# Quantization and martingale couplings

Benjamin Jourdain 33footnotemark: 3 Université Paris-Est, Cermics (ENPC),
INRIA, F-77455 Marne-la-Vallée, France. E-mail<EMAIL_ADDRESS>Gilles Pagès Laboratoire de Probabilités, Statistique et Modélisation, UMR
8001, Campus Pierre et Marie Curie, Sorbonne Université case 158, 4, pl.
Jussieu, F-75252 Paris Cedex 5, France<EMAIL_ADDRESS>research benefited from the support of the “Chaire Risques Financiers”,
Fondation du Risque

###### Abstract

Quantization provides a very natural way to preserve the convex order when
approximating two ordered probability measures by two finitely supported ones.
Indeed, when the convex order dominating original probability measure is
compactly supported, it is smaller than any of its dual quantizations while
the dominated original measure is greater than any of its stationary (and
therefore any of its optimal) quadratic primal quantization. Moreover, the
quantization errors then correspond to martingale couplings between each
original probability measure and its quantization. This permits to prove that
any martingale coupling between the original probability measures can be
approximated by a martingale coupling between their quantizations in
Wassertein distance with a rate given by the quantization errors but also in
the much finer adapted Wassertein distance. As a consequence, while the
stability of (Weak) Martingale Optimal Transport problems with respect to the
marginal distributions has only been established in dimension $1$ so far,
their value function computed numerically for the quantized marginals
converges in any dimension to the value for the original probability measures
as the numbers of quantization points go to $\infty$.

AMS Subject Classification (2010): 60E15, 65C50, 65D32, 60J22, 60G42.

## Introduction

For $d\\!\in{N}^{*}$ and $\mu,\nu$ in the set ${\mathcal{P}}({R}^{d})$ of
probability measures on ${R}^{d}$, we say that $\mu$ is smaller than $\nu$ in
the convex order and denote $\mu\leq_{cvx}\nu$ if

$\forall\varphi:{R}^{d}\to{R}\mbox{ convex
},\;\int_{{R}^{d}}\varphi(x)\mu(dx)\leq\int_{{R}^{d}}\varphi(y)\nu(dy),$ (0.1)

when the integrals make sense (since any real valued convex function is
bounded from below by an affine function $\int_{{R}^{d}}\varphi(x)\mu(dx)$
makes sense in ${R}\cup\\{+\infty\\}$ as soon as
$\int_{{R}^{d}}|x|\mu(dx)<+\infty$). For $p\geq 1$, we denote by
${\mathcal{P}}_{p}({R}^{d})=\\{\mu\\!\in{\mathcal{P}}({R}^{d}):\int_{{R}^{d}}|x|^{p}\mu(dx)<+\infty\\}$
the Wasserstein space with index $p$ over ${R}^{d}$. When
$\mu,\nu\in{\mathcal{P}}_{1}({R}^{d})$, according to the Strassen theorem
[50], $\mu\leq_{cvx}\nu$ if and only if there exists a martingale coupling
between $\mu$ and $\nu$ that is a probability measure $\pi(dx,dy)$ on
${R}^{d}\times{R}^{d}$ with marginals $\int_{y\in{R}^{d}}\pi(dx,dy)$ and
$\int_{x\in{R}^{d}}\pi(dx,dy)$ equal to $\mu(dx)$ and $\nu(dy)$ respectively
such that $\pi(dx,dy)=\mu(dx)\pi_{x}(dy)$ for some Markov kernel $\pi_{x}(dy)$
with the martingale property: $\forall x\\!\in{R}^{d}$, $\pi_{x}\in{\cal
P}_{1}({R}^{d})$ and $\int_{{R}^{d}}y\pi_{x}(dy)=x$. We denote by ${\cal
P}(\mu,\nu)$ the set of probability measures on ${R}^{d}\times{R}^{d}$ with
respective marginals $\mu$ and $\nu$ and by ${\cal M}(\mu,\nu)$ the subset of
${\cal P}(\mu,\nu)$ consisting of martingale couplings.

Let $(\mu,\nu)$ belong to the set ${\cal P}_{\leq}\times{\cal P}_{p}({R}^{d})$
of couples of elements of ${\cal P}_{p}({R}^{d})$ with the first one smaller
than the second in the convex order. In its simplest form, the Martingale
Optimal Transport problem consists in computing

$V_{c}(\mu,\nu)=\inf_{\pi\in{\cal
M}(\mu,\nu)}\int_{{R}^{d}\times{R}^{d}}c(x,y)\pi(dx,dy)$

and the optimal martingale couplings achieving this infimum for some
measurable cost function $c:{R}^{d}\times{R}^{d}\to{R}$. When the interest
rate is zero, for an exotic option written on $d$ traded assets with payoff
given by the function $c$ of their prices at times $s$ and $t$, then
$V_{c}(\mu,\nu)$ (resp. $-V_{-c}(\mu,\nu)$) provides a robust lower (resp.
upper) price bound when $\mu$ and $\nu$ are the respective joint laws of these
$d$ assets at times $s$ and $t$ (for instance obtained by calibration of a
model to vanilla option prices). Since its introduction in [9], this MOT
problem has received recently a great attention in the financial mathematics
literature. In particular, the structure of martingale optimal transport
couplings [11, 13, 15, 22, 28], continuous time formulations [16, 21, 27],
links with the Skorokhod embedding problem [8], numerical methods [2, 1, 14,
25, 26] and stability properties [6, 33, 52] have been investigated. The MOT
problem is a particular instance where the measurable cost function
$C:{R}^{d}\times{\cal P}_{1}({R}^{d})\to{R}$ is linear in the measure
component ($C(x,\eta)=\int_{{R}^{d}}c(x,y)\eta(dy)$) of the Weak Martingale
Optimal Transport problem

$\inf_{\pi\in{\cal M}(\mu,\nu)}\int_{{R}^{d}}C(x,\pi_{x})\mu(dx)$

introduced in [6] by adding the martingale constraint to the Weak Optimal
Transport problem introduced in [23].

To devise a numerical procedure devoted to the computation of the value and of
the optimal couplings in the MOT and WMOT problems, a first natural step
consists in approximating $\mu$ and $\nu$ by finitely supported probability
measures which are still in the convex order. To our best knowledge, few
studies consider the problem of preserving the convex order while
approximating a sequence of probability measures. We mention the thesis of
Baker [7] who proposes the following construction in dimension $d=1$. Let for
$\eta\in{\cal P}({R})$ and $u\in(0,1)$,
$F_{\eta}^{-1}(u)=\inf\\{x\in{R}:\eta((-\infty,x])\geq u\\}$ be the quantile
of $\eta$ of order $u$. For $(\mu,\nu)\in{\cal P}_{\leq}\times{\cal
P}_{1}({R})$ and $N,K\in{N}^{*}$ with $N/K\in{N}^{*}$, one has

$\frac{1}{N}\sum_{i=1}^{N}\delta_{N\int_{\frac{i-1}{N}}^{\frac{i}{N}}F_{\mu}^{-1}(u)du}\leq_{cvx}\frac{1}{K}\sum_{i=1}^{K}\delta_{K\int_{\frac{i-1}{K}}^{\frac{i}{K}}F_{\nu}^{-1}(u)du}.$

Dual (or Delaunay) quantization introduced by Pagès and Wilbertz [43] and
further studied in [44, 45, 46] gives another way to preserve the convex order
in dimension $d=1$ when using the same grid to quantize both probability
measures.

In two recent papers [1, 2], Alfonsi, Corbetta and Jourdain propose to restore
for $(\mu,\nu)\in{\cal P}_{\leq}\times{\cal P}_{1}({R}^{d})$ the convex
ordering from any finitely supported approximations $\tilde{\mu}$ and
$\tilde{\nu}$ of $\mu$ and $\nu$. In dimension $d=1$, one may define the
increasing (resp. decreasing) convex order by adding the constraint that the
test function $\varphi$ is non-decreasing (resp. non-increasing) in (0.1).
According to [1], the convex order restauration can be achieved by keeping
$\tilde{\mu}$ (resp. $\tilde{\nu}$) and replacing $\tilde{\nu}$ (resp.
$\tilde{\mu}$) by the supremum (resp. infimum) between $\tilde{\mu}$ and
$\tilde{\nu}$ for the increasing convex order when
$\int_{R}x\tilde{\nu}(dx)\leq\int_{R}x\tilde{\mu}(dx)$ and the decreasing
convex order when $\int_{R}x\tilde{\nu}(dx)\geq\int_{R}x\tilde{\mu}(dx)$. The
convex, increasing convex and decreasing convex orders are nicely
characterized in terms of the potential function obtained as the anti-
derivative of the cumulative distribution function (or of the quantile
function). The supremum and infimum of two probability measures for one of
these orders can be computed using their potential functions. For a general
dimension $d$, [2] suggests to keep $\tilde{\nu}$ and replace $\tilde{\mu}$ by
its projection on the set of probability measures dominated by $\tilde{\nu}$
for the quadratic Wasserstein distance $W_{2}$ (see (0.2) below for the
definition of this distance). This projection can be computed by solving a
quadratic optimization problem with linear constraints.

In the present paper, when $\nu$ is compactly supported, we investigate the
combined approximation of $\mu$ by some quadratic-optimal primal quantization
and of $\nu$ by some dual quantization. By construction, any quadratic-optimal
primal quantization of $\mu$ satisfies a stationarity property which implies
that it is smaller than $\mu$ in the convex order. On the other hand, any dual
quantization of $\nu$ is greater than this probability measure in the convex
order. Therefore the convex order between $\mu$ and $\nu$ is preserved by this
combined approximation. Notice that, in contrast with the previous approaches,
it cannot be generalized to the convex order preserving approximation of more
than two probability measures. Moreover the dual quantization approximation is
only possible for compactly supported probability measures. In contrast with
these restrictions, we will see that the studied approach proves to provide
robust approximations of (Weak) Martingale Optimal Transport problems even in
dimension $d\geq 2$.

The first section of the paper is devoted to primal (or Voronoi) quantization.
For $\mu\in{\cal P}_{p}({R}^{d})$ with $p\geq 1$, we show that an element of
the set ${\cal P}({R}^{d},N)$ of probability measures on ${R}^{d}$ whose
support contains at most $N$ points is an $L_{p}$-optimal $N$-quantization of
$\mu$ iff it is a $W_{p}$-projection of $\mu$ on ${\cal P}({R}^{d},N)$ where
the Wasserstein distance $W_{p}$ is defined by

$W_{p}(\mu,\nu)^{p}=\inf_{\pi\in{\cal
P}(\mu,\nu)}\int_{{R}^{d}\times{R}^{d}}|x-y|^{p}\pi(dx,dy).$ (0.2)

In the quadratic $p=2$ case, any stationary and therefore any optimal
$N$-quantization $\hat{\mu}^{N}$ of the measure $\mu$ is smaller than $\mu$ in
the convex order. Moreover $W_{2}(\hat{\mu}^{N},\mu)=M_{2}(\hat{\mu}^{N},\mu)$
where

$M_{p}(\eta,\nu)=\inf_{\pi\in{\cal
M}(\eta,\nu)}\int_{{R}^{d}\times{R}^{d}}|x-y|^{p}\pi(dx,dy)\mbox{ for
}(\eta,\nu)\in{\cal P}_{\leq}\times{\cal P}_{1}({R}^{d}).$ (0.3)

This enables us to check that any quadratic-optimal primal quantizer for $\mu$
remains quadratic-optimal for each probability measure smaller than $\mu$ and
greater than its associated quantization for the convex order.

The second section deals with the dual quantization of a compactly supported
probability measure $\mu$ which is obtained by minimizing $M_{p}(\mu,\eta)$
over the subset ${\cal P}_{\geq\mu}({R}^{d},N)$ of ${\cal P}({R}^{d},N)$
consisting in probability measures greater than $\mu$ in the convex order.
Since ${\cal M}(\mu,\eta)\subset{\cal P}(\mu,\eta)$, $M_{p}(\mu,\eta)\geq
W_{p}(\mu,\eta)$ and, in general, the inequality is strict. It turns out,
that, even in the quadratic $p=2$ case, $L_{p}$-optimal dual quantizations of
$\mu$ are not necessarily $W_{p}$-projections of $\mu$ on ${\cal
P}_{\geq\mu}({R}^{d},N)$. Nevertheless, we check that any quadratic-optimal
dual quantizer of $\mu$ remains quadratic-optimal for each probability measure
greater than $\mu$ and smaller than its associated dual quantization in the
convex order.

In the third section, we consider two probability measures $\mu,\nu\in{\cal
P}({R}^{d})$ such that $\mu\leq_{cvx}\nu$ with $\nu$ compactly supported.
Since the quantization errors between $\mu$ and any of its quadratic-optimal
primal $N$-quantization $\hat{\mu}^{N}$ and between $\nu$ and any of its
$L_{p}$-optimal dual $K$-quantization $\check{\nu}^{K}$ correspond to
martingale couplings, we are able to approximate in Wasserstein distance on
${\cal P}({R}^{d}\times{R}^{d})$ any martingale coupling $\pi\in{\cal
M}(\mu,\nu)$ by a martingale coupling $\bar{\pi}^{N,K}\in{\cal
M}(\hat{\mu}^{N},\check{\nu}^{K})$ with a rate given by the quantization
errors. We also check that, as $N,K\to\infty$, $\bar{\pi}^{N,K}$ converges to
$\pi$ for the much finer adapted Wasserstein distance defined in (0.5) below
which captures the temporal structure of probability distributions with two
time marginals. Numerous financial applications of this adapted Wasserstein
distance have been investigated in [3]. According to [4], the topology induced
by this distance is equal to the other adapted topologies which had been
introduced in particular in view of financial applications.

The adapted Wasserstein is particularly well suited to deal with Weak
Martingale Optimal Transport problems. While their stability with respect to
the marginal distributions has only been established in dimension $1$ so far
[10], this enables us to check in Section 4 that their value function computed
numerically for the quantized marginals converge in any dimension to the value
for the original probability measures as the numbers of quantization points go
to $\infty$.

For the reader’s convenience, we now list the notations and basic properties
that have been introduced in the above text.

Definitions and notations.

Let $d\\!\in{N}^{*}$

$\bullet$ $|\cdot|$ denotes the canonical Euclidean norm on ${R}^{d}$.

$\bullet$ ${\rm conv}(A)$ denotes the (closed) convex hull of
$A\subset{R}^{d}$ and ${\rm card}(A)$ or $|A|$ its cardinality (depending on
the context).

$\bullet$ Let ${\cal P}({R}^{d})$ denote the set of probability measures on
${R}^{d}$ endowed with its Borel sigma-field ${\cal B}({R}^{d})$.

$\bullet$ For $p\geq 1$, let ${\cal P}_{p}({R}^{d})=\\{\mu\in{\cal
P}({R}^{d}):\int_{{R}^{d}}|x|^{p}\mu(dx)<\infty\\}$.

$\bullet$ For $p\geq 1$, let ${\cal P}_{\leq}\times{\cal
P}_{p}({R}^{d})=\\{(\mu,\nu)\in{\cal P}_{p}({R}^{d})\times{\cal
P}_{p}({R}^{d}):\mu\leq_{cvx}\nu\\}$.

$\bullet$ For every integer $N\geq 1$, we denote by $\mathcal{P}({R}^{d},N)$
the set of distributions on ${R}^{d}$ whose support contains at most $N$
points.

