mu=\nu$ such
that there exists a convex potential
$\psi:{\mathbb{R}}^{d}\rightarrow{\mathbb{R}}$ satisfying $\nabla\psi=Q$
$\mu$-almost everywhere.

It follows from McCann’s theorem, [44] that, as soon as $\mu$ is absolutely
continuous, such a map $Q$ exists and is unique. Moreover, if $\mu$ and $\nu$
have finite moments of order two, by the result known in the literature as
Brenier’s theorem [10, 16], $Q$ is characterized as the solution of the
following Monge problem of optimal transport,

(2.1) $Q=\mathop{\rm
arg\;min}\limits_{T:T_{\\#}\mu=\nu}\int_{\mathcal{X}}\|u-T(u)\|^{2}d\mu(u).$

Intuitively, the reference distribution $\mu$ must be chosen so that a
relevant notion of quantiles can be derived from it, whereas being the
gradient of a convex function $\psi$ is a generalization of monotonicity. For
instance, one can choose the spherical uniform distribution, denoted by
$\mu=U_{d}$. It is given by the product $R\Phi$ between two independent random
variables $R$ and $\Phi$, being drawn respectively from a uniform distribution
on $[0,1]$ and on the unit sphere. Samples from $U_{d}$ are distributed from
the origin to the outward within the unit ball, so that the balls of radius
$\alpha\in[0,1]$ have probability $\alpha$ while being nested, as $\alpha$
grows. With this in mind, the hyperspheres of radius $\alpha$ are relevant
quantile contours with respect to $\mu$. Being the gradient of a convex
function, $Q$ adequately transports this center-outward ordering towards the
distribution $\nu$. Thus, when $\mu=U_{d}$, we shall refer to $Q$ as the
center-outward quantile function of $\nu$. This property is illustrated in
Figure 2, where radius, in red, and circles, in blue, are transported from the
unit ball to a banana-shaped distribution thanks to the mapping $Q$ obtained
with the computational approach described in Section 5.

Figure 2.  (Left) center-outward quantiles of the spherical uniform
distribution $\mu=U_{d}$, and (right) center-outward quantiles of a discrete
distribution $\nu$ obtained by $Q_{\\#}\mu=\nu$.

This relevant ordering clearly catches the geometry of the support of the
target distribution $\nu$, and it comes with quantile  regions indexed by a
probability level $\alpha\in[0,1]$, by use of the change of variables formula
for push-forward maps. More details are given in [13, 37]. Hereafter, we use
the spherical uniform as the reference distribution, but other distributions
could be chosen, depending on the applications, [18, 27, 31]. Note that if
$\nu$ is an empirical measure based on random observations, it has a finite
second order moment, and the estimation of the quantile map amounts to solve
the OT problem (2.1). Following [37], we assume, without loss of generality,
that  $\psi$ in Definition 2.1 satisfies $\psi(0)=0$ and, for
$u\in{\mathbb{R}}^{d}$ such that $\|u\|=1$,

(2.2) $\psi(u)=\liminf\limits_{\begin{subarray}{c}v\rightarrow u\\\
\|v\|<1\end{subarray}}\psi(v).$

Moreover, we impose, for all $u\in{\mathbb{R}}^{d}$ such that $\|u\|>1$,

(2.3) $\psi(u)=+\infty.$

This being said, the convex potential $\psi$ is uniquely defined over its
domain, which is the closed unit ball
$\\{u|\psi(u)<+\infty\\}=\overline{\mathbb{B}}(0,1)$,  where it is continuous,
[51][Theorem 10.1]. However, the gradient $\nabla\psi$ is only defined almost
everywhere. At every $u$ where $\psi$ is not differentiable, one can still
define the subdifferential

$\partial\psi(u)=\\{z\in{\mathbb{R}}^{d}:\forall
x\in{\mathbb{R}}^{d},\psi(x)-\psi(u)\geq\langle z,x-u\rangle\\}.$

Following the suitable suggestion of an anonymous referee, for all
$u\in\mathbb{B}(0,1)$, we can define $Q(u)$ as the average of
$\partial\psi(u)$ so that $Q$ is defined everywhere. In fact, from [28], as
soon as $\nu$ is a continuous probability measure with non vanishing density
on ${\mathbb{R}}^{d}$, $\psi$ can be shown to be differentiable everywhere on
$\mathbb{B}(0,1)\backslash\\{0\\}$. The same result is showed under milder
assumptions in [20], that is presented hereafter.

######  Assumption 2.1.

Let $\nu$ be an absolutely continuous measure with probability density
function $p$ defined on its support $\mathcal{X}$. For every $R>0$, there
exist  two constants $0<\lambda_{R}<\Lambda_{R}$ such that, for all
$x\in\mathcal{X}\cap\mathbb{B}(0,R)$,

$\lambda_{R}\leq p(x)\leq\Lambda_{R}.$

######  Assumption 2.2.

The support $\mathcal{X}\subset{\mathbb{R}}^{d}$ of $\nu$ is convex.

Under these assumptions, the next theorem is  given in [20].

######  Theorem 2.1 (Regularity of the center-outward quantile function,
[20]).

Under Assumptions 2.1 and 2.2, there exists a compact convex set $K$ with
Lebesgue measure $0$ such that the center-outward quantile function $Q$ is a
homeomorphism from $\mathbb{B}(0,1)\backslash\\{0\\}$ to
$\mathcal{X}\backslash K$, with inverse $Q^{-1}$ the center-outward
distribution function. Moreover, $Q^{-1}=\nabla\psi^{*}$ where $\psi^{*}$ is
the Fenchel-Legendre transform of $\psi$ such that $Q=\nabla\psi$,

$\psi^{*}(x)=\operatornamewithlimits{sup}\limits_{u\in\mathbb{B}(0,1)}\\{\langle
x,u\rangle-\psi(u)\\}.$

Hereafter, this allows us to properly define quantile contours, ranks and
signs. For the sake of simplicity, we define
$\mathcal{P}_{*}({\mathbb{R}}^{d})$ as the set of integrable probability
measures for which $Q$ is a homeomorphism, see Theorem 2.1. This includes any
$\nu$ satisfying assumptions 2.1 and 2.2, that is also integrable, to ensure
finiteness of the center-outward superquantiles. We emphasize that, for any
$\nu\in\mathcal{P}_{*}({\mathbb{R}}^{d})$, $Q$ is defined everywhere on
$\mathbb{B}(0,1)\backslash\\{0\\}$, as opposed to almost everywhere on
$\mathbb{B}(0,1)$. The next definitions, taken from [13, 37], gather the main
concepts that we use in  all the sequel.

######  Definition 2.2 (Quantile contours, ranks and signs).

Let  $\nu\in\mathcal{P}_{*}({\mathbb{R}}^{d})$ with center-outward quantile
function $Q$ and supported on $\mathcal{X}\subset{\mathbb{R}}^{d}$. Then, for
the distribution $\nu$,
(i) the quantile region $\mathds{C}_{\alpha}$ of order $\alpha\in[0,1]$ is the
image by $Q$ of the ball $\mathbb{B}(0,\alpha)$.
(ii) the quantile contour $\mathcal{C}_{\alpha}$ of order $\alpha\in[0,1]$ is
the boundary of $\mathds{C}_{\alpha}$.
(iii) the rank function $\mathcal{R}_{\nu}:\mathcal{X}\rightarrow[0,1]$ is
defined by $\mathcal{R}_{\nu}(x)=\|Q^{-1}(x)\|$.
(iv) the sign function
$\mathcal{D}_{\nu}:\mathcal{X}\rightarrow\mathbb{B}(0,1)$ is defined by
$\mathcal{D}_{\nu}(x)=Q^{-1}(x)/\|Q^{-1}(x)\|$.

A few remarks follow from Definition 2.2. The $\nu$-probability of
$\mathds{C}_{\alpha}$ is $\alpha$, by the change of variables formula for
push-forward maps, which is a first requirement for quantile regions. In
addition, one may note that the rank and sign functions require the
invertibility of $Q$. From [37][Section 2], the continuity and invertibility
of $Q$ outside the origin ensures the crucial fact that the quantile contours
are closed and nested. It is the notion of center-outward ranks that allows to
order points in $\mathcal{X}$ relatively to $\nu$, consistently with Tukey’s
halfspace depth, as highlighted in [13]. It induces the following weak order.

######  Definition 2.3.

For  $\nu\in\mathcal{P}_{*}({\mathbb{R}}^{d})$ and $x,y\in\mathcal{X}$, we
denote $x\geq_{\mathcal{R}}y$ and say that $y$ is deeper than $x$ if

$\|Q^{-1}(x)\|\geq\|Q^{-1}(y)\|.$

The deeper a point is in $\mathcal{X}$, the less extremal it is with respect
to $\nu$. For a fixed $u\in\mathbb{B}(0,1)\backslash\\{0\\}$, we can then
write $x\geq_{\mathcal{R}}Q(u)$ which enables to consider the observations
“beyond” $Q(u)$ in some sense. In a context where the focus is on the  central
observations, $u\mapsto\mathbb{E}[X|X\leq_{\mathcal{R}}Q(u)]$ has been
introduced in [38] as a center-outward Lorenz function for $X$, see Section
6.1.  The reverse conditional expectation
$\mathbb{E}[X|X\geq_{\mathcal{R}}Q(u)]$ might be thought of as a natural
candidate for a superquantile concept. Nevertheless, it cannot be understood
as a “typical” observation within outward areas. Indeed, consider the
following example.

######  Example 2.1.

Suppose that $\nu=U_{d}$, so that $Q$ is the identity. For every
$u\in\mathbb{B}(0,1)\backslash\\{0\\}$,
$\mathbb{E}\left[X|X\geq_{\mathcal{R}}Q(u)\right]=\mathbb{E}\left[X\big{|}\|X\|\geq\|u\|\right]$
is the expectation of a symmetric distribution over an annulus centered at the
origin. Therefore, one has that $\mathbb{E}[X|X\geq_{\mathcal{R}}Q(u)]=0$ and
it cannot be thought of as a typical extreme observation.

This is caused by a lack of information in
$\mathbb{E}[X|X\geq_{\mathcal{R}}Q(u)]$. In fact, the rank function neglects
the directional information of $X$, which lies in the sign function. In order
to overcome this issue, we have to introduce a few notation. For any
$u\in\mathbb{B}(0,1)\backslash\\{0\\}$, let

(2.4) $L_{u}=\Big{\\{}t\frac{u}{\|u\|}:t\in[0,1]\Big{\\}}$

be a parametrization of the radius of $\mathbb{B}(0,1)\backslash\\{0\\}$ whose
direction is $u/\|u\|$. The sign curve $C_{u}$ associated to $u$ is the image
of $L_{u}$ by the center-outward quantile function, namely

$C_{u}=Q(L_{u}).$

Several sign curves are represented in red at the right-hand side of Figure 2.
Averaging observations less deep that $Q(u)$ along the sign curve $C_{u}$
induces a “typical” value “beyond” $Q(u)$ in a meaningful multivariate way.

######  Definition 2.4.

Let $\nu\in\mathcal{P}({\mathbb{R}}^{d})$ with center-outward quantile
function $Q$. The center-outward superquantile function of $\nu$ is the
function $S$ defined, for any $u\in\mathbb{B}(0,1)\backslash\\{0\\}$, by

$S(u)=\frac{1}{1-\|u\|}\int_{\|u\|}^{1}Q\Big{(}t\frac{u}{\|u\|}\Big{)}dt,$

where the above integral is to be understood component-wise.

######  Remark 2.1 (Consistency with the univariate case).

On the real line, there is only one sign curve $C_{1}=\\{Q(t);t\in[0,1]\\}$.
Thus, definition (1.3) can be seen as averaging observations less deep than
$Q(\alpha)$, w.r.t. $\nu$, along the sign curve $C_{1}$. Nonetheless, note
that center-outward quantiles slightly differ from classical quantiles in
dimension $d=1$, because $U([-1,1])\neq U([0,1])$, even if they carry the same
information, [37][Appendix B].

