Gilles Pagès
email: gilles.pages@upmc.fr

Benedikt Wilbertz
email: benedikt.wilbertz@gmx.de

Sharp rate for the dual quantization problem

Keywords: quantization, quantization rate, Zador's Theorem, Pierce's Lemma, dual quantization, Delaunay triangulation, random quantization

, where it has been shown that, at least in a Euclidean setting, dual quantizers are based on a Delaunay triangulation, the dual counterpart of the Voronoi tessellation on which "regular" quantization relies. Moreover, this new approach shares an intrinsic stationarity property, which makes it very valuable for numerical applications.

We establish in this paper the counterpart for dual quantization of the celebrated Zador theorem, which describes the sharp asymptotics for the quantization error when the quantizer size tends to infinity. On the way we establish an extension of the so-called Pierce Lemma by a random quantization argument. Numerical results confirm our choices.

Introduction

Starting with [START_REF] Pagès | Dual quantization for random walks with application to credit derivatives[END_REF] and continued in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF], we introduced a new notion of vector quantization called dual quantization (or Delaunay quantization in a Euclidean framework). We developed in [START_REF] Pagès | Optimal Delaunay and Voronoi quantization schemes for pricing American style options[END_REF] some first applications towards the design of numerical schemes for multi-dimensional optimal stopping and stochastic control problems arising in Finance (see also [START_REF] Bonnans | Energy contracts management by stochastic programming techniques[END_REF]). In general, the principle of dual quantization consists of mapping an R d -valued random vector (r.v.) onto a non-empty finite subset (or grid) Γ ⊂ R d using an appropriate random splitting operator J Γ : Ω 0 × R d → Γ (defined on an exogenous probability space (Ω 0 , S 0 , P 0 )) which satisfies the intrinsic stationarity property

∀ ξ ∈ conv(Γ), E P0 (J Γ (ξ)) = Ω0 J Γ (ω 0 , ξ) P 0 (dω 0 ) = ξ, (1)
where conv(Γ) denotes the convex hull of Γ in R d . Every r.v. X : (Ω, S, P) → conv(Γ) defined on a probability space can be canonically extended to (Ω 0 × Ω, S 0 ⊗ S, P 0 ⊗ P) in order to define dual quantization induced by Γ as X Γ,dual (ω 0 , ω) = J Γ (ω 0 , X(ω)).

As a specific feature inherited from [START_REF] Bonnans | Energy contracts management by stochastic programming techniques[END_REF], it always satisfies the dual or reverse stationary property

E P⊗P0 (J Γ (X) | X) = X.
This can be compared to the more classical Voronoi framework where the Γ-quantization of X is defined from a Borel nearest neighbour projection Proj Γ by X Γ,vor (ω) = Proj Γ (X(ω)).

The stationary property then reads: E(X | X Γ,vor ) = X Γ,vor , except that it holds only for grids which are critical points (typically local minima) of the so-called distortion function (see e.g. [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF]) in a Euclidean framework.

To each quantization is corresponds a functional approximation operator: Voronoi quantization is related to the stepwise constant functional approximation operator f • Proj Γ whereas dual quantization leads to an operator defined for every ξ ∈ conv(Γ) by

J Γ (f )(ξ) = E P0 f (J Γ (ω 0 , ξ)) = x∈Γ f (x)λ x (ξ), (2)
where λ x (ξ) = P 0 (J Γ (., ξ) = x), x ∈ Γ, are barycentric "pseudo-coordinates" of ξ in Γ satisfying λ x (ξ) ∈ [0, 1], x∈Γ λ x (ξ) = 1 and x∈Γ λ x (ξ)x = ξ. The operator J Γ is an interpolation operator which turns out, under appropriate conditions, to be more regular (continuous and stepwise affine, see [START_REF] Pagès | Optimal Delaunay and Voronoi quantization schemes for pricing American style options[END_REF]) than the "Voronoi" one. It is shown in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF][START_REF] Pagès | Dual quantization for random walks with application to credit derivatives[END_REF][START_REF] Pagès | Optimal Delaunay and Voronoi quantization schemes for pricing American style options[END_REF] how we can take advantage of this intrinsic stationary property to produce more accurate error bounds for the resulting cubature formula

E P (f ( X Γ,dual )) = E P (J Γ (f )(X)) = E P⊗P0 f (J Γ (ω 0 , ξ)) = x∈Γ w dual x f (x) (3)
where w dual x = E P (λ x (X)) = P ⊗ P 0 (J Γ (ω 0 , X) = x), x ∈ Γ, regardless of any optimality property Γ with respect to P X . Typically, if f ∈ Lip(R d , R) (Lipschitz continuous function) with coefficient [f ] Lip ,

E P f (X) -E P⊗P0 f ( X Γ,dual ) ≤ [f ] Lip X -X Γ,dual L 1 (P⊗P0) = [f ] Lip E P⊗P0 X -J Γ (ω 0 , X) = [f ] Lip E P⊗P0 E P⊗P0 ( X -J Γ (ω 0 , X) | X)
whereas, if f has Lipschitz continuous differential (the norm on R d is denoted . ), a second order Taylor expansion yields

E P f (X) -E P⊗P0 f ( X Γ,dual ) ≤ f (X) -E P⊗P0 f (J Γ (ω 0 , X)) | X L 1 (P⊗P0) ≤ [Df ] Lip E P⊗P0 X -J Γ (ω 0 , X) 2 ≤ [Df ] Lip E P⊗P0 E P⊗P0 ( X -J Γ (ω 0 , X) 2 | X) (4)
where

E P⊗P0 ( X -J Γ (ω 0 , X) p | X) = x∈Γ λ x (X) X -x p = J Γ ( . 2 )(X), p = 1, 2.
More generally, if one aims at approximating E f (X) | g(Y ) by its dually quantized counterpart E P⊗P0⊗P1 f (J Γ X (ω 0 , X)) | J Γ Y (ω 1 , Y ) (with obvious notations), it is also possible under natural additional assumptions to get error bounds based on both related dual quantization error moduli, see e.g. the proof (Step 2) of Proposition 2.1 in [START_REF] Pagès | Optimal Delaunay and Voronoi quantization schemes for pricing American style options[END_REF].

This suggests to investigate the properties and the asymptotic behaviour of the (Γ, L p )-mean dual quantization error, p ∈ (0, ∞), defined by

X -X Γ,dual p L p (P⊗P0) = X -J Γ (ω 0 , X) p L p (P⊗P0) = E P⊗P0 E P⊗P0 X -J Γ (ω 0 , X) p | X
so as to make it as small as possible. This program can be summed up in four phases:

-The first step is to minimize the above conditional expectation, i.e. E( ξ -J Γ (ω 0 , ξ) p ) for every ξ ∈ conv(Γ), for a fixed grid Γ i.e. to determine the best splitting random operator J Γ . In a regular quantization, this phase corresponds to showing that the nearest neighbour projection on Γ is the best projection on Γ.

-The second step is "optional" . It aims at finding grids which minimize the mean dual quantization error X -J Γ (ω 0 , X)

L p (P⊗P0)
among all grids Γ whose convex hull contains the support of the distribution of X or equivalently such that P(X ∈ conv(Γ)) = 1.

-The third step is to extend dual quantization to r.v.s X with unbounded support while the performances of the resulting cubature formula (see [START_REF] Cohort | Limit theorems for random normalized distortion[END_REF]), having in mind that the stationarity can no longer holds.

The first two steps have been already solved in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF]. We discuss in-depth the third one in Section 2.2). The aim of this paper is to solve the fourth and last step: elucidate is the rate of decay to 0 of the optimal L p -mean dual quantization error modulus, i.e. minimized over all grids Γ of size at most N -as N grows to infinity. This is the to establish in a dual quantization framework the counterpart of Zador's Theorem which rules the convergence rate of optimal "regular" (Voronoi) quantization and is recalled below. To be more precise, we will establish such a theorem, for L ∞ -bounded r.v.s but also, once mean dual quantization error will have been extended in an appropriate way following [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF], to general r.v.s.

Let us now introduce in more formal way the (local and mean) dual quantization error moduli, following [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF]. For a grid Γ ⊂ R d , we define the L p -mean dual quantization error of X induced by the grid Γ by d p (X; Γ) = F p (X; Γ) L p (P) [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF] where F p denotes the local dual quantization error function defined by

F p (ξ; Γ) = inf x∈Γ λ x ξ -x p 1 p , λ x ∈ [0, 1], x∈Γ λ x x = ξ, x∈Γ λ x = 1 (6)
Note that F p (ξ; Γ) < +∞ if and only if ξ ∈ conv(Γ) so that d p (X; Γ) < +∞ if and only if X ∈ conv(Γ) P-a.s.. and that

d p (X; Γ) = X -X Γ,dual p L p (P⊗P0)
. Hence, this notion only makes sense for compactly supported r.v.s. In particular if the support of P X is compact and contains d + 1 affinely independent points, d n,p (X, Γ) = +∞ as long as n ≤ d.This new quantization modulus leads to an optimal dual quantization problem at level N ,

d n,p (X) = inf d n,p (X, Γ), Γ ⊂ R d , |Γ| ≤ n = inf F p (X; Γ) p , Γ ⊂ R d , |Γ| ≤ n . (7)
One important application of quantization in general is the use of quantization grids as numerical cubature formula (see [START_REF] Capéraà | Méthodes et modèles en statistique non paramétrique[END_REF]). The main feature here is the stationarity which allows to derive a second order formula for the integration error. Since, by construction, dual quantization can achieve stationarity only on a compact set, we show in section 2.2 that the extension of dual quantization to non-compactly supported random variables as defined in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF] preserves this second order rate on the whole support of the r.v.

We therefore define the splitting operator J Γ outside conv(Γ) by setting

∀ ξ ∈ R d \ conv(Γ), J Γ (ω 0 , ξ) = Proj conv (Γ)∩∂Γ (ξ)
where Proj conv (Γ)∩∂Γ is a Borel nearest neighbour projection on conv (Γ) ∩ ∂Γ. This choice is not unique: an alternative extension could be to set J Γ (ω 0 , ξ) = Proj conv(Γ) (ξ). But the above choice is tractable in terms of simulation and we will prove that it does not deteriorate the resulting mean error when |Γ| → +∞. Though the stationary property is lost as expected, we point out in Section 2.2 that this operator remains as performing as J Γ is for bounded r.v.s when implementing cubature formulas for unbounded r.v.s. Then, we to derive the extended local dual quantization error function by

Fp (ξ; Γ) := F p (ξ; Γ) 1 conv(Γ) (ξ) + dist(X, Γ) 1 conv(Γ) c (ξ), (8)
and the extended L p -mean dual quantization error of X induced by Γ by dp (X; Γ) = Fp (X; Γ) L p (P) .

Finally, we define the extended L p -mean dual quantization error at level n given by

dn,p (X) = inf dp (X, Γ), Γ ⊂ R d , |Γ| ≤ n . (10)
Finally, we briefly recall a few facts about the (regular) Voronoi optimal quantization problem at level n associated to the nearest neighbour projection Proj Γ : it reads

e n,p (X) = inf dist(X, Γ) L p (P) , Γ ⊂ R d , |Γ| ≤ n (11)
(where dist(x, A) = inf a∈A xa ). It is well-known that e n,p (X) ↓ 0 as soon as n → +∞ and X ∈ L p (P). Moreover, the rate of convergence to 0 of e n,p (X) is ruled by Zador's Theorem (see [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF]).

Theorem 1 (Zador). Let X ∈ L p ′ R d (P), p ′ > p. Let P X = h.λ d + ν, ν ⊥ λ d be the distribution of X where λ d denotes the Lebesgue measure on (R d , Bor(R d )). Then lim n→∞ n 1 d e n,p (X) = Q vq • ,p,d h 1 p d p+d where h d p+d = R d h(ξ) d p+d dξ 1+ p d and Q vq • ,p,d = inf n n 1 d e n,p (U ([0, 1] d )) ∈ (0, ∞).
The above rate depends on d and is known as the curse of dimensionality. Its statement and proof goes back to Zador (PhD, 1954) for the uniform distributions on hypercubes, its extension to absolutely continuous distributions is due to Bucklew and Wise in [START_REF] Bucklew | Multidimensional asymptotic quantization theory with r th power distortion measures[END_REF]. A first general rigor proof (according to mathematical standards) was provided in [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF] in 2000 (see also [6] for a survey of the history of quantization).

