Yating Liu
email: yating.liu@lpsm.paris

Gilles Pagès
email: gilles.pages@sorbonne-universite.fr

Probability measure characterization by L p -quantization error function

Keywords: Probability distribution characterization, Vector quantization, Voronoï diagram, Wasserstein convergence

We establish conditions to characterize probability measures by their L p -quantization error functions in both R d and Hilbert settings. This characterization is two-fold: static (identity of two distributions) and dynamic (convergence for the L p -Wasserstein distance). We first propose a criterion on the quantization level N , valid for any norm on R d and any order p based on a geometrical approach involving the Voronoï diagram. Then, we prove that in the L 2 -case on a (separable) Hilbert space, the condition on the level N can be reduced to N = 2, which is optimal. More quantization based characterization cases in dimension 1 and a discussion of the completeness of a distance defined by the quantization error function can be found at the end of this paper.

Introduction

Vector quantization was originally developed as an optimal discretization method for signal transmission and compression by the Bell laboratories in the 1950s. Many seminal and historical contributions on vector quantization and its connections with information theory were gathered and published later in [START_REF][END_REF]. In the unsupervised learning area, vector quantization has a close connection with the automatic classification (clustering) through the k-means algorithm. More recently, in the 1990s, it became an efficient tool in numerical probability to compute regular and conditional expectations (see [START_REF] Pagès | A space quantization method for numerical integration[END_REF], [START_REF] Bally | A quantization algorithm for solving multi-dimensional discrete-time optimal stopping problems[END_REF] and [START_REF] Pagès | Optimal quadratic quantization for numerics: the Gaussian case[END_REF]) with in view the pricing of derivative products. Thus, a quantization based numerical schemes have been developed for American option pricing (see [START_REF] Bally | A quantization tree method for pricing and hedging multidimensional American options[END_REF]), and for the simulation of Backward Stochastic Differential Equation or nonlinear filtering (see [START_REF] Pagès | Improved error bounds for quantization based numerical schemes for BSDE and nonlinear filtering[END_REF]). For a first mathematically rigorous monograph of various aspects of vector quantization theory, we refer to [START_REF] Graf | Foundations of quantization for probability distributions[END_REF] (and the references therein). For more engineering applications to signal compression, see e.g. [START_REF] Gersho | Vector quantization and signal compression[END_REF] among an extensive literature.

In all these applications, either of probabilistic or statistical nature, vector quantization is used to produce a kind of skeleton of a probability distribution. To be more precise, let (Ω, A, P) denote a probability space and let X be a random variable defined on (Ω, A, P) and valued in (E, | • | E ), where E is R d or a separable Hilbert space H and | • | E denotes respectively the norm on R d or the norm on H induced by the inner product (• | •) H . Let µ denote the probability distribution of X, denoted by P X = µ or Law(X) = µ and assume that µ has a finite p-th moment, p ∈ [1, +∞). The quantization grid (also called codebook in signal compression or cluster center in machine learning theory) is a finite set of points in E, denoted by Γ = {x 1 , ..., x N } ⊂ E. Let us define the distance between a point ξ and a set A in E by d(ξ, A) = min a∈A |ξ -a| E . The L p -mean quantization error of Γ, defined by e p (µ, Γ) := d(X, Γ) p = E min a∈Γ |ξ -a| p E µ(dξ) is called an L p -optimal quantization grid (or optimal grid in short) at level N . We refer to [START_REF] Graf | Foundations of quantization for probability distributions[END_REF][Theorem 4.12] for the existence of such an optimal grid on R d and to [START_REF] Luschgy | Functional quantization of Gaussian processes[END_REF][Proposition 2.1] or [START_REF] Cuesta | The strong law of large numbers for k-means and best possible nets of Banach valued random variables[END_REF] on (separable) Hilbert spaces. There is usually no closed form for optimal grids, however, in the quadratic case (p = 2), it can be computed by the stochastic optimization methods such as the CLVQ algorithm or the randomized Lloyd algorithm (see [START_REF] Pagès | CEM-RACS 2013-modelling and simulation of complex systems: stochastic and deterministic approaches[END_REF][Section 3], [START_REF] Kieffer | Exponential rate of convergence for Lloyd's method[END_REF] and [START_REF] Pagès | Pointwise convergence of the Lloyd I algorithm in higher dimension[END_REF]).

Optimal grids Γ * ,(N ) "carries" the information of the initial measure. For example, let µ ∈ P p+ε (R d ) for some ε > 0, where P p (E) := {µ probability distribution on E s.t. where, for a Polish space S, (S) = = ⇒ denotes the weak convergence of probability measures on S. We refer to [START_REF] Graf | Foundations of quantization for probability distributions[END_REF][Theorem 7.5] for a proof of this result. This weak convergence (1.2) emphasizes that, an absolutely continuous probability measure µ is entirely characterized by the sequence of L p -optimal quantization grids Γ * ,(N ) at levels N , N ≥ 1.

We consider now the L p -mean quantization error function as follows.

Definition 1.1 (Quantization error function). Let µ ∈ P p (R d ), p ∈ [1, +∞). The L p -mean quantization error function of µ at level N , denoted by e N,p (µ, •), is defined by:

e N,p (µ, •) : (R d ) N -→ R + x = (x 1 , . . . , x N ) -→ e N,p (µ, x) = R d min 1≤i≤N |ξ -x i | p µ(dξ) 1 p . (1.
3)

The definition of e N,p (µ, •) obviously depends on the associated norm on R d and the variable of e N,p (µ, •) is a priori an N -tuple in (R d ) N . However, for a finite grid Γ ⊂ R d , if the level N ≥ card(Γ), then for any N -tuple x Γ = (x Γ 1 , . . . , x Γ N ) ∈ (R d ) N such that Γ = {x Γ 1 , . . . , x Γ N }, we have e p (µ, Γ) = e N,p (µ, x Γ ). For example, e p µ, {x 1 , x 2 } = e 2,p µ, (x 1 , x 2 ) = e 3,p µ, (x 1 , x 1 , x 2 ) , etc. Note that e N,p is a symmetric function on (R d ) N and that, owing to the above definition, inf Γ⊂R d ,card(Γ)≤N e p (µ, Γ) = inf x∈(R d ) N e N,p (µ, x).

(1.4) Therefore, throughout this paper, with a slight abuse of notation, we will also denote the L pquantization error at level N for a grid Γ of size at most N by e N,p (µ, Γ).

The equality (1.4) directly shows that the optimal grids are characterized by the L p -mean quantization error functions. Next, we show that the quantization error function e N,p (µ, •) is entirely characterized by the probability distribution µ.

Notice that for any µ ∈ P p (R d ), the function e N,p (µ, •) defined in (1.3) is 1-Lipschitz continuous for every N ≥ 1 since for any

x = (x 1 , . . . , x N ), y = (y 1 , . . . , y N ) ∈ (R d ) N , |e N,p (µ, x) -e N,p (µ, y)| = R d min 1≤i≤N |ξ -x i | p µ(dξ) 1 p - R d min 1≤j≤N |ξ -y j | p µ(dξ) 1 p ≤ R d min 1≤i≤N |ξ -x i | -min 1≤j≤N |ξ -y j | p µ(dξ) 1 p (by the Minkowski inequality) ≤ R d max 1≤i≤N |x i -y i | p µ(dξ) 1 p = max 1≤i≤N |x i -y i | . (1.5)
We recall now the definition of the L p -Wasserstein distance.

Definition 1.2 (L p -Wasserstein distance). Let (S, d) be a Polish space and S = Bor(S, d) be its Borel σ-field. For p ∈ [1, +∞), let P p (S) denote the set of probability measures on (S, S) with a finite p th -moment. The L p -Wasserstein distance W p (µ, ν) between µ, ν ∈ P p (S), denoted by W p (µ, ν), is defined by

W p (µ, ν) = inf π∈Π(µ,ν) S×S d(x, y) p π(dx, dy) 1 p = inf E d(X, Y ) p 1 p , X, Y : (Ω, A, P) → (S, S) with P X = µ, P Y = ν , (1.6)
where in the first line of (1.6), Π(µ, ν) denotes the set of all probability measures on (S 2 , S ⊗2 ) with respective marginals µ and ν.

If we consider e N,p (µ, x) as a function of µ ∈ P p (R d ), then e N,p is also 1-Lipschitz in µ. In fact, let X, Y be two random variables with probability distributions P X = µ and P Y = ν. For every

N -tuple x = (x 1 , . . . , x N ) ∈ (R d ) N , we have e N,p (µ, x) -e N,p (ν, x) = min i=1,...,N |X -x i | p - min i=1,...,N |Y -x i | p ≤ min i=1,...,N |X -x i | -min i=1,...,N |Y -x i | p (by the Minkowski inequality) ≤ max i=1,...,N | |X -x i | -|Y -x i | | p ≤ X -Y p . (1.7)
As this inequality holds for every couple (X, Y ) of random variables with marginal distributions µ and ν, it follows that for every level N ≥ 1,

e N,p (µ, •) -e N,p (ν, •) sup := sup x∈(R d ) N |e N,p (µ, x) -e N,p (ν, x)| ≤ W p (µ, ν). (1.8) Hence, if (µ n ) n≥1 is a sequence in P p (R d ) converging for the W p -distance to µ ∞ ∈ P p (R d ), then e N,p (µ n , •) -e N,p (µ ∞ , •) sup ≤ W p (µ n , µ ∞ ) n→+∞ -----→ 0.
(1.9) Definition 1.1, and the inequalities (1.5), (1.7), (1.8), (1.9) can be directly extended to any separable Hilbert space H. Inequalities (1.8) and (1.9) show that for every N ≥ 1, and p ∈ [1, +∞), the quantization error function e N,p (µ, •) is characterized by the probability distribution µ. Hence, the characterization relations between a probability measure µ, its L p -quantization error function and its optimal grids can be synthesized by the following scheme:

Probability measure µ Quantization error function e N,p (µ, •) Optimal grid Γ * ,(N )
See (1.8) and (1.9)

? argmin see (1.4) If µ ∈ P p+ε (R d ), µ λ d (absolutely continuous) and
if we know the optimal grid for every level N , see (1.2).

The characterization of a probability measure µ by its L p -optimal quantization grids suggests to consider the "reverse" questions of (1.8) and (1.9): When is a probability measure µ ∈ P p (R d ) characterized by its L p -quantization error function e N,p (µ, •)? And if so, does the convergence in an appropriate sense of the L p -quantization error functions characterizes the convergence of their probability distributions for the W p -distance?

These questions can be formalized as follows (the first one in a slightly extended sense):

• Question 1 -Static characterization: If for µ, ν∈ P p (R d ), e N,p (µ, •) = e N,p (ν, •) + C for some real constant C, then do we have µ = ν (and C = 0)? • Question 2 -Characterization of W p -convergence: If for µ n , n ≥ 1, µ ∞ ∈ P p (R d ), e N,p (µ n , •) converges pointwise to e N,p (µ ∞ , •), then do we have W p (µ n , µ ∞ ) n→+∞ -----→ 0? For any N 1 , N 2 ∈ N * with N 1 ≤ N 2 , it is clear that e N2,p (µ, •) = e N2,p (ν, •) resp. e N2,p (µ n , •) n→+∞ -----→ e N2,p (µ ∞ , •) implies e N1,p (µ, •) = e N1,p (ν, •) resp. e N1,p (µ n , •) n→+∞ -----→ e N1,p (µ ∞ , •) .
Hence, beyond these two above questions, we need to determine an as low as possible level N for which both answers are positive. For this purpose, we define The paper is organized as follows. We first recall in Section 1.1 some properties of the Wasserstein distance W p . Then in Section 2, we begin to analyze the problem of probability distribution characterization in a general finite dimensional framework by considering any dimension d, any order p and any norm on R d . We show that a positive answer to Question 1 and 2 follows from the existence of a bounded open Voronoï cell in a Voronoï diagram of size N , which in turn can be derived from a minimal covering of the unit sphere by unit closed balls centered on the sphere. As a consequence, we define for N ≥ N d,p,|•| a quantization based distance Q N,p := e N,p (µ, •) -e N,p (ν, •) sup which we will prove to be topologically equivalent to the Wasserstein distance W p . The results in this section are established for p ≥ 1, but several results can be extended to the case 0 < p < 1 by the usual adaptations of the proofs.

