O Bencheikh
email: oumaima.bencheikh@enpc.fr.

B Jourdain

Approximation rate in Wasserstein distance of probability measures on the real line by deterministic empirical measures

Keywords: deterministic empirical measures, Wasserstein distance, rate of convergence AMS Subject Classification (2010): 49Q22, 60-08

Introduction

Let ρ ≥ 1 and µ be a probability measure on the real line with finite moment of order ρ. We are interested in the rate of convergence to 0 in terms of N ∈ N * of (0.1) e N (µ, ρ)

:= inf W ρ 1 N N i=1 δ xi , µ : -∞ < x 1 ≤ x 2 ≤ • • • ≤ x N < +∞ ,
where W ρ denotes the Wasserstein distance with index ρ. The Hoeffding-Fréchet or comonotone coupling between two probability measures on the real line is optimal for W ρ so that when

-∞ < x 1 ≤ x 2 ≤ • • • ≤ x N < +∞, (0.2) W ρ ρ 1 N N i=1 δ xi , µ = N i=1 ˆi N i-1 N x i -F -1 (u) ρ du,
where F -1 denotes the quantile function of µ defined by F -1 (u) = inf {x ∈ R : F (x) ≥ u} for u ∈ (0, 1) with F (x) = µ ((-∞, x]) for x ∈ R. The motivation for our study is the approximation of the probability measure µ by finitely supported probability measures. Examples of application are provided by the optimal initialization of systems of particles with mean-field interaction [START_REF] Jourdain | Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data[END_REF][START_REF] Bencheikh | Weak and strong error analysis for mean-field rank based particle approximations of one dimensional viscous scalar conservation laws[END_REF], where, to preserve the mean-field feature, it is important to get N points with equal weight 1/N (of course, nothing prevents several of these points to be equal) and by the numerical analysis of restricted Monte Carlo methods which may only use random bits instead of random numbers [START_REF] Giles | Random bit quadrature and approximation of distributions on Hilbert spaces[END_REF]. The optimal approximation in the quadratic case ρ = 2 has been shown by Baker [START_REF] Baker | Quantizations of probability measures and preservation of the convex order[END_REF] to preserve the convex order. This is not the case for the optimal quantization [START_REF] Graf | Foundations of Quantization for Probability Distributions[END_REF] which is obtained by also optimizing over the weights and considering :

inf W ρ N i=1 p i δ xi , µ : -∞ < x 1 ≤ x 2 ≤ • • • ≤ x N < +∞, (p 1 , • • • , p N ) ∈ [0, 1] N , N i=1 p i = 1 .
The optimal quantization was introduced in signal processing [START_REF] Bennet | Spectra of quantized systems[END_REF] but has turned out since to be useful in scientific computing [START_REF] Pagès | Numerical Probability: An Introduction with Applications to Finance[END_REF]. Both the minimization problem with respect to the locations

x 1 ≤ x 2 ≤ • • • ≤ x N
under prescribed but non necessarily uniform weights (p 1 , • • • , p N ) and with respect to the weights (p 1 , • • • , p N ) under prescribed locations x 1 ≤ x 2 ≤ • • • ≤ x N have been studied by Xu and Berger [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF]. The case when the weights are not prescribed but only satisfy a size constraint has been studied in [START_REF] Bouchitté | Asymptotics of a class of optimal location problems[END_REF] while [START_REF] Mak | Support points[END_REF] addresses the minimization with respect to the locations under prescribed uniform weights when the Wasserstein distance is replaced by the energy distance.

According to Corollary 5.12 [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF], the fact that ´R |x| ρ µ(dx) < +∞ ensures that e N (µ, ρ) goes to 0 as N → ∞. The main purpose of the paper is to study the rate at which this convergence occurs. In particular, we would like to give necessary conditions on µ, which, when possible, are also sufficient, to ensure convergence at a rate N -α with α > 0 called the order of convergence.