By the same token, we can define the center-outward expected shortfall
function.

######  Definition 2.5.

Let $\nu\in\mathcal{P}({\mathbb{R}}^{d})$ with center-outward quantile
function $Q$. The center-outward expected shortfall function of $\nu$ is the
function $E$ defined, for any $u\in\mathbb{B}(0,1)\backslash\\{0\\}$, by

$E(u)=\frac{1}{\|u\|}\int_{0}^{\|u\|}Q\Big{(}t\frac{u}{\|u\|}\Big{)}dt,$

where the above integral is to be understood component-wise.

######  Remark 2.2.

For $\nu\in\mathcal{P}_{*}({\mathbb{R}}^{d})$, $Q$ is neither defined at the
origin nor at the boundary of the unit ball, unless the support of $\nu$ is
compact. Hence the integrals in $S(u)$ and $E(u)$ are improper and they shall
be understood respectively as

$\lim\limits_{r\rightarrow
1^{-}}\int_{\|u\|}^{r}Q\Big{(}t\frac{u}{\|u\|}\Big{)}dt\hskip
28.45274pt\mbox{and}\hskip 28.45274pt\lim\limits_{r\rightarrow
0^{+}}\int_{r}^{\|u\|}Q\Big{(}t\frac{u}{\|u\|}\Big{)}dt.$

One may note that they are convergent as soon as

(2.5) $\int_{0}^{1}\left\|Q(t\frac{u}{\|u\|})\right\|dt<\infty.$

By the change of variables formula for push-forwards maps and by definition of
$U_{d}$,
$\mathbb{E}_{\nu}[\|X\|]=\mathbb{E}_{U_{d}}[\|Q(U)\|]=\mathbb{E}_{(R,\Phi)}[\|Q(R\Phi)\|].$
In other words, denoting by ${\mathbb{P}}_{\mathbb{S}}$ the uniform
probability measure on the unit sphere
$\mathbb{S}^{d-1}=\\{\varphi\in{\mathbb{R}}^{d}:\|\varphi\|=1\\}$,

$\mathbb{E}_{\nu}[\|X\|]=\int_{\mathbb{S}^{d-1}}\int_{0}^{1}\|Q(t\varphi)\|dt\,d{\mathbb{P}}_{\mathbb{S}}(\varphi).$

Thus, one can see by contradiction that, as soon as
$\mathbb{E}_{\nu}[\|X\|]<\infty$, (2.5) holds for almost all
$\varphi\in\mathbb{S}^{d-1}$, thus $S$ and $E$ must be finite almost
everywhere. But, under continuity of $Q$ together with
$\mathbb{E}_{\nu}[\|X\|]<\infty$, it is not hard to see that $S$ and $E$ are
continuous, thus they are in fact finite everywhere on
$\mathbb{B}(0,1)\backslash\\{0\\}$.

The following naturally extends Definition 2.2.

######  Definition 2.6 (Superquantile and expected shortfall regions and
contours).

Let $\nu\in\mathcal{P}({\mathbb{R}}^{d})$ with center-outward superquantile
function $S$ and expected shortfall $E$. Then,
(i) the superquantile (resp. expected shortfall) region
$\mathds{C}^{s}_{\alpha}$ (resp. $\mathds{C}^{e}_{\alpha}$) of order
$\alpha\in[0,1]$ is the image by $S$ (resp. $E$) of the ball
$\mathbb{B}(0,\alpha)$.
(ii) the superquantile (resp. expected shortfall) contour
$\mathcal{C}^{s}_{\alpha}$ (resp. $\mathcal{C}^{e}_{\alpha}$) of order
$\alpha\in[0,1]$ is the boundary of $\mathds{C}^{s}_{\alpha}$ (resp.
$\mathds{C}^{e}_{\alpha}$).
(iii) averaged sign curves by $E$ or $S$ are respectively defined by
$E(L_{u})$ and $S(L_{u})$ for any $u\in\mathbb{B}(0,1)\backslash\\{0\\}$.

These concepts describe point clouds through periphery or central areas, and
are illustrated in the numerical experiments carried out in Section 5.

###  2.2. Invariance properties

The two following lemmas are immediate consequences of [31, Lemmas A.7,A.8].

######  Lemma 2.1.

Assume that $X\in{\mathbb{R}}^{d}$ is a random vector. Suppose that $a>0$,
$b\in{\mathbb{R}}^{d}$ and $Y=aX+b$. Denote by $S_{X},S_{Y}$ and $E_{X},E_{Y}$
their center-outward superquantile and expected shortfall functions. Then, for
$u\in\mathbb{B}(0,1)\backslash\\{0\\}$,

$S_{Y}(u)=aS_{X}(u)+b\quad\text{and}\quad E_{Y}(u)=aE_{X}(u)+b.$

###### Proof.

We only detail $S_{Y}$, as one can deduce $E_{Y}$ identically. From [31][Lemma
A.7],

$\displaystyle S_{Y}(u)$
$\displaystyle=\frac{1}{1-\|u\|}\int_{\|u\|}^{1}Q_{Y}(t\frac{u}{\|u\|})dt,$
$\displaystyle=\frac{1}{1-\|u\|}\int_{\|u\|}^{1}\Big{(}aQ_{X}(t\frac{u}{\|u\|})+b\Big{)}dt=aS_{X}(u)+b.\qed$

######  Lemma 2.2.

Assume that $X\in{\mathbb{R}}^{d}$ is a random vector. Let $S_{X},S_{Y}$ and
$E_{X},E_{Y}$ be the center-outward superquantile and expected shortfall
functions associated respectively with $X$ and $Y=AX$, for $A$ an orthonormal
matrix. Then, for $u\in\mathbb{B}(0,1)\backslash\\{0\\}$,

$\displaystyle S_{Y}(u)=AS_{X}(A^{T}u)\quad\text{and}\quad
E_{Y}(u)=AE_{X}(A^{T}u).$

###### Proof.

Again, we only detail $S_{Y}$, as the proof for $E_{Y}$ is identical.
Combining [31][Lemma A.8] with the fact that $\|A^{T}u\|=\|u\|$ for an
orthogonal matrix,

$\displaystyle S_{Y}(u)$
$\displaystyle=\frac{1}{1-\|u\|}\int_{\|u\|}^{1}Q_{Y}(t\frac{u}{\|u\|})dt,$
$\displaystyle=\frac{1}{1-\|u\|}\int_{\|u\|}^{1}AQ_{X}(tA^{T}\frac{u}{\|u\|})dt=AS_{X}(A^{T}u).\qed$

###  2.3. Main results

Quoting Rockafellar & Royset in [52],

> “the superquantile function […] is as fundamental to a random variable as
> the distribution and quantile functions”.

This assertion is partially motivated by the fact that the distribution,
quantile and superquantile functions are uniquely determined one to another,
and by the fact that pointwise convergence of these functions metrizes
convergence in distribution. Such properties hold for our integrated concepts
and are stated hereafter.  First of all, we shall make repeated use of the
fact that the center-outward superquantile and expected shortfall functions
are two sides of the same coin, that is, for
$u\in\mathbb{B}(0,1)\backslash\\{0\\}$,

(2.6) $\int_{0}^{1}Q\Big{(}t\frac{u}{\|u\|}\Big{)}dt=\|u\|E(u)+(1-\|u\|)S(u).$

This is a generalization of the immediate property that, for the univariate
setting given in the introduction with (1.1) and (1.2), we have for all
$\alpha\in]0,1[$,

$\mathbb{E}[X]=\alpha E(\alpha)+(1-\alpha)S(\alpha).$

Our first main result is as follows.

######  Theorem 2.2.

Let $\nu_{1},\nu_{2}\in\mathcal{P}({\mathbb{R}}^{d})$ with respective center-
outward quantile, superquantile and expected shortfall functions denoted by
$Q_{1},S_{1},E_{1}$ and $Q_{2},S_{2},E_{2}$. Then, the following are
equivalent.

  1. (i)

$\nu_{1}=\nu_{2}$

  2. (ii)

$Q_{1}=Q_{2}$ $U_{d}$-a.e.

  3. (iii)

$S_{1}=S_{2}$ $U_{d}$-a.e.

  4. (iv)

$E_{1}=E_{2}$ $U_{d}$-a.e.

###### Proof.

First of all, it is already known that $(i)\Leftrightarrow(ii)$. Indeed, with
the choice of $U_{d}$ as the reference distribution, McCann’s theorem [44]
ensures that $(i)\Rightarrow(ii)$. Obviously, $(ii)\Rightarrow(i)$, from the
very definition of push-forward measures that is
$\nu_{1}(A)=U_{d}(Q_{1}^{-1}(A))$ and $\nu_{2}(A)=U_{d}(Q_{2}^{-1}(A))$.
Hereafter, we proceed through the implication loop
$(ii)\Rightarrow(iii)\Rightarrow(iv)\Rightarrow(ii)$. Let
${\mathbb{P}}_{\mathbb{S}}$ be the uniform probability measure on
$\mathbb{S}^{d-1}=\\{\varphi\in{\mathbb{R}}^{d}:\|\varphi\|_{2}=1\\}$. Let $R$
and $\varphi$ be drawn uniformly on $[0,1]$ and $\mathbb{S}^{d-1}$,
respectively. Suppose that they are independent, so that their joint
distribution is given by
$d{\mathbb{P}}_{(R,\varphi)}(r,\varphi)=d{\mathbb{P}}_{R}(r)d{\mathbb{P}}_{\mathbb{S}}(\varphi)$.
There, by definition, if $f:(r,\varphi)\mapsto r\varphi$ on
$[0,1]\times\mathbb{S}^{d-1}$, then $U_{d}=f_{\\#}{\mathbb{P}}_{(R,\varphi)}$.
Hence, from the change of variables formula for push-forwards, for any Borel
set $A=f(B\times C)$, for $B\subset[0,1]$, $C\subset\mathbb{S}^{d-1}$,

$\displaystyle\int_{A}(S_{1}-S_{2})(u)dU_{d}(u)$
$\displaystyle=\int_{B}\int_{C}(S_{1}-S_{2})(r\varphi)d{\mathbb{P}}_{\mathbb{S}}(\varphi)dr,$
(2.7)
$\displaystyle=\int_{B}\int_{C}\Big{(}\frac{1}{1-r}\int_{r}^{1}(Q_{1}-Q_{2})(t\varphi)dt\Big{)}d{\mathbb{P}}_{\mathbb{S}}(\varphi)dr.$

Nonetheless, for any $r\in B$, by the same change of variables,

(2.8)
$\displaystyle\int_{C}\int_{r}^{1}(Q_{1}-Q_{2})(t\varphi)dtd{\mathbb{P}}_{\mathbb{S}}(\varphi)$
$\displaystyle=\int_{f(C\times[r,1])}(Q_{1}-Q_{2})(u)dU_{d}(u).$

Note that $U_{d}(f(C\times[r,1]))=(1-r){\mathbb{P}}_{\mathbb{S}}(C)$, which is
positive since $r<1$. Obviously, (2.8) vanishes as soon as $Q_{1}=Q_{2}$
$U_{d}$-$a.e.$ It implies that (2.3) also vanishes, which justifies that
$(ii)\Rightarrow(iii)$. Furthermore, we  claim that $(iii)\Rightarrow(iv)$.
Indeed, we have

$\int_{0}^{1}Q_{1}(t\frac{u}{\|u\|})dt=\lim\limits_{r\rightarrow
0^{+}}S_{1}(r\frac{u}{\|u\|})\hskip 18.49411pt\text{and}\hskip
18.49411pt\int_{0}^{1}Q_{2}(t\frac{u}{\|u\|})dt=\lim\limits_{r\rightarrow
0^{+}}S_{2}(r\frac{u}{\|u\|}).$