$\bullet$ For $\mu,\nu\in{\cal P}({R}^{d})$, let

${\cal P}(\mu,\nu)=\left\\{\pi\in{\cal P}({R}^{d}\times{R}^{d}):\;\forall
A\in{\cal B}({R}^{d}),\;\pi(A\times{R}^{d})=\mu(A)\mbox{ and
}\pi({R}^{d}\times A)=\nu(A)\right\\}$

denote the set of couplings between $\mu$ and $\nu$. For $\pi\in{\cal
P}(\mu,\nu)$, we denote by $\pi_{x}(dy)$ the ($\mu(dx)$ a.e. unique) Markov
kernel such that $\pi(dx,dy)=\mu(dx)\pi_{x}(dy)$.

$\bullet$ For $\mu,\nu\in{\cal P}_{1}({R}^{d})$ with $\mu\leq_{cvx}\nu$, let

${\cal M}(\mu,\nu)=\left\\{\pi\in{\cal P}(\mu,\nu):\;\mu(dx)\mbox{ a.e.
}\int_{{R}^{d}}y\pi_{x}(dy)=x\right\\}$

denote the (non empty according to the Strassen theorem) set of martingale
couplings between $\mu$ and $\nu$.

$\bullet$ For $p\geq 1$ and $\mu,\nu\in{\cal P}({R}^{d})$, let

$W_{p}(\mu,\nu)=\inf_{\pi\in{\cal
P}(\mu,\nu)}\left(\int_{{R}^{d}\times{R}^{d}}|x-y|^{p}\pi(dx,dy)\right)^{1/p}\leq\infty$

denote the Wasserstein distance with index $p$. This is a complete metric on
$\mathcal{P}_{p}({R}^{d})$.

$\bullet$ For $p\geq 1$ and $(\mu,\nu)\in{\cal P}_{\leq}\times{\cal
P}_{1}({R}^{d})$, let

$M_{p}(\mu,\nu)=\inf_{\pi\in{\cal
M}(\mu,\nu)}\left(\int_{{R}^{d}\times{R}^{d}}|x-y|^{p}\pi(dx,dy)\right)^{1/p}\leq\infty.$

Since ${\cal M}(\mu,\nu)\subset{\cal P}(\mu,\nu)$, we clearly have,
$W_{p}(\mu,\nu)\leq M_{p}(\mu,\nu)$. In the quadratic $p=2$ case, when
$\mu,\nu\in{\cal P}_{2}({R}^{d})$,

$\displaystyle\forall\pi\in{\cal
M}(\mu,\nu),\;\int_{{R}^{d}\times{R}^{d}}|y-x|^{2}\pi(dx,dy)$
$\displaystyle=\int_{{R}^{d}}|y|^{2}\nu(dy)-2\int_{{R}^{d}}x.\int_{{R}^{d}}y\pi_{x}(dy)\mu(dx)+\int_{{R}^{d}}|x|^{2}\mu(dx)$
$\displaystyle=\int_{{R}^{d}}|y|^{2}\nu(dy)-\int_{{R}^{d}}|x|^{2}\mu(dx)$

so that

$M_{2}^{2}(\mu,\nu)=\int_{{R}^{d}}|y|^{2}\nu(dy)-\int_{{R}^{d}}|x|^{2}\mu(dx).$
(0.4)

$\bullet$ For $\pi\in{\cal P}(\mu,\nu)$ and $\tilde{\pi}\in{\cal
P}(\tilde{\mu},\tilde{\nu})$ we consider the adapted Wasserstein distance with
index $p\geq 1$ between $\pi(dx,dy)=\mu(dx)\pi_{x}(dy)$ and
$\tilde{\pi}(d\tilde{x},d\tilde{y})=\tilde{\mu}(d\tilde{x})\tilde{\pi}_{\tilde{x}}(d\tilde{y})$
:

$\displaystyle AW_{p}(\pi,\tilde{\pi})=\inf_{m\in{\cal
P}(\mu,\tilde{\mu})}\left(\int_{{R}^{d}\times{R}^{d}}(|x-\tilde{x}|^{p}+W_{p}^{p}(\pi_{x},\tilde{\pi}_{\tilde{x}}))m(dx,d\tilde{x})\right)^{1/p}.$
(0.5)

##  1 Primal (Voronoi) quantization and Wasserstein projection

In this subsection, we make a connection between primal quantization and
various projections (in the Wasserstein sense), including, in the quadratic
case, with the one mentioned above in the introduction. Let us first recall
the following basic facts about the (primal) Voronoi quantization of
$\mu\in{\cal P}_{p}({R}^{d})$ with $p\geq 1$ (see [24, 41, 37] among others):

– Let $\Gamma=\\{x_{1},\ldots,x_{{}_{N}}\\}\subset{R}^{d}$ denote a finite
subset of size $N$. The $L^{p}$-quantization error modulus $e_{p}(\Gamma,\mu)$
satisfies

$e_{p}(\Gamma,\mu)^{p}=\int_{{R}^{d}}|x-{\rm Proj}_{\Gamma}(x)|^{p}\mu(dx)$
(1.6)

where ${\rm Proj}_{\Gamma}$ denotes a Borel nearest neighbour projection on
$\Gamma$ satisfying $|x-{\rm Proj}_{\Gamma}(x)|={\rm dist}(x,\Gamma)$. If
$X\sim\mu$, the random variable ${\rm Proj}_{\Gamma}(X)$ with law
$\mu\circ{\rm Proj}_{\Gamma}^{-1}$ is called $\Gamma$-quantization of $X$.

– For any level $N\geq 1$, there exists an optimal grid or $N$-quantizer
$\Gamma_{p,N}$ such that

$e_{p,N}(\mu):=\inf\Big{\\{}e_{p}(\Gamma,\mu):\Gamma\subset{R}^{d},\;|\Gamma|\leq
N\Big{\\}}=e_{p}\big{(}\Gamma_{p,N},\mu\big{)}.$

When ${\rm card}({\rm supp}(\mu))\leq N$ then $\Gamma_{p,N}={\rm supp}(\mu)$
and when ${\rm card}({\rm supp}(\mu))>N$, then $\Gamma_{p,N}$ has exactly $N$
pairwise distinct elements. Moreover, Theorem 4.2 in [24] ensures that

$\mu\Big{(}\\{x\in{R}^{d}:\exists
y\neq\tilde{y}\in\Gamma_{p,N},\;|x-y|=|x-\tilde{y}|={\rm
dist}(x,\Gamma_{p,N})\\}\Big{)}=0,$ (1.7)

so that $\mu\circ{\rm Proj}_{\Gamma_{p,N}}^{-1}$ does not depend on the choice
of the Borel nearest neighbour projection ${\rm Proj}_{\Gamma_{p,N}}$ on
$\Gamma_{p,N}$. We denote this probability measure by
$\hat{\mu}^{\Gamma_{p,N}}$. In the same way, when $X\sim\mu$, ${\rm
Proj}_{\Gamma_{p,N}}(X)$ a.s. does not depend on the Borel nearest neighbour
projection ${\rm Proj}_{\Gamma_{p,N}}$ and is denoted
$\hat{X}^{\Gamma_{p,N}}$.

– In the quadratic case ($p=2$), any optimal quantization grid $\Gamma_{2,N}$
(possibly not unique) and its induced quantization
$\hat{X}^{\Gamma_{2,N}}={\rm Proj}_{\Gamma_{2,N}}(X)$ with distribution
$\hat{\mu}^{\Gamma_{2,N}}$ satisfy a stationarity (or self-consistency)
property (see e.g. [24], [41] or [37], Proposition 5.1 among others) that is

${E}\big{(}X\,|\,\hat{X}^{\Gamma_{2,N}}\big{)}=\hat{X}^{\Gamma_{2,N}}\mbox{ so
that }\hat{\mu}^{\Gamma_{2,N}}\leq_{cvx}\mu.$ (1.8)

Indeed, the support of the distribution of
${E}\big{(}X\,|\,\hat{X}^{\Gamma_{2,N}}\big{)}$ is equal to
$\tilde{\Gamma}_{2,N}:=\\{{E}(X|\hat{X}^{\Gamma_{2,N}}=x),x\in\Gamma_{2,N}\\}$
(when ${\rm card}({\rm supp}(\mu))\leq N$, $\hat{X}^{\Gamma_{2,N}}=X$ and
$\tilde{\Gamma}_{2,N}=\Gamma_{2,N}={\rm supp}(\mu)$), contains at most $N$
points and $e_{2}(\tilde{\Gamma}_{2,N},\mu)^{2}={E}({\rm
dist}(X,\tilde{\Gamma}_{2,N})^{2})\leq{E}(|X-{E}\big{(}X\,|\,\hat{X}^{\Gamma_{2,N}}\big{)}|^{2})$.
As a consequence,

$\displaystyle e_{2,N}(\mu)^{2}$ $\displaystyle\leq
e_{2}(\tilde{\Gamma}_{2,N},\mu)^{2}\leq{E}(|X-{E}\big{(}X\,|\,\hat{X}^{\Gamma_{2,N}}\big{)}|^{2})={E}(|X-\hat{X}^{\Gamma_{2,N}}|^{2})-{E}(|\hat{X}^{\Gamma_{2,N}}-{E}\big{(}X\,|\,\hat{X}^{\Gamma_{2,N}}\big{)}|^{2})$
$\displaystyle=e_{2,N}(\mu)^{2}-{E}(|\hat{X}^{\Gamma_{2,N}}-{E}\big{(}X\,|\,\hat{X}^{\Gamma_{2,N}}\big{)}|^{2}),$

so that the second term in the right-hand side vanishes. Therefore the
distribution of $(\hat{X}^{\Gamma_{2,N}},X)$ belongs to ${\cal
M}(\hat{\mu}^{\Gamma_{2,N}},\mu)$ and

$e_{2,N}^{2}(\mu)={E}[|X-\hat{X}^{\Gamma_{2,N}}|^{2}]={E}[|X|^{2}]-{E}[|\hat{X}^{\Gamma_{2,N}}|^{2}]=M_{2}^{2}(\hat{\mu}^{\Gamma_{2,N}},\mu).$
(1.9)

###### Proposition 1.1

Let $p\\!\in[1,+\infty)$ and $\mu\\!\in{\cal P}_{p}({R}^{d})$.

$(a)$ Let $\Gamma\subset{R}^{d}$ be a finite set and ${\cal P}(\Gamma)$ denote
the subset of $\Gamma$-supported distributions. Then

$W_{p}\big{(}\mu,{\cal P}(\Gamma)\big{)}:=\inf_{\nu\in{\cal
P}(\Gamma)}W_{p}(\mu,\nu)=e_{p}(\Gamma,\mu):=\big{\|}{\rm
dist}(.,\Gamma)\big{\|}_{L^{p}(\mu)}$

and for any Borel nearest neighbour projection ${\rm Proj}_{\Gamma}$ on
$\Gamma$, $\mu\circ{\rm Proj}_{\Gamma}^{-1}$ is a $W_{p}$-projection of $\mu$
on ${\cal P}(\Gamma)$.

$(b)$ The probability measure $\nu\in{\cal P}({R}^{d},N)$ is a
$W_{p}$-projection of $\mu$ on ${\cal P}({R}^{d},N)$ iff
$\nu=\hat{\mu}^{\Gamma_{N}}$ for some $L_{p}$-optimal $N$-quantizer
$\Gamma_{N}$ of $\mu$. Moreover, $W_{p}(\mu,{\cal
P}({R}^{d},N))=e_{p,N}(\mu)$.

$(c)$ Quadratic case ($p=2$). A subset $\Gamma$ of ${R}^{d}$ with cardinality
at most $N$ is a quadratic optimal $N$-quantizer of $\mu$ iff there exists a
probability measure $\nu\in{\cal P}_{\leq\mu}({R}^{d},N)$ such that
$\nu(\Gamma)=1$ and one of the following equivalent conditions is satisfied

  * •

$\nu$ is a $W_{2}$-projection of $\mu$ on ${\cal P}_{\leq\mu}({R}^{d},N)$ i.e.

$W_{2}(\mu,\nu)=W_{2}(\mu,{\cal P}_{\leq\mu}({R}^{d},N)):=\inf_{\eta\in{\cal
P}_{\leq\mu}({R}^{d},N)}W_{2}(\mu,\eta),$

  * •

$\int_{{R}^{d}}|x|^{2}\nu(dx)=\sup_{\eta\in{\cal
P}_{\leq\mu}({R}^{d},N)}\int_{{R}^{d}}|x|^{2}\eta(dx)$.

Moreover, we then have $W_{2}(\nu,\mu)=M_{2}(\nu,\mu)$ and
$\nu=\hat{\mu}^{\Gamma}$.

Apart from the interpretation in terms of $W_{p}$-projection, the first
statement can be found in Lemma 3.4 p.33 [24]. Before proving the proposition,
let us state and check some easy consequence of the necessary and sufficient
condition in $(b)$.

###### Corollary 1.2

Let $\Gamma_{2,N}$ be a quadratic optimal $N$-quantizer of $\mu\in{\cal
P}_{2}({R}^{d})$. Then for any probability measure $\nu$ such that
$\hat{\mu}^{\Gamma_{2,N}}\leq_{cvx}\nu\leq_{cvx}\mu$, $\Gamma_{2,N}$ is a
quadratic optimal $N$-quantizer of $\nu$ and
$\hat{\nu}^{\Gamma_{2,N}}=\hat{\mu}^{\Gamma_{2,N}}$.

Proof of Corollary 1.2. Since $\hat{\mu}^{\Gamma_{2,N}}\in{\cal
P}_{\leq\nu}({R}^{d},N)\subset{\cal P}_{\leq\mu}({R}^{d},N)$ and
$\int_{{R}^{d}}|x|^{2}\hat{\mu}^{\Gamma_{2,N}}(dx)=\sup_{\eta\in{\cal
P}_{\leq\mu}({R}^{d},N)}\int_{{R}^{d}}|x|^{2}\eta(dx)$ by the necessary
condition in Proposition 1.1 $(c)$, one has

$\int_{{R}^{d}}|x|^{2}\hat{\mu}^{\Gamma_{2,N}}(dx)=\sup_{\eta\in{\cal
P}_{\leq\nu}({R}^{d},N)}\int_{{R}^{d}}|x|^{2}\eta(dx).$

Therefore, by the sufficient condition in Proposition 1.1 $(c)$,
$\Gamma_{2,N}$ is a quadratic optimal $N$-quantizer of $\nu$ and
$\hat{\nu}^{\Gamma_{2,N}}=\hat{\mu}^{\Gamma_{2,N}}$. $\Box$

Proof of Proposition 1.1. $(a)$ Let $\nu\\!\in{\cal P}(\Gamma)$ and
$\pi\in{\cal P}(\mu,\nu)$. Then $\mu(dx)$ a.e., $\pi_{x}(\Gamma)=1$ so that
$\pi_{x}(dy)$ a.e. $|x-y|\geq{\rm dist}(x,\Gamma)$. Therefore

$\displaystyle\int_{{R}^{d}\times{R}^{d}}|x-y|^{p}\pi(dx,dy)$
$\displaystyle\geq\int{\rm dist}(x,\Gamma)^{p}\mu(dx)=e_{p}(\Gamma,\mu)^{p}.$

Taking the infimum over $\pi\in{\cal P}(\mu,\nu)$ and $\nu\\!\in{\cal
P}(\Gamma)$, we deduce that $W_{p}\big{(}\mu,{\cal P}(\Gamma)\big{)}^{p}\geq
e_{p}(\Gamma,\mu)^{p}$. Now let ${\rm Proj}_{\Gamma}$ denote a Borel nearest
neighbour projection on $\Gamma$. Since $\mu\circ{\rm
Proj}_{\Gamma}^{-1}\in{\cal P}(\Gamma)$ and $\mu\circ(I_{d},{\rm
Proj}_{\Gamma})^{-1}\in{\cal P}(\mu,\mu\circ{\rm Proj}_{\Gamma}^{-1})$, we
have

$W^{p}_{p}\big{(}\mu,{\cal P}(\Gamma)\big{)}\leq W^{p}_{p}(\mu,\mu\circ{\rm
Proj}_{\Gamma}^{-1})\leq\int_{{R}^{d}}|x-{\rm
Proj}_{\Gamma}(x)|^{p}\mu(dx)=e_{p}(\Gamma,\mu)^{p},$

where the equality follows from (1.6). Therefore the inequalities are
equalities and $\mu\circ{\rm Proj}_{\Gamma}^{-1}$ is a $W_{p}$-projection of
$\mu$ on ${\cal P}(\Gamma)$.