One of course has e N (µ, ρ) ≤ W ρ (µ N , µ) and e N (µ, ρ)

≤ E 1/ρ W ρ ρ (µ N , µ) where µ N = 1 N N i=1 δ Xi
is the usual empirical measure of random variables (X i ) i≥1 i.i.d. according to µ. In the one-dimensional setting of the present paper, the convergence rate of W ρ (µ N , µ) has been studied in [START_REF] Del Barrio | Central limit theorems for the Wasserstein distance between the empirical and the true distributions[END_REF] for ρ = 1 and in [START_REF] Del Barrio | Asymptotics for L2 functionals of the empirical quantile process, with applications to tests of fit based on weighted Wasserstein distances[END_REF] in the quadratic case ρ = 2, the one of E 1/ρ W ρ ρ (µ N , µ) for ρ ≥ 1 in the book [START_REF] Bobkov | One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances[END_REF] by Bobkov and Ledoux. In general dimension, estimations of E 1/ρ W ρ ρ (µ N , µ) and concentration inequalities for W ρ ρ (µ N , µ) are given in [START_REF] Fournier | On the rate of convergence in Wasserstein distance of the empirical measure[END_REF]. In the random case, the largest possible order of convergence (apart from the case when µ is a Dirac mass and the error vanishes) is α = 1/2, which matches the rate of convergence in the standard strong law of large numbers given by the central limit theorem under square integrability.

The rate of convergence of e N (µ, ρ) has already been addressed by Xu and Berger [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF] in the one-dimensional setting of the present paper, by Chevallier [START_REF] Chevallier | Uniform decomposition of probability measures: quantization, classification and rate of convergence[END_REF] in higher finite dimension and by [START_REF] Giles | Random bit quadrature and approximation of distributions on Hilbert spaces[END_REF] along the subsequence (N = 2 n ) n∈N in the quadratic case ρ = 2 when µ is a Gaussian measure on an Hilbert space or the law of the solution to a scalar autonomous stochastic differential equation. In particular, according to Theorem 5.21 (ii) [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF], when the support of µ is bounded, then sup N ≥1 N 1 ρ e N (µ, ρ) < +∞, while Remark 5.22 (ii) [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF] ensures that lim sup N →+∞ N 1 ρ e N (µ, ρ) > 0 when F -1 is discontinuous. According to Theorem 5.20 [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF], apart when µ is a Dirac mass and e N (µ, ρ) vanishes for all N, ρ ≥ 1, lim sup N →∞ N e N (µ, ρ) > 0 so that the order of convergence cannot exceed 1 . It is equal to 1 when the density f of the absolutely continuous with respect to the Lebesgue measure part of µ is dx a.e. positive on {x ∈ R : 0 < F (x) < 1} (or equivalently F -1 is absolutely continuous), since Theorem 5.15 [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF] then ensures that

lim N →+∞ N e N (µ, ρ) = 1 2(ρ + 1) 1/ρ ˆR 1 {0<F (x)<1} f ρ-1 (x) dx 1/ρ
, where, by Theorem 2.4 [START_REF] Bencheikh | Approximation rate in Wasserstein distance of probability measures on the real line by deterministic empirical measures[END_REF], the right-hand side is not smaller than lim inf N →+∞ N e N (µ, ρ) even without the positivity assumption on the density. In [START_REF] Chevallier | Uniform decomposition of probability measures: quantization, classification and rate of convergence[END_REF], Chevallier addresses the multidimensional setting and proves in Theorem 3 that for a probability measure µ on R d with support bounded by r, there exist points x 1 , . . . , x N ∈ The case when the support of µ is not bounded is also considered by Xu and Berger [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF] in the onedimensional setting of the present paper and by [START_REF] Chevallier | Uniform decomposition of probability measures: quantization, classification and rate of convergence[END_REF] in the multidimensional setting. In Corollary 1 [START_REF] Chevallier | Uniform decomposition of probability measures: quantization, classification and rate of convergence[END_REF],

R d such that 1 4r W ρ 1 N N i=1 δ xi , µ ≤ f ρ,d (N ) where f ρ,d (N ) is respectively equal to d d-ρ 1 ρ N -1 d , 1+ln N
1 N N i=1 δ xi , µ = 0 when ´Rd |x| ρ 1-αρ µ(dx) < +∞ for some α ∈ (0, 1/ρ). This generalizes the one-dimensional statement of Theorem 5.21 (i) [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF] : under the same moment condition, lim N →∞ N α e N (µ, ρ) = 0.