Consequently, if we assume that $S_{1}=S_{2}$ $U_{d}$-$a.e.$, we obtain that

(2.9) $\int_{0}^{1}(Q_{1}-Q_{2})(t\frac{u}{\|u\|})dt=0.$

Using (2.6), the desired result follows, $(iii)\Rightarrow(iv)$. Finally,
assume that $E_{1}=E_{2}$ $U_{d}$-a.e. Consequently, for all $r\in]0,1[$ and
for all $\varphi\in\mathbb{S}^{d-1}$,

$\int_{0}^{r}Q_{1}(t\varphi)dt=\int_{0}^{r}Q_{2}(t\varphi)dt.$

Using that $\int_{a}^{b}=\int_{0}^{b}-\int_{0}^{a}$, for any $0\leq a\leq
b\leq 1$,

(2.10) $\int_{a}^{b}Q_{1}(t\varphi)dt=\int_{a}^{b}Q_{2}(t\varphi)dt.$

For any measurable $B\subset\mathbb{S}^{d-1}$, by integrating (2.10) w.r.t.
${\mathbb{P}}_{\mathbb{S}}$,

$\int_{B}\int_{a}^{b}Q_{1}(t\varphi)d{\mathbb{P}}_{R}(t)d{\mathbb{P}}_{\mathbb{S}}(\varphi)=\int_{B}\int_{a}^{b}Q_{2}(t\varphi)d{\mathbb{P}}_{R}(t)d{\mathbb{P}}_{\mathbb{S}}(\varphi).$

By use of $U_{d}=f_{\\#}{\mathbb{P}}_{(R,\varphi)}$, a change of variables
above yields, for any $A\subset\mathbb{B}(0,1)\backslash\\{0\\}$,

$\int_{A}Q_{1}(u)dU_{d}(u)=\int_{A}Q_{2}(u)dU_{d}(u),$

and the result follows.

∎

Interestingly enough, our proposed integrated quantile functions are both
simply related to the Kantorovich potential. Somehow, this development
generalizes the work of [53] where the distribution function is related to the
univariate superquantile by the way of the surexpectation function, which is
nothing more than a particular primitive of the distribution function.

######  Proposition 2.1.

The center-outward expected shortfall function  of
$\nu\in\mathcal{P}({\mathbb{R}}^{d})$ satisfies, for any
$u\in\mathbb{B}(0,1)\backslash\\{0\\}$,

(2.11) $\langle E(u),u\rangle=\psi(u).$

Moreover, the center-outward superquantile function  of
$\nu\in\mathcal{P}_{*}({\mathbb{R}}^{d})$ verifies, for
$u\in\mathbb{B}(0,1)\backslash\\{0\\}$,

(2.12) $\langle
S(u),u\rangle=\frac{\|u\|}{1-\|u\|}\Big{(}\psi(\frac{u}{\|u\|})-\psi(u)\Big{)}.$

###### Proof.

Fix $u\in\mathbb{B}(0,1)\backslash\\{0\\}$ and let $f(t)=\psi(tu)$. Then $f$
is a finite and convex function on $]-a,a[$ for some $a>1$, so that one can
apply [51][Corollary 24.2.1]. Combined with $a.e.$ differentiability,
[51][Theorem 25.3], this yields

(2.13) $\psi(u)=\int_{0}^{1}\langle\nabla\psi(tu),u\rangle dt,$

which can be rewritten as

$\psi(u)=\langle\int_{0}^{1}Q(tu)dt,u\rangle=\langle E(u),u\rangle,$

from a simple change of variables, leading to  our first point (2.11).
Moreover, denote for  any $u\in\mathbb{B}(0,1)\backslash\\{0\\}$,

$f(t)=\psi(t\frac{u}{\|u\|}).$

This function is finite on $[0,1]$ but not on a larger interval anymore. But,
because $\nu\in\mathcal{P}_{*}({\mathbb{R}}^{d})$, $f$ is differentiable
everywhere on $]0,1[$, and we have by the chain rule formula,

$f^{\prime}(t)=\langle Q(t\frac{u}{\|u\|}),\frac{u}{\|u\|}\rangle.$

Once $u$ is fixed, $f$ is in fact differentiable at the boundary points $0$
and $1$ by using only the significant one-sided derivatives, that is

$f^{\prime}(0)=\lim\limits_{t\rightarrow 0^{+}}\langle
Q(t\frac{u}{\|u\|}),\frac{u}{\|u\|}\rangle\hskip 28.45274pt\mbox{and}\hskip
28.45274ptf^{\prime}(1)=\lim\limits_{t\rightarrow 1^{-}}\langle
Q(t\frac{u}{\|u\|}),\frac{u}{\|u\|}\rangle.$

In addition, $f^{\prime}$ is integrable on $[0,1]$ since (2.5) holds for $\nu$
integrable. Thus, it follows from [54][Theorem 8.21] that

(2.14) $\psi(\frac{u}{\|u\|})=\int_{0}^{1}\langle
Q(t\frac{u}{\|u\|}),\frac{u}{\|u\|}\rangle
dt=\langle\int_{0}^{1}Q(t\frac{u}{\|u\|})dt,\frac{u}{\|u\|}\rangle.$

Consequently,  we deduce from (2.6) together with (2.11) and (2.14), that

(2.15) $\psi(\frac{u}{\|u\|})=\psi(u)+(1-\|u\|)\langle
S(u),\frac{u}{\|u\|}\rangle,$

which implies (2.12), completing the proof of Proposition 2.1.

∎

Thanks to this relation between the potential $\psi$ whose gradient gives $Q$
and the center-outward expected shortfall function, we are now able to
characterize the convergence in distribution for a sequence of random vectors
through superquantiles and expected shortfalls. For that purpose, we rely on
existing results on the relation between convergence in distribution and
center-outward quantile functions.

######  Lemma 2.3.

Let $(X,X_{n})$ be a sequence of random vectors with distributions $\nu$ and
$\nu_{n}$ respectively,  in $\mathcal{P}_{*}({\mathbb{R}}^{d})$. Then,

(2.16) $\displaystyle\hskip
28.45274ptX_{n}\overset{\mathcal{L}}{\rightarrow}X$
$\displaystyle\Leftrightarrow$ $\displaystyle{\color[rgb]{0,0,0}\forall
u\in\mathbb{B}(0,1),\,\lim_{n\rightarrow\infty}\psi_{n}(u)=\psi(u),}$ (2.17)
$\displaystyle\Leftrightarrow$
$\displaystyle\lim_{n\rightarrow\infty}\operatornamewithlimits{sup}\limits_{u\in
K}|\psi_{n}(u)-\psi(u)|=0,$ (2.18) $\displaystyle\Leftrightarrow$
$\displaystyle\lim_{n\rightarrow\infty}\operatornamewithlimits{sup}\limits_{u\in
K}\|Q_{n}(u)-Q(u)\|=0,$

for every compact $K\subset\mathbb{B}(0,1)\backslash\\{0\\}$.  In fact,
uniform convergence of $\psi_{n}$ towards $\psi$ even holds on every compact
$K\subset\mathbb{B}(0,1)$.

###### Proof.

On the one hand, assume that $(X_{n})$ converges in distribution to $X$. Then,
the  right-hand side of (2.18) is the main result of [31][Theorem 4.1]  when
$Q_{n}$ and $Q$ are homeomorphisms between convex sets, with uniform
convergence on any compact subset. But, with reference distribution $U_{d}$,
center-outward quantile maps are not, because of the discontinuity at the
origin, see [37][Remark 3.4]. Nonetheless, as highlighted in the proof of
[37][Proposition 3.3], the assumption that
$X_{n}\overset{\mathcal{L}}{\rightarrow}X$ for
$\nu_{n},\nu\in\mathcal{P}_{*}({\mathbb{R}}^{d})$ is sufficient to apply
Theorem 2.8 in [22]. As a consequence, $\psi_{n}(u)$ converges towards
$\psi(u)$ for every $u\in\mathbb{B}(0,1)$, that is the right-hand side of
(2.16). Being finite and convex functions, [51][Theorem 10.8] ensures that
such pointwise convergence implies the right-hand side of (2.17). Since
$\psi_{n}$ and $\psi$ are differentiable on $\mathbb{B}(0,1)\backslash\\{0\\}$
as soon as $\nu_{n},\nu\in\mathcal{P}_{*}({\mathbb{R}}^{d})$, one can apply
[51][Theorem 25.7] on any open and convex set
$K^{\prime}\subset\mathbb{B}(0,1)\backslash\\{0\\}$. This gives us the uniform
convergence of $Q_{n}$ towards $Q$ for any compact $K$ included in an open and
convex set $K^{\prime}\subset\mathbb{B}(0,1)\backslash\\{0\\}$. To extend this
to any compact $K$, we use that, by compacity, one can extract from the open
cover $\\{\mathbb{B}(x,\delta);x\in K\\}$ a finite cover. Consequently, it
exists $N\in\mathbb{N}$ and $x_{1},\cdots,x_{N}\in K$ such that

(2.19) $\sup\limits_{u\in
K}\|Q_{n}(u)-Q(u)\|\leq\sum_{k=1}^{N}\sup\limits_{u\in\mathbb{B}(x_{k},\delta)}\|Q_{n}(u)-Q(u)\|.$

Here, the closure of each ball $\mathbb{B}(x_{k},\delta)$ is a compact set,
which, because of the choice of $\delta$, is a subset of some open and convex
set $K^{\prime}\subset\mathbb{B}(0,1)\backslash\\{0\\}$. But we already know
that $Q_{n}$ uniformly converges towards $Q$ on such a set, thus the right-
hand side of (2.19) vanishes when $n\rightarrow 0$, which yields the right-
hand side of (2.18). It only remains to prove that
$X_{n}\overset{\mathcal{L}}{\rightarrow}X$ as a direct consequence. This last
claim relies on the Portmanteau theorem which says that $(X_{n})$ converges in
distribution to $X$ iff for any bounded and continuous function $f$,

(2.20)
$\lim\limits_{n\rightarrow+\infty}\mathbb{E}[f(X_{n})]=\mathbb{E}[f(X)].$

However, we clearly have for any bounded and continuous function $f$,

$\displaystyle\mathbb{E}[f(X_{n})]$
$\displaystyle=\int_{{\mathbb{R}}^{d}}f(x)d\nu_{n}(x)=\int_{\mathbb{B}(0,1)}f(Q_{n}(u))dU_{d}(u).$

using the change of variables $\nu_{n}=Q_{n\\#}U_{d}$. Hence, as $f\circ
Q_{n}$ is uniformly bounded, the dominated convergence theorem leads to
(2.20). ∎

######  Theorem 2.3.

Let $(X,X_{n})$ be a sequence of random vectors with distributions $\nu$ and
$\nu_{n}$ respectively,  in $\mathcal{P}_{*}({\mathbb{R}}^{d})$. Then,

(2.21) $\displaystyle\hskip
28.45274ptX_{n}\overset{\mathcal{L}}{\rightarrow}X$
$\displaystyle\Leftrightarrow$ $\displaystyle\forall
u\in\mathbb{B}(0,1)\backslash\\{0\\},\lim\limits_{n\rightarrow+\infty}E_{n}(u)=E(u),$
(2.22) $\displaystyle\Leftrightarrow$
$\displaystyle\lim_{n\rightarrow\infty}\operatornamewithlimits{sup}\limits_{u\in
K}\|E_{n}(u)-E(u)\|=0,$

$\text{for every compact }K\subset\mathbb{B}(0,1)\backslash\\{0\\}$.