$(b)$ Let $\Gamma_{p,N}$ be an $L^{p}$-optimal $N$-quantizer of $\mu$. Then,
for any subset $\Gamma$ of ${R}^{d}$ with at most $N$ points,
$e_{p}(\Gamma_{p,N},\mu)=e_{p,N}(\mu)\leq e_{p}(\Gamma,\mu)$. With $(a)$, we
deduce that

$W_{p}(\mu,\hat{\mu}^{\Gamma_{p,N}})\leq W_{p}(\mu,{\cal P}(\Gamma)).$

By taking the infimum over $\Gamma$, we deduce that
$W_{p}(\mu,\hat{\mu}^{\Gamma_{p,N}})\leq W_{p}(\mu,{\cal P}({R}^{d},N))$ and
$\hat{\mu}^{\Gamma_{p,N}}$ is a $W_{p}$-projection of $\mu$ on ${\cal
P}({R}^{d},N)$ so that

$e_{p,N}(\mu)=e_{p}(\mu,\Gamma_{p,N})=W_{p}(\mu,\hat{\mu}^{\Gamma_{p,N}})=W_{p}(\mu,{\cal
P}({R}^{d},N)).$

Conversely, let $\nu$ be a projection of $\mu$ on ${\cal P}({R}^{d},N)$ and
let $\Gamma_{N}=\\{x\in{R}^{d}:\nu(\\{x\\})>0\\}$. The cardinality of
$\Gamma_{N}$ is at most $N$. For any subset $\Gamma$ of ${R}^{d}$ with at most
$N$ points,

$W_{p}(\mu,{\cal P}(\Gamma_{N}))\leq W_{p}(\mu,\nu)=W_{p}(\mu,{\cal
P}({R}^{d},N))\leq W_{p}(\mu,{\cal P}(\Gamma)).$ (1.10)

Therefore, by $(a)$, $e_{p}(\Gamma_{N},\mu)\leq e_{p}(\Gamma,\mu)$ and since
$\Gamma$ is arbitrary, we deduce that $\Gamma_{N}$ is an $L^{p}$-optimal
$N$-quantizer of $\mu$. Moreover, the choice $\Gamma=\Gamma_{N}$ in (1.10)
implies that the first inequality is an equality so that, with $(a)$,
$W^{p}_{p}(\mu,\nu)=\int_{{R}^{d}}{\rm dist}(x,\Gamma_{N})^{p}\mu(dx)$. Hence,
for any $W_{p}$-optimal coupling $\pi\in{\cal P}(\mu,\nu)$,

$\int_{{R}^{d}\times{R}^{d}}(|y-x|^{p}-{\rm
dist}(x,\Gamma_{N})^{p})\pi(dx,dy)=0.$

Since $1=\nu(\Gamma_{N})=\int_{{R}^{d}}\pi_{x}(\Gamma_{N})\mu(dx)$,
$\pi(dx,dy)$ a.e., $|y-x|^{p}\geq{\rm dist}(x,\Gamma_{N})^{p}$. Therefore,
$\pi(dx,dy)$ a.e. $y\in\Gamma_{N}$ and $|y-x|={\rm dist}(x,\Gamma_{N})$. With
(1.7), we conclude that $\mu(dx)$ a.e. there is a unique point
$\hat{x}^{\Gamma_{N}}\in\Gamma_{N}$ such that $|\hat{x}^{\Gamma_{N}}-x|={\rm
dist}(x,\Gamma_{N})$ and $\pi_{x}(dy)=\delta_{\hat{x}^{\Gamma_{N}}}(dy)$.
Therefore the second marginal $\nu$ of $\pi$ is equal to
$\hat{\mu}^{\Gamma_{N}}$.

$(c)$ In the quadratic case ($p=2$), by $(b)$, (1.8) and (1.9), any
$W_{2}$-projection $\nu$ of $\mu$ on ${\cal P}({R}^{d},N)$ is characterized by
the existence of a quadratic optimal $N$-quantizer $\Gamma_{2,N}$ of $\mu$
such that $\nu=\hat{\mu}^{\Gamma_{2,N}}$, belongs to the smaller set ${\cal
P}_{\leq\mu}({R}^{d},N)$ and satisfies $e_{2,N}^{2}(\mu)=M_{2}^{2}(\nu,\mu)$.
Therefore the $W_{2}$-projections of $\mu$ on ${\cal P}({R}^{d},N)$ and on
${\cal P}_{\leq\mu}({R}^{d},N)$ coincide. To conclude the proof, let us check
that $\nu$ is such a projection iff
$\int_{{R}^{d}}|x|^{2}\nu(dx)=\sup_{\eta\in{\cal
P}_{\leq\mu}({R}^{d},N)}\int_{{R}^{d}}|x|^{2}\eta(dx)$. If $\eta\in{\cal
P}_{\leq\mu}({R}^{d},N)$, then, by the comparison between $W_{2}$ and $M_{2}$
given in the introduction and (0.4), one has

$W_{2}^{2}(\mu,\eta)\leq
M_{2}^{2}(\eta,\mu)=\int_{{R}^{d}}|y|^{2}\mu(dy)-\int_{{R}^{d}}|x|^{2}\eta(dx).$
(1.11)

Let $\nu$ be a $W_{2}$-projection of $\mu$ on ${\cal P}_{\leq\mu}({R}^{d},N)$.
Using $(b)$ for the third equality then (1.11) for the last inequality and the
last equality, we obtain that

$\displaystyle\int_{{R}^{d}}|y|^{2}\mu(dy)-\int_{{R}^{d}}|x|^{2}\nu(dx)$
$\displaystyle=M_{2}^{2}(\nu,\mu)=e_{2,N}(\mu)^{2}=W^{2}_{2}\big{(}\mu,{\cal
P}({R}^{d},N)\big{)}$ $\displaystyle=W^{2}_{2}\big{(}\mu,{\cal
P}_{\leq\mu}({R}^{d},N)\big{)}\leq\inf_{\eta\in{\cal
P}_{\leq\mu}({R}^{d},N)}M^{2}_{2}(\eta,\mu)$
$\displaystyle=\int_{{R}^{d}}|y|^{2}\mu(dy)-\sup_{\eta\in{\cal
P}_{\leq\mu}({R}^{d},N)}\int_{{R}^{d}}|x|^{2}\eta(dx).$

Since $\nu\\!\in{\cal P}_{\leq\mu}({R}^{d},N)$, the two inequalities are
equalities. Therefore

$\int_{{R}^{d}}|x|^{2}\nu(dx)=\sup_{\eta\in{\cal
P}_{\leq\mu}({R}^{d},N)}\int_{{R}^{d}}|x|^{2}\eta(dx)$

and since $W^{2}_{2}(\mu,{\cal P}({R}^{d},N)\big{)}\leq W^{2}_{2}(\mu,\nu)\leq
M_{2}^{2}(\nu,\mu)$, these two inequalities are equalities and
$W^{2}_{2}(\mu,\nu)=M_{2}^{2}(\nu,\mu)$. Moreover,

$e_{2,N}(\mu)^{2}=W^{2}_{2}\big{(}\mu,{\cal
P}_{\leq\mu}({R}^{d},N)\big{)}=\int_{{R}^{d}}|y|^{2}\mu(dy)-\sup_{\eta\in{\cal
P}_{\leq\mu}({R}^{d},N)}\int_{{R}^{d}}|x|^{2}\eta(dx).$

If $\nu\in{\cal P}_{\leq\mu}({R}^{d},N)$ is such that
$\int_{{R}^{d}}|x|^{2}\nu(dx)=\sup_{\eta\in{\cal
P}_{\leq\mu}({R}^{d},N)}\int_{{R}^{d}}|x|^{2}\eta(dx)$, the last equality
combined with (1.11) written for $\eta=\nu$ ensures that $\nu$ is a
$W_{2}$-projection of $\mu$ on ${\cal P}_{\leq\mu}({R}^{d},N)$. $\Box$

Remark about uniqueness. As a consequence of Proposition 1.1 $(b)$, it turns
out that the uniqueness of $W_{p}$-projections of $\mu$ on ${\cal
P}({R}^{d},N)$, that of distributions $\hat{\mu}^{N}$ of $L^{p}$-optimal
$N$-quantizations and that of $L^{p}$-optimal $N$-quantizers are equivalent.
In dimension $d=1$, for $p=2$, distributions with $\log$-concave densities
have a unique optimal $N$-quantizer (see Kiefer [35]) hence this projection is
unique. In higher dimension, a general result seems difficult to reach:
indeed, the ${\cal N}(0;I_{d})$ distribution, being invariant under the action
of ${\cal O}(d,{R})$ (orthogonal transforms), so are the (hence infinite) sets
of its optimal quantizers at levels $N\geq 2$.

Let us recall the sharp rate of convergence of the $L^{p}$-quantization error
stated for instance in Theorem 5.2 [37].

######  Theorem 1.3 (Pierce Lemma for primal quantization)

Let $p\geq 1$ and $\eta>0$. For every dimension $d\geq 1$, there exists a real
constant $\widetilde{C}^{vor}_{d,\eta,p}>0$ such that, for every random vector
$X:(\Omega,{\cal A},{P})\to{R}^{d}$,

$e_{p,N}(X)\leq\widetilde{C}^{vor}_{d,\eta,p}N^{-\frac{1}{d}}\sigma_{p+\eta}(X)$
(1.12)

where, for every $r>0$,
$\sigma_{r}(X)=\inf_{a\in{R}^{d}}\|X-a\|_{r}\leq+\infty$.

Example. Let us explicit the quadratic optimal primal $N$-quantizer of
$\mu(dx)=\frac{1_{]0,1[}(x)}{2\sqrt{x}}dx$. In [20], by checking that the
distortion function has a unique critical point, Fort and Pagès prove
uniqueness and derive semi-closed forms for optimal quantizers of three
families of one-dimensional distributions indexed by $\rho>0$ : the
exponential distributions $1_{(0,\infty)}(x)\rho e^{-\rho x}dx$, the power
distributions $\rho 1_{(0,1)}(x)x^{\rho-1}dx$ on the interval $(0,1)$ which
include $\mu$ for $\rho=\frac{1}{2}$ and the power distributions $\rho
1_{(1,\infty)}(x)x^{-(1+\rho)}dx$ on the interval $(1,+\infty)$. For $\mu(dx)$
and $p=2$, their semi-closed form relies on the inductive solution of third
degree equations. Working with a different parametrization, we will end up
with much simpler second degree equations. We have $F_{\mu}(x)=1_{\\{x\geq
0\\}}\sqrt{x}$ and $F_{\mu}^{-1}(u)=u^{2}$ for $u\in(0,1)$. We parametrize the
boundaries of the optimal quadratic Voronoi cells by
$x_{k+\frac{1}{2}}=F_{\mu}^{-1}(q_{k})=q_{k}^{2}$ for $k\in\\{0,\ldots,N\\}$
where $0=q_{0}<q_{1}<q_{2}<\ldots<q_{N-1}<q_{N}=1$. In particular
$x_{0+\frac{1}{2}}=0$ and $x_{N+\frac{1}{2}}=1$. By the stationarity property
(1.8), for $k\in\\{1,\ldots,N\\}$,
$x_{k}=\frac{1}{q_{k}-q_{k-1}}\int_{x_{k-\frac{1}{2}}}^{x_{k+\frac{1}{2}}}x\mu(dx)$
(by (1.7), $\mu(\\{x_{k-\frac{1}{2}}\\})=\mu(\\{x_{k+\frac{1}{2}}\\})=0$ and
we do not need to worry about the inclusion of the boundary points in the
integration interval). The equality
$x_{k+\frac{1}{2}}=\frac{x_{k}+x_{k+1}}{2}$ valid for $k\in\\{1,\ldots,N-1\\}$
also writes

$F_{\mu}^{-1}(q_{k})=\frac{1}{2}\left(\frac{1}{q_{k}-q_{k-1}}\int_{q_{k-1}}^{q_{k}}F_{\mu}^{-1}(u)du+\frac{1}{q_{k+1}-q_{k}}\int_{q_{k}}^{q_{k+1}}F_{\mu}^{-1}(u)du\right).$

For the above choice of $\mu$, we obtain

$q_{k}^{2}=\frac{1}{6}\left(q_{k}^{2}+q_{k}q_{k-1}+q_{k-1}^{2}+q_{k+1}^{2}+q_{k+1}q_{k}+q_{k}^{2}\right)\mbox{
so that }q_{k+1}^{2}+q_{k+1}q_{k}+q_{k}q_{k-1}+q_{k-1}^{2}-4q_{k}^{2}=0.$

We deduce that $q_{k+1}=c_{k+1}q_{1}$, where
$c_{k+1}^{2}+c_{k+1}c_{k}+c_{k}c_{k-1}+c_{k-1}^{2}-4c_{k}^{2}=0$ so that
$c_{k+1}=\frac{\sqrt{17c_{k}^{2}-4c_{k}c_{k-1}-4c_{k-1}^{2}}-c_{k}}{2}$.
Starting from $c_{0}=0$ and $c_{1}=1$, we easily compute inductively the
factors $c_{k}$ (for instance $c_{2}=\frac{\sqrt{17}-1}{2}$). To ensure
$q_{N}=1$, we need $q_{1}=\frac{1}{c_{N}}$. Therefore
$q_{k}=\frac{c_{k}}{c_{N}}$ for $k\in\\{0,\ldots,N\\}$ and the optimal
quadratic primal grid is given by

$x_{k}=\frac{1}{q_{k}-q_{k-1}}\int_{q_{k-1}}^{q_{k}}F_{\mu}^{-1}(u)du=\frac{q_{k}^{2}+q_{k}q_{k-1}+q_{k-1}^{2}}{3}=\frac{c_{k}^{2}+c_{k}c_{k-1}+c_{k-1}^{2}}{3c_{N}^{2}},\;k\in\\{1,\ldots,N\\}.$

Of course for $a<b$, when $\mu(dx)=\frac{1_{]a,b[}(x)}{2\sqrt{(b-a)(x-a)}}dx$,
then $x_{k}=a+(b-a)\frac{c_{k}^{2}+c_{k}c_{k-1}+c_{k-1}^{2}}{3c_{N}^{2}}$ and
when $\mu(dx)=\frac{1_{]a,b[}(x)}{2\sqrt{(b-a)(b-x)}}dx$,
$x_{N+1-k}=b-(b-a)\frac{c_{k}^{2}+c_{k}c_{k-1}+c_{k-1}^{2}}{3c_{N}^{2}}$.