It should be noted that d n,p (X) and dn,p (X) do not coincide even for bounded r.v.s. We will extensively use (see [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF]) that

d n,p (X) ≥ dn,p (X) ≥ e n,p (X).
This paper is entirely devoted to establishing the sharp asymptotics of the optimal dual quantization error moduli d n,p (X) and dn,p (X) as n goes to infinity. The main result is stated in Theorem 2 (Zador's like theorem) (see Section 2.1 below). Proposition 2 (a Pierce like Lemma) is a companion result which provides a non-asymptotic upper bound for the exact rate simply involving moments of the r.v. X (higher than p). Our proof has the same structure as that of the original Zador Theorem (see e.g. [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF] where it has been rigorously completed for the first time), except that the splitting operator J Γ is much more demanding to handle than the plain nearest neighbour projection: it requires more sophisticate arguments borrowed from convex analysis (including dual primal/methods) and geometry, both in a probabilistic framework. In one dimension the exact rate O(n -1 ) for d n,p (X) and dn,p (X) follows from a random quantization argument detailed in Section 4 (extended Pierce Lemma for d n,p (X)). This rate can be transferred in a d-dimensional framework to O(n -1 d ) using a product (dual) quantization argument (see Proposition 1 below and Section 3.2). Finally, the sharp upper bound is obtained in Section 5 by successive approximation procedures of the density of X, whereas the lower bound relies on a new "firewall" Lemma.

Notations: • conv(A) stands for the convex hull of A ⊂ R d , |A| for its cardinality, diam . (A) = sup x,y∈A xy for its diameter and aff.dim(A) for the dimension of the affine subspace of R d spanned by A.

• We denote n i := n! i!(n-i)! , n, i ∈ {0, . . . , n}, n ∈ N. • ⌊x⌋ and ⌈x⌉ will denote the lower and the upper integral part of the real number x respectively; set likewise x ± = max(±x, 0). For two sequences of real numbers (a n ) and

(b n ), a n ∼ b n if a n = u n b n with lim n u n = 1. • For every x = (x 1 , . . . , x d ) ∈ R d , |x| ℓ r = (|x 1 | r + • • • |x d | r ) 1/r denotes the ℓ r -norm or pseudo- norm, 0 < r < +∞ and |x| ℓ ∞ = max 1≤i≤d |x i | denotes the ℓ ∞ -norm. A general norm on R d will be denoted • . • supp(µ) denotes the support of a distribution µ on (R d , Bor(R d )).
2 Main results and motivation for extended dual quantization

Main results

The theorem below establishes for any p > 0 and any norm on R d the counterpart of Zador's Theorem in the framework of dual quantization for both d n,p and dn,p error moduli.

Theorem 2. (a) Let X ∈ L ∞ R d (P). Assume the distribution P X of X reads P X = h.λ d + ν, ν ⊥ λ d . Then lim n→∞ n 1 d d n,p (X) = lim n→∞ n 1 d dn,p (X) = Q dq • ,p,d h 1 p d p+d where Q dq • ,p,d = inf n≥1 n 1 d d n,p (U ([0, 1] d )) ∈ (0, ∞). (b) Let X ∈ L p ′ R d (P), p ′ > p. Assume the distribution P X of X reads P X = h.λ d + ν, ν ⊥ λ d . Then lim n→∞ n 1 d dn,p (X) = Q dq • ,p,d h 1 p d p+d . (c) If d = 1, then d n,p (U ([0, 1])) = 2 (p + 1)(p + 2) 1 p 1 n -1 , which implies Q dq | . |,p,1 = 2 p+1 p+2 1 p Q vq | . |,p,1 .
Moreover, we will also establish in Section 5 an upper bound for the dual quantization coefficient

Q dq • ,p,d when • = |•| ℓ r . Proposition 1 (Product quantization). Let r, p ∈ [1, ∞) with r ≤ p. Then it holds for every d ∈ N Q dq |•| ℓ r ,p,d ≤ d 1 r • Q dq |•|,p,1
where | . | denotes standard absolute value on R.

Since this upper bound achieves the same asymptotic rate as in the case of regular quantization (cf. Corollary 9.4 in [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF]), this suggests the rate O(d 1 r ) to be also the true one for Q dq

• ,p,d as d → ∞. As a step towards the above sharp rate theorem, we also establish a counterpart of the so-called Pierce Lemma (as stated in an operating form e.g. in [START_REF] Luschgy | Functional quantization rate and mean regularity of processes with an application to Lévy processes[END_REF]). In practice, it turns out to be quite useful for applications since it provides non-asymptotic error bounds which only depend on the moments of the r.v. X and the size of the optimal grid as emphasized in [START_REF] Pagès | Optimal Delaunay and Voronoi quantization schemes for pricing American style options[END_REF] (see section 4.1 for the proof).

Proposition 2 (d-dimensional extended Pierce Lemma). (a) Let p, η > 0. There exists a real constant C d,p,η > 0 such that, for every n ≥ 1 and every r.v.

X ∈ L p+η R d (Ω, A, P), dn,p (X) ≤ C d,p,η σ p+η, . (X) n -1/d
where σ p+η, . (X) = inf a∈R d Xa L p+η denotes the L p+η -pseudo-standard deviation of X.

(b) If supp(P X ) is compact then there exists a real constant C ′ d,p,η > 0 such that, for every n ≥ 1

d n,p (X) ≤ C ′ d,p,η diam . (supp(P X )) n -1/d .
2.2 How to use the extended L p -dual quantization error modulus?

We briefly explain why the extended dual quantization error modulus, already been introduced in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF] for non-compactly supported distributions, is the right tool to perform automatically an optimized truncation of non-compactly supported distributions. basically, it uses its additional "outer Voronoi projection" as a penalization term which expands automatically the convex hull of the dually optimal grid at its appropriate "amplitude", making altogether the distribution outside of its convex hull "negligible" and sharing an optimal rate of decay n -1 d as its size n goes to infinity. The specific choice of a Voronoi quantization among other possible solutions for this penalization is motivated by both its theoretical tractability and its simple implementability in stochastic grid optimization algorithms. This feature if of the highest importance for numerical integration or conditional execration approximation. This is the main motivation to introduce and deeply investigate the sharp asymptotics of this L p -mean extended dual quantization error modulus dn,p (X).

We saw in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF] that Euclidean dual quantization of a compactly supported distribution produces stationary (dual) quantizers, namely r.v.s X dual satisfying E( X dual | X) = X, so that (see Proposition ?? in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF]), dual quantization based cubature formula induce on functions f ∈ C 1 Lip (R d , R) (Lipschitz functions with Lipschitz continuous gradient) an error at most equal to [Df ] Lip d 2,n (X) 2 . Taking into account the rate established in Theorem 2(a), this yields a O(n -2 d ) error rate. There is no way to extend dual quantization to (possibly) unbounded r.v.s so that it preserves the above stationarity property. However, with the choice we made (nearest neighbor projection on the grid outside its convex hull), natural heuristic arguments strongly suggest that the above order O(n -2 d ) is still satisfied for functions in C 1 Lip (R d , R). We consider an unbounded Borel distribution µ = P X of an R d -valued r.v. X. Let Γ n be an Euclidean L 2 -optimal extended dual quantization grid of size n for µ (see [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF] or Theorem 4) and X dual the resulting Γ n -valued extended dual quantization of X. Let C n = conv Γ n denote the convex hull of Γ n . It is clear by construction of X dual that X dual = X dual + X vor where, with obvious notations,

1 {X∈Cn} E X dual |X = 1 {X∈Cn} X (dual stationarity) and X vor = Proj Γn∩Cn (X). Hence, if f ∈ C 1 Lip (R d , R), E (Df (X)|X -X dual )|X ∈ C n = 0 and E f X dual |X ∈ C n -E f X |X ∈ C n = E f X dual -f X -Df (X).(X -X dual )|X ∈ C n ≤ [Df ] Lip d n,2 (Γ n , X dual |X ∈ C n ) 2 .
Consequently,

E f X dual 1 {X∈Cn} -E f X 1 {X∈Cn} ≤ [Df ] Lip d n,2 ( X dual , Γ n ) 2 /P(X ∈ C n ) ≤ [Df ] Lip dn,2 (X, Γ n ) 2 /P(X ∈ C n ).
On the other hand,

E f X vor 1 {X / ∈Cn} -E f X 1 {X / ∈Cn} ≤ [f ] Lip e n,2 X, Γ n P X / ∈ C n 1 2 ≤ [f ] Lip dn,2 (X)P X / ∈ C n 1 2 .
Relying on Theorem 2(b), we know that, if

µ = h.λ d ⊥ + ν, then dn,2 (X) ∼ Q dq 2,|.| eucl h 1 p d 2+d n -1 d .
The "outside" contribution will be negligible compared to the "inside" one as soon as

P X / ∈ C n = o dn,2 (X, Γ n ) 2 = o n -2 d . (12)
This condition turns out to be not very demanding and can be checked, at least heuristically, as illustrated below: if X d = N (0; I d ), one may conjecture, taking advantage of the spherical symmetries of the normal distribution, that C n is approximately a sphere centered at 0 with radius ρ n = max a∈Γn |a|. As

P(|X| ≥ ξ) ∼ V d ξ d-2 e -ξ 2 2 as ξ → +∞ (with V d = λ d-1 (S d (0, 1))).
Condition [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF] is satisfied as soon as lim inf n ρn √

log n > 2 √ d (≥ if d = 1, 2).
As an example, one must have in mind that, for optimal Voronoi quantization, this inequality is satisfied since (see [START_REF] Pagès | Asymptotics of the maximal radius of an L r -optimal sequence of quantizers[END_REF])

lim n ρn √ log n = 2(1 + 2/d) > 2 √ d .
More precisely, we have

P X / ∈ C n ∼ κ d (log n) d 2 -1 n -1-2 d so that dn,2 (X)P X / ∈ C n 1 2 = O n -2 d -1 2 (log n) d-2 4
.

Numerical experiments, not reproduced here, carried out with the above N (0; I d ) distribution confirm that the radius of optimal dual quantizers always achieves this asymptotics which makes the above partially heuristic reasoning very likely. Moreover, we also tested the two rates of convergence of P(X ∈ C n ) and dn,2 (X) 2 , this time on the joint distribution of the (W 1 , sup t∈[0,1] W t ), W standard Brownian motion which has less symmetries (see appendix A). They also confirm that the above partially heuristic reasoning is very likely.

Dual quantization: background and basic properties

Throughout the paper, except specific mention, R d is equipped with a norm • .

More background

In the introduction, the definitions related to Voronoi (or regular)and dual quantizations of a r.v. X defined on a probability space (Ω, S, P) have been recalled (see ( 7)-( 10)). The aim of this section is to come back briefly to the origin and the motivations which led us to introduce dual quantization in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF]. On the way, we will also recall several basic results on dual quantization established in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF]. First, we will assume throughout the paper that the r.v. of interest, X, is truly d-dimensional in the sense that aff.dim(supp(P X )) = d.

Let us start by a few practical points. First note that although all these definitions are related to a r.v. X, in fact it only depends on the distribution P = P X , so we will also often write d p (P, Γ) for d p (X, Γ) and d n,p (P). Furthermore, to alleviate notations, we will denote from now on F p , d p and dp , . . . instead of (F p ) p , (d p ) p and ( dp ) p ,. . . Let us come back to the terminology dual quantization: it refers to a canonical example of the intrinsic stationary splitting operator: the dual quantization operator.

To be more precise, let p ∈ [1, +∞) and let Γ = {x 1 , . . . , x n } ⊂ R d be a grid of size n ≥ d + 1 such that aff.dim(Γ) = d i.e. Γ contains at least one d + 1-tuple of affinely independent points. The underlying idea is to "split" ξ ∈ conv(Γ) across at most d + 1 affinely independent points in Γ proportionally to its barycentric coordinates of ξ. There are usually many possible choices of such a Γ-valued (d + 1)-tuple of affinely independent points, so we introduced a minimal inertia based criterion to select the most appropriate one ξ, namely the function F p (ξ; Γ) defined for every ξ as the value of the minimization problem

F p (ξ; Γ) = inf (λ1,...,λn) n i=1 λ i ξ -x i p 1 p , λ i ∈ [0, 1], i λ i x i 1 = ξ 1 . (13)
Owing to the compactness of the constraint set (λ i ≥ 0, i λ i = 1, i λ i x i = ξ), there exists at least one solution λ * (ξ) to the above minimization problem. Moreover, for any such solution, one shows using convex extremality arguments, that the set I * (ξ) := i ∈ {1, . . . , n} s.t. λ * i (ξ) > 0 defines an affinely independent subset {x i , i ∈ I * (ξ)}.