In Section 3, we consider the quadratic case (i.e. the order p=2) and extend the characterization result to probability distributions on a separable Hilbert space H with the norm |•| H induced by the inner product (• | •) H . In this section, we will prove by a purely analytical method that N H,2,|•| H = 21 and the topological equivalence of Wasserstein distance W 2 and the distance

Q H 2,2 (µ, ν) := e 2,2 (µ, •) -e 2,2 (ν, •) sup on P 2 (H).
Section 4 is devoted to the one-dimensional setting. Quantization based characterization not yet covered by the discussion in Section 2 and Section 3 are established. Furthermore, we prove that Q 1,1 is a complete distance on P 1 (R) and give a counterexample to show that the distances Q N,2 , N ≥ 2 are not complete on P 2 (R) in Section 4.2.

Preliminaries on Wasserstein distance

Let (S, d) be a general Polish metric space. The relation between weak convergence and convergence for the Wasserstein distance W p (see Definition 1.2) is recalled in Theorem 1.1. We recall below some useful facts about the L p -Wasserstein distance that will be called upon further on. The first one is that, for every p ∈ [1, +∞), W p is a distance on P p (S) W p p if p ∈ (0, 1) , see e.g. [START_REF] Villani | Topics in optimal transportation[END_REF][Theorem 7.3] for the proof and [START_REF] Berti | Gluing lemmas and Skorohod representations[END_REF] for a recent reference. Next, the metric space P p (S), W p is separable and complete, see e.g. [START_REF] Bolley | Separability and completeness for the Wasserstein distance[END_REF] for the proof. More generally, we refer to [START_REF] Villani | of Grundlehren der Mathematischen Wissenschaften[END_REF][Chapter 6] for an in depth presentation of Wasserstein distance and its properties.

Theorem 1.1. (see [Vil03][Theorem 7.12]) Let µ n ∈ P p (S) for every n ∈ N * ∪ {∞}. Let p ∈ [1, +∞). Then, (a) W p (µ n , µ ∞ ) → 0 if and only if    (α) µ n (S) = = ⇒ µ ∞ (β) ∃ x 0 ∈ S, S d(x 0 , ξ) p µ n (dξ) → S d(x 0 , ξ) p µ ∞ (dξ) . (b) If ∃ x 0 ∈ S, lim R→+∞ sup n≥1 d(x0,ξ) p ≥R d(x 0 , ξ) p µ n (dξ) = 0, (1.11)
then (µ n ) n≥1 is relatively compact for the Wasserstein distance W p .

General quantization based characterizations on R d

This section is devoted to establishing a general criterion that positively answers to Questions 1 and 2 in any dimension d, for any order p and any norm on R d . The idea is to design an approximate identity (ϕ ε ) ε>02 based on the quantization error function e N,p (µ, •). Our construction of (ϕ ε ) ε>0

relies on a purely geometrical idea: it is based on a specified Voronoï diagram containing a bounded open Voronoï cell that we introduce in Section 2.1. The static characterization is established in Theorem 2.1. Furthermore, Theorem 2.2 shows that a pointwise convergence of the quantization error functions is enough to imply the W p -convergence of a P p (R d )-valued sequence.

A review of Voronoï diagram, existence of bounded cells

Let Γ = {x 1 , . . . , x N } be a grid of size N . The Voronoï cell generated by x i ∈ Γ is defined by

V xi (Γ) = ξ ∈ R d : |ξ -x i | = min 1≤j≤N |ξ -x j | , (2.1) and V xi (Γ) 1≤i≤N is called the Voronoï diagram of Γ, which is a finite covering of R d (see [GL00]). A Borel measure partition C xi (Γ) 1≤i≤N is called a Voronoï partition of R d induced by Γ if for every i ∈ {1, . . . , N }, C xi (Γ) ⊂ V xi (Γ).
We also define the open Voronoï cell generated by

x i ∈ Γ by V o xi (Γ) = ξ ∈ R d : |ξ -x i | < min 1≤j≤N,j =i |ξ -x j | . (2.2) If the norm | • | on R d is strictly convex, we have Vxi (Γ) = V o xi (Γ) and V o xi (Γ) = V xi (Γ)
, where Å and A denote the interior and the closure of A. Examples of strictly convex norms are the isotropic r -norms for 1 < r < +∞ defined by (a 1 , . . . , a d ) r = a 1 r + • • • + a d r 1/r . However, this is not true for any norm on R d , typically not for the 1 -norm (see [START_REF] Graf | Foundations of quantization for probability distributions[END_REF][Figure 1.2]) or the ∞ -norm.

We recall that A ⊂ R d is star-shaped with respect to a ∈ A if for every b ∈ A and any λ ∈ [0, 1], a + λ(b -a) ∈ A.

Proposition 2.1. (see [START_REF] Graf | Foundations of quantization for probability distributions[END_REF][Proposition 1.2]) Let Γ = {x 1 , . . . , x N } be a grid of size N ≥ 1. For every i ∈ {1, . . . , N }, V xi (Γ) and V o xi (Γ) are star-shaped relative to x i .

Now we discuss a sufficient condition to obtain a Voronoï diagram containing a bounded open Voronoï cell. The first result in this direction is a rewriting Proposition 1.10 in [START_REF] Graf | Foundations of quantization for probability distributions[END_REF] for Euclidean norms (stated here in view of our applications).

2.3. Let a 1 , . . . , a k ∈ S |•| (0, 1) such that S |•| (0, 1) ⊂ k i=1 B|•| (a i , 1) (such a covering exists since S |•| (0, 1) is compact). If we choose Γ = {0, a 1 , . . . , a k }, then the Voronoï open set V o 0 (Γ) ⊂ B|•| (0, 1) and λ d V o 0 (Γ) > 0. Proof. As S |•| (0, 1) ⊂ k i=1 B|•| (a i , 1), for every ξ ∈ S |•| (0, 1), there exists j ∈ {1, . . . , k} such that |ξ -a j | ≤ 1 = |ξ|. If Γ = {0, a 1 , . . . , a k }, then ∀ξ ∈ S |•| (0, 1), ∃ j ∈ {1, . . . , k} such that ξ ∈ V aj (Γ).
(2.3)

Assume that there exists ξ ∈ V o 0 (Γ) \ B|•| (0, 1). Since V o 0 (Γ)
is star-shaped relatively to 0 and

1 |ξ| ∈ (0, 1), we have ξ |ξ| ∈ S |•| (0, 1) ∩ V o 0 (Γ). This contradicts (2.3) since V o 0 (Γ) ∩ V aj (Γ) = ∅, j = 1, . . . , k. Consequently, V o 0 (Γ) ⊂ B|•| (0, 1). Finally, V o 0 (Γ) is an open set containing 0, therefore, λ d V o 0 (Γ) > 0.
The idea of the above proposition is to cover the unit sphere centered at the origin by a finite number of unit balls centered on the unit sphere. This leads us to introduce the following definition.

Definition 2.1. We define the minimal sphere covering number c(d,

| • |) as follows, c(d, | • |) := min k : ∃{a 1 , . . . , a k } ⊂ S |•| (0, 1) such that S |•| (0, 1) ⊂ k i=1 B|•| (a i , 1) < +∞.
(ii) c(2, | • | 1 ) = 2 and c(2, | • | r ) = 3 for every 1 < r < +∞. (iii) c(d, | • | ∞ ) = 2 for every dimension d. (iv) Let r ≥ 1 such that 2 r ≥ d, then c(d, | • | r ) ≤ 2d.

A general condition for probability measure characterization

Let Γ = {x 1 , . . . , x N } be a grid in which there exists at least an

x i0 ∈ Γ such that the open Voronoï cell V o xi 0 (Γ)
> 0} = V o xi 0 (Γ) so that supp(ϕ) = V o xi 0 (Γ) is compact. Hence, ϕ dλ d ∈ (0, +∞) since ϕ(x i0 ) = d x i0 , Γ \ {x i0 } > 0 and we can normalize ϕ by setting ϕ 1 (ξ) := ϕ(xi 0 +ξ) ϕdλ d . For every ε > 0, we define ϕ ε (ξ) := 1 ε d ϕ 1 ξ ε , then (ϕ ε ) ε>0 is clearly an approximate identity (see [Gra14][Section 1.2.4]).
The following theorem gives conditions on the L p -quantization error function to characterize a probability measure.

Theorem 2.1 (Static characterization). Let p ∈ [1, +∞), let | • | be a norm on R d and let N ≥ c(d, | • |) + 1, or N ≥ d + 2 if | • | is Euclidean.
Then, the answer to Question 1 is positive i.e. if there exists a constant C such that e p N,p (µ,

•) = e p N,p (ν, •) + C, µ, ν ∈ P p (R d ), then µ = ν. The constant C is a posteriori 0.
Proof. Following Proposition 2.2 and 2.3, we choose a grid Γ = {0, a 1 , . . . , a

N -1 } such that V o 0 (Γ) is bounded and λ d V o 0 (Γ) > 0. We define ϕ : R d → R + , by ϕ(ξ) = min a∈Γ\{0} |ξ -a| p - min a∈Γ |ξ -a| p = min a∈Γ\{0} |ξ -a| p -|ξ| p + and (ϕ ε ) ε>0 by ϕ ε (ξ) := 1 Cϕε d ϕ ξ ε , where C ϕ = ϕ dλ d . For any x ∈ R d , ϕ ε * µ(x) = R d ϕ ε (x -ξ)µ(dξ) = R d 1 ε d ϕ( x-ξ ε ) ϕdλ d µ(dξ) = 1 C ϕ ε d R d Å min a∈Γ\{0} x -ξ ε -a p -min a∈Γ x -ξ ε -a p ã µ(dξ) = 1 C ϕ ε d+p ï R d min a∈Γ\{0} |x -εa -ξ| p µ(dξ) - R d min a∈Γ |x -εa -ξ| p µ(dξ) ò .
If we define two N -tuples x and x0 as

x = (x -εa 1 , x -εa 1 , x -εa 2 , . . . , x -εa N -1 ) and x0 = (x, x -εa 1 , x -εa 2 , . . . , x -εa N -1 ), then R d min a∈Γ\{0} |x -εa -ξ| p µ(dξ) = e p N,p (µ, x) and R d min a∈Γ |x -εa -ξ| p µ(dξ) = e p N,p (µ, x0 ). Hence, ϕ ε * µ(x) = 1 Cϕε d+p e p N,p (µ, x) -e p N,p (µ, x0 ) .
The assumption e p N,p (µ, •) = e p N,p (ν, •) + C implies that e p N,p (µ, x) -e p N,p (µ, x0 ) = e p N,p (ν, x)e p N,p (ν, x0 ), so that, for every x ∈ R d and every ε > 0,

ϕ ε * µ(x) = ϕ ε * ν(x).
One can finally conclude that µ = ν by letting ε → 0 since (ϕ ε ) ε>0 is an approximate identity (see [START_REF] Rudin | Functional analysis[END_REF][Theorem 6.32]). Hence C = 0.