##  2 Dual (Delaunay) quantization

We assume throughout this section that $\mu$ is compactly supported. Let
$X:(\Omega,{\cal A},{P})\to{R}^{d}$ be a random vector lying in
$L^{\infty}({P})$ with distribution $\mu$. Optimal dual (or Delaunay)
quantization as introduced in [44] relies on the best approximation which can
be achieved by a discrete random vector $\check{X}$ that satisfies a certain
stationarity assumption on the extended probability space
$(\Omega\times[0,1],{\cal A}\otimes{\cal B}([0,1]),{P}\otimes\lambda)$ where
${\cal B}([0,1])$ and $\lambda$ respectively denote the Borel sigma field and
the Lebesgue measure on the interval $[0,1]$. To be more precise, we define,
for $p\\!\in[1,+\infty)$,

$\displaystyle d_{p,N}(X)$ $\displaystyle=$
$\displaystyle\inf_{\check{X}}\Big{\\{}\big{\|}X-\check{X}\big{\|}_{p}:\check{X}:(\Omega\times[0,1],{\cal
A}\otimes{\cal B}([0,1]),{P}\otimes\lambda)\to{R}^{d},$
$\displaystyle\qquad\qquad\qquad\qquad{\rm
card}\big{(}\check{X}(\Omega\times[0,1])\big{)}\leq N\text{ and
}{E}(\check{X}|X)=X\Big{\\}}.$

For every level $N\geq d+1$, the set of such $\check{X}$ is not empty. Indeed,
one may choose $d+1$ points whose convex hull has a non empty interior and
includes the support of $\mu$. Then the unique probability measure supported
on these points with the same expectation as $\mu$ belongs to the set
${\mathcal{P}}_{\geq\mu}({R}^{d},d+1)$ of distributions dominating $\mu$ for
the convex order and supported by at most $d+1$ elements. By Lemma 2.22 in
[34], we see that for each $\nu\in{\mathcal{P}}_{\geq\mu}({R}^{d},N)$ and each
martingale coupling $\pi\in{\cal M}(\mu,\nu)$, there exists on
$(\Omega\times[0,1],{\cal A}\otimes{\cal B}([0,1]),{P}\otimes\lambda)$ a
random vector $\check{X}$ such that $(X,\check{X})$ is distributed according
to $\pi$ and therefore satisfies ${E}(\check{X}|X)=X$. Hence

$d_{p,N}(X)^{p}=\inf_{\nu\in{\mathcal{P}}_{\geq\mu}({R}^{d},N)}\inf_{\pi\in{\mathcal{M}}(\mu,\nu)}\int_{{R}^{d}\times{R}^{d}}|y-x|^{p}\pi(dx,dy)=\inf_{\nu\in{\mathcal{P}}_{\geq\mu}({R}^{d},N)}M_{p}^{p}(\mu,\nu).$
(2.13)

As a consequence, $d_{p,N}(X)$ only depends on the distribution $\mu$ of $X$
and can subsequently also be denoted $d_{p,N}(\mu)$. Next, one easily checks
that ${\mathcal{P}}_{\geq\mu}({R}^{d},N)=\bigcup_{\Gamma\in{\cal
G}_{N}}{\mathcal{P}}_{\geq\mu}(\Gamma)$ where

$\displaystyle{\cal G}_{N}=\\{\Gamma\subset{R}^{d}\mbox{ with cardinality
$\leq N$ and such that }{\rm supp}(\mu)\subset{\rm conv}(\Gamma)\\},$
$\displaystyle\mbox{ and }{\mathcal{P}}_{\geq\mu}(\Gamma)=\\{\nu\in{\cal
P}({R}^{d}):\mu\leq_{cvx}\nu\mbox{ and }\nu(\Gamma)=1\\}.$

For $\Gamma\in{\cal G}_{N}$, there exists a dual projection ${\rm
Proj}_{\Gamma}^{del}:{\rm conv}\big{(}\Gamma\big{)}\times[0,1]\to\Gamma$, also
called a splitting operator, which satisfies, beyond measurability, the
following stationarity property

$\forall\,y\\!\in{\rm conv}\big{(}\Gamma\big{)},\quad\int_{0}^{1}{\rm
Proj}_{\Gamma}^{del}(y,u)du=y,$ (2.14)

from which one derives the dual stationarity property

${E}\,\big{(}{\rm Proj}^{del}_{\Gamma}(X,U)\,\big{|}\,X\big{)}=X\mbox{ when
$U\sim\mathcal{U}([0,1])$ is independent of $X$}.$ (2.15)

The stationarity property remains valid as soon as $X$ is ${\rm
conv}(\Gamma)$-valued and implies that the distribution of ${\rm
Proj}^{del}_{\Gamma}(X,U)$ belongs to ${\mathcal{P}}_{\geq\mu}(\Gamma)$ which
is therefore non empty.

For $\Gamma\in{\cal G}_{N}$, let

$d_{p}(\mu,\Gamma)^{p}=\inf_{\nu\in{\mathcal{P}}_{\geq\mu}(\Gamma)}M_{p}^{p}(\mu,\nu),$

so that $d_{p,N}(\mu)^{p}=\inf_{\Gamma\in{\cal G}_{N}}d_{p}(\mu,\Gamma)$.

In dimension $d=1$, when $\Gamma=\\{x_{1},x_{2},\ldots,x_{N}\\}$ with
$x_{1}<x_{2}<\ldots<x_{N}$, the probability measure minimizing $\nu\mapsto
M_{p}^{p}(\mu,\nu)$ over ${\mathcal{P}}_{\geq\mu}(\Gamma)$ is the distribution
$\check{\mu}^{\Gamma}$ of ${\rm Proj}_{\Gamma}^{del}(X,U)$ for the splitting
operator

${\rm
Proj}_{\Gamma}^{del}(x,u)=\sum_{i=1}^{N-1}1_{[x_{i},x_{i+1})}(x)\left(1_{\\{u\leq\frac{x_{i+1}-x}{x_{i+1}-x_{i}}\\}}x_{i}+1_{\\{u>\frac{x_{i+1}-x}{x_{i+1}-x_{i}}\\}}x_{i+1}\right)+1_{\\{x=x_{N}\\}}x_{N}.$

Moreover, the coupling minimizing $\int_{{R}\times{R}}|x-y|^{p}\pi(dx,dy)$
over ${\cal M}(\mu,\check{\mu}^{\Gamma})$ is the distribution of $(X,{\rm
Proj}_{\Gamma}^{del}(X,U))$. Last, according to the remark after Proposition
10 in [44], when $\mu\leq_{cvx}\eta$ with $\eta$ compactly supported in
$[x_{1},x_{N}]$, then $\check{\mu}^{\Gamma}\leq_{cvx}\check{\eta}^{\Gamma}$.
This can be seen using the affine interpolation on $\Gamma$

$\check{\varphi}^{\Gamma}(x):=1_{(-\infty,x_{1})\cup[x_{N},+\infty)}(x)\varphi(x)+\sum_{i=1}^{N-1}1_{[x_{i},x_{i+1})}(x)\left(\frac{x_{i+1}-x}{x_{i+1}-x_{i}}\varphi(x_{i})+\frac{x-x_{i}}{x_{i+1}-x_{i}}\varphi(x_{i+1})\right)$

of a convex function $\varphi:{R}\to{R}$. Indeed $\check{\varphi}^{\Gamma}$ is
still convex and one has

$\int_{R}\varphi(x)\check{\mu}^{\Gamma}(dx)=\int_{R}\check{\varphi}^{\Gamma}(x)\mu(dx)\leq\int_{R}\check{\varphi}^{\Gamma}(x)\eta(dx)=\int_{R}\varphi(x)\check{\eta}^{\Gamma}(dx).$

According to the introduction in [2], this convex order preservation does not
generalize to higher dimensions where the minimizers are not so easy to
express.

Whatever the dimension $d\in{N}^{*}$, for every level $N\geq d+1$, there
exists an $L^{p}$-optimal dual quantization grid $\Gamma^{del}_{p,N}$ and a
splitting operator ${\rm Proj}_{\Gamma^{del}_{p,N}}^{del}$ (see [44]) such
that

$d_{p,N}(\mu)=d_{p}(\mu,\Gamma^{del}_{p,N})=\|X-{\rm
Proj}_{\Gamma^{del}_{p,N}}^{del}(X,U)\|_{p}$

and ${\rm Proj}_{\Gamma^{del}_{p,N}}^{del}(X,U)$ takes each value in
$\Gamma^{del}_{p,N}$ with positive probability. For more details on this dual
projection, see [44, 45] where this notion has been developed and analyzed. We
will see in the examples that even in dimension one, the convex order is not
preserved by optimal dual quantization.

Notice that by Proposition 1.1 $(b)$, the inequality $W_{p}^{p}(\mu,\nu)\leq
M_{p}^{p}(\mu,\nu)$ valid for $\nu\in{\cal P}_{\geq\mu}({R}^{d},N)$ and
(2.13),

$e_{p,N}(\mu)=W_{p}(\mu,{\cal P}({R}^{d},N))\leq W_{p}(\mu,{\cal
P}_{\geq\mu}({R}^{d},N))\leq\inf_{\nu\in{\cal
P}_{\geq\mu}({R}^{d},N)}M_{p}(\nu,\mu)=d_{p,N}(\mu).$ (2.16)

We may wonder whether the last inequality is an equality. Combining the
tightness of any sequence of probability measures in
${\mathcal{P}}_{\geq\mu}({R}^{d},N)$ minimizing the $W_{p}$-distance to $\mu$
deduced from the inequality

$\forall\nu\in{\mathcal{P}}({R}^{d}),\;\int_{{R}^{d}}|x|^{p}\nu(dx)\leq
2^{p-1}\left(\int_{{R}^{d}}|x|^{p}\mu(dx)+W_{p}^{p}(\mu,\nu)\right),$

the closedness of ${\mathcal{P}}_{\geq\mu}({R}^{d},N)$ for the weak
convergence topology and the lower semi-continuity of the Wasserstein distance
for this topology (see for instance Remark 6.12 p97 [51]), we obtain the
existence of a $W_{p}$-projection $\tilde{\mu}$ of $\mu$ on
${\mathcal{P}}_{\geq\mu}({R}^{d},N)$. The last inequality in (2.16) is an
equality iff the set

$\left\\{\pi\in{\cal
P}(\mu,\tilde{\mu}):W^{p}_{p}(\mu,\tilde{\mu})=\int_{{R}^{d}\times{R}^{d}}|y-x|^{p}\pi(dx,dy)\right\\}$

of $W_{p}$-optimal couplings between $\mu$ and some $W_{p}$-projection
$\tilde{\mu}$ of $\mu$ on ${\mathcal{P}}_{\geq\mu}({R}^{d},N)$ intersects
${\mathcal{M}}(\mu,\tilde{\mu})$. Moreover, $\tilde{\mu}$ is then the
distribution of an $L^{p}$-optimal dual $N$-quantization of $\mu$. But there
is no reason why the intersection should be non empty. We also may wonder
whether, by a somewhat naive symmetry with the situation described in
Proposition 1.1 $(b)$ for the Voronoi quantization, the distribution of an
$L^{p}$-optimal dual $N$-quantization of $\mu$ coincides with a
$W_{p}$-projection $\tilde{\mu}$ of $\mu$ on
${\mathcal{P}}_{\geq\mu}({R}^{d},N)$. According to the examples below, this
property holds when $\mu$ is the uniform distribution on the interval $[0,1]$
(nevertheless $W_{p}({\mathcal{U}}[0,1],{\cal
P}_{\geq{\mathcal{U}}[0,1]}({R}^{d},N))<d_{p,N}({\mathcal{U}}[0,1])$) but is
not true in general.

Note that, in the quadratic case $p=2$, for $\nu\in{\cal
P}_{\geq\mu}({R}^{d},N)$, since
$M_{2}^{2}(\mu,\nu)=\int_{{R}^{d}}|y|^{2}\nu(dy)-\int_{{R}^{d}}|x|^{2}\mu(dx)$,

$d_{2,N}(\mu)^{2}=\inf_{\nu\in{\mathcal{P}}_{\geq\mu}({R}^{d},N)}\int_{{R}^{d}}|y|^{2}\nu(dy)-\int_{{R}^{d}}|x|^{2}\mu(dx).$
(2.17)

###### Proposition 2.1

In the quadratic case $p=2$, any optimal dual quantization grid
$\Gamma^{del}_{2,N}$ remains optimal for each probability measure $\eta$ such
that $\mu\leq_{cvx}\eta\leq_{cvx}\check{\mu}_{N}$ where $\check{\mu}_{N}$
denotes the distribution of ${\rm Proj}_{\Gamma^{del}_{2,N}}^{del}(X,U)$.

Proof. Let $\eta$ be such that $\mu\leq_{cvx}\eta\leq_{cvx}\check{\mu}_{N}$.
Since
$\inf_{\nu\in{\mathcal{P}}_{\geq\mu}({R}^{d},N)}\int_{{R}^{d}}|y|^{2}\nu(dy)$
is attained for $\nu=\check{\mu}_{N}$, so is
$\inf_{\nu\in{\mathcal{P}}_{\geq\eta}({R}^{d},N)}\int_{{R}^{d}}|y|^{2}\nu(dy)$.
Therefore, (2.17) written with $\eta$ replacing $\mu$ implies that

$d_{2,N}(\eta)^{2}=\int_{{R}^{d}}|y|^{2}\check{\mu}_{N}(dy)-\int_{{R}^{d}}|x|^{2}\eta(dx)=M_{2}^{2}(\eta,\check{\mu}_{N}).$

With the definition of $d_{2}(\eta,\Gamma^{del}_{2,N})$, we deduce that
$d_{2,N}(\eta)^{2}\geq d_{2}(\eta,\Gamma^{del}_{2,N})^{2}$. Therefore
$\Gamma^{del}_{2,N}$ is an optimal dual quadratic quantization grid for
$\eta$. $\Box$

Examples. $(a)$ Let $\mu={\cal U}[0,1]$ where, for two real numbers $a<b$,
${\cal U}[a,b]$ denotes the uniform distribution on $[a,b]$ with density
$\frac{1_{[a,b]}(x)}{b-a}$ with respect to the Lebesgue measure. We consider
its approximation by probability measures in ${\cal P}({R},N)$. A generic
element of ${\cal P}({R},N)$ writes

$\nu_{N}=\sum_{k=1}^{N}p_{k}\delta_{x_{k}}$

with $x_{1}\leq x_{2}\leq\ldots\leq x_{N}$ and
$(p_{1},\ldots,p_{N})\in[0,1]^{N}$ satisfying $\sum_{k=1}^{N}p_{k}=1$. We will
consider the particular choices
$\hat{\mu}_{N}=\frac{1}{N}\sum_{k=1}^{N}\delta_{\frac{2k-1}{2N}}$ and
$\check{\mu}_{N}=\frac{1}{2(N-1)}\delta_{0}+\frac{1}{N-1}\sum_{k=2}^{N-1}\delta_{\frac{k-1}{N-1}}+\frac{1}{2(N-1)}\delta_{1}$
of the respective distributions of the optimal primal and dual quantizations
of $\mu$ on $N$ points. For $p\geq 1$, let us recover that $\hat{\mu}_{N}$ is
the $W_{p}$-projection of ${\cal U}[0,1]$ on ${\cal P}({R},N)$ (consequence of
Proposition 1.1 $(b)$) and check that $\check{\mu}_{N}$ is the
$W_{p}$-projection of ${\cal U}[0,1]$ on ${\cal P}_{\geq{\cal
U}[0,1]}({R},N)$. The image of $\mu$ by $(0,1)\ni u\mapsto
F_{\nu_{N}}^{-1}(u)-u$ is equal to

$\eta_{N}:=p_{1}{\cal U}[x_{1}-p_{1},x_{1}]+p_{2}{\cal
U}[x_{2}-(p_{1}+p_{2}),x_{2}-p_{1}]+\ldots+p_{N}{\cal
U}[x_{N}-1,x_{N}-(p_{1}+\ldots+p_{N-1})].$