If, for every ξ ∈ conv(Γ), this solution is unique, the dual quantization operator is simply defined on conv(Γ) by

∀ ξ ∈ conv(Γ), ∀ ω 0 ∈ Ω 0 , J * Γ (ω 0 , ξ) = i∈I(ξ) * x i 1 { i-1 j=1 λ * j (ξ)≤U (ω0)< i j=1 λ * j (ξ)} , (14)
where U denotes a random variable uniformly distributed over [0, 1] on an exogenous probability space (Ω 0 , S 0 , P 0 ). This operator J * Γ is then measurable (see [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF]). The above uniqueness assumption is not so stringent, especially for applications. Thus, in a purely Euclidean quadratic framework: . = | . | ℓ 2 (canonical Euclidean norm) and p = 2 and if Γ is said in "general position" ( 1 ), then {ξ s.t. I * (ξ) = I}, |I| ≤ d + 1 makes up a Borel partition of conv(Γ) (with possibly empty elements), known in 2-dimension as the Delaunay triangulation of Γ (see [START_REF] Rajan | Optimality of the delaunay triangulation in R d[END_REF] for the connection with Delaunay triangulations).

In a more general framework, we refer to [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF] for a construction of dual quantization operators. Such operators are splitting operators since, by construction, they satisfy the stationarity property (1).

One must have in mind that the dual quantization operators J * Γ (ω 0 , ξ) play the role of the nearest neighbour projections for regular Voronoi quantization. One checks that, by construction,

∀ ξ ∈ conv(Γ), J * Γ (ξ) -ξ L p (P0) = F p (ξ; Γ) L p (P0) 1 no d + 2 points of Γ lie on a sphere in R d .
so that, as soon as supp(P X ) ⊂ Γ (or equivalently P(X ∈ conv(Γ)) = 1),

d p (X; Γ) = J * Γ (X) -X L p (P0⊗P) = F p (X; Γ) L p (P0⊗P) .
At this stage, it appears naturally that the the second step of the optimization process is to find (at least) one grid which optimally "fits" (the distribution of) X for this criterion i.e. which is the solution to the second level optimization problem

d n,p (X) = inf J * Γ (X) -X L p (P0⊗P) , J * Γ : Ω 0 × conv(Γ) → Γ, conv(Γ) ⊃ supp(P X ), |Γ| ≤ n . Note that if X ∈ L ∞ R d (P), d n,p (X) < +∞ if and only if n ≥ d + 1 (whereas it is identically infinite if X is not essentially bounded).
The existence of an optimal grid (or dual quantizer) has been established in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF] (see below).

The error modulus d n,p (X) can also be characterized as the lowest L p -mean approximation error by a r.v. having at most n values and satisfying the intrinsic stationarity property as established in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF] (Theorem 2, precisely recalled in Theorem 3 below). It should be compared to the well-known property satisfied by the mean (regular) quantization error modulus e n,p (X), namely e n,p (X) = inf X -

X L p (P) , | X(Ω)| ≤ n .
A stochastic optimization procedure based on a stochastic gradient approach has been devised in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF] to compute optimal dual quantization grids w.r.t. various distributions (so far, uniform over

[0, 1] 2 , normal, (W 1 , sup t∈[0,1] W t ), W standard Brownian motion in a purely Euclidean framework).
Let us conclude by two results established in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF]. The first one is the characterization of dual quantization operator in terms in terms of best L p -approximation (see [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF], Theorem 2). Theorem 3. Let X : Ω, S, P) → R d be a r.v. such that aff.dim(supp(P X )) = d and let n ∈ N,

n ≥ d + 1. Then d n,p (X) = inf E X -J Γ (X) L p : J Γ : Ω 0 × R d → Γ, intrinsic stationary, supp(P X ) ⊂ conv(Γ), |Γ| ≤ n = inf E X -X L p : X : (Ω 0 × Ω, S 0 ⊗ S, P 0 ⊗ P) → R d , | X(Ω 0 × Ω)| ≤ n, E( X|X) = X ≤ +∞.
This quantity is finite if and only if X ∈ L ∞ (Ω, S, P).

Finally, the following existence result for optimal dual quantizers at level n ∈ N and the L pnorm with p ∈ (1, ∞) is established in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF]. Although we will not use it in our proofs, this result is recalled for the reader's convenience.

Theorem 4 (Existence of optimal quantizers). Let X ∈ L p (P) for some p ∈ (1, ∞).

(a) If supp(P X ) is compact, then there exists for every n ∈ N a grid Γ * n ⊂ R d , |Γ * n | ≤ n such that d p (X; Γ * n ) = d n,p (X).
(b) If P X is strongly continuous in the sense that it assigns no mass to hyperplanes of R d , then there exists for every n ∈ N a grid

Γ * n ⊂ R d , |Γ * n | ≤ n such that dp (X; Γ * n ) = dn,p (X).
If furthermore |supp(P X )| ≥ n, then the above statements hold with |Γ * n | = n.

Local properties of the dual quantization functional

We establish or recall in this paragraph some first general properties of the local L p -dual quantization functional F p , which will be needed for the final proof of Theorem 2.

Proposition 3. Let Γ 1 , Γ 2 ⊂ R d be finite grids and let ξ ∈ R d . Then Γ 1 ⊂ Γ 2 =⇒ F p (ξ; Γ 2 ) ≤ F p (ξ; Γ 1 ). Proof. First note that the set {λ ∈ R n | [ x1 ... xm 1 ... 1 ] λ = ξ 1 } is clearly a compact set on which the continuous function λ → n i=1 λ i ξ -x i p attains a minimum. Assume Γ 1 = {x 1 , . . . , x m } and Γ 2 = {x 1 , . . . , x m , x m+1 , . . . , x n }. Then F p (ξ; Γ 2 ) = min λ∈R n n i=1 λ i ξ -x i p s.t. [ x1 ... xn 1 ... 1 ]λ= ξ 1 , λ≥0 ≤ min λ∈R n ,λm+1=•••=λn=0 n i=1 λ i ξ -x i p s.t. [ x1 ... xm 1 ... 1 ]λ= ξ 1 , λ≥0 = min λ∈R m m i=1 λ i ξ -x i p s.t. [ x1 ... xm 1 ... 1 ]λ= ξ 1 , λ≥0
= F p (ξ; Γ 1 ).

We will also make use of the following three properties established in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF] (Propositions 11, 12, 13 respectively). In particular, the third claim yields a first upper bound for the asymptotics of the local L p -dual quantization error when the size of the grid goes to infinity.

Let Γ = {x 1 , . . . , x n } ⊂ R with x 1 ≤ . . . ≤ x n . Then ∀ξ ∈ [x 1 , x n ], F p (ξ; Γ) ≤ max 1≤i≤n-1 x i+1 -x i 2 p . (b) Local product Quantization: Let • = | • | ℓ p and let Γ = 1≤j≤d Γ j for some Γ j ⊂ R. Then ∀ ξ ∈ R d , F p,|.| ℓ p (ξ; Γ) = d j=1 F p (ξ j ; Γ j ) 1 p
and the same holds true with Fp,ℓ p on R d . (c) Product Quantization:

Let C = a + L [0, 1] d , a = (a 1 , . . . , a d ) ∈ R d , L > 0,
be a hypercube, with edges parallel to the coordinate axis with common edge-length L. Let Γ be the product quantizer of size (m + 1) d defined by

Γ = d k=1 a j + iL m , i = 0, . . . , m .
There exists a positive real constant C . ,p = sup |x| ℓ p =1 x p > 0 such that

∀ ξ ∈ C, F p (ξ; Γ) ≤ d C • ,p • L 2 p • m -p . ( 15
)
4 Extended Pierce lemma and applications

The aim of this section is to provide a non-asymptotic "universal" upper-bound for the optimal (extended) L p -mean dual quantization error in the spirit of [START_REF] Pierce | Asymptotic quantizing error for unbounded random variables[END_REF]: it achieves nevertheless the optimal rate of convergence when the size n goes to infinity. Like for Voronoi quantization this upper-bound deeply relies on a random quantization argument and will be a key in the proof of the sharp rate (step 2 of the proof of Theorem 2). For every integer n ≥ 1, we define the set of "non-decreasing" n-tuples of R n by

I n := {(x 1 , . . . , x n ) ∈ R n , -∞ < x 1 ≤ x 2 ≤ • • • ≤ x n < +∞}.
Let (x 1 , . . . , x n ) ∈ I n (so that Γ = {x 1 , . . . , x n } has at most n elements) and let ξ ∈ R. When d = 1, it is clear that the minimization problem ( 6) always has a unique solution when ξ ∈ [x 1 , x n ] so that, for every

ω 0 ∈ Ω 0 = [0, 1], one has J * (x1,...,xn) (ω 0 , ξ) = n-1 i=1 x i 1 {ω0≤ x i+1 -ξ x i+1 -x i } + x i+1 1 {ω0≥ x i+1 -ξ x i+1 -x i } 1 [xi,xi+1) (ξ) +x 1 1 (-∞,x1) (ξ) + x n 1 [xn,+∞) (ξ).
It follows from ( 8) that

F p n (ξ, x 1 , . . . , x n ) = E P0 ξ -J * (x1,...,xn) (ω 0 , ξ) p = n-1 i=1 (x i+1 -ξ) p (ξ -x i ) x i+1 -x i + (x i+1 -ξ)(ξ -x i ) p x i+1 -x i 1 [xi,xi+1) (ξ) (16) +(x 1 -ξ) p 1 (-∞,x1) (ξ) + (ξ -x n ) p 1 [xn,+∞) (ξ)
(the subscript n is temporarily added to the functional F p , Fp , etc, to emphasize that they are defined on I n × R). The functionals F p n share three important properties extensively used in what follows:

• Additivity: Let (x 1 , . . . , x i0 , . . . , x n ) ∈ I n . Then for every ξ ∈ R

F p n (ξ, x 1 , . . . , x n ) = F p i0 (ξ, x 1 , . . . , x i0 )1 (-∞,xi 0 ) (ξ) + F p n-i0+1 (ξ, x i0 , . . . , x n )1 [xi 0 ,+∞) (ξ).
• Consistency and monotony: Let (x 1 , . . . , x n ) ∈ I n and x i ∈ [x i , x i+1 ] for an i ∈ {1, . . . , n-1}.

For every ξ ∈ R,

F p n+1 (ξ, x 1 , . . . , x i-1 , x i , x i , x i+1 , . . . , x n ) ≤ F p n (ξ, x 1 , . . . , x i-1 , x i , x i+1 , . . . , x n ). When ξ ∈ [x 1 , x n ], F p n (ξ; x 1 , . . . , x n ) coincides with F p (ξ, {x 1 , . . . , x n }
) and this inequality is a consequence of the definition of F p as the value function of the minimization problem (6). Outside, the above inequality holds as an equality since it amounts to the nearest distance of

ξ to [x 1 , x n ]. As a consequence, n -→ dn,p (X) = inf (x1,...,xn)∈In Fp,n (X, x 1 , . . . , x n ) L p is non-increasing, (17)
More generally, for every fixed

x 0 ∈ R, both n -→ inf (x 0 ,x2,...,xn)∈In Fp,n (X, x 0 , x 2 , . . . , x n ) L p and n -→ inf (x1,x2,...,xn-1,x 0 )∈In Fp,n (X, x 1 , . . . , x n-1 , x 0 ) L p (18) are non-increasing. • Scaling: ∀ ω ∈ Ω 0 , ∀ (x 1 , . . . , x n ) ∈ I n , ∀ ξ ∈ R, ∀ α ∈ R + , ∀ β ∈ R, (i) F p n (α ξ + β, α x 1 + β, . . . , α x n + β) = α F p n (ξ, x 1 , . . . , x n ), (ii) F p n (ξ, x 1 , . . . , x n ) = F p n (-ξ, -x n , . . . , -x 1 ).
Theorem 5. Let p, η > 0. There exists a real constant C p,η > 0 such that for every random variable X : (Ω, A, P) → R,

∀ n ≥ 1, inf (x1,...,xn)∈In Fp,n (X, x 1 , . . . , x n ) L p ≤ C p,η X L p+η n -1 .
The proof below relies on a random quantization argument involving an n-sample of the Pareto(δ)-distribution on [1, +∞). Though significantly more demanding, it plays the same crucial role in establishing the sharp rate result as the so-called Pierce Lemma established in [START_REF] Luschgy | Functional quantization rate and mean regularity of processes with an application to Lévy processes[END_REF] (see also [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF]) for Voronoi quantization to prove the original Zador Theorem.