The following theorem shows that the pointwise convergence of the L p -mean quantization error function is a necessary and sufficient condition for W p -convergence of probability distributions in

P p (R d ). Theorem 2.2 (W p -convergence characterization). Let p ∈ [1, +∞) and let | • | be any norm on R d . Let µ n ∈ P p (R d ) for n ∈ N * ∪{∞}.
The following properties are equivalent:

(i) W p (µ n , µ ∞ ) n→+∞ ------→ 0, (ii) ∀N ≥ 1, e N,p (µ n , •) n→+∞ ------→ e N,p (µ ∞ , •) uniformly on R d , (iii) ∃ N ≥ c(d, |•|)+1 or N ≥ d+2 if |•| is Euclidean such that, e N,p (µ n , •) n→+∞ ------→ e N,p (µ ∞ , •) pointwise on R d . Proof of Theorem 2.2. (i) ⇒ (ii) is obvious from (1.9). (ii) ⇒ (iii) is obvious. (iii) ⇒ (i) First of all, it follows from the convergence e N,p (µ n , •) n→+∞ -----→ e N,p (µ ∞ , •) that e p N,p (µ n , 0) n→+∞ -----→ e p N,p (µ ∞ , 0) i.e. R d |ξ| p µ n (dξ) n→+∞ -----→ R d |ξ| p µ ∞ (dξ) < +∞, (2.4)
where 0 = (0, . . . , 0). In particular, the sequence

R d |ξ| p µ n (dξ) n≥1
is bounded. Hence, the sequence of probability measures (µ n ) n≥1 is tight.

Let µ ∞ be a weak limiting probability distribution of (µ n ) n≥1 i.e. there exists a subsequence

α(n) of n such that µ α(n) (R d ) ==⇒ µ ∞ as n → +∞. Let x = (x 1 , . . . , x N ) be any N -tuple in (R d ) N . We define a continuous function f x : R d → R by f x (ξ) := min 1≤i≤N |ξ -x i | p -|ξ| p .
Hence, owing to the elementary inequality v p -u p ≤ pv p-1 (v-u)

for any 0 ≤ u ≤ v < +∞, we derive

f x (ξ) ≤ max i∈{1,...,N } p |ξ| + |x i | p-1 |x i | ≤ C x,p (1 + |ξ| p-1 ), (2.5)
where C x,p is a constant depending on x and p.

Owing to (2.4) and (2.5), the sequence

f p p-1 x dµ n n≥1 is bounded, hence f x is uniformly integrable with respect to (µ n ) n≥1 since p p-1 > 1, so that f x is uniformly integrable with respect to any subsequence (µ α(n) ) n≥1 . It follows that R d f x (ξ)µ α(n) (dξ) → R d f x (ξ) µ ∞ (dξ), as n → +∞, where R d f x (ξ)µ α(n) (dξ) = R d min i∈{1,...,N } |ξ -x i | p -|ξ| p µ α(n) (dξ) = e p N,p (µ α(n) , x) -e p N,p (µ α(n) , 0), and
R d f x (ξ) µ ∞ (dξ) = e p N,p ( µ ∞ , x) -e p N,p ( µ ∞ , 0).
On the other hand, e p N,p (µ α(n) , x) -e p N,p (µ α(n) , 0) converges to e p N,p (µ ∞ , x) -e p N,p (µ ∞ , 0) owing to the pointwise convergence in (iii) at 0 = (0, . . . , 0) and x = (x 1 , . . . , x N ).

Therefore,

e p N,p ( µ ∞ , x) -e p N,p ( µ ∞ , 0) = e p N,p (µ ∞ , x) -e p N,p (µ ∞ , 0), which implies that for every x ∈ (R d ) N , e p N,p ( µ ∞ , x)-e p N,p (µ ∞ , x) = C, where C = e p N,p ( µ ∞ , 0)-e p N,p (µ ∞ , 0) is a real constant. It follows from Theorem 2.1 that µ ∞ = µ ∞ ,
which implies that µ ∞ is the only limiting distribution of (µ n ) n≥1 for the weak convergence and consequently µ n (R d ) ==⇒ µ. We have already proved that

R d |ξ| p µ n (dξ) n→+∞ -----→ R d |ξ| p µ ∞ (dξ) from (2.4), which finally shows that W p (µ n , µ ∞ ) n→+∞ -----→ 0 owing to Theorem 1.1.
A careful reading of the proof shows that the following "à la Paul Lévy" characterization result holds for limiting functions of L p -quantization error functions.

Corollary 2.1. Let p ∈ [1 + ∞). Let (µ n ) n≥1 be a P p (R d )-valued sequence. If e N,p (µ n , •) n→+∞ -----→ f pointwise for some N such that static characterization holds true (Question 1), then there exists µ ∞ ∈ P p (R d ) such that µ n (R d ) ==⇒ µ ∞ as n → +∞ and f p = e p N,p (µ ∞ , • ) + lim n R d |ξ| p µ n (dξ) - R d |ξ| p µ ∞ (dξ).

Now we will take advantage of what precedes to introduce a quantization based distance on

P p (R d ). Let C b (R d ) N , R denote the space of bounded R-valued continuous functions defined on (R d ) N equipped with the sup norm • sup . Let p ∈ [1, +∞). If µ ∈ P p (R d ), e N,p (µ, •) -e N,p (δ 0 , •) ∈ C b (R d ) N , R (note that e N,p δ 0 , (x 1 , . . . , x N ) = min i=1,...,N |x i |) since inequality (1.8) implies that e N,p (µ, •) -e N,p (δ 0 , •) sup ≤ W p (µ, δ 0 ) = R d |ξ| p µ(dξ) 1/p
< +∞. Then, we define a function

Q N,p on P p (R d ) by (µ, ν) -→ Q N,p (µ, ν) := e N,p (µ, •) -e N,p (δ 0 , •) -e N,p (ν, •) -e N,p (δ 0 , •) sup = e N,p (µ, •) -e N,p (ν, •) sup . (2.6) For any µ, ν ∈ P p (R d ), inequality (1.8) implies Q N,p (µ, ν) ≤ W p (µ, ν) < +∞ so that Q N,p (µ, ν) ∈ [0, +∞).
Combining Theorems 2.1 and 2.2 implies the following result.

Corollary 2.2. Let p ∈ [1, +∞). (a) N d,p,|•| ≤ c(d, | • |) + 1 for any norm and N d,p,|•| ≤ d + 2 if | • | is Euclidean. (b) If N ≥ c(d, | • |) + 1 or N ≥ d + 2 if | • | is Euclidean, then Q N,
= L 2 [0, T ], dt and X = (X t ) t∈[0,T ] a bi-measurable process such that T 0 EX 2 t dt < +∞.
For more information about functional data analysis with an L 2 -setup, we refer to [START_REF] Hsing | Theoretical foundations of functional data analysis, with an introduction to linear operators[END_REF] among others.

We first prove in the quadratic case (p = 2), that both static (see further Proposition 3.1) and W 2 -convergence (see further Theorem 3.1) characterizations can be obtained at level N = 2 by an analytical method. Then we will show that N H,2 := N H,2,|•| H = 2 and for any µ, ν ∈ P 2 (H), Q 2,2 (µ, ν) := e 2,2 (µ, •) -e 2,2 (ν, •) sup is a well-defined distance on P 2 (H) which is topologically equivalent to W 2 . Proofs of quadratic quantization based characterizations rely on the following lemma.

Lemma 3.1. (a) Let µ, ν ∈ P 2 (H). If for every u ∈ H, |u| = 1, µ • ξ → (ξ | u) -1 = ν • ξ → (ξ | u) -1 , then µ = ν. (b) Let µ n ∈ P 2 (H) for every n ∈ N * ∪ {∞}. If H |ξ| 2 µ n (dξ) n→+∞ ------→ H |ξ| 2 µ ∞ (dξ) and for every u ∈ H, |u| = 1, µ n • ξ → (ξ | u) -1 (R) = = ⇒ µ ∞ • ξ → (ξ | u) -1 , then W 2 (µ n , µ ∞ ) → 0. Proof. As (H, | • |) is separable, let (h k ) k≥1 be a countable orthonormal basis of (H, | • |).
(a) Let X, Y be random variables with respective distributions µ and ν and let λ ∈ H. We define for every m ≥ 1,

X (m) := m k=1 (X | h k )h k , Y (m) := m k=1 (Y | h k )h k and λ (m) := m k=1 (λ | h k )h k . For m ≥ 1, let u (m) = λ (m) |λ (m) | (convention 0 |0| = 0), then we have (λ | X (m) ) = +∞ k=1 (λ | h k )(X (m) | h k ) = m k=1 (λ | h k )(X | h k ) = λ (m) X u (m) . Similarly, (λ | Y (m) ) = λ (m) Y u (m) . Let i be such that i 2 = -1. It follows that E e i(λ|X (m) ) = E e i|λ (m) |(X|u (m) ) = H e i |λ (m) | ξ µ • ξ → (u (m) | ξ) -1 (dξ) = H e i |λ (m) | ξ ν • ξ → (u (m) | ξ) -1 (dξ) = E e i(λ|Y (m) ) .
Since we can arbitrarily choose λ, we have for every m ≥ 1, Law(X (m) ) = Law(Y (m) ). Let F : H → R be a bounded continuous function. Then, for every

m ≥ 1, E F (X (m) ) = E F (Y (m) ) which implies E F (X) = E F (Y ) by letting m → +∞. Hence, µ = ν.
(b) For every n ≥ 1, let X n be random variables with distribution µ n and let X ∞ be a random variable with distribution µ ∞ . We define for every n ≥ 1 and for every m ≥ 1, X

:= m i=1 (X n |h i )h i and X (m) ∞ := m i=1 (X ∞ |h i )h i . (m) n
Following the lines of item (a), we get for every m ≥ 1, X . For every (temporarily

) fixed m ≥ 1, lim n E F (X n ) -E F (X ∞ ) ≤ lim n E F (X n ) -E F (X (m) n ) + lim n E F (X (m) n ) -E F (X (m) ∞ ) + E F (X (m) ∞ ) -E F (X ∞ ) ≤ lim n E F (X n ) -E F (X (m) n ) + 0 + E F (X (m) ∞ ) -E F (X ∞ ) (since X (m) n (H) = = ⇒ X (m) ∞ ).
Then, for every n ≥ 1,

E F (X n ) -E F (X (m) n ) ≤ E F (X n ) -F (X (m) n ) ≤ [F ] Lip E X n -X (m) n ≤ [F ] Lip X n -X (m) n 2 .
Similarly, we also have

E F (X (m) ∞ ) -E F (X∞) ≤ [F ]Lip X∞ -X (m) ∞ 2 .
It follows from Fatou's Lemma for the weak convergence and the convergence assumption made on

E|X n | 2 that lim sup n X n -X (m) n 2 2 = lim sup n E X n -X (m) n 2 = lim sup n E X n 2 -E X (m) n 2 = E X ∞ 2 -lim inf n E X (m) n 2 ≤ E X ∞ 2 -E X (m) ∞ 2 = X ∞ -X (m) ∞ 2 2 . (3.1)
Hence, for every m ≥ 1,

lim n E F (X n ) -E F (X ∞ ) ≤ lim sup n [F ] Lip X n -X (m) n 2 + [F ] Lip X ∞ -X (m) ∞ 2 ≤ 2[F ] Lip X ∞ -X (m) ∞ 2 . Then, X ∞ -X (m) ∞ 2 → 0 as m → +∞ by the Lebesgue dominated convergence theorem since X ∞ -X (m) ∞ ≤ X ∞ ∈ L 2 (P) so that E F (X n ) → E F (X ∞ ) as n → +∞. Thus, X n (H)
==⇒ X ∞ and we can conclude that W p (µ n , µ ∞ ) → 0 by applying Theorem 1.1.