When $\nu_{N}=\hat{\mu}_{N}$ (resp. $\nu_{N}=\check{\mu}_{N}$) then
$\eta_{N}=\hat{\eta}_{N}:={\cal U}[-\frac{1}{2N},\frac{1}{2N}]$ (resp.
$\eta_{N}=\check{\eta}_{N}:={\cal U}[-\frac{1}{2(N-1)},\frac{1}{2(N-1)}]$).
Since $\hat{\eta}_{N}=N1_{[-\frac{1}{2N},\frac{1}{2N}]}(x)dx$ and $\eta_{N}$
has a density with respect to the Lebesgue measure with values in
$\\{0,1,2,\ldots,N\\}$, $\left(\eta_{N}-\hat{\eta}_{N}\right)^{+}$ is
supported on the complement of $[\frac{-1}{2N},\frac{1}{2N}]$ where
$\left(\hat{\eta}_{N}-\eta_{N}\right)^{+}$ is supported. Since both measures
share the same mass, we deduce that for $p\geq 1$,

$\int_{{R}}|x|^{p}\left(\eta_{N}-\hat{\eta}_{N}\right)^{+}(dx)\geq\int_{R}|x|^{p}\left(\hat{\eta}_{N}-\eta_{N}\right)^{+}(dx)\mbox{
i.e.}\int_{R}|x|^{p}\eta_{N}(dx)\geq\int_{R}|x|^{p}\hat{\eta}_{N}(dx).$

Using that in dimension $d=1$, the comonotonous coupling is $W_{p}$-optimal,
we conclude that

$\forall\nu_{N}\in{\cal P}({R},N),\;W_{p}^{p}({\cal
U}[0,1],\nu_{N})=\int_{{R}}|x|^{p}\eta_{N}(dx)\geq\int_{R}|x|^{p}\hat{\eta}_{N}(dx)=W_{p}^{p}({\cal
U}[0,1],\hat{\nu}_{N}),$

with strict inequality unless $\eta_{N}=\hat{\eta}_{N}$. When $\eta_{N}$ is
supported on $[-\frac{1}{2N},\frac{1}{2N}]$ (which is clearly equivalent to
$\eta_{N}=\hat{\eta}_{N}$), then for each $k\in\\{1,\ldots,N\\}$,
$\frac{-1}{2N}\leq x_{k}-(p_{1}+\ldots+p_{k})$ and
$x_{k}-(p_{1}+\ldots+p_{k-1})\leq\frac{1}{2N}$ so that $p_{k}\leq\frac{1}{N}$.
With the normalisation, we deduce that $p_{\ell}=\frac{1}{N}$ for each
$\ell\in\\{1,\ldots,N\\}$, which plugged in the two last inequalities implies
that $x_{k}=\frac{2k-1}{2N}$ for each $k\in\\{1,\ldots,N\\}$ so that
$\nu_{N}=\hat{\mu}_{N}$. Therefore, for all $p\geq 1$, $\hat{\mu}_{N}$ is the
$W_{p}$-projection of ${\cal U}[0,1]$ on ${\cal P}({R},N)$ and

$W_{p}({\cal U}[0,1],{\cal P}({R},N))=W_{p}({\cal
U}[0,1],\hat{\mu}_{N})=\left(\int_{{R}}|x|^{p}\hat{\eta}_{N}(dx)\right)^{1/p}=\frac{1}{2(p+1)^{1/p}N}.$

If $\nu_{N}\geq_{cvx}{\cal U}[0,1]$, then $x_{1}\leq 0$ and $x_{N}\geq 1$ so
that
$[x_{1}-p_{1},x_{1}]\cap[x_{N}-1,x_{N}-(p_{1}+\ldots+p_{N-1})]\subset\\{0\\}$
and $\eta_{N}$ has a density with respect to the Lebesgue measure with values
in $\\{0,1,\ldots,N-1\\}$. By repeating the previous argument with
$\hat{\eta}_{N}$ replaced by $\check{\eta}_{N}$, we obtain that

$\displaystyle\forall\nu_{N}\in{\cal P}_{\geq{\cal
U}[0,1]}({R},N),\;W_{p}^{p}({\cal
U}[0,1],\nu_{N})=\int_{{R}}|x|^{p}\eta_{N}(dx)\geq\int_{{R}}|x|^{p}\check{\eta}_{N}(dx)=W_{p}^{p}({\cal
U}[0,1],\check{\mu}_{N}),$

with strict inequality unless $\eta_{N}=\check{\eta}_{N}$. When
$\nu_{N}\geq_{cvx}{\cal U}[0,1]$ and $\eta_{N}$ is supported on
$[-\frac{1}{2(N-1)},\frac{1}{2(N-1)}]$ (which is clearly equivalent to
$\eta_{N}=\check{\eta}_{N}$), then for each $k\in\\{1,\ldots,N\\}$,
$\frac{-1}{2(N-1)}\leq x_{k}-(p_{1}+\ldots+p_{k})$ and
$x_{k}-(p_{1}+\ldots+p_{k-1})\leq\frac{1}{2(N-1)}$ so that
$p_{k}\leq\frac{1}{N-1}$ and with the reinforced inequalities $x_{1}\leq 0$
and $x_{N}\geq 1$ due to the convex order, $p_{1}\leq\frac{1}{2(N-1)}$ and
$p_{N}\leq\frac{1}{2(N-1)}$. With the normalisation, we deduce that
$p_{1}=p_{N}=\frac{1}{2(N-1)}$ and $p_{\ell}=\frac{1}{N-1}$ for each
$\ell\in\\{2,\ldots,N-1\\}$, which plugged in the two (reinforced)
inequalities implies that $x_{1}=0$, $x_{k}=\frac{k-1}{N-1}$ for each
$k\in\\{2,\ldots,N-1\\}$ and $x_{1}=1$ so that $\nu_{N}=\check{\mu}_{N}$.
Therefore, for all $p\geq 1$, $\check{\mu}_{N}$ is the $W_{p}$-projection of
${\cal U}[0,1]$ on ${\cal P}_{\geq{\cal U}[0,1]}({R},N)$ and

$\displaystyle W_{p}({\cal U}[0,1],{\cal P}_{\geq{\cal U}[0,1]}({R},N))$
$\displaystyle=W_{p}({\cal
U}[0,1],\check{\mu}_{N})=\left(\int_{{R}}|x|^{p}\check{\eta}_{N}(dx)\right)^{1/p}=\frac{1}{2(p+1)^{1/p}(N-1)}$
$\displaystyle<\left(\frac{2}{(p+1)(p+2)}\right)^{1/p}\frac{1}{N-1}=d_{2,N}({\mathcal{U}}[0,1]).$

Notice that when $\nu_{N}\geq_{cvx}{\cal U}[0,1]$, then the measures
$(\eta_{N}-\check{\eta}_{N})^{+}$ supported on
$[-\frac{1}{2(N-1)},\frac{1}{2(N-1)}]^{c}$ and
$(\check{\eta}_{N}-\eta_{N})^{+}$ supported on
$[-\frac{1}{2(N-1)},\frac{1}{2(N-1)}]$ share the same mass and barycenter so
that for each convex function $\varphi:{R}\to{R}$,
$\int_{R}\varphi(x)(\eta_{N}-\check{\eta}_{N})^{+}(dx)\geq\int_{R}\varphi(x)(\check{\eta}_{N}-\eta_{N})^{+}(dx)$
and $\eta_{N}\geq_{cvx}\check{\eta}_{N}$.

The probability distribution ${\cal U}[0,1]$ is smaller in the convex order
than its dual quantization
$\mu_{6}:=\frac{1}{5}\delta_{0}+\frac{7}{30}\delta_{\frac{2}{5}}+\frac{1}{15}\delta_{\frac{7}{15}}+\frac{1}{15}\delta_{\frac{8}{15}}+\frac{7}{30}\delta_{\frac{3}{5}}+\frac{1}{5}\delta_{1}$
on the grid
$\left\\{0,\frac{2}{5},\frac{7}{15},\frac{8}{15},\frac{3}{5},1\right\\}$. Note
that $\mu_{6}$ is not comparable with
$\check{\mu}_{6}=\frac{1}{10}\delta_{0}+\frac{1}{5}\left(\delta_{\frac{1}{5}}+\delta_{\frac{2}{5}}+\delta_{\frac{3}{5}}+\delta_{\frac{4}{5}}\right)+\frac{1}{10}\delta_{1}$
for the convex order. Indeed, we have

$\displaystyle\mu_{6}=\frac{2}{5}\left(\frac{1}{2}\delta_{0}+\frac{1}{2}\delta_{\frac{2}{5}}\right)+\frac{1}{5}\left(\frac{1}{6}\delta_{\frac{2}{5}}+\frac{1}{3}\delta_{\frac{7}{15}}+\frac{1}{3}\delta_{\frac{8}{15}}+\frac{1}{6}\delta_{\frac{3}{5}}\right)+\frac{2}{5}\left(\frac{1}{2}\delta_{\frac{3}{5}}+\frac{1}{2}\delta_{1}\right)$
$\displaystyle\check{\mu}_{6}=\frac{2}{5}\left(\frac{1}{4}\delta_{0}+\frac{1}{2}\delta_{\frac{1}{5}}+\frac{1}{4}\delta_{\frac{2}{5}}\right)+\frac{1}{5}\left(\frac{1}{2}\delta_{\frac{2}{5}}+\frac{1}{2}\delta_{\frac{3}{5}}\right)+\frac{2}{5}\left(\frac{1}{4}\delta_{\frac{3}{5}}+\frac{1}{2}\delta_{\frac{4}{5}}+\frac{1}{4}\delta_{1}\right)$
$\displaystyle\mbox{ with
}\frac{1}{2}\delta_{0}+\frac{1}{2}\delta_{\frac{2}{5}}\geq_{cvx}\frac{1}{4}\delta_{0}+\frac{1}{2}\delta_{\frac{1}{5}}+\frac{1}{4}\delta_{\frac{2}{5}},\;\frac{1}{6}\delta_{\frac{2}{5}}+\frac{1}{3}\delta_{\frac{7}{15}}+\frac{1}{3}\delta_{\frac{8}{15}}+\frac{1}{6}\delta_{\frac{3}{5}}\leq_{cvx}\frac{1}{2}\delta_{\frac{2}{5}}+\frac{1}{2}\delta_{\frac{3}{5}}$
$\displaystyle\mbox{ and
}\frac{1}{2}\delta_{\frac{3}{5}}+\frac{1}{2}\delta_{1}\geq_{cvx}\frac{1}{4}\delta_{\frac{3}{5}}+\frac{1}{2}\delta_{\frac{4}{5}}+\frac{1}{4}\delta_{1}$

so that
$\int_{{R}}\varphi(x)\mu_{6}(dx)>\int_{{R}}\varphi(x)\check{\mu}_{6}(dx)$
(resp.
$\int_{{R}}\varphi(x)\mu_{6}(dx)<\int_{{R}}\varphi(x)\check{\mu}_{6}(dx)$)
when the convex function $\varphi:{R}\to{R}$ is strictly convex on
$[0,\frac{2}{5}]$ and affine on $[\frac{2}{5},1]$ (resp. affine on
$[0,\frac{2}{5}]$, strictly convex on $[\frac{2}{5},\frac{3}{5}]$ and affine
on $[\frac{3}{5},1]$). Since $\mu_{6}$ is clearly equal to its $L^{p}$-optimal
dual $6$-quantization, this shows that the convex order is not preserved by
optimal dual quantization.

$(b)$ Let now $\mu(dx)=2x\mbox{\bf 1}_{[0,1]}(x)dx$. We look for
$\nu\in{\mathcal{P}}_{\geq\mu}({R},3)$ minimizing either
$\int_{{R}}y^{2}\nu(dy)$ to compute the law of the optimal quadratic dual
quantization of $\mu$ on $N=3$ points or $W_{2}^{2}(\mu,\nu)$ to compute
$\tilde{\mu}$. Since $d=1$, $W_{2}^{2}(\mu,\nu)$ is equal to the integral
$\int_{0}^{1}(F_{\mu}^{-1}(u)-F_{\nu}^{-1}(u))^{2}du$ of the squared
difference between the quantile functions of $\mu$ and $\nu$. For the first
criterion, we are going to check that it is equivalent to minimize over the
following parametric subset of ${\mathcal{P}}_{\geq\mu}({R},3)$

$\left\\{\nu_{u}(dy)=\frac{u}{3}\delta_{0}(dy)+\frac{1+\sqrt{u}}{3}\delta_{\sqrt{u}}(dy)+\frac{2-\sqrt{u}-u}{3}\delta_{1}(dy):u\in(0,1)\right\\}.$

One has $\int y^{2}\nu_{u}(dy)=\frac{2+u^{3/2}-\sqrt{u}}{3}$ and the infimum
is attained for $u=1/3$ so that
$\Gamma^{del}_{2,3}=\\{0,\frac{1}{\sqrt{3}},1\\}$,
$d_{2,3}(\mu)^{2}=d_{2}(\mu,\Gamma^{del}_{2,3})^{2}=\int
y^{2}\nu_{1/3}(dy)-\int x^{2}\mu(dx)=\frac{1}{6}-\frac{2}{3^{5/2}}$. On the
other hand,

$\displaystyle W_{2}^{2}(\mu,\nu_{u})$
$\displaystyle=\int_{0}^{u/3}(0-\sqrt{v})^{2}dv+\int_{u/3}^{(1+\sqrt{u}+u)/3}(\sqrt{u}-\sqrt{v})^{2}dv+\int_{(1+\sqrt{u}+u)/3}^{1}(1-\sqrt{v})^{2}dv$
$\displaystyle=-\frac{1}{6}+\frac{u^{3/2}-\sqrt{u}}{3}+4\frac{(1-\sqrt{u})(1+\sqrt{u}+u)^{3/2}+u^{2}}{3^{5/2}}.$

One easily checks that $\frac{d}{du}W_{2}^{2}(\mu,\nu_{u})|_{u=1/3}>0$ and
that $W_{2}^{2}(\mu,\nu_{u})$ is minimal for $u\simeq 0.326$ so that
$\tilde{\mu}\neq\nu_{1/3}$. Moreover, since ${\cal
P}_{\geq\mu}(\Gamma^{del}_{2,3})=\\{v\nu_{1/3}+(1-v)\nu_{0}:v\in[0,1]\\}$
contains $\nu_{1/3}$,

$W_{2}^{2}(\mu,{\cal P}_{\geq\mu}(\Gamma^{del}_{2,3}))\leq
W^{2}_{2}(\mu,\nu_{1/3})\simeq 0.0199758<0.0383666\simeq
d_{2}(\mu,\Gamma^{del}_{2,3})^{2}.$

Let us finally check that
$\inf_{\nu\in{\mathcal{P}}_{\geq\mu}({R},3)}\int_{R}y^{2}\nu(dy)\geq\inf_{u\in(0,1)}\int_{R}y^{2}\nu_{u}(dy)$
and that $\nu_{u}\in{\mathcal{P}}_{\geq\mu}({R},3)$ for each $u\in(0,1)$.
First note that for each $u\in(0,1)$, the mean $2/3$ of $\nu_{u}$ is equal to
the one of $\mu$. According to the characterization of the convex order in
terms of potential functions, we have $\nu\geq_{cvx}\mu$ iff
$\int_{R}x\nu(dx)=\frac{2}{3}$ and

$\forall
x\in{R},\;\varphi_{\nu}(x):=\int_{-\infty}^{x}\nu((-\infty,y])dy\geq\varphi_{\mu}(x)=1_{[0,1]}(x)\frac{x^{3}}{3}+1_{\\{x>1\\}}\left(x-\frac{2}{3}\right).$
(2.18)