In the proof, we will make use of the Γ and B functions defined by Γ(a) = Proof.

Step 1. We first assume that X is [1, +∞)-valued and n ≥ 2. Let (Y n ) n≥1 be a sequence of i.i.d. Pareto(δ)-distributed random variables (with probability density f Y (y) = δy -δ-1 1 {y≥1} ) defined on a probability space (Ω ′ , A ′ , P ′ ). Let δ = δ(η, p) ∈ (0, η ⌈p⌉ ) be chosen so that ℓ = ℓ(p, η) = p δ is an integer and ℓ ≥ 2. For every n ≥ ℓ(p, η), set n = nℓ + 2 ∈ N, n ≤ n. It follows from the monotony property (18) that inf

(1,x2,...,xn)∈In Fp,n (X, 1, x 2 , . . . , x n ) L p ≤ inf (1,x2,...,x n )∈I n Fp, n (X, 1, x 2 , . . . , x n ) L p ≤ Fp, n (X, Y (n) 0 , Y (n) 1 , . . . , Y (n) n-1 ) L p (Ω×Ω ′ ,P⊗P ′ )
where, for every n ≥ 1,

Y (n) = (Y (n) 1 , . . . , Y (n)
n ) denotes the standard order statistics of the first n terms of the sequence (Y k ) k≥1 and Y (n) 0 = 1. On the other hand, we recall (see e.g. [START_REF] Capéraà | Méthodes et modèles en statistique non paramétrique[END_REF]) that the joint distribution of (Y

(n) i , Y (n) i+1 ), 1 ≤ i ≤ n -1, is given by P ′ (Y (n) i ,Y (n) i+1 ) (du, dv) = δ 2 n! (i -1)!(n -i -1)! (1 -u -δ ) i-1 v -δ(n-i-1) (uv) -δ-1 du dv.
Step 2. Assume that n ≥ 3. Since X and (Y 1 , . . . , Y 0 ) are independent and X ≥ 1

Fp, n (X, Y (n) 0 , Y (n) 1 , . . . , Y (n) n-1 ) p L p (Ω×Ω ′ ,P⊗P ′ ) = [1,+∞) Fp, n (ξ, Y (n) 0 , Y (n) 1 , . . . , Y (n)
n-1 ) p L p (Ω ′ ,P ′ ) P X (dξ).

Relying on the expression (16) of the functional F p n , we set for every i = 0, . . . , nℓ and ξ ≥ 1

(a) i := E (Y (n) i+1 -ξ) p (ξ -Y (n) i ) Y (n) i+1 -Y (n) i 1 {Y (n) i <ξ≤Y (n) i+1 } , (b) i := E (Y (n) i+1 -ξ)(ξ -Y (n) i ) p Y (n) i+1 -Y (n) i 1 {Y (n) i <ξ≤Y (n) i+1 } and (c) n-1 := E ξ -Y (n) n-ℓ+1 p 1 {ξ≥Y (n) n-ℓ+1 } . We will first inspect the sum n-ℓ i=0 ( ) i , = a, b successively. Let i ∈ {1, . . . , ñ -1}. It follows from the above expression of the distribution of (Y (n) i , Y (n) i+1 ) that (a) i = δ 2 1≤u≤ξ≤v (v -ξ) p (ξ -u) v -u (1 -u -δ ) i-1 v -δ(n-i-1) (uv) -δ-1 du dv n! (i -1)!(n -i -1)! .
The change of variable v = ξ(w + 1) yields

(a) i = n(n-1) n-2 i-1 δ 2 ξ 1 du (ξ-u)(1-u -δ ) i-1 u -δ-1 ξ p-δ(n-i) +∞ 0 dw w p ξ(w + 1) -u (w+1) -δ(n-i)-1 .
Noting that

ξ-u ξ(w+1)-u ≤ 1 w+1 then leads to (a) i ≤ n(n -1) n-2 i-1 δ 2 n(n -1)ξ p-δ(n-i) ξ 1 (1 -u -δ ) i-1 u -δ-1 du × +∞ 0 w p (1 + w) -δ(n-i)-2 dw.
The change of variable w = 1 y -1 shows that +∞ 0 w p (1+w) -δ(n-i)-2 dw = B(δ(n-i)-p+1, p+1)

whereas ξ 1 (1 -u -δ ) i-1 u -δ-1 du = (1 -ξ -δ ) i δi so that (a) i ≤ δn n-1 i (1 -ξ -δ ) i ξ p-δ(n-i) Γ(p + 1)Γ(δ(n -i) -p + 1) Γ(δ(n -i) + 2)
where we used the standard identity n-

1 i = n-1 i n-2 i-1 . When i = 0, noting that the density of Y (n) 1 = min 1≤i≤n Y i is δny -δn-1 1 {y≥1} , we get (a) 0 = E (Y (n) 1 -ξ) p (ξ -1) Y (n) 1 -1 1 {1≤ξ≤Y (n) 1 } = δn +∞ ξ (ξ -1) (v -ξ) p v -1 v -δn-1 dv = δnξ p-δn +∞ 0 (ξ -1) ξ(w + 1) -1
w p (w + 1) -δn-1 dw where we set v = ξ(w + 1)

≤ δnξ p-δn B(δn -p + 1, p + 1)
where we used in the last line that

ξ-1 ξ(w+1)-1 ≤ 1 w+1 . As a consequence n-ℓ i=0 (a) i ≤ δ n Γ(p + 1) n-ℓ i=0 n-1 i ξ p-δ(n-i) (1 -ξ -δ ) i Γ(δ(n -i) -p + 1 Γ(δ(n -i) + 2 ≤ δ n Γ(p + 1)ξ p (1 -ξ -δ ) n n j=ℓ n-1 j-1 (ξ δ -1) -j Γ(δj -p + 1) Γ(δj + 2) .
Now using that for every a > 0, Γ(x+a) Γ(x) ∼ x a as x → ∞, we derive the existence of a real constants κ(0) p,δ , κ

p,δ > 0 such that

∀ j ≥ 0, Γ(δj -p + 1) Γ(δj + 2) ≤ κ(0) p,δ j -(p+1) ≤ κ (0) p,δ j ⌈p⌉-p j(j + 1) • • • (j + ⌈p⌉) .
In turn, using that

n+⌈p⌉ j+⌈p⌉ = (n + ⌈p⌉) • • • n (j + ⌈p⌉) • • • j n-1 j-1 ,
we finally obtain

n-ℓ i=0 (a) i ≤ κ (0) p,δ nΓ(p + 1)ξ p δ(1 -ξ -δ ) n 1 (n + ⌈p⌉) • • • (n + 1)n n j=ℓ n+⌈p⌉ j+⌈p⌉ (ξ δ -1) -j j ⌈p⌉-p ≤ κ (0) p,δ Γ(p + 1)ξ p δ(1 -ξ -δ ) n n ⌈p⌉-p (n + ⌈p⌉) • • • (n + 1) (ξ δ -1) ⌈p⌉ 1 + (ξ δ -1) -1 n+⌈p⌉ . Now (1 -ξ -δ ) n ξ p (ξ δ -1) ⌈p⌉ 1 + (ξ δ -1) -1 n+⌈p⌉ = ξ p+δ⌈p⌉
so that, using that ξ ≥ 1 and δ < η ⌈p⌉ , we get ξ p+δ⌈p⌉ ≤ ξ p+η which in turn implies

n-ℓ i=0 (a) i ≤ κ (0) p,δ δ Γ(p + 1)ξ p+η 1 n p .
Let us pass now to the second sum involving (b) i . First note that, on the event

Y (n) i ≤ ξ ≤ Y (n) i + Y (n) i+1 2 (which is clearly included in Y (n) i ≤ ξ ≤ Y (n) i+1 ), one has (ξ -Y (n) i ) p (Y (n) i+1 -ξ) ≤ (ξ -Y (n) i )(Y (n)
i+1ξ) p so that, owing to what precedes, we can focus on

n-ℓ i=0 ( b) i where ( b) i := E (ξ -Y (n) i ) p 1 Y (n) i +Y (n) i+1 2 ≤ξ≤Y (n) i+1 ≥ E (Y (n) i+1 -ξ)(ξ -Y (n) i ) p Y (n) i+1 -Y (n) i 1 Y (n) i +Y (n) i+1 2 ≤ξ≤Y (n) i+1
. This time we will analyze successively the sum over i = 1, . . . , nℓ and the case i = 0.

n-ℓ i=1 ( b) i = δ 2 n(n -1) {1≤u≤ξ≤v≤2ξ-u} du dv (uv) -δ-1 (ξ -u) p n-ℓ i=1 n-2 i-1 v -δ(n-2-(i-1)) (1 -u -δ ) i-1 ≤ δ 2 n(n -1) {1≤u≤ξ≤v≤2ξ-u} du dv(uv) -δ-1 (ξ -u) p (1 -u -δ + v -δ ) n-2 ≤ δ 2 n(n -1) ξ 1 duu -δ-1 (ξ -u) p 2ξ-u ξ dv v -δ-1 e -(n-2)(u -δ -v -δ ) = δ 2 n(n -1) ξ 1 duu -δ-1 (ξ -u) p e -(n-2)u -δ 2ξ-u ξ dv v -δ-1 e (n-2)v -δ
where we used in the in the second line that nℓ -

1 ≤ n -2 since ℓ ≥ 1. Setting v = y -1 δ yields 2ξ-u ξ v -δ-1 e (n-2)v -δ dv = 1 δ ξ -δ (2ξ-u) -δ e (n-2)y dy ≤ 1 δ ξ -δ -(2ξ -u) -δ e (n-2)ξ -δ ≤ (ξ -u)ξ -δ-1 e (n-2)ξ -δ
where we used in the last line the fundamental formula of Calculus. Consequently,

n-ℓ i=1 ( b) i ≤ n(n -1)δ 2 ξ -δ-1 ξ 1 u -δ-1 (ξ -u) p+1 e -(n-2)(u -δ -ξ -δ ) du = n(n -1)ξ -δ-1 δ (n-2)(1-ξ -δ ) 0 ξ - x n -2 + ξ -δ -1 δ p+1 e -x dx n -2
where we put u = x n-2 + ξ -δ -1 δ . Now, applying again fundamental formula of Calculus to the function z -1 δ yields,

ξ - x n -2 + ξ -δ -1 δ = (ξ -δ ) -1 δ - x n -2 + ξ -δ -1 δ ≤ x δ(n -2) ξ δ+1 so that n-ℓ i=1 ( b) i ≤ n(n -1) (n -2) p+2 δ -p ξ (p+1)(δ+1)-(δ+1) (n-2)(1-ξ -δ ) 0 x p+1 e -x dx ≤ κ (1)
p,δ Γ(p + 2)n -p ξ p(δ+1)

for some constant κ

p,δ > 0. When i = 0, keeping in mind that Y

(n) 1 = min 1≤i≤n Y i , ( b) 0 ≤ (ξ -1) p P(ξ ≤ Y (n) 1 ≤ 2ξ -1) = (ξ -1) p ξ -nδ -(2ξ -1) -nδ ≤ nδ(ξ -1) p+1 ξ -nδ-1 = nδξ p(1+δ) g(1/ξ)
where g(u) = (1u) p+1 u (n+p)δ , u ∈ (0, 1). One checks that g attains its maximum over (0, 1] at u * = (n+p)δ (n+p)δ+p+1 so that sup u∈(0,1]

g(u) = g(u * ) = p + 1 (n + p)δ + p + 1 p+1 (u * ) (n+p)δ ≤ 1 1 + n+p p+1 δ p+1 .
Finally, there exists a real constant κ

(2) p,δ > 0 such that ( b) 0 ≤ ξ p(δ+1) δn (1 + n+p p+1 δ) p+1 ≤ κ (2)
p,δ ξ p(δ+1) n -p .