Proposition 3.1 (Static characterization). Let µ, ν ∈ P 2 (H). If e 2 2,2 (µ, •) = e 2 2,2 (ν, •)+C for some real constant C, then µ = ν and C = 0. Proof. Let a, b ∈ H, then e 2 2,2 µ, (a, b) = H |ξ -a| 2 ∧ |ξ -b| 2 µ(dξ). As e 2 2,2 µ, (a, b) = e 2 2,2 ν, (a, b) +C for every (a, b) ∈ H 2 , in particular, if a = b, H |ξ -a| 2 µ(dξ) = H |ξ -a| 2 ν(dξ) + C. Hence, using that (x -y) + = x -x ∧ y, we have ∀a, b ∈ H, H |ξ -a| 2 -|ξ -b| 2 + µ(dξ) = H |ξ -a| 2 -|ξ -b| 2 + ν(dξ). (3.2) Note that |ξ -a| 2 -|ξ -b| 2 = 2 b -a ξ -a+b 2
. Hence, if we take a = λu and b = λ u with λ, λ ∈ R, λ > λ for some common u ∈ H with |u| = 1, we obtain

|ξ -a| 2 -|ξ -b| 2 + = 2(λ -λ) Å (ξ | u) - λ + λ 2 ã + .
As a consequence of (3.2), we derive that

∀λ, λ ∈ R, λ > λ, H Å (ξ | u) - λ + λ 2 ã + µ(dξ) = H Å (ξ | u) - λ + λ 2 ã + ν(dξ).
In turn, this implies, by letting λ → λ,

∀u ∈ H, |u| = 1, ∀λ ∈ R, H (ξ | u) -λ + µ(dξ) = H (ξ | u) -λ + ν(dξ). (3.3)
The function λ → (ξ | u) -λ + is right differentiable with 1 (ξ|u)>λ as a right derivative and µintegrable. Hence, by the Lebesgue differentiation theorem, we can right differentiate the equality (3.3) which yields for every u ∈ H, |u| = 1 and for every λ ∈ R, µ

(ξ | u) > λ = ν (ξ | u) > λ . Hence, for every u ∈ H, |u| = 1, µ • ξ → (ξ | u) -1 = ν • ξ → (ξ | u)
-1 since they have the same survival function. We conclude by Lemma 3.1 (a) that µ = ν and C = 0.

The following theorem shows the equivalence of W 2 -convergence of (µ n ) n≥1 in P 2 (H) and the pointwise convergence of quadratic quantization error function e 2,2 (µ n , •) n≥1 .

Theorem 3.1 (W 2 -convergence characterization). Let µ n ∈ P 2 (H) for every n ∈ N * ∪ {∞}. The following properties are equivalent:

(i) W 2 (µ n , µ ∞ ) n→+∞ ------→ 0, (ii) e 2,2 (µ n , •) n→+∞ ------→ e 2,2 (µ ∞ , •) uniformly, (iii) e 2,2 (µ n , •) n→+∞ ------→ e 2,2 (µ ∞ , •) pointwise.
Before proving Theorem 3.1, we recall the convergence of left and right derivatives of a converging sequence of convex functions. Let ∂ -f (respectively ∂ + f ) denote the left derivative (resp. right derivative) of a convex function f . Lemma 3.2. (See e.g. [Lac82][Theorems 2.5]) Let f n : R → R, n ∈ N * , be a sequence of convex functions converging pointwise to a function f : R → R.

Let G := {x ∈ R | ∂ -f (x) = ∂ + f (x)}. Then for every point x ∈ R \ G, lim n ∂ + f n (x) = lim n ∂ -f n (x) = f (x). Proof of Theorem 3.1. (i) ⇒ (ii) is obvious from (1.9). (ii) ⇒ (iii) is obvious. (iii) ⇒ (i) For every (a, b) ∈ H 2 , e 2 2,2 µ n , (a, b) = H |ξ -a| 2 ∧|ξ -b| 2 µ n (dξ) n→+∞ -----→ e 2 2,2 µ ∞ , (a, b) = H |ξ -a| 2 ∧|ξ -b| 2 µ ∞ (dξ).
In particular, ∀a ∈ H, H |ξ -a| 2 µ n (dξ) n→+∞ -----→ H |ξ -a| 2 µ ∞ (dξ). Hence, using that (x-y) + =

x -x ∧ y, we get

∀a, b ∈ H, H |ξ -a| 2 -|ξ -b| 2 + µ n (dξ) n→+∞ -----→ H |ξ -a| 2 -|ξ -b| 2 + µ ∞ (dξ).
Following the lines of the proof of Proposition 3.1, we get

∀λ ∈ R, ∀u ∈ H, |u| = 1, H (ξ | u) -λ + µ n (dξ) n→+∞ -----→ H (ξ | u) -λ + µ ∞ (dξ). (3.4)
For µ ∈ P 2 (H) and u ∈ S |•| (0, 1), we define the real-valued convex function φ µ by φ µ : λ → (ξ | u) -λ + µ(dξ). It follows from (3.4) that (φ µn ) n≥0 converges pointwise to φ µ∞ . Moreover, φ µn , φ µ∞ are right-differentiable and their right derivatives are given by ∂ + φ µn (λ) = µ n (ξ | u) > λ and ∂ + φ µ∞ (λ) = µ ∞ (ξ | u) > λ respectively. Note that the functions 1-∂ + φ µn and 1-∂ + φ µ∞ are the cumulative distribution functions of the probability distributions

µ n • ξ → (ξ | u) -1 and µ ∞ • ξ → (ξ | u) -1
and that the set of discontinuity points of 1 -∂ + φ µ∞ and ∂ + φ µ∞ , is

G = {λ : µ ∞ {ξ : (ξ | u) = λ} > 0}.
We know from Lemma 3.2 that for every

λ ∈ R \ G, ∂ + φ µn (λ) n→+∞ -----→ ∂ + φ µ∞ (λ) and that ∂ -φ µ∞ is continuous on R \ G. Hence ∀u ∈ H, |u| = 1, µ n • ξ → (ξ | u) -1 (R) ==⇒ µ ∞ • ξ → (ξ | u) -1 . (3.5)
Moreover, e 2,2 µ n , (0, 0) converges to e 2,2 µ ∞ , (0, 0) , which also reads

H |ξ| 2 µ n (dξ) → H |ξ| 2 µ ∞ (dξ).
Consequently, it follows from Lemma 3.1-(b) that W 2 (µ n , µ ∞ ) → 0 as n → +∞.

Remark We conclude this section by an "à la Paul Lévy" characterization of a limit of quantization errors functions.

Theorem 3.2 ( À la Paul Lévy characterization). Let (H, | • | H ) be a separable Hilbert space. Let (µ n ) n≥1 be a P 2 (H)-valued sequence and let f :

H → R + be such that e 2,2 (µ n , •) n→+∞ -----→ f pointwise.
Then there exists µ ∞ ∈ P 2 (H) such that µ n (Hw) =⇒ µ ∞ (where (H w ) stands for the weak topology on H) and

f 2 = e 2,2 (µ ∞ , •) 2 + lim n H |ξ| 2 µ n (dξ) - H |ξ| 2 µ ∞ (dξ).
Proof. The sequence e 2,2 µ n , (0, 0) 2 = H |ξ| 2 µ n (dξ), n ≥ 1, is bounded, hence the sequence (µ n ) n≥1 is tight for the weak topology (H w ) on H, which generates the same Borel σ-field as the strong one. Consequently there exists a subnet µ ϕ(n)

(Hw)
=⇒ µ ∞ ∈ P 2 (H) since the mapping ξ → |ξ| 2 is weakly lower semi-continuous and non-negative (see [START_REF] Topsoe | Compactness and tightness in a space of measures with the topology of weak convergence[END_REF][Lemma 2.3 and Theorem 3.1] and [START_REF] Kelley | General topology[END_REF][Chapter 2] for the definition of subnet). Now note that, for a fixed

x = (x 1 , x 2 ) ∈ H 2 , the mapping ξ → min |ξ -x 1 | 2 , |ξ -x 2 | 2 -|ξ| 2 = min |x 1 | 2 -2(x 1 |ξ), |x 2 | 2 -2(x 2 |ξ) is weakly continuous and (µ n ) n≥1 -uniformly integrable since it is sublinear. Hence e 2 2,2 (µ ϕ(n) , x) -→ H min |x 1 | 2 -2(x 1 |ξ), |x 2 | 2 -2(x 2 |ξ) µ ∞ (dξ) + f 2 (0, 0) as n → +∞ = e 2 2,2 (µ ∞ , x) + f 2 (0, 0) - H |ξ| 2 µ ∞ (dξ).
For two such limiting distributions µ ∞ and µ ∞ it follows from what precedes that e 2 2,2 (µ

∞ , •) = e 2 2,2 (µ ∞ , •) + C ∞ for some real constant C ∞ . Hence µ ∞ = µ ∞ by Proposition 3.1, which in turn implies that µ n (Hw) =⇒ µ ∞ .
4 Further quantization based characterizations on R Let | • | denote the absolute value on R. Results from Section 2 (Theorem 2.1 and 2.2, Proposition 2.4-(i)) imply that N 1,p := N 1,p,|•| ≤ 3 for any p ≥ 1. Moreover, Proposition 3.1 and Theorem 3.1 imply that N 1,2 = 2. Other quantization based characterizations are developed in Section 4.1. Then we discuss the completeness of the distance Q 1,1 defined in (2.6) on P 1 (R) and of Q 2,2 on P 2 (R) with opposite answers in Section 4.2.

Quantization based characterization on R

(i) W 1 (µ n , µ ∞ ) n→+∞ ------→ 0, (ii) e 1,1 (µ n , •) n→+∞ ------→ e 1,1 (µ ∞ , •) uniformly, (iii) e 1,1 (µ n , •) n→+∞ ------→ e 1,1 (µ ∞ , •) pointwise.
(c) The distance Q 1,1 and W 1 are topologically equivalent on P 1 (R) and N 1,1 = 1.

Proof. (a)

The function e 1,1 (µ, •) reads x → R |ξ -x| µ(dξ), hence it is convex and its right derivative is given by x

→ -1+2µ ]-∞, x] . So if e 1,1 (µ, •) = e 1,1 (ν, •)+C, we have µ ]-∞, x] = ν ] -∞, x] for all x ∈ R, which implies µ = ν (and C = 0). (b) It is obvious that (i) ⇒ (ii) and (ii) ⇒ (iii). Now we prove (iii) ⇒ (i).
For every n ≥ 1, e 1,1 (µ n , •) can also be written as a → R |ξ -a| µ n (dξ), which is convex with right derivative at a given by -1

+ 2µ n ] -∞, a] . Consequently, if e 1,1 (µ n , •) converges pointwise to e 1,1 (µ ∞ , •) on R, then µ n ] -∞, a] converges pointwise to µ ∞ ] -∞, a] for all a ∈ R such that µ ∞ ( a )
(i) W p (µ n , µ ∞ ) n→+∞ ------→ 0, (ii) e 2,p (µ n , •) n→+∞ ------→ e 2,p (µ ∞ , •) uniformly, (iii) e 2,p (µ n , •) n→+∞ ------→ e 2,p (µ ∞ , •) pointwise.
(c) The distances Q 2,p and W p are topologically equivalent on P p (R) and N 1,p = 2.