If $\nu$ weights at most three points, then the convex function
$\varphi_{\nu}$ is piecewise affine with at most three changes of slope, the
left-most slope being equal to $0$ and the right-most equal to $1$. Therefore
if $\nu\in{\mathcal{P}}_{\geq\mu}({R},3)$ then $\nu(\\{a\\})>0$ for some
$a\leq 0$ and $\nu(\\{b\\})>0$ for some $b\geq 1$. If, moreover,
$\nu(\\{\sqrt{u}\\})>0$ for some $u\in(0,1)$, then since the slope of
$\varphi_{\nu}$ is constant on $(0,\sqrt{u})$ and on $(\sqrt{u},1)$, for all
$x\in{R}$,

$\varphi_{\nu}(x)\geq
1_{(0,\sqrt{u}]}(x)\frac{u^{3/2}}{3}\times\frac{x}{\sqrt{u}}+1_{(\sqrt{u},1]}(x)\left(\frac{u^{3/2}}{3}\times\frac{1-x}{1-\sqrt{u}}+\frac{1}{3}\times\frac{x-\sqrt{u}}{1-\sqrt{u}}\right)+1_{\\{x>1\\}}\left(x-\frac{2}{3}\right)=\varphi_{\nu_{u}}(x),$

so that, by convexity of the square function,
$\int_{R}y^{2}\nu(dy)\geq\int_{R}y^{2}\nu_{u}(dy)$. If, on the other hand,
$\nu((0,1))=0$, then $\varphi_{\nu}$ has a constant slope on $(0,1)$ and we
even have

$\int_{R}y^{2}\nu(dy)\geq\sup_{u\in(0,1)}\int_{R}y^{2}\nu_{u}(dy).$

Therefore
$\inf_{\nu\in{\mathcal{P}}_{\geq\mu}({R},3)}\int_{R}y^{2}\nu(dy)\geq\inf_{u\in(0,1)}\int_{R}y^{2}\nu_{u}(dy)$.
To conclude that this inequality is an equality, it is enough to check that
$\nu_{u}\in{\mathcal{P}}_{\geq\mu}({R},3)$ for each $u\in(0,1)$. This follows
from the inequality $\varphi_{\nu_{u}}(x)\geq\varphi_{\mu}(x)$ valid for all
$x\in{R}$ and all $u\in[0,1]$ since the graph of the convex function
$\varphi_{\mu}$ is under its chords.

We finally recall the main result on convergence rate of dual quantization for
bounded random vectors established in [46].

######  Theorem 2.2 (Pierce Lemma for dual quantization)

Let $p\geq 1$ and $\eta>0$. For every dimension $d\geq 1$, there exists a real
constant $\widetilde{C}^{del}_{d,\eta,p}>0$ such that, for every random vector
$X:(\Omega,{\cal A},{P})\to{R}^{d}$, $L^{\infty}({P})$-bounded,

$d_{p,N}(X)\leq\widetilde{C}^{del}_{d,\eta,p}N^{-\frac{1}{d}}\sigma_{p+\eta}(X)$
(2.19)

where, for every $r>0$, $\sigma_{r}(X)=\inf_{a\in{R}^{d}}\|X-a\|_{r}<+\infty$.

Remark. Note that this claim and the one in Theorem 1.3 remain true if the
support of ${P}_{{}_{X}}$ does not span ${R}^{d}$ as an affine space, but
$A_{\mu}$ with dimension $d^{\prime}<d$. However, if such is the case, then
(1.12) and (2.19) hold with factor $N^{-1/d^{\prime}}$ so that $N^{-1/d}$ is
suboptimal.

##  3 Quantized approximations of martingale couplings

Let $\nu\in{\mathcal{P}}({R}^{d})$ be compactly supported and $\mu\in{\cal
P}_{2}({R}^{d})$. For $K,N\geq 1$, let $\check{\nu}^{K}$ be an $L^{p}$-optimal
dual $K$-quantization of $\nu$ with grid $\Gamma^{del}_{p,K}$ and
$\hat{\mu}^{N}$ be a quadratic optimal primal $N$-quantization of $\mu$ with
grid $\Gamma_{2,N}$. The two quantization errors correspond to martingale
quasi-metrics (in comparison to Wasserstein metrics (0.2), only martingale
couplings are considered in the minimization defining (0.3)) between $\nu$
(resp. $\hat{\mu}_{N}$) and $\check{\nu}^{K}$ (resp. $\mu$) :

$d_{p,K}^{p}(\nu)=M_{p}^{p}(\nu,\check{\nu}^{K})\mbox{ and
}e_{2,N}^{2}(\mu)=M_{2}^{2}(\hat{\mu}^{N},\mu).$ (3.20)

In contrast with dual quantization where the martingale quasi-metric appears
from the very beginning of the construction, the optimization in primal
quantization relies on Wasserstein metrics. But in the quadratic $p=2$ case,
the stationarity property (1.8) satisfied at optimality implies that
$W_{2}(\hat{\mu}_{N},\mu)=M_{2}(\hat{\mu}_{N},\mu)$.

Let $U\sim{\mathcal{U}}[0,1]$ and $q_{y}(d\check{y})$ denote the law of ${\rm
Proj}^{del}_{\Gamma^{del}_{p,K}}(y,U)$ when $y\in{\rm
Conv}(\Gamma^{del}_{p,K})$ and $\delta_{y}(d\check{y})$ otherwise. When
$\mu\leq_{cvx}\nu$, we are now going to exploit (3.20) to approximate any
$\pi\in{\mathcal{M}}(\mu,\nu)$ by

$\bar{\pi}^{N,K}(d\hat{x},d\check{y})=\int_{(x,y)\in{R}^{d}\times{R}^{d}}\delta_{{\rm
Proj}_{\Gamma_{2,N}}(x)}(d\hat{x})\pi(dx,dy)q_{y}(d\check{y}).$ (3.21)

For $(X,Y)\sim\pi$ independent from the random variable $U$ uniformly
distributed on $[0,1]$, the random vector $({\rm
Proj}_{\Gamma_{2,N}}(X),X,Y,{\rm Proj}_{\Gamma^{del}_{p,K}}^{del}(Y,U))$ is
distributed according to $\delta_{{\rm
Proj}_{\Gamma_{2,N}}(x)}(d\hat{x})\pi(dx,dy)q_{y}(d\check{y})$ and therefore
$({\rm Proj}_{\Gamma_{2,N}}(X),{\rm Proj}_{\Gamma^{del}_{p,K}}^{del}(Y,U))$ is
distributed according to $\bar{\pi}^{N,K}$. Therefore, the first marginal of
$\bar{\pi}^{N,K}$ is the distribution $\hat{\mu}^{N}$ of ${\rm
Proj}_{\Gamma_{2,N}}(X)$ and its second marginal is the law $\check{\nu}^{K}$
of ${\rm Proj}_{\Gamma^{del}_{p,K}}^{del}(Y,U)$. Moreover, using that $U$ is
independent from $(X,Y)$ and (2.14) for the second equality, then that $\pi$
is a martingale coupling for the fourth equality and the stationarity property
(1.8) for the last one, we obtain

$\displaystyle{E}[{\rm Proj}_{\Gamma^{del}_{p,K}}^{del}(Y,U))|{\rm
Proj}_{\Gamma_{2,N}}(X)]$ $\displaystyle={E}[{E}[{\rm
Proj}_{\Gamma^{del}_{p,K}}^{del}(Y,U))|(X,Y)]|{\rm
Proj}_{\Gamma_{2,N}}(X)]={E}[Y|{\rm Proj}_{\Gamma_{2,N}}(X)]$
$\displaystyle={E}[{E}[Y|X]|{\rm Proj}_{\Gamma_{2,N}}(X)]={E}[X|{\rm
Proj}_{\Gamma_{2,N}}(X)]={\rm Proj}_{\Gamma_{2,N}}(X).$

so that $\bar{\pi}^{N,K}\in{\cal M}(\hat{\mu}^{N},\check{\nu}^{K})$.

###### Theorem 3.1

Let $p\geq 1$, $\mu,\nu\in{\mathcal{P}}({R}^{d})$ be such that
$\mu\leq_{cvx}\nu$ with $\nu$ compactly supported and for $N,K\geq 1$, let
$\hat{\mu}^{N}$ be a quadratic optimal primal $N$-quantization of $\mu$ and
$\check{\nu}^{K}$ be an $L^{p}$-optimal dual $K$-quantization of $\nu$. Then,
for each $\pi\in{\mathcal{M}}(\mu,\nu)$, the couplings
$\bar{\pi}^{N,K}\in{\mathcal{M}}(\hat{\mu}^{N},\check{\nu}^{K})$ defined in
(3.21) are such that

$W_{p}^{p}(\bar{\pi}^{N,K},\pi)\leq\begin{cases}e^{p}_{2,N}(\mu)+d^{p}_{p,K}(\nu)\mbox{
if $p\leq 2$}\\\ CN^{-p/d}+2^{\frac{(p-2)}{2}}d^{p}_{p,K}(\nu)\mbox{ for
$C<\infty$ not depending on $N,K$ if $2<p<2+d$}\end{cases}.$

Moreover, when $p\geq 2$,

$W_{2}^{2}(\bar{\pi}^{N,K},\pi)\leq e^{2}_{2,N}(\mu)+d_{p,K}^{2}(\nu).$

Last, for any $p\geq 1$,

$\limsup_{N\to\infty}AW_{p}(\bar{\pi}^{N,K},\pi)\leq d_{p,K}(\nu).$

According to Theorems 1.3 and 2.2, $\sup_{N\geq 1}N^{1/d}e_{2,N}(\mu)<\infty$
and $\sup_{K\geq 1}K^{1/d}d_{p,K}(\nu)<\infty$.

Proof. We have

$\displaystyle W_{p}^{p}(\bar{\pi}^{N,K},\pi)$
$\displaystyle\leq\int_{{R}^{d}\times{R}^{d}\times{R}^{d}\times{R}^{d}}\left(|\hat{x}-x|^{2}+|y-\check{y}|^{2}\right)^{p/2}\delta_{{\rm
Proj}_{\Gamma_{2,N}}(x)}(d\hat{x})\pi(dx,dy)q_{y}(d\check{y})$
$\displaystyle\leq
2^{\frac{(p-2)^{+}}{2}}\int_{{R}^{d}\times{R}^{d}\times{R}^{d}\times{R}^{d}}\left(|\hat{x}-x|^{p}+|y-\check{y}|^{p}\right)\delta_{{\rm
Proj}_{\Gamma_{2,N}}(x)}(d\hat{x})\pi(dx,dy)q_{y}(d\check{y})$
$\displaystyle=2^{\frac{(p-2)^{+}}{2}}\int_{{R}^{d}}|{\rm
Proj}_{\Gamma_{2,N}}(x)-x|^{p}\mu(dx)+2^{\frac{(p-2)^{+}}{2}}{E}\left[\left|Y-{\rm
Proj}^{del}_{\Gamma^{del}_{p,K}}(Y,U)\right|^{p}\right]$
$\displaystyle=2^{\frac{(p-2)^{+}}{2}}\int_{{R}^{d}}|{\rm
Proj}_{\Gamma_{2,N}}(x)-x|^{p}\mu(dx)+2^{\frac{(p-2)^{+}}{2}}d_{p,K}^{p}(\nu).$

When $p\leq 2$, $\int_{{R}^{d}}|{\rm
Proj}_{\Gamma_{2,N}}(x)-x|^{p}\mu(dx)\leq\left(\int_{{R}^{d}}|{\rm
Proj}_{\Gamma_{2,N}}(x)-x|^{2}\mu(dx)\right)^{p/2}=e^{p}_{2,N}(\mu)$. On the
other hand, when $2<p<2+d$, since $\mu$ is compactly supported, we may apply
the $L^{2}-L^{p}$\- distortion mismatch Theorem 4.3 [38] to obtain that
$\sup_{N\geq 1}N^{p/d}\int_{{R}^{d}}|{\rm
Proj}_{\Gamma_{2,N}}(x)-x|^{p}\mu(dx)<\infty$, which completes the proof of
the first inequality. In a similar way, when $p\geq 2$

$\displaystyle W_{2}^{2}(\bar{\pi}^{N,K},\pi)$
$\displaystyle\leq\int_{{R}^{d}\times{R}^{d}\times{R}^{d}\times{R}^{d}}\left(|\hat{x}-x|^{2}+|y-\check{y}|^{2}\right)\delta_{{\rm
Proj}_{\Gamma_{2,N}}(x)}(d\hat{x})\mu(dx)\pi_{x}(dy)q_{y}(d\check{y})$
$\displaystyle=e_{2,N}^{2}(\mu)+{E}\left[\left|Y-{\rm
Proj}^{del}_{\Gamma^{del}_{p,K}}(Y,U)\right|^{2}\right]\leq
e_{2,N}^{2}(\mu)+{E}\left[\left|Y-{\rm
Proj}^{del}_{\Gamma^{del}_{p,K}}(Y,U)\right|^{p}\right]^{2/p}$
$\displaystyle=e_{2,N}^{2}(\mu)+d_{p,N}^{2}(\nu).$

Let now

$\check{\pi}^{K}(dx,d\check{y})=\int_{y\in{R}^{d}}\pi(dx,dy)q_{y}(d\check{y})=\int_{y\in{R}^{d}}\mu(dx)\pi_{x}(dy)q_{y}(d\check{y}).$

We have $\check{\pi}^{K}\in{\cal M}(\mu,\check{\nu}^{K})$. Using the identity
coupling $\mu(dx)\delta_{x}(d\tilde{x})$ between $\mu$ and $\mu$ in the
definition of the adapted Wasserstein distance then the coupling
$\pi_{x}(dy)q_{y}(d\check{y})$ between $\check{\pi}^{K}_{x}(d\check{y})$ and
$\pi_{x}(dy)$ in the definition of the usual Wasserstein distance, we obtain
that

$\displaystyle AW^{p}_{p}(\check{\pi}^{K},\pi)$
$\displaystyle\leq\int_{{R}^{d}}W^{p}_{p}(\check{\pi}^{K}_{x},\pi_{x})\mu(dx)\leq\int_{{R}^{d}}\int_{{R}^{d}}\int_{0}^{1}|{\rm
Proj}^{del}_{\Gamma^{del}_{p,K}}(y,u)-y|^{p}du\pi_{x}(dy)\mu(dx)$
$\displaystyle\leq\int_{{R}^{d}}\int_{0}^{1}|y-{\rm
Proj}^{del}_{\Gamma^{del}_{p,K}}(y,u)|^{p}du\nu(dy)={E}|Y-{\rm
Proj}^{del}_{\Gamma^{del}_{p,K}}(Y,U)|^{p}=d^{p}_{p,K}(\nu).$

With the triangle inequality, we deduce that to check the last statement in
the theorem, it is enough to prove that
$\lim_{N\to\infty}AW_{p}(\bar{\pi}^{N,K},\check{\pi}^{K})=0$.