As concerns the (c) n-ℓ+1 term, we proceed as follows.

E ξ -Y (n) n-ℓ+1 p 1 {ξ≥Y (n) n-ℓ+1 } ≤ ξ p P(ξ ≥ Y (n) n-ℓ+1 ) ≤ ξ p(1+δ) E(Y (n) n-ℓ+1 ) -pδ . Note that E(Y (n) n-ℓ+1 ) -pδ = Γ(n + 1) Γ(n -ℓ + 1)Γ(ℓ) 1 0 (1 -v) n-ℓ v ℓ+p-1 dv = Γ(n + 1) Γ(ℓ) Γ(ℓ + p) Γ(n + p + 1) ∼ Γ(ℓ + p) Γ(ℓ) n -p = O(n -p ).
Finally, for every ξ ≥ 1, (c) n-ℓ+1 ≤ κ

p,δ ξ p(1+δ) n -p . Consequently, there exists a real constant κ p,η = max j=0,...,3 κ

(j) p,δ > 0 such that for every n ≥ n p,η = ℓ(η, p) ∨ 3, ∀ ξ ≥ 1, n p E F p n (ξ, Y (n) 0 , . . . , Y (n)
n+1 ) ≤ κ p,η ξ p+η since p δ ≤ η. Hence for every r.v. X, we derive by integrating in ξ ∈ [1, +∞) with respect to P X (dξ):

n p inf (1,x2,...,xn)∈In E F p n (X, 1, x 2 , . . . , x n ) ≤ n p E F p n (X, Y (n) 0 , . . . , Y (n) n+1 ) ≤ κ p,η E X p+η .
Step 3. If X is a non-negative random variable, applying the second step to X + 1 and using the scaling property (i) satisfied by F p,n yields for n ≥ n p,η (as defined in Step 2), inf

(0,x2,...,xn)∈In Fp,n (X, 0, x 2 , . . . , x n ) L p = inf (1,x2,...,xn)∈In Fp,n (X + 1, 1, . . . , x n ) L p ≤ κ 1/p p,η 1 + X 1+ η p L p+η n ≤ C (0) p,η (1 + X 1+ η p L p+η ) n with C (0) p,η = (2 1+η κ p,η ) 1 p .
We may assume that X L p+η ∈ (0, ∞). Then, applying the above bound to the non-negative random variable X = X X L p+η taking again advantage of the scaling property (i), we obtain inf (0,x2,...,xn)∈In

Fp,n (X, 0, x 2 , . . . ,

x n ) L p = X L p+η inf (0,x2,...,xn)∈In Fp,n ( X, 0, x 2 , . . . , x n ) L p ≤ X L p+η C (0) p,η 1 + 1 n = 2C (0) p,η X L p+η 1 n .
Step 4. Let X be a real-valued random variable and let for every integer n ≥ 1, x 1 , . . . , x n ∈ (-∞, 0), x n+1 = 0 and x n+2 , . . . , x 2n+1 ∈ (0, +∞). It follows from the additivity property that that

F p 2n+1 (X, x 1 , . . . , x 2n+1 ) = F p n+1 (X + , x n+1 , . . . , x 2n+1 )1 {X≥0} + F p n+1 (-X -, x 1 , . . . , x n+1 )1 {X<0} = F p n+1 (X + , x 1 , . . . , x n+1 )1 {X≥0} + F p n+1 (X -, -x n+1 , . . . , -x 1 )1 {X<0} .
Consequently, using that X + × X -≡ 0 and that x n+1 = 0, we get inf Fp,2n+1 (X,

(
x 1 , . . . , x 2n+1 ) p L p ≤ X - p L p+η + X + p L p+η 2C (0) p,η n + 1 p .
Now using that (a + b) ≤ 2 1-1 q (a q + b q ) 1 q , a, b ≥ 0, with q = 1 + η p ≥ 1, we derive that

X - p L p+η + X + p L p+η ≤ 2 η p+η X - p+η L p+η + X + p+η L p+η p p+η = 2 η p+η X p L p+η
since X -× X + ≡ 0. Now, the monotonicity property (17) implies that, for every n ≥ 2 n p,η , dn,p (X) = inf

(x1,...,xn)∈In Fp,n (X, x 1 , . . . , x n ) L p ≤ 2 η p(p+η) 2C (0) p,η X L p+η n .
Still calling upon (17), we note that, for every n ∈ {1, . . . , 2n p,η }, dn,p (X) ≤ d1,p (X) = inf x∈R Xx 1 L p ≤ X L p so that dn,p (X) ≤ 2n p,η X L p+η n which completes the proof by setting C p,η = max 2n p,η , 2 1+ η p(p+η) C (0) p,η .

A d-dimensional non-asymptotic upper-bound for the dual quantization error

Now, combining Theorem 5 and Proposition 4(b), we are in position to show Proposition 2 (the d-dimensional version of the extended Pierce Lemma) which provides a non-asymptotic upperbound at the exact rate for dual quantization error moduli.

Proof of Proposition 2. (a) First note that dn,p (X) = dn,p (Xa), a ∈ R d (invariance by translation) so we may assume that X is L p+η -centered i.e. σ p+η, . (X) = X L p+η . When d = 1, Theorem 5 solves the problem.

Let d ≥ 2. Let X = (X 1 , . . . , X d ) (X i components of X). It follows form Proposition 4 that, if Γ = 1≤i≤d Γ i , with Γ i ⊂ R, |Γ i | = n i with n 1 • • • n d ≤ n. Then for every ξ = (ξ 1 , . . . , ξ d ) ∈ R d F p . (ξ; Γ) ≤ C p, .
X j p L p+η ⌊n 1 d ⌋ -p ≤ C p, . C p,η sup k≥2 k 1 d k 1 d -1 p n -p d d j=1 X j p L p+η ≤ C p, . C p,η 2 p n -p d d η p+η E|X| p+η ℓ p+η ≤ d η C p, . C p,η 2 p C . ,p+η X p+η L p+η n -p d
where C . ,r = sup x =1 |x| r ℓ r , r > 0. (b) Let C be the smallest hypercube withe edges parallel to the coordinate axis containing conv(Supp(P X )). Up to a translatation, which leaves d n,p (X) invariant, we may assume that C = [0, L] d where 0 ≤ L ≤ diam . (Supp(P X )). The conclusion follows by integrating Inequality (15) with respect to P X (dξ) with m = ⌊n 1 d ⌋ and following the lines of the proof of claim (a).

Proof of the sharp rate theorem

On the way to proving the sharp rate theorem, we have to establish few additional propositions.

Proposition 5 (Sub-linearity). Let

P = m i=1 s i P i where s 1 , . . . , s m ∈ [0, 1], m i=1 s i = 1 and let n 1 , . . . , n m ∈ N such that m i=1 n i ≤ n. Then d p n (P) ≤ m i=1 s i d p ni (P i ).
Proof. For ε > 0 and every

i = 1, . . . , m, choose Γ i ⊂ R d , |Γ i | ≤ n i such that d p (P i ; Γ i ) ≤ (1 + ε) d p ni (P i ). Set Γ = m i=1 Γ i ; from Proposition 3 we get d p n (P) ≤ d p n (P; Γ) = m i=1 s i F p (ξ; Γ) P i (dξ) ≤ m i=1 s i F p (ξ; Γ i ) P i (dξ) ≤ (1 + ε) m i=1 s i d p ni (P i ).
Letting ε → 0 completes the proof.

Remark. Proposition 5 does not hold for dp n since F p is not decreasing for the inclusion order on grids. This induces substantial difficulties in the proof of the sharp rate compared to the regular quantization setting.

Proposition 6 (Scaling property). Let

C = a + ρ[0, 1] d (a ∈ R d , ρ > 0) be a d-dimensional
hypercube, with edges parallel to the coordinate axis and edge-length ρ > 0. Then

d n,p (U (C)) = ρ • d n,p (U [0, 1] d ). Proof. Keeping in mind that λ d ([0, ρ] d ) = ρ d , it holds that d p (U (C); {a + ρx 1 , . . . , a + ρx n }) = [0,ρ] d min λ∈R n n i=1 λ i ξ -ρx i p s.t. [ ρx1 ... ρxn 1 ... 1 ]λ= ξ 1 , λ≥0 λ d (dξ) λ d [0, ρ] d = [0,1] d min λ∈R n n i=1 λ i ρu -ρx i p s.t. [ ρx1 ... ρxn 1 ... 1 ]λ=[ ρu 1 ], λ≥0 λ d (du) = ρ p [0,1] d min λ∈R n n i=1 λ i u -x i p s.t. [ x1 ... xn 1 ... 1 ]λ=[ u 1 ], λ≥0 λ d (du) = ρ p • d p (U [0, 1] d ; {x 1 , . . . , x n }).
The following lemma shows that also for dn,p the convex hull spanned by a sequence of "semioptimal" quantizers asymptotically covers the interior of supp(P X ). This fact is trivial for d n,p if X has a compact support. Lemma 1. Let K = conv{a 1 , . . . , a k } ⊂ ˚ supp(P) be a set with K = ∅ and let Γ n be a sequence of quantizers such that dn,p (P, Γ n ) → 0 as n → +∞. Then there exists n 0 ∈ N such that for all n ≥ n 0 K ⊂ conv(Γ n ).

Proof. Set a 0 = 1 k k i=1 a i and define for every ρ > 0 K(ρ) = conv{ã 1 (ρ), . . . , ãk (ρ)} with ãi (ρ) = a 0 + (1 + ρ)(a ia 0 ). Since K ⊂ ˚ supp(P) there exists ρ 0 > 0 such that K = K(ρ 0 ) ⊂ supp(P). From now on, we denote ãi (ρ 0 ) by ãi . Since moreover ãi ∈ supp(P), there exists a sequence (a n i ) n≥1 having values in conv(Γ n ) and converging to ãi . Otherwise there would exist ε 0 > 0 and a subsequence (n

′ ) such that B(ã i , ε 0 ) ⊂ (conv(Γ n ′ )) c . Then dp n ′ (X, Γ n ′ ) ≥ E dist(X, Γ n ′ ) p 1 {X∈B(ãi,ε0/2)} ε 0 2 p P(B(ã i , ε 0 /2)) > 0
since ãi ∈ supp(P). This contradicts the assumption on the sequence (Γ n ) n≥1 .

Since K has a nonempty interior, it follows that aff. dim{a 1 , . . . , a k } = aff. dim{ã 1 , . . . , ãk } = d. Consequently, we may choose a subset I * ⊂ {1, . . . , k}, |I * | = d + 1, so that {ã j : j ∈ I * } is an affinely independent system in R d and furthermore there exists n 0 ∈ N such that the same holds for {a n j : j ∈ I * }, n ≥ n 0 . Hence, we may write for n ≥ n 0 ãi =

j∈I * µ n,i j a n j , j∈I * µ n,i j = 1, i = 1, . . . , k. (19)
This linear system has the unique asymptotic solution µ ∞,i j = δ ij (Kronecker symbol), which implies µ n,i j → δ ij when n → +∞. Now let ξ ∈ K ⊂ K and write

ξ = k i=1 λ i a i for some λ i ≥ 0, k i=1 λ i = 1.
One easily checks that it also holds

ξ = k i=1 λi ãi with λi = ρ 0 k(1 + ρ 0 ) + λ i 1 + ρ 0 ≥ ρ 0 k(1 + ρ 0 )
> 0 and

k i=1 λi = 1.
Furthermore, we may choose n 1 ≥ n 0 such that, for every n ≥ n 1 ,

µ n,i i > 1 2 and ∀j = i, |µ n,i j | ≤ ρ 0 4k(1 + ρ 0 )
.

Using (19), this leads to

ξ = j∈I * k i=1 λi µ n,i j a n j and k i=1 λi µ n,i j > λj µ n,j j - k i=1,i =j λi |µ n,i j | > ρ 0 2k(1 + ρ 0 ) - ρ 0 4k(1 + ρ 0 ) = ρ 0 4k(1 + ρ 0 ) > 0, j ∈ I * .
Finally, one completes the proof by noting that

j∈I * k i=1 λi µ n,i j = k i=1 λi j∈I * µ n,i j = 1.
As already said, Proposition 5 does not hold anymore for dn,p . As a consequence we have to establish a "firewall Lemma", which will be a useful tool to overcome this problem in the non-compact setting.