The proof of Proposition 4.2 is based on the following lemma. Hence, the function e p 2,p µ, (a, b) is continuously differentiable in a, since, for any even number p ≥ 2, we have ∂|ξ-a| p f (ξ)

∂a = p(a -ξ) p-1 f (ξ) and sup a ∈(a-1,a+1) p(a -ξ) p-1 f (ξ) ≤ p2 p-1 f (ξ) |a + 1| p-1 ∨ |a -1| p-1 + |ξ| p-1 ∈ L 1 (λ) since R |ξ| p f (ξ)dξ < +∞. Likewise, e p 2,p µ, (a, b) is continuously differentiable in b with partial derivatives ∂e p 2,p µ, (a, b) ∂a = p a+b 2 -∞ (a -ξ) p-1 f (ξ)dξ and ∂e p 2,p µ, (a, b) ∂b = p +∞ a+b 2 (b -ξ) p-1 f (ξ)dξ. Moreover, we have ∂(a-ξ) p-1 f (ξ) ∂a = (p -1)(a -ξ) p-2 f (ξ) and sup a ∈(a-1,a+1) (p -1)(a -ξ) p-2 f (ξ) ≤ (p -1)2 p-2 f (ξ) |a + 1| p-2 ∨ |a -1| p-2 + |ξ| p-2 ∈ L 1 (dξ) since R |ξ| p f (ξ)dξ < +∞.
By a similar reasoning, one derives that e p 2,p µ, (a, b) is continuously twice differentiable with second order partial derivatives

∂ 2 e p 2,p ∂a 2 µ, (a, b) = p a+b 2 -∞ (p -1)(a -ξ) p-2 f (ξ)dξ - 1 2 p (b -a) p-1 f ( a + b 2 ) , ∂ 2 e p 2,p ∂b 2 µ, (a, b) = p +∞ a+b 2 (p -1)(b -ξ) p-2 f (ξ)dξ - 1 2 p (b -a) p-1 f ( a + b 2 ) , ∂ 2 e p 2,p ∂a∂b µ, (a, b) = ∂ 2 e p 2,p ∂b∂a µ, (a, b) = -p 1 2 p (b -a) p-1 f a + b 2 .
Hence, for every (a, b) ∈ R 2 such that a < b,

∂ 2 e p 2,p ∂a 2 µ, (a, b) + ∂ 2 e p 2,p ∂b 2 µ, (a, b) -2 ∂ 2 e p 2,p ∂a∂b µ, (a, b) = p(p -1)e p-2 2,p-2 µ, (a, b) .

Proof of Proposition 4.2. (a)

Step 1: µ and ν are absolutely continuous with continuous density functions. Note that e p 2,p (µ,

•) = e p 2,p (ν, •) + C implies either µ = ν by Proposition 3.1 if p = 2, or, if p > 2 e p-2 2,p-2 (µ, •) = e p-2
2,p-2 (ν, •) (after differentiation) by Lemma 4.1. We can conclude by induction.

Step 2 (General case). Let X,Y be two random variables with the respective distributions µ and ν, such that

∀(a, b) ∈ R 2 , e p 2,p X, (a, b) = e p 2,p Y, (a, b) + C. (4.2)
Let Z be a random variable with probability distribution P Z = N (0, 1), independent of X and Y . Moreover, the random variables X + εZ and Y + εZ have distributions N (0, ε 2 ) * µ and N (0, ε 2 ) * ν respectively, both with continuous densities. It follows from Step 1 that Law(X + εZ) = Law(Y + εZ) for every ε > 0 so that Law(X)=Law(Y ) by letting ε → 0.

(b) It is obvious that (i) ⇒ (ii) and (ii) ⇒ (iii). Now we prove (iii) ⇒ (i). It follows from Lemma 4.1 that e 2,p (µ n , •)

n→+∞ -----→ e 2,p (µ ∞ , •) implies e 2,p-2 (µ n , •) n→+∞ -----→ e 2,p-2 (µ ∞ , •
) and, by induction, yields e 2,2 (µ n , •) n→+∞ -----→ e 2,2 (µ ∞ , •), so that Theorem 3.1 and Theorem 1.1 imply that µ n converges weakly to µ ∞ . The convergence of the p-th moment follows from e 2,p (µ n , 0)

n→+∞ -----→ e 2,p (µ ∞ , 0). Hence W p (µ n , µ ∞ ) n→+∞ -----→ 0 by Theorem 1.1.
(c) The claim (a) and (b) directly imply that if p is an even integer, p ≥ 2, the distances Q 2,p and W p are topologically equivalent on P p (R) and N 1,p ≤ 2. Now we prove that N 1,p = 2. Note that for every

x ∈ R, e p 1,p (µ, x) = R |ξ -x| p µ(dξ) = R (ξ 2 -2ξx + x 2 ) p 2 µ(dξ)
, which is polynomial in x and whose coefficients are the k-th moments of µ, k ∈ {1, ..., p}. Thus, as soon as two different distributions µ and ν have the same first p moments, e p 1,p (µ, •) = e p 1,p (ν, •). This implies N 1,p > 1.

About completeness of P

1 (R), Q 1,1 and P 2 (R), Q N,2
We know from [START_REF] Bolley | Separability and completeness for the Wasserstein distance[END_REF] that for p ≥ 1, (P p (R), W p ) is a complete space and we have proved that Q 1,1 (respectively Q 2,2 ) is topologically equivalent to W 1 (resp. W 2 ) on P 1 (R) (resp. P 2 (R)). Now we discuss whether Q 1,1 and Q 2,2 are complete distances.

Proposition 4.3. The metric space P 1 (R), Q 1,1 is complete.

Proof. The inequality (1.8) directly implies that a Cauchy sequence in

P 1 (R), W 1 is also a Cauchy sequence in P 1 (R), Q 1,1 . Now let (µ n ) n≥1 be a Cauchy sequence in P 1 (R), Q 1,1 . It follows from the definition of Q 1,1 that e 1,1 (µ n , •) -e 1,1 (δ 0 , •) n≥1 is a Cauchy sequence in C b (R, R), • sup . As C b (R, R), • sup is complete, there exists a function g ∈ C b (R, R) such that e 1,1 (µ n , •) -e 1,1 (δ 0 , •) -g sup n→+∞ -----→ 0.
(4.5)

Note that for any a ∈ R, e 1,1 (δ 0 , a) = |a|. The sequence e 1,1 (µ n , 0) -e 1,1 (δ 0 , 0) = e 1,1 (µ n , 0) is also a Cauchy sequence in R. Therefore, e 1,1 (µ n , 0) n≥1 = R |ξ| µ n (dξ) n≥1 is bounded, which implies that (µ n ) n≥1 is tight. It follows from Prohorov's theorem that there exists a subsequence (µ ϕ(n) ) n≥1 weakly converging to µ ∞ . Moreover, by Fatou's lemma in distribution,

µ ∞ ∈ P 1 (R) since R |ξ| µ ∞ (dξ) ≤ lim inf n R |ξ| µ ϕ(n) (dξ) < +∞.
Now, we prove that g = e 1,1 ( µ, •)-e 1,1 (δ 0 , •). First, let us define a function f a (ξ) := |ξ -a|-|ξ|. For every a ∈ R, f a is bounded and continuous. Hence, the weak convergence of (µ

ϕ(n) ) n≥1 implies that R f a (ξ)µ ϕ(n) (dξ) n→+∞ -----→ R f a (ξ) µ ∞ (dξ). Besides, R f a (ξ)µ ϕ(n) (dξ) = R |ξ -a| -|ξ| µ ϕ(n) (dξ) = e 1,1 (µ ϕ(n) , a
) -e 1,1 (µ ϕ(n) , 0), which converges to g(a) + e 1,1 (δ 0 , a) -g(0) + e 1,1 (δ 0 , 0) as n → +∞ by (4.5). Hence, for every a ∈ R,

g(a) + e 1,1 (δ 0 , a) -g(0) + e 1,1 (δ 0 , 0 =0 ) = R f a (ξ) µ ∞ (dξ) = e 1,1 ( µ ∞ , a) -e 1,1 ( µ ∞ , 0), i.e. e 1,1 ( µ ∞ , a) -e 1,1 (δ 0 , a) -g(a) = e 1,1 ( µ ∞ , 0) -g(0). Setting C = e 1,1 ( µ ∞ , 0) -g(0), we derive that for every a ∈ R, e 1,1 ( µ ∞ , a) -e 1,1 (δ 0 , a) -g(a) = C. (4.6)
Now we prove that C = 0. Generally, for any ν ∈ P 1 (R), one has

lim a→+∞ e 1,1 (ν, a -e 1,1 (δ 0 , a) = lim a→+∞ e 1,1 (ν, a -|a| = lim a→+∞ e 1,1 (ν, a -a = lim a→+∞ R |ξ -a| ν(dξ) -a = lim a→+∞ {ξ≥a} (ξ -a)ν(dξ) + {ξ<a} (a -ξ)ν(dξ) -a = lim a→+∞ {ξ≥a} ξν(dξ) -2 {ξ≥a} aν(dξ) + {ξ<a} (-ξ)ν(dξ) .
As ν ∈ P 1 (R) i.e. R |ξ| ν(dξ) < +∞, we derive that lim a→+∞ ξ<a (-ξ)ν(dξ) = R (-ξ)ν(dξ) and lim a→+∞ {ξ≥a} ξν(dξ) = 0. This implies 0 ≤ lim

C = C + R ξ µ ∞ (dξ) = -C + R ξ µ ∞ (dξ)
, which in turn implies C = 0, i.e. g = e 1,1 ( µ ∞ , •) -e 1,1 (δ 0 , •). Then it follows from (4.5) that

e 1,1 (µ n , •) -e 1,1 (δ 0 , •) -e 1,1 ( µ ∞ , •) -e 1,1 (δ 0 , •) = e 1,1 (µ n , •) -e 1,1 ( µ ∞ , •) sup n→+∞ -----→ 0
Hence, W 1 (µ n , µ ∞ ) → 0 by applying Proposition 4.1, that is, (µ n ) n≥1 is a Cauchy sequence in (P 1 (R), W 1 ). The completeness of (P 1 (R), W 1 ) implies immediately that (P 1 (R), Q 1,1 ) is complete.

Theorem 4.1. For any N ≥ 2, the metric space P 2 (R), Q N,2 is not complete.

We will build a sequence on P 2 (R) which is Cauchy for Q N,2 but not for W 2 . First, we have the following result.