###### Proof.

On the one hand, assume that $(X_{n})$ converges in distribution to $X$. Then,
it follows from (2.18) that $(Q_{n})$ converges uniformly to $Q$ on any
compact $K\subset\mathbb{B}(0,1)\backslash\\{0\\}$.  For any
$u\in\mathbb{B}(0,1)\backslash\\{0\\}$, we have from Definition 2.5 that

$\displaystyle\left\|E_{n}(u)-E(u)\right\|$
$\displaystyle\leq\frac{1}{\|u\|}\int_{0}^{\|u\|}R_{n}(t\frac{u}{\|u\|})dt,$

where $R_{n}(v)=\left\|Q_{n}(v)-Q(v)\right\|.$ Fix a compact
$K\subset\mathbb{B}(0,1)\backslash\\{0\\}$ such that for every $u\in K$,
$C<\|u\|<D$ for some positive constants $C,D$. Then, we clearly have

$\displaystyle\sup_{u\in K}\left\|E_{n}(u)-E(u)\right\|$
$\displaystyle\leq\frac{1}{C}\int_{0}^{D}\sup_{u\in
K}R_{n}(t\frac{u}{\|u\|})dt.$

Consequently, for any $\xi\in{\mathbb{R}}$ such that $0\leq\xi<D$,

(2.23) $\displaystyle\sup_{u\in K}\left\|E_{n}(u)-E(u)\right\|$
$\displaystyle\leq\frac{1}{C}\left(\int_{0}^{\xi}\sup_{u\in
K}R_{n}(t\frac{u}{\|u\|})dt+\int_{\xi}^{D}\sup_{u\in
K}R_{n}(t\frac{u}{\|u\|})dt\right).$

On the one hand, for all $t\in[\xi,D]$ and $u\in K$, $tu/\|u\|$ lies inside a
compact $K^{\prime}\subset\mathbb{B}(0,1)\backslash\\{0\\}$, so that the
second term in (2.23) satisfies

(2.24) $\int_{\xi}^{D}\sup_{u\in
K}R_{n}(t\frac{u}{\|u\|})dt\leq(D-\xi)\sup_{v\in
K^{\prime}}R_{n}(v)\leq\sup_{v\in K^{\prime}}R_{n}(v),$

which vanishes when $n\rightarrow+\infty$. On the other hand,

(2.25) $\int_{0}^{\xi}\sup_{u\in
K}R_{n}(t\frac{u}{\|u\|})dt\leq\xi\sup_{v\in\overline{\mathbb{B}}(0,D)}R_{n}(v).$

We now claim that $\sup_{v\in\overline{\mathbb{B}}(0,D)}R_{n}(v)$ is bounded
by a constant. Because $R_{n}$ vanishes when $n\rightarrow+\infty$ uniformly
over any compact subset of $\mathbb{B}(0,1)\backslash\\{0\\}$, it is
sufficient to bound the set $R_{n}(\\{0\\})$. Recall that $Q_{n}(0)$ and
$Q(0)$ can be defined by any point in the subdifferentials $\psi_{n}(\\{0\\})$
and $\psi(\\{0\\})$, respectively. However, combining the uniform convergence
of $(\psi_{n})$ by Lemma 2.3 with [31][Lemma D.1], we find that for all
$\delta>0$, it exists $n_{0}\in\mathbb{N}$ such that for all $n\geq n_{0}$,

(2.26)
$\partial\psi_{n}(\\{0\\})\subset\bigcup_{y\in\partial\psi(\\{0\\})}\mathbb{B}(y,\delta).$

Because $\nu\in\mathcal{P}_{*}({\mathbb{R}}^{d})$, $\partial\psi(\\{0\\})$ is
bounded, see Theorem 2.1, and so is $\partial\psi_{n}(\\{0\\})$, for $n$ large
enough, from (2.26). Therefore the right-hand side of (2.25) can be bounded by
$\xi M$ for $M>0$. Then it follows from (2.23) and (2.24) that, for any
$\xi\in[0,D[$, it exists $n_{1}\in\mathbb{N}$ such that, for all $n\geq
n_{1}$,

$\sup_{u\in K}\left\|E_{n}(u)-E(u)\right\|\leq\frac{1}{C}\Big{(}\xi
M+\sup_{v\in K^{\prime}}R_{n}(v)\Big{)}.$

Thus, for any $\xi\in[0,D[$,

$\lim\limits_{n\rightarrow+\infty}\sup_{u\in
K}\left\|E_{n}(u)-E(u)\right\|\leq\frac{\xi M}{C}.$

which leads to the right-hand side of (2.22) as $\xi$ goes to zero. In
addition, the right-hand side of (2.22) immediately implies the right-hand
side of (2.21). On the other hand, assume that the right-hand side of (2.21)
holds. Then, we obtain from (2.11) that $\forall u\in\mathbb{B}(0,1)$,

$\lim\limits_{n\rightarrow+\infty}\psi_{n}(u)=\psi(u).$

Thus, the desired result follows from Lemma 2.3, which completes the proof of
Theorem 2.3.  ∎

Our last result requires some uniform integrability assumption on the sequence
$(X_{n})$.

######  Theorem 2.4.

Let $(X,X_{n})$ be a sequence of random vectors with distributions $\nu$ and
$\nu_{n}$ respectively,  in $\mathcal{P}_{*}({\mathbb{R}}^{d})$. In addition,
suppose that there exists a positive random variable $Z$ such that
$\mathbb{E}[Z\ln(Z)]<+\infty$ and for all $n\in\mathbb{N}$,

(2.27) $\|X_{n}\|\leq Z\hskip 28.45274pta.s.$

Then,

(2.28)
$X_{n}\overset{\mathcal{L}}{\rightarrow}X\Leftrightarrow\lim_{n\rightarrow\infty}S_{n}(u)=S(u),$

for almost all $u\in\mathbb{B}(0,1)$.

######  Remark 2.3.

One can observe that if the support of every $\nu_{n}$ is bounded by some
constant $M$, (2.27) is no longer needed. Under this restrictive assumption,
there is no mass going out to infinity in any direction and one can easily
check that

$X_{n}\overset{\mathcal{L}}{\rightarrow}X\Leftrightarrow\lim_{n\rightarrow\infty}\operatornamewithlimits{sup}\limits_{u\in
K}\|S_{n}(u)-S(u)\|=0,$

for every compact $K\subset\mathbb{B}(0,1)\backslash\\{0\\}$.

###### Proof.

On the one hand, assume that $(X_{n})$ converges in distribution to $X$. We
clearly have from (2.27) that

$\mathbb{E}[\;\operatornamewithlimits{sup}\limits_{n\in\mathbb{N}}\|X_{n}\|\ln(\|X_{n}\|)]\leq\mathbb{E}[Z\ln(Z)]<+\infty.$

Hence, using the same change of variables as in the proof of Theorem 2.2, we
obtain that

(2.29)
$\mathbb{E}[\;\operatornamewithlimits{sup}\limits_{n\in\mathbb{N}}\|X_{n}\|\ln(\|X_{n}\|)]=\int_{\mathbb{S}^{d-1}}\int_{0}^{1}\operatornamewithlimits{sup}\limits_{n\in\mathbb{N}}\|Q_{n}(t\varphi)\|\ln(\|Q_{n}(t\varphi)\|)dtd{\mathbb{P}}_{\mathbb{S}}(\varphi)<+\infty,$

where we recall that
$\mathbb{S}^{d-1}=\\{\varphi\in{\mathbb{R}}^{d}:\|\varphi\|_{2}=1\\}$.
Consequently, we deduce from (2.29) that for almost all
$\varphi\in\mathbb{S}^{d-1}$,

(2.30)
$\operatornamewithlimits{sup}\limits_{n\in\mathbb{N}}\int_{0}^{1}\|Q_{n}(t\varphi)\|\ln(\|Q_{n}(t\varphi)\|)dt\leq\int_{0}^{1}\operatornamewithlimits{sup}\limits_{n\in\mathbb{N}}\|Q_{n}(t\varphi)\|\ln(\|Q_{n}(t\varphi)\|)dt<+\infty.$

It clearly leads to the uniform integrability of $(Q_{n})$ along sign curves,
see [9][Theorem 4.5.9]. However, we already saw from Lemma 2.3 that the
convergence in distribution of $(X_{n})$ to $X$ implies the pointwise
convergence of $Q_{n}$ to $Q$ on $\mathbb{B}(0,1)\backslash\\{0\\}$.
Therefore, it follows from the Lebesgue-Vitali theorem [9][Theorem 4.5.4] that
for  almost all $\varphi\in\mathbb{S}^{d-1}$,

$\lim\limits_{n\rightarrow+\infty}\int_{0}^{1}Q_{n}(t\varphi)dt=\int_{0}^{1}Q(t\varphi)dt,$

which means that for  almost all $u\in\mathbb{B}(0,1)\backslash\\{0\\}$,

(2.31)
$\lim\limits_{n\rightarrow+\infty}\int_{0}^{1}Q_{n}\Big{(}t\frac{u}{\|u\|}\Big{)}dt=\int_{0}^{1}Q\Big{(}t\frac{u}{\|u\|}\Big{)}dt.$

Hereafter, we already saw from (2.6) that

(2.32)
$S_{n}(u)-S(u)=\frac{1}{1-\|u\|}\left(\int_{0}^{1}(Q_{n}-Q)\Big{(}t\frac{u}{\|u\|}\Big{)}dt-\|u\|\Big{(}E_{n}(u)-E(u)\Big{)}\right).$

Finally, the right-hand side of (2.28) follows from (2.21) together with
(2.31) and (2.32). On the other hand, assume that the right-hand side of
(2.28) holds. We have for  almost all $u\in\mathbb{B}(0,1)\backslash\\{0\\}$,

(2.33)
$\lim\limits_{n\rightarrow+\infty}\int_{\|u\|}^{1}Q_{n}\Big{(}t\frac{u}{\|u\|}\Big{)}dt=\int_{\|u\|}^{1}Q\Big{(}t\frac{u}{\|u\|}\Big{)}dt.$

Hence, we obtain from (2.33)  that for almost every $r\in]0,1[$ and
$\varphi\in\mathbb{S}^{d-1}$,

(2.34)
$\lim\limits_{n\rightarrow+\infty}\int_{r}^{1}Q_{n}(t\varphi)dt=\int_{r}^{1}Q(t\varphi)dt.$

Obviously, for  almost all $r\in]0,1[$ and  almost all
$\varphi\in\mathbb{S}^{d-1}$,

$\displaystyle\lim\limits_{n\rightarrow+\infty}\int_{0}^{1}(Q_{n}-Q)(t\varphi)dt$
$\displaystyle=\lim\limits_{n\rightarrow+\infty}\left(\int_{0}^{r}(Q_{n}-Q)(t\varphi)dt+\int_{r}^{1}(Q_{n}-Q)(t\varphi)dt\right),$
$\displaystyle=\lim\limits_{n\rightarrow+\infty}\int_{0}^{r}(Q_{n}-Q)(t\varphi)dt.$

In addition, this integral is always finite  since (2.5) holds for integrable
probability measures. As the left-hand side does not depend on $r$,

(2.35)
$\lim\limits_{n\rightarrow+\infty}\int_{0}^{1}(Q_{n}-Q)(t\varphi)dt=\lim\limits_{r\rightarrow
0}\lim\limits_{n\rightarrow+\infty}\int_{0}^{r}(Q_{n}-Q)(t\varphi)dt.$

From the uniform integrability of $(Q_{n})$ along sign curves, we obtain that

$\lim\limits_{r\rightarrow
0}\lim\limits_{n\rightarrow+\infty}\int_{0}^{r}\|Q_{n}(t\varphi)-Q(t\varphi)\|dt=0.$

Hence, by use of (2.35), we find that

$\lim\limits_{n\rightarrow+\infty}\int_{0}^{1}Q_{n}(t\varphi)dt=\int_{0}^{1}Q(t\varphi)dt.$