Lemma 2 (Firewall). Let K ⊂ R d be compact and convex with K = ∅. Moreover, let ε > 0 be small enough so that

K ε = {x ∈ K : dist ℓ ∞ (x, K c ) ≥ ε} = ∅.
Let Γ α,ε be a subset of the lattice αZ d with edge-length α > 0 satisfying

K \ K ε ⊂ conv(Γ α,ε ) and ∀ x ∈ K \ K ε , dist • (x, Γ α,ε ) ≤ C • α
where C • > 0 is a real constant only depending on the norm • . Then, for every grid Γ ⊂ R d containing K and every η ∈ (0, 1), it holds

∀ ξ ∈ K ε , F p (ξ; Γ) ≥ 1 (1 + η) p+d+1 F p (ξ; (Γ ∩ K) ∪ Γ α,ε ) -(1 + η) -d-1 η -p (d + 1) C p • α p .
Remark. The lattice Γ α,ε and its size will be carefully defined and estimated for the specified compact sets K when calling upon the firewall lemma in what follows.

Proof. Let Γ = {x 1 , . . . , x n } and let ξ ∈ K ε . Then we may choose

I = I(ξ) ⊂ {1, . . . , n}, |I| ≤ d + 1 such that F p (ξ; Γ) = i∈I λ j ξ -x i p , i∈I λ i x i = ξ, λ i ≥ 0, i∈I λ i = 1.
If for every x i ∈ Γ \ K λ i = 0 then F p (ξ, Γ) = F p (Γ ∩ K) and our claim is trivial. Therefore, let J(ξ) = {i :

x i ∈ Γ \ K, λ i > 0} ⊂ I(ξ) and choose one fixed i 0 ∈ J(ξ). Let θ = θ(i 0 ) ∈ (0, 1) such that xi0 = ξ + θ(x i0 -ξ) ∈ K \ K ε and θ p∧1 θ + λ i0 (1 -θ) ≤ 1 + η (when p ≥ 1 the right constraint is empty). Setting λ0 i = λ i θ θ + λ i0 (1 -θ) , i ∈ I \ {i 0 }, λ0 i0 = λ i0 θ + λ i0 (1 -θ) we arrive at λ0 i0 xi0 + i∈I\{i0} λ0 i x i = ξ, λ0 i ≥ 0, i∈I λ0 i = 1. Consequently λ0 i0 ξ -xi0 p + j∈I\{i0} λ0 i ξ -x i p = λ i0 θ p θ + λ i0 (1 -θ) ξ -x i0 p + i∈I\{i0} λ i θ θ + λ i0 (1 -θ) ξ -x i p ≤ θ p∧1 θ + λ i0 (1 -θ) i∈I λ i ξ -x i p ≤ (1 + η) i∈I λ i ξ -x i p .
Repeating the procedure for every i ∈ J(ξ) finally yields by induction the existence of xi ∈ K \ K ε and λi , i ∈ I such that

i∈I:xi / ∈ K λi xi + i∈I:xi∈ K λi x i = ξ, λi ≥ 0, i∈I λi = 1 and (1 + η) |J(ξ)| F p (ξ; Γ) ≥ i∈I:xi / ∈ K λi ξ -xi p + i∈I:xi∈ K λi ξ -x i p . (20)
Let us denote Γ α,ε = {a 1 , . . . , a m } and let i 0 ∈ J(ξ) so that xi0 is a "modified" x i0 (originally lying in Γ \ K). By construction xi0 ∈ K \ K ε ⊂ conv(Γ α,ε ) and there is J i0 ⊂ {1, . . . , m} such that

F p (x i0 , Γ α,ε ) = j∈Ji 0 µ i0 j xi0 -a j p , j∈Ji 0 µ i0 j x j = xi0 , µ i0 j ≥ 0, j∈Ji 0 µ i0 j = 1 and ∀ j ∈ J i0 , xi0 -a j ≤ C • α.
Using the elementary inequality

∀ p > 0, ∀η > 0, ∀ u, v ≥ 0, (u + v) p ≤ (1 + η) p u p + 1 + 1 η p v p ,
we derive that for every j

∈ J i0 ξ -a j p ≤ ξ -xi0 + xi0 -a j p ≤ (1 + η) p ξ -xi0 p + 1 + 1 η p C p • α p .
As a consequence,

j∈Ji 0 µ i0 j ξ -a j p ≤ (1 + η) p ξ -xi0 p + 1 + 1 η p C p • α p which in turn implies ξ -xi0 p ≥ 1 (1 + η) p j∈Ji 0 µ i0 j ξ -a j p -η -p C p • α p .
Plugging this inequality in (20) yields and using that |J(ξ)| ≤ d + 1, we finally get

(1 + η) |J(ξ)| F p (ξ; Γ) ≥ i∈I:xi∈ K λi ξ -x i p + 1 (1 + η) p i∈I:xi / ∈ K λi j∈Ji µ i j ξ -a j p -|J(ξ)|η -p d C p • α p ≥ 1 (1 + η) p F p ξ; (Γ ∩ K}) ∪ Γ α,ε -η -p (d + 1) C p • α p .
Now we can establish the sharp rate for the uniform distribution U ([0, 1] d ).

Proposition 7 (Uniform distribution). For every p ≥ 1,

Q dq • ,p,d := inf n≥0 n 1/d d n,p U [0, 1] d = lim n→∞ n 1/d d n,p U [0, 1] d . Proof. Let n, m ∈ N, m < n and set k = k(n, m) = n m 1/d ≥ 1. Covering the unit hypercube [0, 1] d by k d translates C 1 , . . . , C k d of the hypercube 0, 1 k d , we arrive at U [0, 1] d = k -d k d i=1 U (C i ).
Hence, Proposition 5 yields

d p n,p U [0, 1] d ≤ k -d k d i=1 d p m (U (C i )).
Furthermore, Proposition 6 implies

d m,p (U (C i )) = k -1 d m,p U [0, 1] d ,
so that we may conclude for all n, m ∈ N, m < n,

d n,p U [0, 1] d ≤ k -1 d m,p U [0, 1] d .
Thus, we get

n 1/d d n,p U [0, 1] d ≤ k -1 n 1/d d m,p U [0, 1] d ≤ k + 1 k m 1/d d m,p U [0, 1] d ,
which yields for every fixed integer m ≥ 1

lim sup n→∞ n 1/d d n,p U [0, 1] d ≤ m 1/d d m,p U [0, 1] d , since lim n→∞ k(n, m) = +∞. This finally implies lim n→∞ n 1/d d n,p U [0, 1] d = inf m≥0 m 1/d d m,p U [0, 1] d .
Proposition 8. For every p ≥ 1,

Q dq • ,p,d = lim n→∞ n 1/d d n,p U [0, 1] d = lim n→∞ n 1/d dn,p U [0, 1] d
Proof. Since for every compactly supported distribution P we have dn,p (P) ≤ d n,p (P) it remains to show

Q dq • ,p,d ≤ lim inf n→∞ n 1/d dn,p U [0, 1] d . For 0 < ε < 1/2 let C ε = (1/2, . . . , 1/2) + 1-ε 2 [-1
, 1] d be the centered hypercube in [0, 1] d with edge-length 1ε and midpoint (1/2, . . . , 1/2). Moreover let (Γ n ) be a sequence of quantizers such that, for every n ≥ 1,

dp (U [0, 1] d ; Γ n ) ≤ (1 + ε) dn,p (U [0, 1] d ).
Owing to Lemma 1, as

C ε ⊂ (1, 1) d , there is an integer n ε ∈ N such that ∀n ≥ n ε , C ε ⊂ conv(Γ n ).
We therefore get for any n ≥ n ε

(1 + ε) d dp n U [0, 1] d ≥ dp U [0, 1] d ; Γ n ≥ Cε F p (ξ, Γ n ) p dξ = Cε F p (ξ, Γ n ) p dξ = λ d (C ε )d p U (C ε ), Γ n ≥ (1 -ε) d d p n U (C ε ) = (1 -ε) d+p d p n U [0, 1] d
where we used the scaling property (Proposition 6) in the last line. Hence, we obtain for all 0 < ε < 1/2

lim inf n→∞ n 1/d dn,p U [0, 1] d ≥ (1 -ε) 1+d/p (1 + ε) d/p Q dq • ,p,d ,
so that letting ε → 0 completes the proof.

Proposition 9. Let P = m i=1 s i U (C i ), m i=1 s i = 1, s i > 0, i = 1, . . . , m, where C i = a i + [0, l] d , i = 1, . . . , m, are pairwise disjoint hypercubes in R d with common edge-length l. Set h := dP dλ d = m i=1 s i l -d ✶ Ci .
Then lim

n→∞ n 1/d d n,p (P) = lim n→∞ n 1/d dn,p (P) = Q dq • ,p,d • h 1 p d/(d+p) .
Proof. Since d n,p (P) ≥ dn,p (P) it suffices to show that lim sup

n→∞ n 1/d d n,p (P) ≤ Q dq • ,p,d • h 1 p d/(d+p) and lim inf n→∞ n 1/d dn,p (P) ≥ Q dq • ,p,d • h 1 p d/(d+p) .
For n ∈ N, set

t i = s d/(d+p) i m j=1 s d/(d+p) j and n i = ⌊t i n⌋, 1 ≤ i ≤ m.
Then, by Proposition 5 and Proposition 6, we get for every n ≥ max 1≤i≤m (1/t i )

d p n (P) ≤ m i=1 s i d p n (U (C i )) = l p m i=1 s i d p ni (U [0, 1] d ).
Proposition 7 then yields

n p d d p ni (U [0, 1] d ) = n n i p d n p d i d p ni (U [0, 1] d ) -→ t -p d i Q dq • ,p,d as n → +∞. Noting that h d/(d+p) = l p s d/(d+p) i (d+p)/d , we get lim sup n→∞ n p d d p n,p (P) ≤ Q dq • ,p,d l p m i=1 s i t -p d i = Q dq • ,p,d • h d/(d+p) .
(b) Let ε ∈ (0, l/2) and let C i,ε denote the closed hypercube with the same center as C i but with edge-length lε. For α ∈ (0, ε/2), we set α = l ⌈l/α⌉ and we define the lattice

Γ α,ε,i = a i + αZ d ∩ (C i \ C i,ε {vertices of C i }. It is clear that conv(Γ α,ε,i ) = C i ⊂ C i \ C i,ε since it contains the vertices of C i . Moreover, for every ξ ∈ C i \ C i,ε , dist ℓ ∞ (ξ, Γ α,ε,i ) ≤ α so that there exists a real constant C • > 0 only depending on the norm . such that dist . (ξ, Γ α,ε,i ) ≤ C • α.
Consequently the lattice Γ α,ε,i satisfies the assumption of the firewall lemma (Lemma 2).