Lemma 4.2. Let (µ n ) n≥1 be a P 2 (R d )-valued sequence which converges weakly to µ ∞ and, for n ∈ N * ∪ {∞}, let X n denote a µ n -distributed random variable . Assume that lim n E |X n | 2 exists and is finite. Then

sup a∈R d e 2,2 µ n , (a, a) - » e 2 2,2 µ ∞ , (a, a) + C 0 n→+∞ -----→ 0, (4.8)
where

C 0 = lim n E |X n | 2 -E |X ∞ | 2 ∈ [0, +∞).
Proof of Lemma 4.2. An elementary computation shows that

e 2 2,2 µ n , (a, a) = R d |ξ -a| 2 µ n (dξ) = R d |ξ| 2 µ n (dξ) -2 R d ξµ n (dξ) a + |a| 2 . As R d |ξ| 2 µ n (dξ) n≥1 is bounded and µ n (R d ) = == ⇒ µ ∞ , we have R d ξµ n (dξ) → R d ξµ ∞ (dξ). It follows that e 2 2,2 µ n , (a, a) = R d |ξ| 2 µ n (dξ) -2 R d ξµ n (dξ) a + |a| 2 n→+∞ ------→ R d |ξ| 2 µ ∞ (dξ) + C 0 -2 R d ξµ ∞ (dξ) a + |a| 2 = e 2 2,2 µ ∞ , (a, a) + C 0 .
Therefore, for every compact set K in R d , we have

≤ E |X n | 2 -2(a | X n ) + |a| 2 -E |X ∞ | 2 -2(a | X ∞ ) + |a| 2 -C 0 X n -a 2 + X ∞ -a 2 ≤ 2 |(a | EX ∞ -EX n )| + E |X n | 2 -E |X ∞ | 2 -C 0 X n -a 2 + X ∞ -a 2 ≤ 2 |a| |EX ∞ -EX n | + E |X n | 2 -E |X ∞ | 2 -C 0 X n 2 -|a| + X ∞ 2 -|a| . (4.10) Let A := 2 sup n∈N∪{∞} E |X n | 2 , then sup |a|>A e 2,2 µ n , (a, a) - » e 2 2,2 (µ ∞ , (a, a)) + C 0 ≤ sup |a|>A 2 |a| |EX ∞ -EX n | + E |X n | 2 -E |X ∞ | 2 -C 0 |a| -X n 2 + |a| -X ∞ 2 ≤ sup |a|>A 2 |a| |EX ∞ -EX n | + E |X n | 2 -E |X ∞ | 2 -C 0 2 |a| -A ≤ sup |a|>A 2 |EX ∞ -EX n | + E |X n | 2 -E |X ∞ | 2 -C 0 A n→+∞ -----→ 0 (4.11)
Hence, (4.9) and (4.11) imply that sup

a∈R d e 2,2 µ n , (a, a) - » e 2 2,2 µ ∞ , (a, a) + C 0 n→+∞ -----→ 0.
Let Z : Ω → R be N (0, 1)-distributed. We define for every n ∈ N,

X n := e n 2 Z-n 2 4 .
(4.12)

For n ≥ 1, let µ n denote the probability distribution of X n . It is obvious that X n converges a.s. to X ∞ = 0, so that µ ∞ = δ 0 . Moreover, for every p > 0, E X p n = e pn 2 8 (p-2) . Hence,

E X n = e -n 2 8 -→ 0 = E X ∞ as n → +∞ so that W 1 (µ n , µ ∞ ) → 0 whereas E X 2 n = 1 for every n ∈ N.
Hence EX 2 n does not converge to E X 2 ∞ = 0, which entails that µ n does not converge to µ ∞ for the Wasserstein distance W 2 and thus µ n is not a W 2 -Cauchy sequence. We first prove (µ n ) n≥1 is a Cauchy sequence in P 2 (R), Q 2,2 . The proof relies on the following three lemmas.

Lemma 4.3. Let Z : Ω → R be N (0, 1)-distributed. Then, ∀z > 0, P(Z ≥ z) ≤ e -z 2 2 z √ 2π . Proof. P(Z ≥ z) = +∞ z 1 √ 2π e -x 2 2 dx ≤ +∞ z x z 1 √ 2π e -x 2 2 dx = e -z 2 2 z √
2π . Lemma 4.4. Define (X n ) as in (4.12), then sup K≥0 K E(X n -K) + → 0 as n → +∞.

Proof. We have

K E(X n -K) + = K ∞ 0 P (X n -K) + ≥ u du = K +∞ 0 P(X n > u + K)du = K +∞ K P(X n ≥ v)dv = K +∞ K P e n 2 Z-n 2 4 ≥ v dv = K +∞ K P Z ≥ n 2 + 2 n ln v dv = K ∞ ln K P Z ≥ n 2 + 2 n u e u du (setting u = ln v).
By Lemma 4.3,

P Z ≥ n 2 + 2 n u ≤ 1 √ 2π e -1 2 ( n 2 + 2 n u) 2 n 2 + 2 n u = 1 √ 2π e -n 2 8 -2 n 2 u 2 -u n 2 + 2 n u
. It follows that,

K E(X n -K) + ≤ K ∞ ln K e -n 2 8 -2 n 2 u 2 -u n 2 + 2 n u e u du √ 2π ≤ Ke -n 2 8 n 2 + 2 n ln K ∞ ln K e -2 n 2 u 2 du √ 2π = Ke -n 2 8 n 2 + 2 n ln K ∞ 2 n ln K e -w 2 2 n 2 dw √ 2π (by setting w = 2 n u) = Ke -n 2 8 n 2 + 2 n ln K n 2 P Z ≥ 2 n ln K ≤ nKe -n 2 8 2( n 2 + 2 n ln K) e -1 2 4 n 2 (ln K) 2 √ 2π 2 n ln K (by Lemma 4.3) = n 2 √ 2π e -n 2 8 Ke -2 n 2 (ln K) 2 (1 + 4 n 2 ln K) ln K = n 2 √ 2π e -n 2 8 e ln K(1-2 n 2 ln K) (1 + 4 n 2 ln K) ln K . (4.13)
Since the function u → u(1 -2 n 2 u) attains its maximum at u = n 2 4 with maximum value n 2 8 , we will discuss the value of K E(X n -K) + in the following three cases:

(i) K ≥ e n 2 4 , (ii) e ρ n 2 4 ≤ K ≤ e n 2 4 , (iii) 0 ≤ K ≤ e ρ n 2 4 ,
with the same fixed ρ ∈ (0, 1 2 ) in both (ii) and (iii).

Case (i): K ≥ e n 2 4 , then ln K ≥ n 2 4 . It follows that K E(X n -K) + ≤ n 2 √ 2π e -n 2 8 e ln K(1-2 n 2 ln K) (1 + 4 n 2 ln K) ln K ≤ n 2 √ 2π e -n 2 8 e n 2 8 (1 + 4 n 2 × n 2 4 ) n 2 4 = 1 n √ 2π → 0. Case (ii): e ρ n 2 4 ≤ K ≤ e n 2 4 with a fixed ρ ∈ (0, 1 2 ), then ρ n 2 4 ≤ ln K ≤ n 2 4 . It follows that K E(X n -K) + ≤ ne -n 2 8 2 √ 2π e ln K(1-2 n 2 ln K) (1 + 4 n 2 ln K) ln K ≤ ne -n 2 8 2 √ 2π e n 2 8 (1 + 4 n 2 × ρ n 2 4 )ρ n 2 4 = 2 n(1 + ρ)ρ √ 2π → 0.
Case (iii): 0 ≤ K ≤ e ρ n 2 4 with the same ρ ∈ (0, 1 2 ) as in the situation (ii), then

K E(X n -K) + ≤ e ρ 4 n 2 EX n = e ρ 4 n 2 • e -n 2 8 = e 1 4 (ρ-1 2 )n 2 n→+∞ -----→ 0. Therefore, sup K>0 K E(X n -K) + n→+∞ -----→ 0. By Lemma 4.2, sup a∈R d e 2,2 µ n , (a, a) - » e 2 2,2 µ ∞ , (a, a) + C 0 n→+∞ -----→ 0. Consequently, it
is reasonable to guess that e N,2 (µ n , •) ≤ sup We discuss now the value of sup (a1,...,a N +1 )∈R N +1

• sup ------→ n→+∞ » e 2 N,2 (µ ∞ , •) + 1 so that (µ n ) n∈N is a Cauchy sequence in (P 2 (R d ), Q N,2
(i) 0 ≤ a ≤ b, (ii) a ≤ 0 ≤ b, ® (ii, α) a ≤ 0 ≤ b with |a| ≤ 1 2 |b| (ii, β) a ≤ 0 ≤ b with 1 2 |b| ≤ |a| ≤ |b| , (iii) b ≤ 0 ≤ a, with |a| ≤ |b|, (iv
Case (i): 0 ≤ a ≤ b. We have sup (a,b)∈(i) e 2,2 µ n , (a, b) -g 2 (a, b) ≤ sup (a,b)∈(i) e 2,2 µ n , (a, b) -e 2,2 µ n , (a, a) + e 2,2 µ n , (a, a) -g 2 (a, a) ≤ sup (a,b)∈(i) R |ξ -a| 2 ∧ |ξ -b| 2 µ n (dξ) - R |ξ -a| 2 µ n (dξ) + e 2,2 µ n , (a, a) -g 2 (a, a) ≤ sup (a,b)∈(i) R |ξ -a| 2 -|ξ -a| 2 ∧ |ξ -b| 2 µ n (dξ) + e 2,2 µ n , (a, a) -g 2 (a, a) (since √ α -β ≤ β -α for β > α > 0) ≤ sup (a,b)∈(i) R |ξ -a| 2 -|ξ -b| 2 + µ n (dξ) + e 2,2 µ n , (a, a) -g 2 (a, a) ≤ sup (a,b)∈(i) R 2(b -a) ξ - b + a 2 + µ n (dξ) + e 2,2 µ n , (a, a) -g 2 (a, a) ≤ sup (a,b)∈(i) 2 R b 2 ξ - b 2 + µ n (dξ) + e 2,2 µ n , (a, a) -g 2 (a, a) ≤ 2 … sup K≥0 K E(X n -K) + + sup a∈R e 2,2 µ n , (a, a) -g 2 (a, a) n→+∞ -----→ 0. Case (ii,α): a ≤ 0 ≤ b, with |a| ≤ 1 2 |b|. We have sup (a,b)∈(ii,α) e 2,2 µ n , (a, b) -g 2 (a, b) ≤ sup (a,b)∈(ii,α) e 2,2 µ n , (a, b) -e 2,2 µ n , (a, a) + e 2,2 µ n , (a, a) -g 2 (a, a) ≤ sup (a,b)∈(ii,α) R 2(b -a) ξ - b + a 2 + µ n (dξ) + e 2,2 µ n , (a, a) -g 2 (a, a) ≤ sup (a,b)∈(ii,α) R 3 • b ξ - b 4 + µ n (dξ) + e 2,2 µ n , (a, a) -g 2 (a, a) ≤ 2 √ 3 • … sup K≥0 K E(X n -K) + + sup a∈R e 2,2 µ n , (a, a) -g 2 (a, a) n→+∞ -----→ 0. Case (ii,β): a ≤ 0 ≤ b, with 1 2 |b| ≤ |a| ≤ |b|. One has sup (a,b)∈(ii,β) e 2,2 µ n , (a, b) -g 2 (a, b) ≤ sup
(a,b)∈(ii,β) 2(b -a) E X n -b+a 2 + X n -a 2 + sup a∈R e 2,2 µ n , (a, a) -g 2 (a, a) . As X n -a 2 = EX 2 n =1 -2a EX n ≥0 + |a| 2 1/2 ≥ » 1 + |a| 2 ,
e N +1,2 µ n , (a 1 , . . . , a N +1 ) -g N +1 (a 1 , . . . , a N +1 ) in the following cases: The second term on the right hand side of (4.15) converges to 0 as n → +∞ owing to the assumption on the level N .