It ensures that for  almost all $u\in\mathbb{B}(0,1)\backslash\\{0\\}$,

(2.36)
$\lim\limits_{n\rightarrow+\infty}\int_{0}^{1}Q_{n}\Big{(}t\frac{u}{\|u\|}\Big{)}dt=\int_{0}^{1}Q\Big{(}t\frac{u}{\|u\|}\Big{)}dt.$

Consequently, we deduce from (2.32) and (2.36) that for  a.e.
$u\in\mathbb{B}(0,1)\backslash\\{0\\}$,

(2.37) $\lim\limits_{n\rightarrow+\infty}E_{n}(u)=E(u)$

Finally, it follows from (2.37) together with (2.21) that $(X_{n})$ converges
in distribution to $X$, which completes the proof of Theorem 2.4. ∎

##  3\. A class of reference measures

The choice of the reference distribution $\mu$ is a major tool to adapt MK
quantiles, ranks and signs to any task encountered. Notably, the uniform
distribution over the unit hypercube has advantages concerning marginal
independence, [19, 31], whereas orthogonal invariance only holds with a
spherical reference distribution, [31, 38]. $U_{d}$ plays a specific role for
the interpretation of quantile regions indexed by a probability content level.
This section deals with an alternative class of reference measures defined by
$\alpha$-level sets on the $p$-unit ball, that preserve the interpretability
of $U_{d}$. Denote by $\mathbb{S}^{d,p}$ the unit sphere
$\mathbb{S}^{d,p}=\\{\varphi\in{\mathbb{R}}^{d}:\|\varphi\|_{p}=1\\}$ with

$\|\varphi\|_{p}^{p}=\sum_{k=1}^{d}|\varphi_{k}|^{p},$

and $\mathbb{S}^{d,p}_{+}=\mathbb{S}^{d,p}\cap{\mathbb{R}}^{d}_{+}$.

######  Definition 3.1.

The $p$-spherical uniform distribution $U_{d,p}$ is defined as the product
$R\Phi$ between two independent random variables $R$ and $\Phi$ being drawn
from the uniform distribution on $[0,1]$ and the uniform distribution on
$\mathbb{S}^{d,p}$, respectively.

The MK quantile function $Q_{p}$ can always be defined almost everywhere from
a convex potential $\psi$ by Definition 2.1, for which we can always assume
that $\psi(0)=0$. At any point $u$ of non differentiability of $\psi$, one can
still define $Q_{p}(u)$ as the average of the subdifferential
$\partial\psi(u)$, so that $Q_{p}$ is defined everywhere on the $p$-unit ball.
The interpretability of quantile concepts remains the same among this class.
Crucially for us, our definitions and properties of superquantiles and
expected shortfalls naturally adapt. Definition 2.4 as well as Definition 2.5
extend, for all $u\in\mathbb{B}(0,1)$, to

$S_{p}(u)=\frac{1}{1-\|u\|_{p}}\int_{\|u\|_{p}}^{1}Q_{p}(t\frac{u}{\|u\|_{p}})dt$

and

$E_{p}(u)=\frac{1}{\|u\|_{p}}\int_{0}^{\|u\|_{p}}Q_{p}(t\frac{u}{\|u\|_{p}})dt.$

######  Remark 3.1.

Note that the uniform distribution of the radial part $\|U\|$ for $U\sim
U_{d,p}$ simplifies the presentation here, as the integrals along sign curves
correspond to classical integration. One could consider other radial parts, as
done for instance in [18], but the integration should be written with respect
to the probability measure of $\|U\|$.

We now develop on how to sample uniformly on $p$-spheres. Let $\Gamma$ stand
for the Euler Gamma function and denote by $\mathcal{L}_{p}$ the generalized
Gamma distribution with parameters $(1,1,p)$ on ${\mathbb{R}}$ with
probability density function

$f_{p}(x)=\frac{p}{\Gamma(p^{-1})}\exp(-x^{p})I_{{\mathbb{R}}_{+}}(x).$

A random variable $X$ on ${\mathbb{R}}_{+}$ is drawn from $\mathcal{L}_{p}$ if
and only if $X^{p}$ follows a Gamma distribution with shape parameter $1/p$,
and scale parameter $1$. The following lemma indicates how to sample on
$p$-spheres when $p>0$.

######  Lemma 3.1 ([4, 55]).

For any real $p>0$, the components of a random vector
$X\in{\mathbb{R}}^{d}_{+}$ are independent with distribution $\mathcal{L}_{p}$
if and only if the random variables $\|X\|_{p}$ and $\|X\|_{p}^{-1}X$ are
independent where

  1. -

$\|X\|_{p}^{-1}X$ is uniformly distributed on $\mathbb{S}^{d,p}_{+}$,

  2. -

$\|X\|_{p}^{p}$ has a Gamma distribution with shape $d/p$ and scale $1$.

For $p=2$, $\mathcal{L}_{p}$ corresponds to $|Z|$ where $Z$ is drawn from a
$\mathcal{N}(0,1)$ distribution. The MK distribution function with reference
$U_{d}$ (or $U_{d,2}$) has been obtained in [13][Section 2.4]. Here, we use
similar arguments to find gradient-of-convex maps for our alternative class of
reference measures. We emphasize that such maps are invariant to shifts, see
[31][Lemma A.7], whereas invariance to rotations requires a spherically
symmetric reference measure, [31][Lemma A.8].

######  Proposition 3.1.

Suppose that the components of a random vector $X\in{\mathbb{R}}^{d}_{+}$ are
independent with distribution $\mathcal{L}_{p}$, for $p>1$. For $q=p/(p-1)$,
consider $U_{d,q}^{+}$ the restriction of the $q$-spherical uniform
distribution to the convex cone ${\mathbb{R}}^{d}_{+}$. Then, the MK
distribution function with respect to $U_{d,q}^{+}$ has the explicit
formulation

(3.1) $F_{p}(x)=(\|x\|_{p}^{-1}x)^{\otimes(p-1)}G(\|x\|_{p}),$

where $\otimes$ stands for the component-wise exponent and $G$ is the
univariate distribution function of $\|X\|_{p}$. Its inverse, the MK quantile
function, is given, for all $u\in{\mathbb{R}}^{d}_{+}$ such that
$\|u\|_{q}\leq 1$, by

(3.2) $Q_{p}(u)=(\|u\|_{q}^{-1}u)^{\otimes(q-1)}G^{-1}(\|u\|_{q}).$

The MK superquantile and expected shortfall functions are respectively given,
for all $u\in{\mathbb{R}}^{d}_{+}$ such that $\|u\|_{q}\leq 1$, by

(3.3) $S_{p}(u)=(\|u\|_{q}^{-1}u)^{\otimes(q-1)}\overline{S}(\|u\|_{q}),$
(3.4) $E_{p}(u)=(\|u\|_{q}^{-1}u)^{\otimes(q-1)}\overline{E}(\|u\|_{q}),$

where $\overline{S}$ and $\overline{E}$ are the univariate superquantile and
expected shortfall functions associated with the distribution of $\|X\|_{p}$.

###### Proof.

Because $\|X\|_{p}^{p}$ follows a Gamma distribution with parameters
$(d/p,1)$, that is a generalized Gamma distribution with parameters
$(1,d/p,1)$, $\|X\|_{p}$ has a generalized Gamma distribution with parameters
$(1,d,p)$. For $G$ its distribution function, let

$\Psi(z)=\int_{-\infty}^{z}G(t)dt,$

for $z\in{\mathbb{R}}$, and $\varphi(x)=\Psi(\|x\|_{p})$. Then, $\varphi$ must
be convex by the composition between the non-decreasing function $\Psi$ and
$x\mapsto\|x\|_{p}$, both convex, see [52][Theorem 5.1]. Moreover, we retrieve
$F_{p}$ in (3.1) through $\nabla\varphi(x)=F_{p}(x)$, and we have that
$F_{p}(X)\sim U_{d,p}^{+}$ from Lemma 3.1. To get the quantile function, it
remains to invert (3.1). If $u=(\|x\|_{p}^{-1}x)^{\otimes(p-1)}G(\|x\|_{p})$,

$\|u\|_{q}^{q}=G(\|x\|_{p})^{q}\sum_{i=1}^{d}\Big{(}\frac{x}{\|x\|_{p}}\Big{)}^{p}.$

Thus, $\|u\|_{q}=G(\|x\|_{p})$ which yields
$\|u\|_{q}^{-1}u=(\|x\|_{p}^{-1}x)^{\otimes(p-1)}$. This rewrites
$\|u\|_{q}^{-1}u\|x\|_{p}^{p-1}=x^{\otimes(p-1)}$, where
$\|x\|_{p}=G^{-1}(\|u\|_{q})$, so

$\|u\|_{q}^{-1}uG^{-(p-1)}(\|u\|_{q})=x^{\otimes(p-1)}.$

Finally, (3.2) follows by applying the exponent $q-1=1/(p-1)$.∎

Figure 3 illustrates Proposition 3.1. Each column contains the reference
measure $U_{d,q}^{+}$ for $q=p/(p-1)$ in the first line, and the associated
distribution with $i.i.d.$ components $X_{i}\sim\mathcal{L}_{p}$ below.
Reference contours of orders $0.25,0.5,0.75$ are represented as well as the
explicit MK quantile contours. Whereas the center-outward ordering intrinsic
to $U_{d}$ is natural for elliptical models, [13][Section 2.4], the left-to-
right ordering may be more relevant under generalized Gamma models in
${\mathbb{R}}^{d}_{+}$.

(a) $p=1.5$

(b) $p=2$

(c) $p=3$

(d) $p=1.5$

(e) $p=2$

(f) $p=3$

Figure 3. (First line) Reference measures $U_{d,q}^{+}$ for indices $q$
conjugate with $p$. (Second line) Generalized Gamma distributions
$\mathcal{L}_{p}$ and explicit quantile contours and sign curves.

##  4\. Multivariate values and vectors at risk

Considering several univariate risks separately leads to neglecting the
correlation structure between the components of a random vector. Hence,
studying a notion of risk in a multivariate way is of major importance. We
refer to [26][Section 2.1] for motivations for intrinsically multivariate risk
analysis problems.  The risk framework considers a vector of dependent losses
$X\in{\mathbb{R}}^{d}_{+}$, where each component is a positive measure whose
unit may be different from one to another. Naturally, the larger the value of
a component, the greater the associated risk.  On the one hand, we argue that
real-valued risk measures are not sufficient in  such setting. With  real-
valued random variables, real-valued risk measures are able to catch all the
information needed. But, with the increase of the dimension, one may need
vectors in ${\mathbb{R}}^{d}$ to get the directional information of the
multivariate tails. With this in mind, we make here a proposal for vector-
valued risk measures, namely Vectors-at-Risk and Conditional-Vectors-at-Risk,
which aim to sum up the relevant information contained in the center-outward
quantiles and superquantiles. Related to these vector-valued measures, we
define multivariate values-at-risk and multivariate conditional-values-at-risk
which are measures of the form of
$\rho:{\mathbb{R}}^{d}_{+}\rightarrow{\mathbb{R}}$ able to compare $X$ and $Y$
by $\rho(X)>\rho(Y)$. Importantly, the computation of these real-valued
measures gives the vector-valued ones, hitting two targets with one shot
without added complexity.  On the other hand, a multivariate extension of the
concepts of VaR and CVaR shall have the same interpretation  as in
${\mathbb{R}}$. In dimension $d=1$, the VaR of some risk at a level $\alpha$
is simply the quantile of order $\alpha$. Accordingly, the CVaR is the
superquantile of order $\alpha$. These are to be understood respectively as
the worst risk encountered with $\nu$-probability $\alpha$, and as the
averaged risk beyond this quantile. Being an observation from the underlying
distribution, it shall be vector-valued in our multivariate framework.