To do so, we denote by $(x_{i})_{1\leq i\leq|\Gamma_{2,N}|}$ the
$|\Gamma_{2,N}|\leq N$ points in $\Gamma_{2,N}$ and by $(y_{j})_{1\leq
j\leq|\Gamma^{del}_{p,K}|}$ the $|\Gamma^{del}_{p,K}|\leq K$ points in
$\Gamma^{del}_{p,K}$ (as soon as the support of $\mu$ (resp. $\nu$) is not
restricted to less than $N$ (resp. $K$) points, $|\Gamma_{2,N}|=N$ (resp.
$|\Gamma^{del}_{p,K}|=K)$). Let $C_{i}=\\{x\in{R}^{d}:|x-x_{i}|<\min_{1\leq
k\leq|\Gamma_{2,N}|,k\neq i}|x-x_{k}|\\}$,
$i\in\\{1,\ldots,|\Gamma_{2,N}|\\}$, denote the open Voronoi cells induced by
$\Gamma_{2,N}$. Using (1.7) for the equality, we have

$\displaystyle AW^{p}_{p}(\bar{\pi}^{N,K},\check{\pi}^{K})$
$\displaystyle\leq\int_{{R}^{d}\times{R}^{d}}(|\hat{x}-x|^{p}+W_{p}^{p}(\bar{\pi}^{N,K}_{\hat{x}},\check{\pi}^{K}_{x}))\delta_{{\rm
Proj}_{\Gamma_{2,N}}(x)}(d\hat{x})\mu(dx)$
$\displaystyle=\sum_{i=1}^{|\Gamma_{2,N}|}\int_{C_{i}}(|x_{i}-x|^{p}+W_{p}^{p}(\bar{\pi}^{N,K}_{x_{i}},\check{\pi}^{K}_{x}))\mu(dx).$
(3.22)

Since $\check{\nu}^{K}(\Gamma^{del}_{p,K})=1$, $\mu(dx)$ a.e.
$\check{\pi}^{K}_{x}(d\check{y})=\sum_{j=1}^{|\Gamma^{del}_{p,K}|}q_{j}(x)\delta_{y_{j}}(d\check{y})$
for some measurable functions $q_{j}$ with values in $[0,1]$ and such that
$\sum_{j=1}^{|\Gamma^{del}_{p,K}|}q_{j}=1$. Moreover, for
$i\in\\{1,\ldots,|\Gamma_{2,N}|\\}$,

$\bar{\pi}^{N,K}_{x_{i}}(d\check{y})=\frac{1}{\mu(C_{i})}\sum_{j=1}^{|\Gamma^{del}_{p,K}|}\int_{C_{i}}q_{j}(\xi)\mu(d\xi)\delta_{y_{j}}(d\check{y}).$

As a consequence, we have

$\displaystyle{\rm
TV}(\bar{\pi}^{N,K}_{x_{i}},\check{\pi}^{K}_{x})=\sum_{j=1}^{|\Gamma^{del}_{p,K}|}\left|q_{j}(x)-\frac{1}{\mu(C_{i})}\int_{C_{i}}q_{j}(\xi)\mu(d\xi)\right|.$

With Theorem 6.15 p.103 [51], we deduce that

$\displaystyle
W_{p}^{p}(\bar{\pi}^{N,K}_{x_{i}},\check{\pi}^{K}_{x})=2^{p-1}\max_{1\leq
j\leq|\Gamma^{del}_{p,K}|}|y_{j}|^{p}{\rm
TV}(\hat{\pi}^{N,K}_{x_{i}},\hat{\pi}^{K}_{x})\leq 2^{p-1}\max_{1\leq
j\leq|\Gamma^{del}_{p,K}|}|y_{j}|^{p}\sum_{j=1}^{|\Gamma^{del}_{p,K}|}\left|q_{j}(x)-\frac{1}{\mu(C_{i})}\int_{C_{i}}q_{j}(\xi)\mu(d\xi)\right|.$

Plugging this inequality in (3.22), we deduce that

$\displaystyle
AW^{p}_{p}(\bar{\pi}^{N,K},\check{\pi}^{K})\leq\sum_{i=1}^{|\Gamma_{2,N}|}\int_{C_{i}}|x_{i}-x|^{p}\mu(dx)+2^{p-1}\max_{1\leq
j\leq|\Gamma^{del}_{p,K}|}|y_{j}|^{p}\sum_{j=1}^{|\Gamma^{del}_{p,K}|}\sum_{i=1}^{|\Gamma_{2,N}|}\int_{C_{i}}\left|q_{j}(x)-\frac{1}{\mu(C_{i})}\int_{C_{i}}q_{j}(\xi)\mu(d\xi)\right|\mu(dx).$
(3.23)

Using Jensen’s inequality in the case $p\in[1,2]$, the $L^{2}-L^{p}$\-
distortion mismatch Theorem 4.3 [38] when $2<p<2+d$ and Hölder’s inequality
when $p\geq 2+d$, we get

$\sum_{i=1}^{|\Gamma_{2,N}|}\int_{C_{i}}|x_{i}-x|^{p}\mu(dx)=\int_{{R}^{d}}|{\rm
Proj}_{\Gamma_{2,N}}(x)-x|^{p}\mu(dx)\leq\begin{cases}e^{p}_{2,N}(\mu)\,\mbox{
if }p\in[1,2]\\\ CN^{-p/d}\;\mbox{ if }2<p<2+d\\\ 2^{(p-3)\vee 0}\max_{1\leq
j\leq|\Gamma^{del}_{p,K}|}|y_{j}|^{p-2}e^{2}_{2,N}(\mu)\\\ \hskip
113.81102pt\mbox{ if }p\geq 2+d\end{cases},$

where the constant $C$ does not depend on $N$. With Theorem 1.3, we deduce
that the first term in the right-hand side of (3.23) goes to $0$ as
$N\to\infty$. Since $\int_{{R}^{d}}|q_{j}(x)|\mu(dx)\leq 1$, by Theorem 3.14
p.69 [49], there exists a sequence $(q^{n}_{j})_{n\in{N}}$ of continuous and
compactly supported functions on ${R}^{d}$ such that
$\lim_{n\to\infty}\int_{{R}^{d}}|q_{j}^{n}(x)-q_{j}(x)|\mu(dx)=0$. Since
$q_{j}$ takes its values in the interval $[0,1]$, we suppose that so do the
functions $q_{j}^{n}$ up to replacing them by $0\vee q^{n}_{j}\wedge 1$ which
affects neither the continuity and compact support property nor the
convergence. We have

$\displaystyle\bigg{|}\sum_{i=1}^{|\Gamma_{2,N}|}$
$\displaystyle\int_{C_{i}}\left|q_{j}(x)-\frac{1}{\mu(C_{i})}\int_{C_{i}}q_{j}(\xi)\mu(d\xi)\right|\mu(dx)-\sum_{i=1}^{|\Gamma_{2,N}|}\int_{C_{i}}\left|q^{n}_{j}(x)-\frac{1}{\mu(C_{i})}\int_{C_{i}}q^{n}_{j}(\xi)\mu(d\xi)\right|\mu(dx)\bigg{|}$
$\displaystyle\leq\sum_{i=1}^{|\Gamma_{2,N}|}\int_{C_{i}}\left|q_{j}(x)-q_{j}^{n}(x)-\frac{1}{\mu(C_{i})}\int_{C_{i}}(q_{j}(\xi)-q_{j}^{n}(\xi))\mu(d\xi)\right|\mu(dx)$
$\displaystyle\leq\sum_{i=1}^{|\Gamma_{2,N}|}\int_{C_{i}}|q_{j}(x)-q_{j}^{n}(x)|\mu(dx)+\sum_{i=1}^{|\Gamma_{2,N}|}\left|\int_{C_{i}}(q_{j}(\xi)-q_{j}^{n}(\xi))\mu(d\xi)\right|\leq
2\int_{{R}^{d}}|q_{j}(x)-q_{j}^{n}(x)|\mu(dx),$

where the right-hand side goes to $0$ as $n\to\infty$. We deduce that to prove
that the second term in the right-hand side of (3.23) goes to $0$ as
$N\to\infty$, it is enough to check that so does
$\sum_{i=1}^{|\Gamma_{2,N}|}\int_{C_{i}}\left|q_{j}(x)-\frac{1}{\mu(C_{i})}\int_{C_{i}}q^{n}_{j}(\xi)\mu(d\xi)\right|\mu(dx)$
for any fixed $n\in{N}$. For $X\sim\mu$ and $\hat{X}^{N}={\rm
Proj}_{\Gamma_{2,N}}(X)$,

$\displaystyle\sum_{i=1}^{|\Gamma_{2,N}|}$
$\displaystyle\int_{C_{i}}\left|q^{n}_{j}(x)-\frac{1}{\mu(C_{i})}\int_{C_{i}}q^{n}_{j}(\xi)\mu(d\xi)\right|\mu(dx)={E}\left|q^{n}_{j}(X)-{E}[q^{n}_{j}(X)|\hat{X}^{N}]\right|$
$\displaystyle\leq{E}^{1/2}\left[\left(q^{n}_{j}(X)-{E}[q^{n}_{j}(X)|\hat{X}^{N}]\right)^{2}\right]\leq{E}^{1/2}\left[\left(q^{n}_{j}(X)-q^{n}_{j}(\hat{X}^{N})\right)^{2}\right]$

where, for the last inequality, we used that the conditional expectation given
$\hat{X}^{N}$ is the best quadratic approximation of a random variable by a
measurable function of $\hat{X}^{N}$. Let $\varepsilon>0$. Since $q_{j}^{n}$
is continuous and compactly supported, this function is uniformly continuous.
Since it takes its values in the interval $[0,1]$, we deduce that there exists
$\eta>0$ such that for all $x,y\in{R}^{d}$,
$|q^{n}_{j}(x)-q^{n}_{j}(y)|\leq\varepsilon
1_{\\{|x-y|\leq\eta\\}}+1_{\\{|x-y|>\eta\\}}$. Therefore

$\displaystyle{E}\left[\left(q^{n}_{j}(X)-q^{n}_{j}(\hat{X}^{N})\right)^{2}\right]\leq\varepsilon^{2}+{P}(|X-\hat{X}^{N}|\geq\eta)\leq\varepsilon^{2}+\frac{{E}[|X-\hat{X}^{N}|^{2}]}{\eta^{2}}=\varepsilon^{2}+\frac{e_{2}(\mu,N)^{2}}{\eta^{2}}.$

With Theorem 1.3, we deduce that the left-hand side goes to $0$ as
$N\to\infty$ and conclude that so does
$AW_{p}(\bar{\pi}^{N,K},\check{\pi}^{K})$.$\hfill\quad{}_{\Box}$

##  4 Application to weak martingale optimal transport problems

We endow ${\cal P}_{\leq}\times{\cal
P}_{1}({R}^{d})=\left\\{(\mu,\nu):\mu,\nu\in{\mathcal{P}}_{1}({R}^{d})\mbox{
and }\mu\leq_{cvx}\nu\right\\}$ with the metric
$W_{1}(\mu,\tilde{\mu})+W_{1}(\nu,\tilde{\nu})$ between $(\mu,\nu)$ and
$(\tilde{\mu},\tilde{\nu})$ and ${R}^{d}\times{\mathcal{P}}_{1}({R}^{d})$ with
the metric obtained as the sum of the Euclidean distance on ${R}^{d}$ and the
Wasserstein distance $W_{1}$ on ${\cal P}_{1}({R}^{d})$. For a cost function
$C:{R}^{d}\times{\mathcal{P}}_{1}({R}^{d})\to{R}$ Borel measurable, the Weak
Martingale Optimal Transport problem introduced in [6] consists in computing
for $(\mu,\nu)\in{\cal P}_{\leq}\times{\cal P}_{1}({R}^{d})$

$V(\mu,\nu)=\inf_{\pi\in{\cal M}(\mu,\nu)}\int_{{R}^{d}}C(x,\pi_{x})\mu(dx)$
(4.24)

and the minimal couplings $\pi\in{\cal M}(\mu,\nu)$. For the choice

$\tilde{C}(x,\eta)=\begin{cases}C(x,\eta)\mbox{ if
}\int_{{R}^{d}}y\eta(dy)=x\\\ +\infty\mbox{ otherwise}\end{cases},$ (4.25)

it can be seen as a particular case of the Weak Optimal Transport problem

$\tilde{V}(\mu,\nu)=\inf_{\pi\in{\cal
P}(\mu,\nu)}\int_{{R}^{d}}\tilde{C}(x,\pi_{x})\mu(dx)\mbox{ for
}(\mu,\nu)\in{\cal P}_{1}({R}^{d})\times{\cal P}_{1}({R}^{d})$ (4.26)

introduced by Gozlan, Roberto, Samson and Tetali in [23] and studied by
Backhoff-Veraguas, Beiglböck and Pammer in [5]. Indeed, when
$(\mu,\nu)\in{\cal P}_{\leq}\times{\cal P}_{1}({R}^{d})$ then for each
$\pi\in{\cal P}(\mu,\nu)\setminus{\cal M}(\mu,\nu)$,
$\int_{{R}^{d}}\tilde{C}(x,\pi_{x})\mu(dx)=+\infty$ and for each $\pi\in{\cal
M}(\mu,\nu)$,
$\int_{{R}^{d}}\tilde{C}(x,\pi_{x})\mu(dx)=\int_{{R}^{d}}C(x,\pi_{x})\mu(dx)$,
which implies that $\tilde{V}(\mu,\nu)=V(\mu,\nu)$.

The martingale optimal transport problem corresponds to the particular case of
the WMOT problem when the cost function is linear in the measure component :
$C(x,\eta)=\int_{{R}^{d}}c(x,y)\eta(dy)$ for a Borel measurable function
$c:{R}^{d}\times{R}^{d}\to{R}$ with at most linear growth in its second
variable.

The existence of minimal couplings in the WMOT problem (4.24) and the lower
semi-continuity of the value function $V$ are deduced from Theorem 2.6 and
Proposition 5.8 $(b)$ [10].

###### Proposition 4.1

Assume that $C:{R}^{d}\times{\mathcal{P}}_{1}({R}^{d})\to{R}$ is lower semi-
continuous, convex in the measure argument (for all $x\in{R}^{d}$, ${\cal
P}_{1}({R}^{d})\ni\eta\mapsto C(x,\eta)$ is convex) and such that

$\sup_{(x,\eta)\in{R}^{d}\times{\mathcal{P}}_{1}({R}^{d})}\frac{|C(x,\eta)|}{1+|x|+\int_{{R}^{d}}|y|\eta(dy)}<+\infty.$

Then for each $(\mu,\nu)\in{\cal P}_{\leq}\times{\cal P}_{1}({R}^{d})$, there
exists $\pi^{\star}\in{\cal M}(\mu,\nu)$, unique if $C$ is strictly convex in
the measure argument, such that
$V(\mu,\nu)=\int_{{R}^{d}}C(x,\pi^{\star}_{x})\mu(dx)$ and $(\mu,\nu)\mapsto
V(\mu,\nu)$ is lower semi-continuous on ${\cal P}_{\leq}\times{\cal
P}_{1}({R}^{d})$.

Theorem 2.6 [10] also ensures convergence of the optimal couplings under
convergence of the value function, a property which holds in dimension $d=1$.