For the first term on the right hand side of (4.15), we have

≤ sup 0≤a1≤a2≤•••≤a N +1 R |ξ -a 1 | 2 -|ξ -a N +1 | = sup 0≤a1≤a2≤•••≤a N +1 R 2(a N +1 -a 1 ) ξ - a 1 + a N +1 2 + µ n (dξ) ≤ sup 0≤a1≤a2≤•••≤a N +1 R 2 • a N +1 ξ - a N +1 2 + µ n (dξ) ≤ 2 • … sup K≥0 K E(X n -K) + n→+∞ -----→ 0.
Case (iii, α): e N +1,2 µ n , (a 1 , . . . , a N +1 ) -e 2,2 µ n , (a 1 , a 1 ) + sup a1∈R e 2,2 µ n , (a 1 , a 1 ) -g N (a 1 , a 1 ) .

a 1 ≤ 0 ≤ a 2 ≤ • • • ≤ a N+1 with |a 1 | ≤ 1 2 |a N+1 |.
(4.17)

The second part of (4.17), sup a1∈R e 2,2 µ n , (a 1 , a 1 ) -g N (a 1 , a 1 ) converges to 0 as n → +∞ owing to Lemma 4.2. Then for the first part of (4.17

1p

  , is used to describe the accuracy level of representing the probability measure µ by Γ. Let N ≥ 1. A quantization grid Γ * ,(N ) satisfying e p (µ, Γ * ,(N )

  E |ξ| p E µ(dξ) < +∞}. Let µ = h • λ d be an absolutely continuous distribution (λ d denotes Lebesgue measure). If for every level N ≥ 1, Γ * ,(N ) is an optimal quantization grid of µ at level N , then 1 N x∈Γ * ,(N ) δ x (R d ) = == ⇒ µ = h d/(d+p) (ξ) h d/(d+p) dλ d λ d (dξ), as N → +∞, (1.2)

N

  d,p,|•| := min{N ∈ N * such that answers to Questions 1 and 2 for e N,p are positive}. (1.10)

  Proposition 2.2 (| • | Euclidean norm). Let (b 1 , . . . , b d+1 ) be an affine basis of R d and let b 0 ∈ ˚ Conv({b 1 , . . . , b d+1 }) = ∅. Set Γ = {0, b 1 -b 0 , . . . , b d+1 -b 0 }. Then, the open Voronoï cell V o 0 (Γ) generated by 0 is bounded. Let us provide now a geometrical criterion for a general norm | • | on R d , let B|•| (x, r) denote the closed ball centered at x with radius r and let S |•| (x, r) denote its sphere.

Proposition

  The index c(d, | • |) is finite since the unit sphere is a compact set in R d . Among all the possible norms, we will focus on the isotropic r -norms on R d . We show some examples of the minimal covering number c(d, | • | r ) in the following proposition (whose proof is postponed to Appendix). Proposition 2.4. (i) c(1, | • |) = 2, where | • | denotes the absolute value.

  is bounded and non-empty. Based on such a grid, one can construct an approximate identity as follows. Let ϕ : R d → R + be the function defined by ϕ(ξ) = min a∈Γ\{xi 0 } |ξ -a| pmin a∈Γ |ξ -a| p . The function ϕ is clearly nonnegative, continuous and {ϕ

)∞

  as n → +∞, since the convergence of characteristic function implies weak convergence. Now, let F : H → R be a Lipschitz continuous function with Lipschitz coefficient [F ] Lip := sup x,y∈H |F (x)-F (y)| |x-y|

  Proposition 4.1 (p = 1). (a) Let µ, ν ∈ P 1 (R). If e 1,1 (µ, •) = e 1,1 (ν, •) + C for some real constant C, then µ = ν and C = 0. (b) If µ n ∈ P 1 (R), n ∈ N * ∪ {∞}, the following properties are equivalent:

  = 0 by Lemma 3.2. This implies µ n (R) = = ⇒ µ ∞ . The convergence of the first moment follows from e 1,1 (µ n , 0) n→+∞ -----→ e 1,1 (µ ∞ , 0). Hence, we conclude that W 1 (µ n , µ ∞ ) n→+∞ -----→ 0 by Theorem 1.1. (c) The claim (c) is a direct result from (a) and (b).

  Proposition 4.2 (Even integer p ≥ 2). Let p be an even integer, p ≥ 2. (a) Let µ, ν ∈ P p (R) such that e p 2,p (µ, •) = e p 2,p (ν, •) + C for some real constant C. Then µ = ν. (b) If µ n ∈ P p (R), n ∈ N * ∪ {∞}, the following properties are equivalent:

Lemma 4. 1 .

 1 Let p be an even number, p ≥ 2. Let µ ∈ P p (R) be absolutely continuous with density f i.e. µ(dξ) = f (ξ)dξ. If f is continuous, then for any a, b ∈ R with a < b, e p-2 2,p-2 µ, (a, b) Proof of Lemma 4.1. Assume that a < b, then e p 2,p µ, (a, b) = a+b 2 -∞ |ξ -a| p f (ξ)dξ+ +∞ a+b 2 |ξ -b| p f (ξ)dξ.

For every ε > 0, e p 2

 2 ,p X + εZ, (a, b) = min x∈{a,b} |ξ + εz -x| p µ(dξ)P Z (dz) = e p 2,p X, (a, b) -εz P Z (dz). (4.3) We derive from (4.2) and (4.3) that ∀(a, b) ∈ R 2 , e p 2,p X + εZ, (a, b) = e p 2,p Y + εZ, (a, b) + C. (4.4)

  a→+∞ {ξ≥a} a ν(dξ) ≤ lim a→+∞ {ξ≥a} ξ ν(dξ) = 0. After a similar calculation with lim a→-∞ e 1,1 (ν, a -e 1,1 (δ 0 , a) , we get lim a→+∞ e 1,1 (ν, a -e 1,1 (δ 0 , a) = R (-ξ)ν(dξ) and lim a→-∞ e 1,1 (ν, a -e 1,1 (δ 0 , a) = R ξν(dξ).

  6) and (4.7) with ν = µ ∞ shows thatlim a→+∞ g(a) = -C -R ξ µ ∞ (dξ) and lim a→-∞ g(a) = -C + R ξ µ ∞ (dξ).On the other hand, for every n ≥ 1, (4.7) applied to ν = µ ϕ(n) implies lim a→±∞ e 1,1 (µ ϕ(n) , a) -e 1,1 (δ 0 , a) = ∓ R ξµ ϕ(n) (dξ).Up to a new extraction of µ ϕ(n) , still denoted by µ ϕ(n) , we may assume thatR ξµ ϕ(n) (dξ) → ‹ C ∈ R as n → +∞ since e 1,1 (µ n , 0) n≥1 = R |ξ| µ n (dξ) n≥1 is bounded.Now the uniform convergence (4.5) implies that lim n lim a→±∞ e 1,1 (µ ϕ(n) , a) -e 1,1 (δ 0 , a) -g(a) = 0 so that ‹

sup a∈K e 2

 2 ,2 µ n , (a, a) -» e 2 2,2 µ ∞ , (a, a) + C 0 n→+∞ -----→ 0, (4.9) owing to Arzelá-Ascoli theorem, since all functions e N,p are 1-Lipschitz continuous (see (1.5)). On the other hand, we have e 2,2 µ n , (a, a) -» e 2 2,2 (µ ∞ , (a, a)) + C 0 = e 2 2,2 µ n , (a, a) -e 2 2,2 µ ∞ , (a, a) + C 0 e 2,2 µ n , (a, a) + » e 2 2,2 µ ∞ , (a, a) + C 0

  ) b ≤ a ≤ 0. Cases (iii) and (iv): b < 0 and a+b 2 < 0. The random variables X n are positive so that |x -a| ≤ |x -b|. Hence e 2,2 µ n , (a, b) = e 2,2 µ n , (a, a) . With a slight abuse of notation, we will write in what follows (a, b) ∈ (iii) for (a, b) ∈ {(a, b) ∈ R 2 | b ≤ 0 ≤ a, and |a| ≤ |b|}. We will adopt the same notation for other cases too. Then for the case (iii) and (iv), it is obvious by applying Lemma 4.2 that sup (a,b)∈(iii)∪(iv) e 2,2 µ n , (a, b) -g 2 (a, b) = sup (a,b)∈(iii)∪(iv) e 2,2 µ n , (a, a) -g 2 (a, a) n→+∞ -----→ 0.

(

  a,b)∈(ii,β) e 2,2 µ n , (a, b) -e 2,2 µ n , (a, a) + e 2,2 µ n , (a, a) -g 2 (a, a) ≤ sup (a,b)∈(ii,β) e 2 2,2 µ n , (a, b) -e 2 2,2 µ n , (a, a) e 2,2 µ n , (a, b) + e 2,2 µ n , (a, a) + e 2,2 µ n , (a, a) -g 2 (a, a) ≤ sup (a,b)∈(ii,β) R 2(b -a) ξ -b+a 2 + µ n (dξ) X n -a 2 + sup a∈R e 2,2 µ n , (a, a) -g 2 (a, a)

  we have sup (a,b)∈(ii,β) e 2,2 µ n , (a, b) -g 2 (a, b)≤ sup (a,b)∈(ii,β) 2(b + |a|)E X n -2 µ n , (a, a) -g 2 (a, a) 2 µ n , (a, a) -g 2 (a, a) . ≤ 8 EX n + sup a∈R e 2,2 µ n , (a, a) -g 2 (a, a) n→+∞ -----→ 0. £ From Nto N +1. Assume now that sup (a 1 ,...,a N )∈R N eN,2 µn, (a1, . . . , aN ) -gN (a1, . . . , aN ) goes 0 as n → +∞. Then, for the level N + 1, we assume without loss of generality that |a 1 | ≤ |a 2 | ≤ • • • ≤ |a N +1 | since g N +1 and e N,2 (µ n , •) are symmetric. Under this assumption, gN+1(a1, . . . , aN+1) = g2(a1, a1) = » |a1| 2 + 1. (4.14)

(

  i) ∃ i ∈ {2, . . . , N + 1} such that ai < 0, (ii) 0 ≤ a1 ≤ a2 ≤ • • • ≤ aN+1, (iii) a1 ≤ 0 ≤ a2 ≤ • • • ≤ aN+1, ® (iii, α) a1 ≤ 0 ≤ a2 ≤ • • • ≤ aN+1, with |a1| ≤ 1 2 |aN+1| (iii, β) a1 ≤ 0 ≤ a2 ≤ • • • ≤ aN+1, with 1 2 |aN+1| ≤ |a1| ≤ |aN+1|.Case (i): ∃ i ∈ {2, . . . , N + 1} such that a i < 0. For every n ≥ 1, X n is a.s. positive. Hence, |X n -a 1 | ≤ |X n -a i | a.s. since we assume that |a 1 | ≤ |a 2 | ≤ • • • ≤ |a N +1 |. Therefore, e N +1,2 µ n , (a 1 , . . . , a N +1 ) = e N,2 µ n , (a 1 , . . . , a i-1 , a i+1 , . . . , a N +1 ) .It follows from (4.14) that sup(a1,...,a N +1 )∈R N +1 e N +1,2 µ n , (a 1 , . . . , a N +1 ) -g N +1 (a 1 , . . . , a N +1 ) = sup (a1,...,ai-1,ai+1,...,a N +1 )∈R N e N,2 µ n , (a 1 , . . . , a i-1 , a i+1 , . . . , a N +1 ) -g N (a 1 , . . . , a i-1 , a i+1 , . . . , a N +1 ) ,which converges to 0 as n → +∞ owing to the assumption on the level N .Case (ii): 0 ≤ a 1 ≤ a 2 ≤ • • • ≤ a N+1 . sup 0≤a1≤a2≤•••≤a N +1e N +1,2 µ n , (a 1 , . . . , a N +1 ) -g N +1 (a 1 , . . . , a N +1 )≤ sup 0≤a1≤a2≤•••≤a N +1e N +1,2 µ n , (a 1 , . . . , a N +1 ) -e N,2 µ n , (a 1 , . . . , a N )+ sup 0≤a1≤a2≤•••≤a N +1e N,2 µ n , (a 1 , . . . , a N ) -g N (a 1 , . . . , a N ) . (4.15)