By construction, the  MK quantile contour of order $\alpha\in[0,1]$ contains
the points having the most outward position with $\nu$-probability $\alpha$.
Hence, we argue that a Vector-at-Risk of order $\alpha$ should belong to this
contour.  Even if it is always adapted to the geometry of $\nu$, a MK quantile
contour can either describe a central or a bottom-left area, by changing the
reference measure. We explore both these possibilities, leaving it to the
reader to decide which tool to adopt. In what follows, we rely on $U_{d}$ and
$U_{d}^{+}$, its restriction to ${\mathbb{R}}^{d}_{+}$. Of course, other
$p$-spherical uniform distributions could be used, but we do not know if this
would induce any benefit. Starting from the choice between central or bottom-
left areas for MK quantile contours, the worst vectors of losses are the
furthest from the origin  in our context. Then, we suggest to select points
from the center-outward quantile contour of order $\alpha$ with maximal norm,
using Definition 2.2. Conditional-Vectors-at-Risk (CVaRs) are defined in the
same way, but considering the center-outward superquantiles, with Definition
2.6.

######  Definition 4.1.

The Vector-at-Risk at level $\alpha$ is defined by

$\mbox{VaR}_{\alpha}(X)\in\operatornamewithlimits{argsup}\limits\left\\{\;\|X\|_{1}\;;\;X\in\mathcal{C}_{\alpha}\;\right\\}.$

Similarly, the Conditional-Vector-at-Risk is defined by

$\mbox{CVaR}_{\alpha}(X)\in\operatornamewithlimits{argsup}\limits\left\\{\;\|X\|_{1}\;;\;X\in\mathcal{C}_{\alpha}^{s}\;\right\\}.$

The choice of $\|\cdot\|_{1}$ in the above definition depends on how to
compare different risks.  We emphasize that this choice is very distinct to
the one of the reference measure : this consideration takes place once the
contours are fixed.  Without added information, we believe that two
observations of same $1$-norm shall be considered of same importance.
Intuitively, when comparing multivariate risks, the risks per component add
up. For example, if

$x_{1}=\begin{pmatrix}1\\\ 0\end{pmatrix}\quad\text{and}\quad
x_{2}=\begin{pmatrix}0.5\\\ 0.5\end{pmatrix}$

belong to the same quantile contour of level $\alpha$, choosing
$\|\cdot\|_{2}$ instead of $\|\cdot\|_{1}$ in Definition 4.1 would induce to
consider that $x_{1}$ is worst than $x_{2}$. This choice is coherent with the
common practice of computing univariate VaR and CVaR on the random variable
$\|X\|_{1}$ in financial applications, but a major difference is that it
encodes the multivariate joint probability of the vector $X$ before applying
the sum. Thus, it takes into account the correlations, while providing more
information, because typical values for component-wise risks can be retrieved
through our Vectors-at-Risk. With this in mind, the Vector-at-Risk of order
$\alpha$ is to be understood as the “worst” risk encountered with probability
$\alpha$, whereas the Conditional-Vector-at-Risk is the average risk beyond
this “worst” observation, where the notion of “worst”  is characterized here
by $\|\cdot\|_{1}$ and the focus is either on central or on bottom-left areas.
Thus, this generalizes the understanding of univariate VaR and CVaR.  We
displayed in Figure 4 the $\mbox{VaR}_{\alpha}$ and $\mbox{CVaR}_{\alpha}$
with quantile and superquantile contours estimated via the numerical scheme
presented in Section 5.1 for $\varepsilon=10^{-3}$. Each red point is the
worst observation inside the corresponding quantile or superquantile regions.

(a) Center-outward reference $U_{d}$

(b) Left-to-right reference $U_{d}^{+}$

Figure 4.  Gaussian distribution with $\mbox{VaR}_{\alpha}$ and
$\mbox{CVaR}_{\alpha}$ in red and associated MK quantile and superquantile
contours in blue, for $\alpha=0.5$.

Intuitively, Definition 4.1 already gives a real-valued risk measure. Note
that the set $\left\\{\;\|X\|_{1}\;;\;X\in\mathcal{C}_{\alpha}\;\right\\}$ is
the same as $\left\\{\;\|Q_{X}(u)\|_{1}\;;\;\|u\|_{2}=\alpha\;\right\\}$ for
both the reference distribution $U_{d}$ and its restriction $U_{d}^{+}$ to
${\mathbb{R}}^{d}_{+}$.

######  Definition 4.2.

The multivariate value-at-risk at level $\alpha$ is defined as

$\rho_{\alpha}^{Q}(X)=\operatornamewithlimits{sup}\limits_{u}\left\\{\;\|Q_{X}(u)\|_{1}\;;\;\|u\|_{2}=\alpha\;\right\\}.$

Similarly, the multivariate conditional value-at-risk is defined as

$\rho_{\alpha}^{S}(X)=\operatornamewithlimits{sup}\limits_{u}\left\\{\;\|S_{X}(u)\|_{1}\;;\;\|u\|_{2}=\alpha\;\right\\}.$

These multivariate (conditional) values-at-risk indicate on the spreadness of
the point cloud drawn from a vector of losses, and contain information about
its centrality and the correlations between its components. Empirical
experiments have been conducted in Section 5.2 where they are compared with
the maximal correlation risk measure from [5].

##  5\. Numerical experiments.

###  5.1. On the choice of the empirical quantile map.

Hereafter, we chose to use the entropic map as a regularized empirical map, as
advocated in [8, 12, 43]. Regularizing the quantile function enables to reach
a smooth function $T:u\mapsto T(u)$ in practice, and has been the concern of
several works, including [5, 37]. Our regularization choice is thus motivated
by the known computational advantages of entropic optimal transport, [17, 30],
with a very recent line of work focusing on the entropic map [33, 47, 48, 50].
A relaxed version of Monge problem (2.1) is the Kantorovich problem, see the
introduction of [60]. This allows easier computations, even more so when one
uses entropically regularized optimal transport. We only state here the semi-
dual formulation of Kantorovich regularized problem, that one can find for
instance in [30]. The notation differ from traditional ones in optimal
transport theory, to be consistent with the context of center-outward
quantiles.  Let $\mathcal{C}(\mathcal{X})$ be the space of continuous
functions from $\mathcal{X}$ to ${\mathbb{R}}$. Let $\varepsilon>0$ be the
regularization parameter, which must be low in order to approximate better the
true OT. For $\mu,\nu\in\mathcal{P}_{2}({\mathbb{R}}^{d})$ with finite second-
order moments and with respective supports $\mathcal{U},\mathcal{X}$, and for
a given $x_{0}\in\mathcal{X}$, the semi-dual problem has a unique solution and
writes

(5.1)
$\max_{v\in\mathcal{C}(\mathcal{X}):v(x_{0})=0}\int_{\mathcal{U}}v^{c,\epsilon}(u)d\mu(u)+\int_{\mathcal{X}}v(x)d\nu(x)-\epsilon,$

where $v^{c,\epsilon}\in\mathcal{C}(\mathcal{U})$ is the smooth c-transform of
$v$

(5.2) $\forall
u\in\mathcal{U},v^{c,\epsilon}(u)=-\epsilon\log\Big{(}\int_{\mathcal{X}}\exp\Big{(}\frac{v(x)-\frac{1}{2}\|u-x\|^{2}}{\epsilon}\Big{)}d\nu(x)\Big{)}.$

In practice, $\nu$, the law  whose we consider the quantiles, is not known and
is approached by its empirical counterpart, so it always has a finite moment
of order $2$. On the contrary, $\mu$, the reference law, is known and
absolutely continuous. In this semi-discrete setting, one can use stochastic
algorithms [7, 8, 30] to solve (5.1) and obtain the optimal Kantorovich
potential $v$. From this, one can deduce the entropic map, [48], an
approximation of the Monge map partially legitimated by an entropic analog of
Brenier’s theorem, [10, 16]

$Q_{\varepsilon}(u)=\nabla\Big{(}\frac{1}{2}\|u\|^{2}-v^{c,\epsilon}(u)\Big{)}.$

Additionally, neither Assumption 2.1 nor Assumption 2.2 are necessary to its
definition or to ensure its continuity. Using (5.2), one has the analytic
formula

(5.3) $Q_{\epsilon}(u)=\int_{\mathcal{X}}xg_{\epsilon}(u,x)d\nu(x),$

where

$g_{\epsilon}(u,x)=\frac{\exp\left(\frac{v_{\epsilon}(x)-\frac{1}{2}\|u-x\|^{2}}{\epsilon}\right)}{\int_{\mathcal{X}}\exp\left(\frac{v_{\epsilon}(z)-\frac{1}{2}\|u-z\|^{2}}{\epsilon}\right)d\nu(z)}=\exp\Big{(}\frac{v_{\epsilon}^{c,\epsilon}(u)+v_{\epsilon}(x)-\frac{1}{2}\|u-x\|^{2}}{\epsilon}\Big{)}.$

This formula can be read as a conditional expectation of $\nu$ w.r.t. the
conditional law associated with the transport plan
$d\pi_{\varepsilon}(u,x)=g_{\epsilon}(u,x)d\nu(x)d\mu(u)$, which appears to be
solution of the Kantorovich regularized problem, [30]. Plugging
$Q_{\varepsilon}$ into Definitions 2.4 and 2.5 induces the entropic analogs

(5.4)
$S_{\epsilon}(u)=\frac{1}{1-\|u\|}\int_{\|u\|}^{1}Q_{\varepsilon}\Big{(}t\frac{u}{\|u\|}\Big{)}dt$

and

(5.5)
$E_{\epsilon}(u)=\frac{1}{\|u\|}\int_{0}^{\|u\|}Q_{\varepsilon}\Big{(}t\frac{u}{\|u\|}\Big{)}dt.$

The empirical counterparts $\widehat{Q}_{\varepsilon,n}$,
$\widehat{S}_{\varepsilon,n}$ and $\widehat{E}_{\varepsilon,n}$ of definitions
(5.3) and (5.4) are obtained by plug-in estimators of the problem (5.1) for
the empirical measure
$\widehat{\nu}_{n}=\frac{1}{n}\sum_{i=1}^{n}\delta_{X_{i}}$ and  for the
reference distribution $\mu$. In the experiments of the next section, we use
the stochastic Robbins-Monro algorithm taken from [7]. The entropic map
$\widehat{Q}_{\varepsilon,n}$ defined in (5.3) is considered as a regularized
empirical quantile function with $\varepsilon=10^{-3}$, and the integrals in
(5.4) are estimated by Riemann sums.  Because the entropic map is continuous
and it is the gradient of a convex function, even empirically, it shall
provide nested and smooth contours. Moreover, estimating Monge maps is known
to suffer from the curse of dimensionality, see [41, 48]. But, when the
objective is the entropic map, for any $\varepsilon>0$, convergence rates
independent from $d$ were obtained in [50]. Furthermore, the empirical version
of the entropic map approaches the true Monge map $(\varepsilon=0)$ with a
rate independent from $d$ if $\nu$ is assumed to be discrete, see [47]. Note
that, even for low $\varepsilon$, the entropic map does not exactly push-
forward  the reference distribution onto $\nu$, and is rather an approximation
of it.

###  5.2. Empirical study on simulated data

####  5.2.1. Descriptive plots for quantiles and superquantiles

Descriptive plots associated with Definition 2.6 are given in Figure 5, to
describe all the information contained in our new  concepts. The continuous
reference distribution $U_{d}$ is transported to the discrete banana-shaped
measure $\nu$ with support of size $n=5000$, via our empirical center-outward
expected shortfall (resp. superquantile) function. Note how the data splits
into a central area and a periphery area by the range of points covered by
both maps. These are satisfying estimators for the well-suited
$\varepsilon=10^{-3}$, that is to say the first column of Figure 5. As the
regularization parameter $\varepsilon$ for $E_{\varepsilon}$ and
$S_{\varepsilon}$ grows, the contours concentrate around the mean vector of
$\nu$, which is a known feature of entropic optimal transport. One can observe
that our regularized approach yields smooth interpolation between image
points. For visualization purposes, the red points of averaged sign curves are
linked by straight paths. The blue points are not, to illustrate how the
contours capture the empty space in the middle of the non convex point cloud.