On the other hand, easy combinatorial arguments show that number of points

m i of Γ α,ε,i falling in C i satisfies ⌈ l α ⌉ d ≤ m i ≤ ⌈ l α ⌉ + 1 d + 2 d whereas the number m i,ε of points falling in C i,ε satisfies ⌈ l-ε α ⌉ -1 d ≤ m i,ε ≤ ⌈ l-ε α ⌉ + 1 d so that l α d - l -ε α + 1 d ≤ |Γ α,ε,i | ≤ l α + 1 d + 2 d - l -ε α -1 d .
We define for every ε ∈ (0, l/2), α ∈ (0, ε/2)

g l,ε (α) = α d |Γ α,ε,i |.
Since α α → 1 and 2α ε/2 α → ε as α → 0, we conclude from the above inequalities that ∀ε ∈ (0, l/2), lim

α→0 g l,ε (α) = l d -(l -ε) d . ( 21
)
Let η ∈ (0, 1) and denote by Γ n a sequence of n-quantizers such that dp (P; Γ n ) ≤ (1 + η)d p n (P). It follows from Proposition 2 that dp (P; Γ n ) → 0 for n → ∞ so that Lemma 1 yields the existence of n ε ∈ N such that for any n ≥ n ε 1≤i≤m

C i,ε ⊂ conv(Γ n ).
We then derive from Lemma 2 (firewall)

dp (U (C i ); Γ n ) = l -d Ci F p (ξ; Γ n ) λ d (dξ) ≥ l -d Ci,ε F p (ξ; Γ n ) λ d (dξ) = l -d Ci,ε F p (ξ; Γ n ) λ d (dξ) ≥ l -d (l -ε) d (1 + η) p+d+1 d p U (C i,ε ); (Γ n ∩ Ci ) ∪ Γ α,ε,i -l -d (l -ε) d (1 + η) -d-1 η p (d + 1)C • • α p .
At this stage, we set for every ρ > 0

α n = α n (ρ) = m ρn 1/d (22)
and denote

n i = |(Γ n ∩ Ci ) ∪ Γ αn,ε,i |. Proposition 6 yields d ni,p (U (C i,ε )) = (l -ε)d ni,p (U [0, 1] d ), so that we get n p d d p n (P) ≥ 1 1 + η m i=1 s i n p d dp (U (C i ); Γ n ) ≥ l -d (l -ε) d (1 + η) p+d+2 m i=1 s i n p d d p U (C i,ε ); (Γ n ∩ Ci ) ∪ Γ αn,ε,i -l -d (l -ε) d (1 + η) -d-2 η p m i=1 s i (d + 1) C • • α p • n p d ≥ l -d (l -ε) d+p (1 + η) p+d+2 m i=1 s i n p d d p ni U [0, 1] d -l -d (l -ε) d (1 + η) -d-2 η p (d + 1) C • m ρ p d . (23)
Since

n i n ≤ |Γ n ∩ Ci | n + g l,ε (α n ) nα d n = |Γ n ∩ Ci | n + ρ m g l,ε (α n ),
we conclude from ( 21) and ( 22) that lim sup

n→∞ m i=1 n i n ≤ 1 + ρ l d -(l -ε) d .
We may choose a subsequence (still denoted by (n)), such that

n 1/d dn,p (P) → lim inf n→∞ n 1/d d n,p (P) and n i n → v i ∈ [0, 1 + ρ(l d -(l -ε) d )].
As a matter of fact, v i > 0, for every i = 1, . . . m: otherwise Proposition 7 would yield

n p d dp n,p (P) ≥ l -d (l -ε) d+p (1 + η) p+d+2 m i=1 s i n i n -p d n p d i d p ni U [0, 1] d -l -d (l -ε) d (1 + η) p-d-2 η p (d + 1)C • • m ρ p d
→ +∞ as n → +∞ which contradicts (a). Consequently, we may normalize the v i 's by setting

v i = v i 1 + ρ(l d -(l -ε) d ) , i = 1, . . . , m, so that m i=1 v i ≤ 1. We derive from Proposition 7 that lim inf n→∞ m i=1 s i n p d d p ni U [0, 1] d ≥ m i=1 s i v -p d i n p d i d p ni U [0, 1] d = Q dq • ,p,d (1 + ρ(l d -(l -ε) d ) -p d m i=1 s i v -p d i ≥ Q dq • ,p,d (1 + ρ(l d -(l -ε) d ) -p d inf i yi≤1,yi≥0 m i=1 s i y -p d i = Q dq • ,p,d (1 + ρ(l d -(l -ε) d ) -p d m i=1 s d/(d+p) i (d+p)/d
.

Hence, we derive from (23)

lim inf n→∞ n p d dp n,p (P) ≥ l -d (l -ε) d+p (1 + η) p+d+2 1 + ρ(l d -(l -ε) d ) p d Q dq • ,p,d m i=1 s d/(d+p) i (d+p)/d -l -d (l -ε) d (1 + η) -d-2 η p (d + 1) C • • m ρ p d . Letting ε → 0 implies lim inf n→∞ n p d dp n,p (P) ≥ l p (1 + η) p+d+2 Q dq • ,p,d m i=1 s d/(d+p) i (d+p)/d - (1 + η) -d-2 η p (d + 1) C • m ρ p d = 1 (1 + η) p+d+2 Q dq • ,p,d • h d/(d+p) - (1 + η) -d-2 η p d C • m ρ p d
and, finally, letting successively ρ go to +∞ and η go to 0 completes the proof.

C i = d r=1 - l 2 + i r l k , - l 2 + (i r + 1)l k .
Then, set

h = dP dλ d and P k = i∈Z d 0≤ir<k P(C i ) U (C i ), h k = dP k dλ d = i∈Z d 0≤ir<k P(C i ) λ d (C i ) ✶ Ci , k ≥ 1. (24)
By differentiation of measures we obtain h k → h, λ d -a.s. as k → ∞. Which in turn implies, owing to Scheffé's Lemma, lim We get likewise by Proposition 9 that, for every k ∈ N, For this product quantizer it follows from Proposition 4 that, for all ξ ∈ C,

k→+∞ h k -h 1 = 0.

Furthermore, lim

lim n→∞ n 1/d d n,p ( Pk ) = Q dq • ,p,d • hk 1 p d/(d+p) . (26)
F p (ξ; Γ 1 ) ≤ C • d i=1 l 2m p ≤ C • ,p,d l p (εn) p d . For n 2 = ⌊(1-ε)n⌋ let Γ 2 be an n 2 -quantizer such that d p (P k ; Γ 2 ) ≤ (1+ε)d p n2 (P k ). We clearly have |Γ 1 ∪ Γ 2 | ≤ n and n p d F p (ξ; Γ 1 ∪ Γ 2 )dP k (ξ) -F p (ξ; Γ 1 ∪ Γ 2 )dP(ξ) ≤ n p d F p (ξ; Γ 1 ∪ Γ 2 )|h k (ξ) -h(ξ)|dλ d ξ ≤ C • ,p,d l p ε p d h k -h 1 = c 1,ε h k -h 1 for k ∈ N and n ≥ max 2 d ǫ , 1 1-ε . This implies n p d d p n (P) ≤ n p d F p (ξ; Γ 1 ∪ Γ 2 )dP(ξ) ≤ n p d F p (ξ; Γ 1 ∪ Γ 2 )dP k (ξ) + c 1 h k -h 1 ≤ n p d F p (ξ; Γ 2 )dP k (ξ) + c 1 h k -h 1 ≤ (1 + ε) n p d d p n2 (P k ) + c 1,ε h k -h 1 ,
so that we can conclude from (25) that lim sup

n→∞ n p d d p n (P) ≤ 1 + ε (1 -ε) p d (Q dq • ,p,d ) p h k d/(d+p) + c 1,ε h k -h 1 .
Letting first k go to infinity and then letting ε go to zero yields lim sup n→∞ n

F p (ξ; Γ 1 ∪ Γ 3 )d Pk (ξ) -F p (ξ; Γ 1 ∪ Γ 3 )dP(ξ) ≤ c 2,ε hk -h 1 . (27)
Moreover, Lemma 1 yields for every k ∈ N the existence of n k,ε ∈ N such that, for all n ≥ n k,ε ,

(1 + ε) dp n2 (P) ≥ dp (P; Γ 3 ) ≥ conv(Γ3) F p (ξ; Γ 3 )dP(ξ) ≥ C k F p (ξ; Γ 3 )dP(ξ) ≥ C k F p (ξ; Γ 1 ∪ Γ 3 )dP(ξ).
Thus, we derive from (27) that, for every n ≥ max n k,ε ,

2 d ε , 1 1-ε , (1 + ε) n p d dp n2 (P) ≥ n p d C k F p (ξ; Γ 1 ∪ Γ 3 )dP(ξ) ≥ n p d C k F p (ξ; Γ 1 ∪ Γ 3 )d Pk (ξ) -c 2,ε hk -h 1 ≥ n p d d p n ( Pk ) -c 2,ε hk -h 1 ,
which yields, once combined with (26),

1 + ε (1 -ε) p d lim inf n→∞ n p d 2 dp n2,p (P) ≥ Q dq • ,p,d hk d/(d+p) -c 2,ε hk -h 1 .
Letting first k go to ∞ and then letting ε go to 0, we get lim inf For every k ∈ {1, . . . , m}, we consider the lattice of C k of size n d i defined by

Let C k = d i=1 [c k,i , c k,i + ℓ i ), k ∈ N,
d i=1 c k,i + r i n k -1 ℓ i , r i = 0, . . . , n k -1, i = 1, . . . , d .
Then, one defines likewise the lattice of C of size

n d m+1 ≤ n/2 d i=1 c k,i + r i n m+1 -1 ℓ i , r i = 0, . . . , n m+1 -1, i = 1, . . . , d .
The grid Γ is made up with all the points of the m + 1 above finite lattices. Now let ξ ∈ A. It is clear from the definition of the function F p that

F p (ξ; Γ) ≤ C . ℓ k /n k p if ξ ∈ m k=1 C k C . ℓ/n m+1 p if ξ ∈ C \ m k=1 C k
where C . > 0 is a real constant only depending on the norm. As a consequence

d p n (P) = m k=1 C k F p (ξ; Γ)dP(ξ) + C\ m k=1 C k F p (ξ; Γ)dP(ξ) ≤ C . m k=1 (ℓ k /n k ) p P(C k ) + (ℓ/n m+1 ) p P(C \ m k=1 C k ) .
Set for every k ∈ {1, . . . , m}, n k = ℓ k (n/2)

1 d ( d k ′ =1 ℓ d k ′ ) 1 d
and n m+1 = ⌊(n/2)

1 d ⌋. Note that d k ′ =1 ℓ d k ′ = m k=1 λ d (C k ) ≤ λ d (O) ≤ ε.
Elementary computations show that for large enough n, all the integers n k are greater than 1 and that m k=1

(ℓ k /n k ) p P(C k ) + (ℓ/n m+1 ) p P(C \ m k=1 C k ) ≤ ( d k ′ =1 ℓ d k ′ ) p d (n/2) -p d P ∪ 1≤k≤m C k + +(n/2) -p d ℓ p P C \ m k=1 C k so that lim sup n n p d d p n (P) ≤ C . (ε/2) p d
which in turn implies, by letting ε go to 0, that lim sup

) ≤ ρ(1-ε) -p d Q dq • ,p,d • h ρ 1 p d/(d+p)
Letting ε go to 0 completes the proof. Furthermore, part (c) was derived in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF], Section 5.1. Hence, it remains to prove (b)

Proof.

Step 1. (Lower bound) If X is compactly supported, the assertion follows from Proposition 10. Otherwise, set for every R ∈ (0, ∞),

C R = [-R, R] d and P(•|C k ) = h✶ C k P(C k ) λ d , k ∈ N.
Proposition 10 yields again

lim n→∞ n 1 d dn,p (P(•|C k )) = Q dq • ,p,d • h✶ C k /P(C k ) 1 p d/(d+p) , (28)
so that dp n,p (P)

≥ P(•|C k ) dp n,p (P(•|C k )) implies for all k ∈ N lim inf n→∞ n 1 d dn,p (P) ≥ Q dq • ,p,d • h✶ C k 1 p d/(d+p) .
Sending k to infinity, we get at

lim inf n→∞ n 1 d dn,p (P) ≥ Q dq • ,p,d • h 1 p d/(d+p) .
Step 2 (Upper bound, supp(P) = R d ). Let ρ ∈ (0, 1). Set K = C k+ρ and K ρ = C k . Let Γ k,α,ρ be the lattice grid associated to K \ K ρ with edge α > 0 as defined in the proof of Proposition 9.

It is straightforward that there exists a real constant C > 0 such that

∀ k ∈ N, ∀ ρ ∈ (0, 1), ∀ α ∈ (0, ρ) : |Γ α,ρ | ≤ Cdρk d-1 α -d .
Let ε ∈ (0, 1). For every n ≥ 1, set α n = α0 n -1 d where α0 ∈ (0, 1) is a real constant and

n 0 = |Γ k,αn,ρ |, n 1 = ⌊(1 -ε)(n -n 0 )⌋, n 2 = ⌊ε(n -n 0 )⌋,
so that α n ∈ (0.ρ), n 0 + n 1 + n 2 ≤ n and n i ≥ 1 for large enough n.

For every ξ ∈ K ρ = C k , for every grid Γ ⊂ R d containing K ρ , we know by the "firewall" Lemma 2 that which finally proves the assertion by sending m → +∞.