###### Proposition 4.2

Assume that $C:{R}^{d}\times{\mathcal{P}}_{1}({R}^{d})\to{R}$ is lower semi-
continuous, convex in the measure argument and such that

$\sup_{(x,\eta)\in{R}^{d}\times{\mathcal{P}}_{1}({R}^{d})}\frac{|C(x,\eta)|}{1+|x|+\int_{{R}^{d}}|y|\eta(dy)}<+\infty.$

Let for each $k\in{N}$, $(\mu_{k},\nu_{k})\in{\cal P}_{\leq}\times{\cal
P}_{1}({R}^{d})$ and $\pi^{\star}_{k}$ be an optimal coupling for
$V(\mu_{k},\nu_{k})$, the existence of which is a consequence of Theorem 4.1.
If $((\mu_{k},\nu_{k}))_{k\in{N}}$ converges to $(\mu,\nu)$ in ${\cal
P}_{\leq}\times{\cal P}_{1}({R}^{d})$ as $k\to\infty$ and
$V(\mu,\nu)=\lim_{k\to\infty}V(\mu_{k},\nu_{k})$, then all the accumulation
points of $(\pi^{\star}_{k})_{k\in{N}}$ for the weak convergence topology are
minimizers for $V(\mu,\nu)$. If $C$ is moreover strictly convex in the measure
argument, then the sequence $(\pi^{\star}_{k})_{k\in{N}}$ converges in
$AW_{1}$ to the unique optimal coupling $\pi^{\star}$ between $\mu$ and $\nu$.
If $d=1$ and either $C$ is continuous or $C$ is continuous in its second
argument and for each Borel subset $A$ of ${R}^{d}$, $(\mu_{k}(A))_{k\in{N}}$
converges to $\mu(A)$ as $k\to\infty$, then
$V(\mu,\nu)=\lim_{k\to\infty}V(\mu_{k},\nu_{k})$.

When $(\mu_{k},\nu_{k})=(\hat{\mu}^{N_{k}},\check{\nu}^{K_{k}})$ with
$\hat{\mu}^{N_{k}}$ a quadratic optimal primal $N_{k}$-quantization of $\mu$
and $\check{\nu}^{K_{k}}$ an $L^{p}$-optimal dual $K_{k}$-quantization of
$\nu$, then Theorem 3.1 ensures that it is possible to approximate in $AW_{1}$
distance any optimal martingale coupling $\pi^{\star}$ between $\mu$ and $\nu$
by martingale couplings between $\hat{\mu}^{N_{k}}$ and $\check{\nu}^{K_{k}}$
and we deduce the upper-semicontinuity of the value function along this
sequence, whatever the dimension $d$.

###### Lemma 4.3

Let $p\geq 1$, $\mu,\nu\in{\mathcal{P}}({R}^{d})$ be such that
$\mu\leq_{cvx}\nu$ with $\nu$ compactly supported and for $N,K\geq 1$,
$\hat{\mu}^{N}$ be a quadratic optimal primal $N$-quantization of $\mu$ and
$\check{\nu}^{K}$ an $L^{p}$-optimal dual $K$-quantization of $\nu$. If
$C:{R}^{d}\times{\mathcal{P}}_{1}({R}^{d})\to{R}$ is continuous, then
$V(\mu,\nu)$ is finite. If moreover, $x\mapsto\sup_{\eta\in{\cal
P}_{1}({R}^{d})}\frac{|C(x,\eta)|}{1+\int_{{R}^{d}}|y|\eta(dy)}$ is locally
bounded on ${R}^{d}$, then

$\limsup_{N,K\to\infty}V(\hat{\mu}^{N},\check{\nu}^{K})\leq V(\mu,\nu).$

If $C:{R}^{d}\times{\mathcal{P}}_{1}({R}^{d})\to{R}$ is Lipschitz continuous
with constant ${\rm Lip}(C)$, then

$\limsup_{N\to\infty}V(\hat{\mu}^{N},\check{\nu}^{K})\leq V(\mu,\nu)+{\rm
Lip}(C)d_{p,K}(\nu).$

With Propositions 4.1 and 4.2, we easily deduce the following corollary.

###### Corollary 4.4

Let $p\geq 1$, $\mu,\nu\in{\mathcal{P}}({R}^{d})$ be such that
$\mu\leq_{cvx}\nu$ with $\nu$ compactly supported and for $N,K\geq 1$,
$\hat{\mu}^{N}$ be a quadratic optimal primal $N$-quantization of $\mu$ and
$\check{\nu}^{K}$ an $L^{p}$-optimal dual $K$-quantization of $\nu$. If
$C:{R}^{d}\times{\mathcal{P}}_{1}({R}^{d})\to{R}$ is continuous, convex in the
measure argument and such that

$\sup_{(x,\eta)\in{R}^{d}\times{\mathcal{P}}_{1}({R}^{d})}\frac{|C(x,\eta)|}{1+|x|+\int_{{R}^{d}}|y|\eta(dy)}<+\infty,$

then $\lim_{N,K\to\infty}V(\hat{\mu}^{N},\check{\nu}^{K})=V(\mu,\nu)$.
Moreover, all accumulation points as $N,K\to\infty$ of sequences
$(\pi^{\star}_{N,K})_{N,K}$ of minimizers for
$V(\hat{\mu}^{N},\check{\nu}^{K})$ are minimizers for $V(\mu,\nu)$. If $C$ is
also strictly convex in the measure argument, then
$\lim_{N,K\to\infty}AW_{1}(\pi^{\star}_{N,K},\pi^{\star})=0$, where
$\pi^{\star}$ is the unique optimal coupling between $\mu$ and $\nu$.

Proof of Lemma 4.3. We first suppose that
$C:{R}^{d}\times{\mathcal{P}}_{1}({R}^{d})\to{R}$ is continuous. Let $\bar{B}$
be some closed ball centered at the origin with sufficiently large radius so
that $\nu(\bar{B})=1$ and ${\cal P}_{\bar{B}}({R}^{d})=\\{\eta\in{\cal
P}({R}^{d}):\eta(\bar{B})=1\\}$. The set ${\cal P}_{\bar{B}}({R}^{d})$ is a
compact subset of ${\cal P}_{1}({R}^{d})$ and therefore the continuous cost
function $C$ is bounded on the compact subset $\bar{B}\times{\cal
P}_{\bar{B}}({R}^{d})$ of ${R}^{d}\times{\cal P}_{1}({R}^{d})$. Since
$\hat{\mu}^{N}\leq_{cvx}\mu\leq_{cvx}\nu$,
$\hat{\mu}^{N}(\bar{B})=\mu(\bar{B})=1$. Moreover for $\pi\in{\cal
P}(\mu,\nu)$, $\int_{{R}^{d}}\pi_{x}(\bar{B})\mu(dx)=\nu(\bar{B})=1$ so that
$(x,\pi_{x})\in\bar{B}\times{\cal P}_{\bar{B}}({R}^{d})$ $\mu(dx)$ a.e..
Therefore

$\inf_{(x,\eta)\in\bar{B}\times{\cal P}_{\bar{B}}({R}^{d})}C(x,\eta)\leq
V(\mu,\nu)\leq\sup_{(x,\eta)\in\bar{B}\times{\cal
P}_{\bar{B}}({R}^{d})}C(x,\eta)$

and $V(\mu,\nu)$ is finite. Let $\varepsilon>0$. By the continuity and the
growth assumption satisfied by the cost function $C$ and the compactness of
$\bar{B}\times{\cal P}_{\bar{B}}({R}^{d})$ ,

$\displaystyle\exists\alpha>0,$
$\displaystyle\;\forall(x,\eta,\tilde{x},\tilde{\eta})\in\bar{B}\times{\cal
P}_{\bar{B}}({R}^{d})\times{R}^{d}\times{\cal P}_{1}({R}^{d})\mbox{ s.t.
}|x-\tilde{x}|+{W}_{1}(\eta,\tilde{\eta})\leq\alpha,\;|C(x,\eta)-C(\tilde{x},\tilde{\eta})|\leq\varepsilon$
$\displaystyle\mbox{ and
}\forall(x,\eta,\tilde{x},\tilde{\eta})\in\bar{B}\times{\cal
P}_{\bar{B}}({R}^{d})\times\bar{B}\times{\cal
P}_{1}({R}^{d}),\;|C(x,\eta)|+|C(\tilde{x},\tilde{\eta})|\leq\frac{1}{\alpha}\left(1+\int_{{R}^{d}}|y|\tilde{\eta}(dy)\right).$

Let $\pi\in{\cal M}(\mu,\nu)$ be such that
$\int_{{R}^{d}}C(x,\pi_{x})\mu(dx)\leq V(\mu,\nu)+\varepsilon$. By Theorem
3.1, there exists $\bar{\pi}^{N,K}\in{\cal M}(\hat{\mu}^{N},\check{\nu}^{K})$
such that $\limsup_{N\to\infty}AW_{1}(\bar{\pi}^{N,K},\pi)\leq d_{p,K}(\nu)$
and $\lim_{N,K\to\infty}AW_{1}(\bar{\pi}^{N,K},\pi)=0$. Let $m^{N,K}\in{\cal
P}(\mu,\hat{\mu}^{N})$ be an optimal coupling for
$AW_{1}(\pi,\bar{\pi}^{N,K})$. We have

$\displaystyle\left|\int_{{R}^{d}}C(x,\pi_{x})\mu(dx)-\int_{{R}^{d}}C(\tilde{x},\bar{\pi}^{N,K}_{\tilde{x}})\hat{\mu}^{N}(d\tilde{x})\right|\leq\int_{{R}^{d}\times{R}^{d}}\left|C(x,\pi_{x})-C(\tilde{x},\bar{\pi}^{N,K}_{\tilde{x}})\right|m^{N,K}(dx,d\tilde{x})$
$\displaystyle\leq\varepsilon+\int_{{R}^{d}\times{R}^{d}}\left|C(x,\pi_{x})-C(\tilde{x},\bar{\pi}^{N,K}_{\tilde{x}})\right|1_{\\{|x-\tilde{x}|+{W}_{1}(\pi_{x},\bar{\pi}^{N,K}_{\tilde{x}})\geq\alpha\\}}m^{N,K}(dx,d\tilde{x})$
$\displaystyle\leq\varepsilon+\frac{1}{\alpha}\int_{{R}^{d}\times{R}^{d}}\left(1+\int_{{R}^{d}}|y|\bar{\pi}^{N,K}_{\tilde{x}}(dy)\right)1_{\\{|x-\tilde{x}|+{W}_{1}(\pi_{x},\bar{\pi}^{N,K}_{\tilde{x}})\geq\alpha\\}}m^{N,K}(dx,d\tilde{x}).$
(4.27)

By Markov inequality,
$\int_{{R}^{d}\times{R}^{d}}1_{\\{|x-\tilde{x}|+{W}_{1}(\pi_{x},\bar{\pi}^{N,K}_{\tilde{x}})\geq\alpha\\}}m^{N,K}(dx,d\tilde{x})\leq\frac{AW_{1}(\pi,\bar{\pi}^{N,K})}{\alpha}$.
Moreover, since

$\int_{(x,\tilde{x})\in{R}^{d}\times{R}^{d}}\bar{\pi}^{N,K}_{\tilde{x}}(dy)m^{N,K}(dx,d\tilde{x})=\int_{\tilde{x}\in{R}^{d}}\bar{\pi}^{N,K}_{\tilde{x}}(dy)\hat{\mu}^{N}(d\tilde{x})=\int_{\tilde{x}\in{R}^{d}}\bar{\pi}^{N,K}(d\tilde{x},dy)=\check{\nu}^{K}(dy),$

there is a Markov kernel $q_{y}(dx,d\tilde{x})$ such that
$\bar{\pi}^{N,K}_{\tilde{x}}(dy)m^{N,K}(dx,d\tilde{x})=\check{\nu}^{K}(dy)q_{y}(dx,d\tilde{x})$
and

$\displaystyle\int_{{R}^{d}\times{R}^{d}}\int_{{R}^{d}}|y|\bar{\pi}^{N,K}_{\tilde{x}}(dy)1_{\\{|x-\tilde{x}|+{W}_{1}(\pi_{x},\bar{\pi}^{N,K}_{\tilde{x}})\geq\alpha\\}}m^{N,K}(dx,d\tilde{x})=\int_{{R}^{d}}|y|\beta(y)\check{\nu}^{K}(dy)$

where the function
$\beta(y):=\int_{{R}^{d}\times{R}^{d}}1_{\\{|x-\tilde{x}|+{W}_{1}(\pi_{x},\bar{\pi}^{N,K}_{\tilde{x}})\geq\alpha\\}}q_{y}(dx,d\tilde{x})$
is $[0,1]$-valued and such that

$\int_{{R}^{d}}\beta(y)\check{\nu}^{K}(dy)=\int_{{R}^{d}\times{R}^{d}}1_{\\{|x-\tilde{x}|+{W}_{1}(\pi_{x},\bar{\pi}^{N,K}_{\tilde{x}})\geq\alpha\\}}m^{N,K}(dx,d\tilde{x})\leq\frac{AW_{1}(\pi,\bar{\pi}^{N,K})}{\alpha}.$

Since
$\lim_{N,K\to\infty}AW_{1}(\pi,\bar{\pi}^{N,K})=0=\lim_{K\to\infty}W_{1}(\nu,\check{\nu}^{K})$,
with Lemma.. BJMP, we deduce that the second term in the right-hand side of
(4.27) tends to $0$ as $N,K\to\infty$. Hence

$\displaystyle\limsup_{N,K\to\infty}V(\hat{\mu}^{N},\check{\nu}^{K})$
$\displaystyle\leq\limsup_{N,K\to\infty}\int_{{R}^{d}}C(\tilde{x},\bar{\pi}^{N,K}_{\tilde{x}})\hat{\mu}^{N}(d\tilde{x})$
$\displaystyle\leq\int_{{R}^{d}}C(x,\pi_{x})\mu(dx)+\limsup_{N,K\to\infty}\left|\int_{{R}^{d}}C(x,\pi_{x})\mu(dx)-\int_{{R}^{d}}C(\tilde{x},\bar{\pi}^{N,K}_{\tilde{x}})\hat{\mu}^{N}(d\tilde{x})\right|$
$\displaystyle\leq V(\mu,\nu)+2\varepsilon.$

Since $\varepsilon$ is arbitrary, we conclude that
$\limsup_{N,K\to\infty}V(\hat{\mu}^{N},\check{\nu}^{K})\leq V(\mu,\nu)$.

We now suppose that $C:{R}^{d}\times{\mathcal{P}}_{1}({R}^{d})\to{R}$ is
Lipschitz continuous with constant ${\rm Lip}(C)$. We then have

$\displaystyle\left|\int_{{R}^{d}}C(x,\pi_{x})\mu(dx)-\int_{{R}^{d}}C(\tilde{x},\bar{\pi}^{N,K}_{\tilde{x}})\hat{\mu}^{N}(d\tilde{x})\right|\leq\int_{{R}^{d}\times{R}^{d}}\left|C(x,\pi_{x})-C(\tilde{x},\bar{\pi}^{N,K}_{\tilde{x}})\right|m^{N,K}(dx,d\tilde{x})$
$\displaystyle\leq{\rm
Lip}(C)\int_{{R}^{d}\times{R}^{d}}\left(|x-\tilde{x}|+W_{1}(\pi_{x},\bar{\pi}^{N,K}_{\tilde{x}})\right)m(dx,d\tilde{x})={\rm
Lip}(C)AW_{1}(\pi,\bar{\pi}^{N,K}).$

Therefore

$\displaystyle
V(\hat{\mu}^{N},\check{\nu}^{K})\leq\int_{{R}^{d}}C(\tilde{x},\bar{\pi}^{N,K}_{\tilde{x}})\hat{\mu}^{N}(d\tilde{x})$
$\displaystyle\leq\int_{{R}^{d}}C(x,\pi_{x})\mu(dx)+{\rm
Lip}(C)AW_{1}(\pi,\bar{\pi}^{N,K})$ $\displaystyle\leq
V(\mu,\nu)+\varepsilon+{\rm Lip}(C)AW_{1}(\pi,\bar{\pi}^{N,K}).$

Since $\varepsilon$ is arbitrary, we deduce that
$\limsup_{N\to\infty}V(\hat{\mu}^{N},\check{\nu}^{K})\leq V(\mu,\nu)+{\rm
Lip}(C)d_{p,K}(\nu)$. $\hfill\quad{}_{\Box}$

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