(a) Expected shortfalls, $\varepsilon=0.001$

(b) Expected shortfalls, $\varepsilon=0.005$

(c) Expected shortfalls, $\varepsilon=0.01$

(d) Superquantiles, $\varepsilon=0.001$

(e) Superquantiles, $\varepsilon=0.005$

(f) Superquantiles, $\varepsilon=0.01$

Figure 5.  Center-outward expected shortfall and superquantile contours of
levels $\alpha$ in $\\{0.1,0.3,0.5,0.7,0.9\\}$ in blue and averaged sign
curves $C_{u}$ in red.

####  5.2.2. Risk measurements on toy examples

(a) Same  Gaussians, reduced covariance matrix.

(b) Same  Gaussians, added outliers.

(c) Same  Gaussians with shift.

(d) Gaussians pointed in the vertical or horizontal direction

Figure 6. VaRs ($+$) and CVaRs ($\times$) on toy examples. Third column :
real-valued risk measurements for the point clouds of the same line.

In dimension $d=2$, an empirical distribution can be represented with a
scatter plot and the riskiest observations are visible with the naked eye:
they are located furthest from the origin. From this principle, we selected
easy-to-handle situations to evaluate our risk measures in Figure 6. Each row
refers to a situation where a blue scatter plot (first column) is to be
compared with an orange one (second column). For each situation, our
(Conditional) Vectors-at-Risk of order $\alpha=0.75$ is illustrated on each
scatter plot. Moreover, the associated $\rho_{\alpha}^{Q}$ and
$\rho_{\alpha}^{S}$ are computed and compared to the maximal correlation risk
mesure $\rho_{C}$ from [5] in the third column. The higher the bar, the
riskier the corresponding vector of losses. In view of their comparison, the
measurements $\rho\in\\{\rho_{\alpha}^{Q},\rho_{\alpha}^{S},\rho_{C}\\}$ are
rescaled. For $Y_{1}$ the distribution of the blue scatter plot and $Y_{2}$
the orange one, one considers, for the height of the bars,

$\rho(Y_{1})/\max\Big{(}\rho(Y_{1}),\rho(Y_{2})\Big{)}\quad\text{and}\quad\rho(Y_{2})/\max\Big{(}\rho(Y_{1}),\rho(Y_{2})\Big{)}.$

For each situation, our VaR and CVaR provide typical observations in the
multivariate tails. The maximal-correlation risk measure $\rho_{C}$ performs
as well as expected, while our CVaR succeeds in more situations. Indeed,
$\rho_{C}$ benefits from several theoretical properties but only measures the
riskiness of $X-\mathbb{E}(X)$, hence it neglects the shift effects.  Figure
6(a) contains two Gaussian distributions with identical mean vectors and
covariance matrices related through the multiplication by a positive real.
This is well tackled by each real-valued risk measurement. The existence of
more outliers must be taken into account similarly, as in figure 6(b), where
the two scatter plots originate from the same underlying distribution, but
some outliers are added to the orange one. The CVaR is much more sensitive to
these outliers than the VaR, as in dimension $d=1$. Note that $\rho_{C}$
ignores the outliers and leads to the wrong decision in the sense than the
blue distribution is considered as the riskiest one. In the situation of
Figure 6(c), the orange scatterplot is identical to the blue one, but shifted
to the right side, so that it must be the riskiest. As $\rho_{C}$ ignores this
shift, it induces the wrong decision. Finally, in the last example of Figure
6(d), making a decision on the relative risk between the underlying
distributions requires a preference for one of the components.  Here, the two
situations reveal risks of same intensity but directed towards different
directions.  To inform on the underlying directional information, vector-
valued risk measures such as our VaRs and CVaRs are needed in complement.

###  5.3. Risk measurements on wind gusts data

In this section, we illustrate our new multivariate risk measures on the
analysis of a real dataset provided by the ExtremalDep R package, [6], and
dedicated to the study of strong wind gusts. This dataset has previously been
studied in [25, 32, 42] in a context of risk measurement. The three variables
are hourly wind gust (WG) in meters per second, wind speed (WS) in meters per
second, and air pressure at sea level (DP) in millibars, recorded at Parcay-
Meslay (France) between July 2004 and July 2013. We consider the 1450 weekly
maximum of each measurement. Because the variables are of different nature, it
is the precise framework where multivariate risk analysis is useful, rather
than the aggregation of several variables.

Figure 7. Three dimensional wind gust data set.

Figure 7 represents our three-dimensional dataset with pair scatterplots under
the diagonal and Pearson correlation values above. The diagonal represents
empirical density functions of each variable. Upper-right dependence can be
observed and has physical explanations. Strong wind gusts occur with stormy
weather, during which strong wind speed and high air pressure are frequently
recorded.

(a) Risk level 0.25

(b) Risk level 0.5

(c) Risk level 0.75

Figure 8. Vectors-at-Risk and Conditional-Vectors-at-Risk.

Figure 8 represents the three dimensional empirical distribution in red
together with our vectorial risk measures. With the increase of the dimension,
such representative plots are no longer convenient. Rather, these measurements
can be retrieved as in Table 1.

| $VaR_{0.25}$ | $VaR_{0.5}$ | $VaR_{0.75}$
---|---|---|---
WS | 8.21 | 9.98 | 12.36
WG | 15.06 | 18.45 | 21.58
DP | 11.92 | 13.65 | 17.84

| $CVaR_{0.25}$ | $CVaR_{0.5}$ | $CVaR_{0.75}$
---|---|---|---
WS | 11.40 | 12.37 | 14.26
WG | 21.28 | 22.43 | 26.24
DP | 16.68 | 19.86 | 23.15

Table 1. Components of (Conditional) Vectors-at-Risk for several risk levels.

This summarizes the targeted information contained in the dataset. For
instance, with a given probability 0.25, 0.5 or 0.75, one shall expect, at
worst, wind gusts of respective speed 15.06, 18.45, 21.58 m/s. With
respectively same probability, averaged observations beyond these worst cases
shall lie around 21.28, 22.43, 26.24 m/s.  As in [42], we interpret these
results thanks to the Beaufort scale. Note that this scale does not capture
the speed of wind gusts, as it usually averages over 10 minutes, by
convention. Values between 13.9 and 17.1 m/s can be considered as high winds.
Strong winds begin with 17.2 m/s, with severely strong winds above 20.7 m/s up
to 24.4 m/s. Severely strong winds can cause slight structural damage, but
less than storms for values around 24.5-28.4 m/s. Above and up to 32.6 m/s,
violent storms are very rarely experienced and cause widespread damage. Wind
speeds greater than 32.7 m/s correspond to hurricanes.  Thanks to Table 1,
with probability 0.75, the worst scenarios in Parcay-Meslay for wind gusts are
severely strong winds. Moreover, along the tail events corresponding to 25% of
occurrences, one shall expect wind gusts of same speed as storms.  Also, to
illustrate the fact that considering univariate measures leads to
underestimating the risk, we display in Figure 2 traditional univariate
Values-at-Risk of WG, the wind gusts. For example, its median is 11.80,
(strong breeze), and must be compared with 18.45, (strong winds), the second
coordinate of $VaR_{0.5}$.

Min | 1st quartile | Median | 3rd quartile | Max
---|---|---|---|---
4.10 | 9.80 | 11.80 | 14.90 | 34
Table 2. Univariate quantiles of our variable WG.

Such differences have statistical foundings. The median depends on the
univariate empirical distribution of WG to describe a probability of $1/2$.
Conversely, our multivariate $VaR_{0.5}$ encodes the multivariate joint
probability of the whole point cloud (WG,WS,DP). This can be summarized by the
fact that univariate risk measures neglect the correlations, which legitimates
the use of multivariate risk analysis.

##  6\. Conclusion and perspectives

In this paper, we provided new concepts of superquantiles and expected
shortfalls, in a meaningful multivariate way, based on the Monge-Kantorovich
quantiles. We expect that our definitions allow the extension of the
univariate applications to the multivariate case. Also, we introduced new
transport-based risk measures, extending the concepts of Value-at-Risk and
Conditional-Value-at-Risk to the dimension $d>1$. The definitions of this
paper are mainly motivated by practical concerns and the coherence of the
proposed concepts with respect to their interpretation in dimension $d=1$.
Interestingly, from our center-outward expected shortfall function, one can
retrieve Lorenz functions defined in [38]. This is discussed in Section 6.1,
and other perspectives are also presented in Section 6.2.

###  6.1. On the class of integrated quantile functions

The role of integrated quantile functions in dimension $d=1$ has been
highlighted in [35]. Obviously, we belong to the line of works trying to
extend such functions to the setting $d>1$. In Section 4 of [38], two
different approaches are discussed, in view of generalizing

(6.1) $\alpha\mapsto\int_{0}^{\alpha}Q(t)dt.$

Hereafter, we bridge their concepts with ours, namely Definition 2.5 that
extends (6.1). Recall that
$\mathbb{S}^{d-1}=\\{\varphi\in{\mathbb{R}}^{d}:\|\varphi\|_{2}=1\\}$ and
${\mathbb{P}}_{S}$ is the uniform probability measure on $\mathbb{S}^{d-1}$.
There, the (absolute) center-outward Lorenz function from [38][Definition 4]
writes as

(6.2)
$L_{\textbf{X}\pm}:\alpha\mapsto\mathbb{E}[X\mathds{1}_{X\in\mathds{C}_{\alpha}}]=\int_{\mathbb{S}^{d-1}}\alpha
E(\alpha\varphi)d{\mathbb{P}}_{S}(\varphi).$

This being said, we believe that our proposed concepts of integration along
sign curves contain more information, namely directional. When characterizing
the contributions of central regions to the expectation, $L_{\textbf{X}\pm}$
provides meaningful concepts of Lorenz curves, but this approach is
insufficient for superquantiles and multivariate tails, as illustrated in
Example 2.1.

Another important generalization of (6.1) is the one from [27] about
multivariate Lorenz curves. A main difference between the concepts of [27] and
[38] is the reference distribution, either the uniform on the unit hypercube
or the spherical uniform. Here, Lorenz curves aim to visualize inequalities
within a given population. In this context, one could focus either on the
contribution of middle classes as in [38], or on the one of the bottom of the
population as in [27], in a way that these works are in fact complementary.
Furthermore, another approach is proposed in [38][Definition 6], that is real-
valued, namely the (absolute) center-outward Lorenz potential function, for
$\varphi\in\mathbb{S}^{d-1}$,

(6.3) $\alpha\mapsto\mathbb{E}_{\varphi}[\psi(\alpha\varphi)].$

This is quite natural because, at the core of the univariate considerations of
[35], one can retrieve the property that the quantile function is the gradient
of a convex potential, that is the definition of the center-outward quantile
function $Q$. Also, it has a physical interpretation, with a measurement of
the work of the quantile function. There, one might note that our center-
outward expected shortfall function draws a connection between [38][Definition
4] and [38][Definition 6], through (6.2) and (2.11). Somehow, this strengthens
the relation between $L_{\textbf{X}\pm}$ and the potential function $\psi$.
Studying deeper such connections between existing notions might lead to
interesting results, but is left for further work.