F p (ξ; (Γ ∩ K) ∪ Γ α,ρ ) ≤ (1 + η) p F p (ξ; Γ) + (1 + η) p (1 + 1/η) p C . α p . Let Γ 1 = Γ 1 (n 1 ,

Concluding remarks and prospects

This result does not complete the theoretical investigations about dual quantization (beyond the existence of optimal dual quantizers in the case p = 1, left open in [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF]): the first one is to elucidate the asymptotic behaviour of the constant Q dq . ,p,d coming out in Theorem 2 as d goes to infinity, most likely by showing that lim d→+∞ Q dq . ,p,d Q vq . ,p,d = 1. From a practical point of view, is it possible to evaluate the mean dual quantization error induced by an optimal Voronoi quantization grid? An answer to that question would be very valuable for applications since many optimal quantization grids have been computed for various distributions (see e.g. [START_REF] Pagès | www.quantize.maths-fi.com. website devoted to quantization[END_REF] for Gaussian distributions).

Many natural questions solved in the optimal Voronoi quantization theory remain open. Among others "Is there a counterpart to the empirical measure theorem for (asymptotically) optimal quantizers?" (see Theorem 7.5, p.96 in [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF])? "How does dual quantization behave with respect to empirical distribution of i.i.d. n-samples of a given distribution?". Is it possible to develop an infinite dimensional "functional" dual quantization?

A Numerical results for dn,2 (X) 2 In order to support the heuristic argumentation on the intrinsic and rate optimal growth limitation of the truncation error P X / ∈ C n induced by the extended dual quantization error modulus, we consider the two dimensional random variable X = (W T , sup 0≤t≤T W t ), where (W t ) 0≤t≤T is a Brownian Motion.

This example is motivated by the pricing of exotic options, where this joint distribution plays an important role.

Using a variant of the CVLQ algorithm (see [START_REF] Pagès | Intrinsic stationarity for vector quantization: foundation of dual quantization[END_REF]) adapted for the dual quantization modulus inside C n and the nearest neighbor mapping outside, we have computed a sequence of optimal grids together with the squared dual quantization error dn,2 (X) 2 and the truncation error P X / ∈ C n .

These results are reported in Table A. The distortion rate dn,2 (X) 2 shows here an absolute stable convergence rate (least-squares fit of exponent yields -1.07192) which is consistent with the theoretical optimal rate of n -2 d . Moreover, the truncation error P X / ∈ C n outperforms also in this case the heuristically derived rate of n -1 and also outperforms the squared "inside" quantization error, which means that also for such an un-symmetric and non-spherical distribution of the Brownian motion and its supremum, an second order rate can be achieved.

This confirms again the motivation of the extended dual quantization error as the correction penalization constraint on growth of the convex hull in order to preserve second order stationarity.

Proposition 4 .

 4 (a) Scalar bound:

+∞ 0 u 1 0

 01 a-1 e -u du, a > 0, and B(a, b) = u a-1 (1u) b-1 du, a, b > 0, respectively, and satisfying B(a, b) = Γ(a)Γ(b) Γ(a+b) .

F 1 d

 1 p (ξ j , Γ j ) where C p, . = sup |ξ| ℓ p =1 ξ p . Integrating with respect to the distribution of X yields dp (X, Γ) ≤ C p, . d j=1 dp (X j , Γ j ) which in turn easily implies dp n (X) ≤ C p, . d j=1 dp nj (X j ). Now set n j = ⌊n ⌋, j = 1, . . . , d. It follows from Theorem 5 that dp n (X) ≤ C p p, . C p,η d j=1

  k→+∞ h k d/(d+p) = h d/(d+p) since h kh d/(d+p) ≤ λ d (C) p d h kh 1 by Jensen's Inequality applied to the probability measure λ d |C λ d (C) . Moreover, by Proposition 9 we have lim n→∞ n 1/d d n,p (P k ) = Q dq • ,p,d h k 1 p d/(d+p) .(25)Likewise, we define an inner approximation of P: denote by the hypercubes C i lying in the interior of supp(P)i ) U (C i ) and hk = d Pkdλ d = h k ✶ C k ,we have as above that hk → h, λ d -a.s. as k → +∞.Consequently we also have lim k→∞ hkh 1 = 0 and lim k→∞ hk d/(d+p) = h d/(d+p) .

( a )

 a Let 0 < ε < 1 and n ≥ 2 d /ε. If we divide each edge of the hypercube C into m = (εn) 1/d -1 intervals of equal length l/m, the interval endpoints define m + 1 grid points on each edge. Denoting by Γ 1 = Γ 1 (ε, n) the product quantizer made up by this procedure, we clearly have |Γ 1 | = (m + 1) d = (εn) 1/d d =: n 1 .

n→∞ n 1 d

 1 dn,p (P) ≥ Q dq • ,p,d h 1 p d/(d+p) . Proposition 11 (Singular distribution). Assume that P is singular with respect to λ d and has compact support. Then lim sup n→∞ n p d dn,p (P) = 0. Proof. Let A be a Borel set such that P(A) = 1 and λ d (A) = 0. Let ε > 0; by the outside regularity of λ d , there exists an open set O = O(ε) ⊃ A such that λ d (O) ≤ ε (and P(O) = 1). Let C be an open hypercube with edges parallel to the coordinate axis, edge-length ℓ and containing the closure of A.

2 :

 2 Claim (a) follows directly from Propositions 10 , 11 and Proposition 5: Assume P = ρP a + (1ρ)P s where P a = h ρ λ d and P s denote the absolutely continuous and singular part of P respectively. The following inequalities hold true ρ dn,p (P a ) ≤ dn,p (P) ≤ ρ dn1,p (P a ) + (1ρ) dn2,p (P s ) for every triplet of integers (n 1 , n 2 , n) withn 1 + n 2 ≤ n. Set n 1 = n 1 (n) = ⌊(1εn)⌋, n 2 = n 2 (n) = ⌊εn⌋.Then we derive that ρQ dq • ,p,d •

F 1 d+} p L p+δ 1 ε( 1 -. 1 r - 1 p

 11111 k) be an n 1 quantizer such that d p n1 (P(.|C k ); Γ 1 ) ≤ (1 + η)d p n1 (P(.|C k )). Set Γ ′ 1 = ((Γ 1 ∩ Ck+ρ ) ∪ Γ k,αn,ρ ). One has Γ ′ 1 ⊂ C k+2ρ for large enough n (so that α n < ρ). Let moreover Γ 2 = Γ 2 (n 2 , k) be an n 2 quantizer such that dp n2 (P(.|C c k ); Γ 2 ) ≤ (1+η) dp n2 (P(.|C c k )). For n ≥ n ρ , we may assume that C k+2ρ ⊂ conv Γ 2 owing to Lemma 1 since C k+2ρ = conv(C k+2ρ \ C k+ 3 2 ρ ) and C k+2ρ \ C k+ 3 2 ρ ⊂ ˚ suppP(.|C c k ). As a consequence Γ ′ 1 ⊂ conv(Γ 2 ) so that conv(Γ ′ 1 ) ⊂ conv(Γ 2 ) = conv(Γ) where Γ = Γ ′ 1 ∪ Γ 2 and C k+ρ ⊂ conv(Γ) = conv(Γ 2 ). Now dp n (P) ≤ C k F p (ξ; Γ)1 {ξ∈conv(Γ2)} + d(ξ, Γ) p 1 {ξ / p (ξ; Γ)1 {ξ∈conv(Γ2)} + d(ξ, Γ) p 1 {ξ / ∈conv(Γ2)} dP(ξ).Using that, for every ξ ∈ C k ,F p (ξ; Γ) ≤ F p (ξ; Γ ′ 1 ) ≤ (1 + η) p F p (ξ; Γ 1 ) + (1 + 1/η) p C . α p n implies dp n (P) ≤ P(C k )(1 + η) p (1 + η) d p n1 (P(.|C k )) + (1 + 1/η) p C . α0 n -+P(C c k ) (1 + η) dp n2 (P(.|C c k )). Consequently n p d dp n (P) ≤ P(C k )(1 + η) p (1 + η) n1 (P(.|C k )) + (1 + 1/η) p C .α0 implies, using Proposition 10 for the modulus d n,p and the d-dimensional version of the extended Pierce Lemma (Proposition 2) for dn,p , n (P) ≤ P(C k )(1 + η) p (1 + η) -p/d (1ε)(1 -Cdρk d-1 α-d 0 ) 1/η) p C . α0 +P(C c k ) (1 + η) C p,d X1 {X∈C c k Cdρk d-1 α-d 0 ) p dOne concludes by letting successively ρ, α0 , η go to 0, k → ∞ and finally ε to 0.Step 3. (Upper bound: general case). Let ρ ∈ (0, 1). Set P ρ = ρP + (1ρ)P 0 where P 0 = N (0; I d ) (d-dimensional normal distribution). It is clear from the very definition of dn,p that dn,p (P) ≤ 1 ρ dn,p (P ρ ) since P ≤ 1 ρ P ρ . The distribution P ρ has h ρ = ρh + (1ρ)h 0 a density (with obvious notations) and one concludes by noting that lim ρ→0 h ρ d/(d+p) = h d/(d+p) owing to the Lebesgue dominated convergence Theorem. Proof of Proposition 1: Using Hölder's inequality one easily checks that for 0 ≤ r ≤ p and x ∈ R d it holds |x| ℓ r ≤ d |x| ℓ p . Moreover, for m ∈ N set n = m d and let Γ ′ be an optimal quantizer for d m,p (U [0, 1] ) (or at least (1 + ε)-optimal for ε > 0). Denoting Γ = d i=1 Γ ′ , it then follows from Proposition 4(b) that n p d d p n U [0, 1] d ≤ n p d d p U [0, 1] d ; Γ = m p d i=1 d p U [0, 1] ; Γ ′ = d m p d p m U [0, 1] . Combining both results and reminding that Q dq • ,p,d holds as an infimum, we obtain for r ∈ [0, •| ℓ p (U [0, 1] d ) ≤ d p r m p d p m U [0, 1] ,

Figure 1 :

 1 Figure 1: log-log plot of dn,2 (X) 2 and P X / ∈ C n with respect to the grid size n

  Proposition 10. Assume that P is absolutely continuous w.r.t. λ d with compact support. Then Preliminary step. Let C = [-l/2, l/2] d be a closed hyper hypercube centered at the origin, parallel to the coordinate axis with edge-length l, such that supp(P) ⊂ C. For k ∈ N consider the tessellation of C into k d closed hypercubes with common edge-length l/k. To be precise, for every i = (i 1 , . . . , i d ) ∈ Z d , we set

			lim n→∞	n	p d d n,p (P) = lim inf n→∞	n	p d dn,p (P) = Q dq • ,p,d • h	1 p d/(d+p)
	Proof. Since d n,p (P) ≥ dn,p (P) it suffices to show that
	lim sup n→∞	n	p d d n,p (P) ≤ Q dq • ,p,d • h	1 p d/(d+p) and lim inf n→∞	n	p d dn,p (P) ≥ Q dq • ,p,d • h	1 p d/(d+p) .

  Assume now that Γ 3 is an n 2 -quantizer such that dp (P; Γ 3 ) ≤ (1 + ε) dp n2 (P). Again it holds |Γ 1 ∪ Γ 3 | ≤ n and we derive as above

		1/d d p n (P) ≤ Q dq • ,p,d h k	1 p d/(d+p) .
	(b) n	p d

  be a countable partition of O consisting of nonempty half-open hypercubes, still with edges parallel to the coordinate axis (see, e.g. Lemma 1.4.2 in [4]). Let n ∈ N, n ≥ 2 d+1 and let n 1 , . . . , n d ≥ 2 be integers such that the product n d 1 +• • •+n d m ≤ n/2. One designs a grid Γ as follows.

Let m = m(ε) ∈ N such that k≥m+1 P(C k ) ≤ ε p d ℓ -p .

Table 1 :

 1 Numerical results for the dual quantization X

	n	dn,2 (X) 2 P X / ∈ C n
	50	0.04076	0.01784
	100 0.01966	0.00795
	150 0.01236	0.00412
	200 0.00931	0.00141

Furthermore we see in figure

1

a log-log plot for the convergence of the two rates dn,2 (X) 2 and P X / ∈ C n .