  sup 0≤a1≤a2≤•••≤a N +1 e N +1,2 µ n , (a 1 , . . . , a N +1 ) -e N,2 µ n , (a 1 , . . . , a N ) = sup 0≤a1≤a2≤•••≤a N +1 R min 1≤i≤N |ξ -a i | 2 µ n (dξ) -R min 1≤i≤N |ξ -a| 2 ∧ |ξ -a N +1 | 2 µ n (dξ) ≤ sup 0≤a1≤a2≤•••≤a N +1 R min 1≤i≤N |ξ -a i | 2 -min 1≤i≤N |ξ -a i | 2 ∧ |ξ -a N +1 | 2 µ n (dξ) = sup 0≤a1≤a2≤•••≤a N +1 R min 1≤i≤N |ξ -a i | 2 -|ξ -a N +1 | 2 + µ n (dξ)

  sup (a1,...,a N +1 )∈(iii,α) e N +1,2 µ n , (a 1 , . . . , a N +1 ) -g N +1 (a 1 , . . . , a N +1 )≤ sup (a1,...,a N +1 )∈(iii,α) e N +1,2 µ n , (a 1 , . . . , a N +1 ) -e N,2 µ n , (a 1 , . . . , a N ) + sup (a1,...,a N +1 )∈(iii,α) e N,2 µ n , (a 1 , . . . , a N ) -g N (a 1 , . . . , a N ) . (4.16)Like in Case (ii), the second term on the right hand side of (4.16) converges to 0 as n → +∞.For the first term of the right hand side of (4.16), we have sup(a1,...,a N +1 )∈(iii,α) e N +1,2 µ n , (a 1 , . . . , a N +1 ) -e N,2 µ n , (a 1 , . . . , a N ) ≤ sup (a1,...,a N +1 )∈(iii,α) R 2(a N +1 -a 1 ) ξ -a 1 + a N +1 2 + µ n (dξ) ≤ sup (a1,...,a N +1 )∈(iii,α) R 3 • a N +1 ξ -a N +1 4 + µ n (dξ) ≤ 2 √ 3 • … sup K≥0 K E(X n -K) + -→ 0.Case (iii, β):a 1 ≤ 0 ≤ a 2 ≤ • • • ≤ a N+1 with 1 2 |a N+1 | ≤ |a 1 | ≤ |a N+1 |.Since we assume |a 1 | ≤ |a 2 | ≤ • • • ≤ |a N +1 |, then for any i ∈ {2, . . . , N + 1}, we have 1 2 |a i | ≤ |a 1 | ≤ |a i |. It follows that sup (a1,...,a N +1 )∈(iii,β) e N +1,2 µ n , (a 1 , . . . , a N +1 ) -g N +1 (a 1 , . . . , a N +1 ) ≤ sup (a1,...,a N +1 )∈(iii,β)

  ), we have sup (a1,...,a N +1 )∈(iii,β)e N +1,2 µ n , (a 1 , . . . , a N +1 ) -e 2,2 µ n , (a 1 , a 1 ) = sup (a1,...,a N +1 )∈(iii,β) e 2 2,2 µ n , (a 1 , a 1 ) -e 2 N +1,2 µ n , (a 1 , . . . , a N +1 ) e N +1,2 µ n , (a 1 , . . . , a N +1 ) + e 2,2 µ n , (a 1 , a 1 ) ≤ sup (a1,...,a N +1 )∈(iii,β) R |ξ -a 1 | 2 -min 1≤i≤N +1 |ξ -a i | 2 µ n (dξ) X n -a 1 2 ≤ sup (a1,...,a N +1 )∈(iii,β) R |ξ -a 1 | 2 -min 2≤i≤N +1 |ξ -a 2 | 2 + µ n (dξ) X n -a 1 2

  H denote the norm on H induced by (• | •) H . When there is no ambiguity, we drop the index H and write (• | •) and | • |. The separable Hilbert space is a very common setup for applications, for example in functional data analysis: one can set H

p defined by (2.6) is a distance on P p (R d ) and Q N,p is topologically equivalent to the Wasserstein distance W p .

Comments on optimality. If we consider only the quadratic case p = 2 and a norm | • | induced by an inner product, the result in Corollary 2.2-(a) is in fact not optimal. In the next section, we will prove that in such a setting, N d,2,|•| = 2 and this result can also be extended to any separable (possibly infinite-dimensional) Hilbert space.

3 Quadratic quantization based characterization on a separable Hilbert space: N H,2 = 2

Let H denote a separable Hilbert space with the inner product (

• | •) H . Let | • |

  The distances Q H 2,2 and W 2 are topologically equivalent on P 2 (H).

	Thus, if µ, ν ∈ P 2 (H) are such that
	H	|ξ| 2 H µ(dξ) =	H	|ξ|	2 H ν(dξ) and	H	ξµ(dξ) =	H	ξν(dξ),	(3.6)
	then we have e 1,2 (µ, •) = e 1,2 (ν, •). But condition (3.6) is clearly not sufficient to have µ = ν.
	Consequently, N H,2,|•| 2 = 2.
	Like what we did in Section 2.2, we define a function Q H 2,2 on P 2 (H)	2 by (µ, ν) → Q H 2,2 (µ, ν) =
	e 2,2 (µ, •) -e 2,2 (ν, •) sup . Then inequality (1.8) implies that Q H 2,2 (µ, ν) ∈ [0, +∞). Moreover,
	Proposition 3.1 and Theorem 3.1 lead the following corollary.
	Corollary 3.1.
	3.1. Proposition 3.1 and Theorem 3.1 directly imply that N H,2,|•| 2 ≤ 2. In fact, for every
	a ∈ H,
	e 2 1,2 (µ, a) =							2 H .

H |ξ -a| 2 H µ(dξ) = H |ξ| 2 H µ(dξ) -2 H ξµ(dξ) a H + |a|

  ). Let g N : R N → R + be defined by(a 1 , . . . , a N ) → g N (a 1 , . . . , a N ) := » e 2 N,2 µ ∞ , (a 1 , . . . , a N ) + 1 = N,2 µ n , (a 1 , . . . , a N ) -g N (a 1 , . . . , a N ) Therefore, (µ n ) n∈N is a Cauchy sequence in (P 2 (R), Q N,2 ) by the definition of Q N,2 .Proof. We proceed by induction.£ N = 2.Since the functions g 2 and e 2,2 (µ n , •) are symmetric, it is only necessary to show that sup (a,b)∈R 2 , |a|≤|b| e 2,2 µ n , (a, b) -g 2 (a, b)

			…	min 1≤i≤N	|a i | 2 + 1.
	Proposition 4.4. For every N ≥ 2,
		sup	n→+∞ -----→ 0.
		(a1,...,a N )∈R N
			n→+∞ -----→ 0. Note that when |a| ≤ |b|, g 2 (a, b) =
	»	|a| 2 + 1 = g

e 2 (a, a). We discuss now the value of e 2,2 µ n , (a, b) -g 2 (a, b) in the following four cases,

Since the dimension of the Hilbert space that we discuss in this section can be finite or infinite, we write directly H instead of d in the subscript of N d,p,|•| .

By approximate identity we mean ϕε∈ L 1 R d , B(R d ), λ d , ε > 0, such that R d ϕεdλ d = 1, sup ε>0 R d |ϕε| dλ d < +∞and lim ε→0 {|ξ|>η} ϕε(ξ)λ d (ξ) = 0 for every η > 0.

+ µ n (dξ)

≤ sup (a1,...,a N +1 )∈(iii,β)

for i ∈ {2, . . . , N + 1}. Consequently, sup (a1,...,a N +1 )∈(iii,β)

This completes the proof.

Proof of Theorem 4.1. Let µ n be the probability distribution of X n defined in (4.12). If for some N ≥ 2, P 2 (R), Q N,2 were complete, then there exists a probability measure 1). Now we prove that c(2, | • | r ) = 3 for every r ∈ (1, +∞).

We choose a 1 = (0, 1),

the last inequality is due to the fact that the function

Consequently, it attains its maximum either at -1 or at 1 2 . Hence, (x, y) -a 2 r r is upper bounded by 1 since

This implies that (x, y) ∈ B|•|r (a 2 , 1).

• If -1 ≤ y ≤ 1 2 and x ≤ 0, then (x, y) ∈ B|•|r (a 3 , 1) by the symmetry of the unit sphere.

Next, we will show c(2, | • | r ) > 2 for every 1 < r < +∞. Let a 1 and a 2 denote the two centers of balls on the sphere S |•| (0, 1). Since the r -ball is centrally symmetric with respect to (0, 0), we fix

1 r ] and x r + y r = 1.

We first prove that if

x r + y r = 1, then (x + y) r > 1. Let q = r -1, then q > 0 and (x + y) r = (x + y) 1+q = (x + y)(x + y) q = x(x + y) q + y(x + y) q > xx q + yy q = x r + y r = 1.

• Case 1. We choose a 2 such that a 2 is centrally symmetric to a 1 with respect to the center (0, 0), i.e. a 2 = (-x, -y).

We prove

which is the straight line (with respect to the Euclidean distance) across the origin and a 1 . Then between z 1 and z 2 , there exists at least one point which is not in the same side of H a1 as a 2 , and this point can not be covered by ∪ i=1,2 B|•|r (a i , 1). (iii) Let a 1 = (-1, 0, . . . , 0) and a 2 = (1, 0, . . . , 0). We will show that S |•|∞ (0, 1) ⊂ i=1,2 B|•|∞ (a i , 1). Let x = (x 1 , . . . , x d ) ∈ S |•|∞ (0, 1). There exists i 0 such that max 1≤i≤d |x i | ≤ |x i0 | = 1.

• If i 0 = 1, and x 1 = -1, then x -a 1 ∞ = x 1 + 1 ∨ max i={2,...,d} x i ≤ 1, that is, x ∈ B|•|∞ (a 1 , 1).

• If i 0 = 1, and x 1 = 1, then x-a 2 ∞ = x 1 -1 ∨max i={2,...,d} x i ≤ 1, that is, x ∈ B|•|∞ (a 2 , 1).

• If i 0 ≥ 2, and

• If i 0 ≥ 2, and

(iv) Let a i = (0, . . . , 1, . . . , 0) -the i th coordinate of a i is equal to 1 and the others equal to 0. We

For any x = (x 1 , . . . , x d ) ∈ S |•|r (0, 1), then there exists i 0 ∈ {1, . . . , d} such that x i0 ≥ 1 2 . Otherwise 1 = 1≤i≤d x i r < d × 2 -r ≤ 1, which yields a contradiction.

• If x i0 ≥ 1 2 , then x -a i0 r = (1 -x i0 ) r + i =i0 x i r = (1 -x i0 ) r + 1 -(x i0 ) r . As x i0 ≤ 1 2 , we have (1 -x i0 ) r -(x i0 ) r ≤ 0, so that x -a i0 r ≤ 1, which implies that x ∈ B|•|r (a i0 , 1).

• If x i0 ≤ -1 2 , one can similarly prove that x ∈ B|•|r (-a i0 , 1).

Consequently, we can conclude that S |•|r (0, 1) ⊂ d i=1 B|•|r (a i , 1) ∪ B|•|r (-a i , 1) .