###  6.2. Other perspectives

Perspectives are both theoretical and practical. Statistical tests associated
with our real-valued measures would be of great interest, to automatically
detect when to use the vector-valued ones. Also, these risk measures appeal
for being used in practical settings.  Extending our definitions to extreme
quantile levels and building related procedures is another line of study. For
instance, [18][Theorem 4.3] ensures the existence of a MK quantile map which
is stable as moving further to the tail contours, which is not the case with
the spherical uniform $U_{d}$. Our proposed center-outward superquantiles can
be defined with other spherical reference measures, thus one can also
calibrate these superquantiles as the work of [18] suggests.  In addition, the
many applications of the univariate superquantile function in other fields
than risk measurement, such as superquantile regression or  optimization with
a superquantile loss, appeal for further work, even more so with Theorems 2.3
and 2.4.  Finally, the convergence of empirical entropic optimal transport
towards its population counterpart is an active field, [21, 34, 45, 48], and
it remains to study to what extent existing results adapt to the convergence
of $\widehat{S}_{\varepsilon,n}$ and $\widehat{E}_{\varepsilon,n}$.

## References

  * [1] C. Acerbi, C. Nordio, and C. Sirtori.  Expected shortfall as a tool for financial risk management.  arXiv:cond-mat/0102304, 2001.
  * [2] C. Acerbi and D. Tasche.  On the coherence of expected shortfall.  Journal of Banking & Finance, 26(7):1487–1503, 2002.
  * [3] S. Armaut, R. Diel, and T. Laloë.  On some depth-based risk measurement for high losses.  hal-03933578, 2023.
  * [4] F. Barthe, O. Guédon, S. Mendelson, and A. Naor.  A probabilistic approach to the geometry of the $\ell_{p}^{n}$-ball.  The Annals of Probability, 33(2):480 – 513, 2005.
  * [5] J. Beirlant, S. Buitendag, E. del Barrio, M. Hallin, and F. Kamper.  Center-outward quantiles and the measurement of multivariate risk.  Insurance: Mathematics and Economics, 95(C):79–100, 2020.
  * [6] B. Beranger, S. Padoan, and G. Marcon.  Extremaldep: extremal dependence models.  R Package, 2021.
  * [7] B. Bercu and J. Bigot.  Asymptotic distribution and convergence rates of stochastic algorithms for entropic optimal transportation between probability measures.  The Annals of Statistics, 49(2):968 – 987, 2021.
  * [8] B. Bercu, J. Bigot, and G. Thurin.  Stochastic optimal transport in banach spaces for regularized estimation of multivariate quantiles.  arXiv:2302.00982, 2023.
  * [9] V. I. Bogachev and M. A. S. Ruas.  Measure theory, volume 1.  Springer, 2007.
  * [10] Y. Brenier.  Polar factorization and monotone rearrangement of vector-valued functions.  Communications on Pure and Applied Mathematics, 44:375–417, 1991\.
  * [11] J. Cai, H. Jia, and T. Mao.  A multivariate cvar risk measure from the perspective of portfolio risk management.  Scandinavian Actuarial Journal, 2022(3):189–215, 2022.
  * [12] G. Carlier, V. Chernozhukov, G. De Bie, and A. Galichon.  Vector quantile regression and optimal transport, from theory to numerics.  Empirical Economics, 62(1):35–62, 2022.
  * [13] V. Chernozhukov, A. Galichon, M. Hallin, and M. Henry.  Monge–Kantorovich depth, quantiles, ranks and signs.  The Annals of Statistics, 45(1):223 – 256, 2017.
  * [14] A. Cousin and E. Di Bernardino.  On multivariate extensions of value-at-risk.  Journal of Multivariate Analysis, 119:32–46, 2013.
  * [15] A. Cousin and E. Di Bernardino.  On multivariate extensions of conditional-tail-expectation.  Insurance: Mathematics and Economics, 55:272–282, 2014.
  * [16] J. A. Cuesta and C. Matrán.  Notes on the wasserstein metric in hilbert spaces.  Annals of Probability, 17:1264–1276, 1989.
  * [17] M. Cuturi.  Sinkhorn distances: Lightspeed computation of optimal transport.  Advances in Neural Information Processing Systems, 26, 2013.
  * [18] C. F. de Valk and J. Segers.  Stability and tail limits of transport-based quantile contours.  LIDAM Discussion Papers ISBA 2018031, Université catholique de Louvain, Institute of Statistics, Biostatistics and Actuarial Sciences (ISBA), 2018.
  * [19] N. Deb and B. Sen.  Multivariate rank-based distribution-free nonparametric testing using measure transportation.  Journal of the American Statistical Association, 118(541):192–207, 2023.
  * [20] E. Del Barrio, A. González-Sanz, and M. Hallin.  A note on the regularity of optimal-transport-based center-outward distribution and quantile functions.  J. Multivar. Anal., 180:104671, 2020.
  * [21] E. del Barrio, A. Gonzalez-Sanz, J.-M. Loubes, and J. Niles-Weed.  An improved central limit theorem and fast convergence rates for entropic transportation costs.  SIAM Journal on Mathematics of Data Science, 3(5):639–669, 2023\.
  * [22] E. del Barrio and J.-M. Loubes.  Central limit theorems for empirical transportation cost in general dimension.  Ann. Probab., 47(2):926–951, 2019.
  * [23] E. del Barrio, A. G. Sanz, and M. Hallin.  Nonparametric multiple-output center-outward quantile regression.  arXiv:2204.11756, 2022.
  * [24] E. Di Bernardino, T. Laloë, V. Maume-Deschamps, and C. Prieur.  Plug-in estimation of level sets in a non-compact setting with applications in multivariate risk theory.  ESAIM: Probability and Statistics, 17:236–256, 2013.
  * [25] E. Di Bernardino and C. Prieur.  Estimation of the multivariate conditional tail expectation for extreme risk levels: Illustration on environmental data sets.  Environmetrics, 29(7):e2510, 2018.
  * [26] I. Ekeland, A. Galichon, and M. Henry.  Comonotonic measures of multivariate risks.  Mathematical Finance, 22(1):109–132, 2012.
  * [27] Y. Fan, M. Henry, B. Pass, and J. A. Rivero.  Lorenz map, inequality ordering and curves based on multidimensional rearrangements.  arXiv:2203.09000, 2022.
  * [28] A. Figalli.  On the continuity of center-outward distribution and quantile functions.  Nonlinear Analysis, 117:413–421, 2018.
  * [29] A. Galichon and M. Henry.  Dual theory of choice with multivariate risks.  Journal of Economic Theory, 147(4):1501–1516, 2012.
  * [30] A. Genevay, M. Cuturi, G. Peyré, and F. Bach.  Stochastic Optimization for Large-scale Optimal Transport.  Advances in neural information processing systems, 29, 2016.
  * [31] P. Ghosal and B. Sen.  Multivariate ranks and quantiles using optimal transport: Consistency, rates, and nonparametric testing.  The Annals of Statistics, 50(2):1012–1037, 2022.
  * [32] Y. Goegebeur, A. Guillou, and J. Qin.  Dependent conditional tail moments for extreme levels.  hal-03948352v3, 2023.
  * [33] Z. Goldfeld, K. Kato, G. Rioux, and R. Sadhu.  Limit theorems for entropic optimal transport maps and the sinkhorn divergence.  arXiv:2207.08683, 2022.
  * [34] A. Gonzalez-Sanz, J.-M. Loubes, and J. Niles-Weed.  Weak limits of entropy regularized optimal transport : potentials, plans and divergences.  arXiv:2207.07427, 2022.
  * [35] A. A. Gushchin and D. A. Borzykh.  Integrated quantile functions: properties and applications.  Modern Stochastics: Theory and Applications, 4(4):285–314, 2017\.
  * [36] M. Hallin.  Measure Transportation and Statistical Decision Theory.  Annual Review of Statistics and Its Application, 9:401–424, 2022\.
  * [37] M. Hallin, E. del Barrio, J. Cuesta-Albertos, and C. Matrán.  Distribution and quantile functions, ranks and signs in dimension d: A measure transportation approach.  The Annals of Statistics, 49(2):1139 – 1165, 2021.
  * [38] M. Hallin and G. Mordant.  Center-outward multiple-output lorenz curves and gini indices a measure transportation approach.  Working papers ecares, ULB – Universite Libre de Bruxelles, 2022.
  * [39] M. Hallin, D. Vecchia, and H. Liu.  Rank-based testing for semiparametric var models: A measure transportation approach.  Bernoulli, 29(1):229–273, 02 2023.
  * [40] J. E. Heffernan and J. A. Tawn.  A conditional approach for multivariate extreme values (with discussion).  Journal of the Royal Statistical Society: Series B (Statistical Methodology), 66(3):497–546, 2004.
  * [41] J.-C. Hütter and P. Rigollet.  Minimax estimation of smooth optimal transport maps.  The Annals of Statistics, 49(2):1166 – 1194, 2021.
  * [42] G. Marcon, P. Naveau, and S. Padoan.  A semi-parametric stochastic generator for bivariate extreme events.  Stat, 6(1):184–201, 2017.
  * [43] S. B. Masud, M. Werenski, J. M. Murphy, and S. Aeron.  Multivariate soft rank via entropic optimal transport: sample efficiency and generative modeling.  Journal of Machine Learning Research, 24(160):1–65, 2023.
  * [44] R. J. McCann.  Existence and uniqueness of monotone measure-preserving maps.  Duke Mathematical Journal, 80(2):309 – 323, 1995.
  * [45] G. Mena and J. Niles-Weed.  Statistical bounds for entropic optimal transport: sample complexity and the central limit theorem.  In Advances in Neural Information Processing Systems, volume 32, 2019.
  * [46] Z. Niu and B. B. Bhattacharya.  Distribution-free joint independence testing and robust independent component analysis using optimal transport.  arXiv:2211.15639, 2022.
  * [47] A.-A. Pooladian, V. Divol, and J. Niles-Weed.  Minimax estimation of discontinuous optimal transport maps: The semi-discrete case.  arXiv:2301.11302, 2023.
  * [48] A.-A. Pooladian and J. Niles-Weed.  Entropic estimation of optimal transport maps.  arXiv:2109.12004, 2021.
  * [49] A. Prékopa.  Multivariate value at risk and related topics.  Annals of Operations Research, 193(1):49–69, 2012.
  * [50] P. Rigollet and A. J. Stromme.  On the sample complexity of entropic optimal transport.  arXiv:2206.13472, 2022.
  * [51] R. T. Rockafellar.  Convex analysis.  Princeton University Press, 1970.
  * [52] R. T. Rockafellar and J. O. Royset.  Superquantiles and their applications to risk, random variables and regression.  In Theory Driven by Influential Applications, pages 151–167. 2013\.
  * [53] R. T. Rockafellar and J. O. Royset.  Random variables, monotone relations, and convex analysis.  Mathematical Programming, 148:297–331, 2014.
  * [54] W. Rudin.  Real and Complex Analysis, volume 2 of Higher Mathematics Series.  McGraw-Hill, 1974.
  * [55] G. Schechtman and J. Zinn.  On the volume of the intersection of two $l^{n}_{p}$ balls.  Proceedings of the American Mathematical Society, 110(1):217–224, 1990.
  * [56] H. Shi, M. Drton, M. Hallin, and F. Han.  Distribution-free tests of multivariate independence based on center-outward quadrant, spearman, kendall, and van der waerden statistics.  arXiv:2111.15567v4, 2021.
  * [57] H. Shi, M. Drton, and F. Han.  Distribution-free consistent independence tests via center-outward ranks and signs.  Journal of the American Statistical Association, 117(537):395–410, 2022.
  * [58] R. Torres, R. E. Lillo, and H. Laniado.  A directional multivariate value at risk.  Insurance: Mathematics and Economics, 65:111–123, 2015.
  * [59] K. Uğurlu.  A new coherent multivariate average-value-at-risk.  Optimization, 72(2):493–519, 2023.
  * [60] C. Villani.  Topics in optimal transportation, volume 58 of Graduate Studies in Mathematics.  American Mathematical Society, 2003.