In the main contribution of the present paper, Theorem 2.2, we refine this result by stating the following necessary and sufficient condition

∀α ∈ (0, 1/ρ) , lim x→+∞ x ρ 1-αρ F (-x) + 1 -F (x) = 0 ⇔ lim N →+∞ N α e N (µ, ρ) = 0.
We also check that

∀α ∈ (0, 1/ρ) , sup x≥0 x ρ 1-αρ F (-x) + 1 -F (x) < +∞ ⇔ sup N ≥1
N α e N (µ, ρ) < +∞, a condition under which, the order of convergence α of the minimal error e N (µ, ρ) is preserved by choosing

x 1 = F -1 1 N ∧ (-N 1 ρ -α ), x N = F -1 N -1 N ∨ N 1 ρ -α and any x i ∈ [F -1 ( i-1 N +), F -1 ( i N )] for 2 ≤ i ≤ N -1.
We also show that for (N α e N (µ, ρ)) N ≥1 to be bounded for α > 1/ρ, the support of µ has to be bounded. Then we address the boundary case α = 1/ρ : among Weibull distributions, we exhibit probability measures µ with unbounded support such that, for ρ > 1, lim N →+∞ N 1/ρ e N (µ, ρ) = 0 and give a necessary condition for N 1/ρ e N (µ, ρ) N ≥1 to be bounded, which unfortunately is not sufficient but ensures the boundedness of

N 1/ρ 1+ln N e N (µ, ρ) N ≥1
. These results are summarized together with the ones obtained by Xu and Berger [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF] in Table 1. They are stated and proved in the second section of the paper.

The first section is devoted to preliminary results. We first recall that, when ´R |x| ρ µ(dx) < ∞, then the infimum in (0.1) is attained:

(0.3) e N (µ, ρ) = W ρ 1 N N i=1 δ x N i , µ = N i=1 ˆi N i-1 N x N i -F -1 (u) ρ du for some points x N i ∈ [F -1 ( i-1 N +), F -1 ( i N )
]∩R which are unique as long as ρ > 1 and explicit when ρ ∈ {1, 2}. To circumvent the otherwise lack of explicit formula for the optimal point x N i when studying the order of convergence of e N (µ, ρ) to 0, we also derive bounds where x N i does not appear for the i-th contribution in the right-hand side of (0.3). We then give an alternative expression of each contribution in terms of the cumulative distribution function F in place of the quantile function F -1 , before taking advantage of the induced alternative formula for e N (µ, ρ) to recover that the error goes to 0 as N → ∞ when ´R |x| ρ µ(dx) < ∞. We also precise Theorem 5.21 (ii) [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF] which states that the boundedness of the support of µ implies that of the sequence N 1/ρ e N (µ, ρ) N ≥1 , by remarking that when F -1 is moreover continuous, then the sequence goes to 0 as N → ∞ for ρ > 1, with an order of convergence arbitrarily low as examplified by the beta distributions. Last, when µ is not a Dirac mass, we complement in a non-asymptotic way the positivity of lim sup N →∞ N e N (µ, ρ) proved by Xu and Berger in Theorem 5.20 [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF].

The proofs of two technical lemmas are given in the appendix.

Notation :

• We set F (x) = µ ((-∞, x]) for x ∈ R and denote F -1 (u) = inf {x ∈ R : F (x) ≥ u} for u ∈ (0, 1). We have u ≤ F (x) ⇔ F -1 (u) ≤ x.
The quantile function F -1 is left-continuous and non-decreasing and we denote by F -1 (u+) its right-hand limit at u ∈ [0, 1) (in particular

F -1 (0+) = lim u→0+ F -1 (u) ∈ [-∞, +∞)) and set F -1 (1) = lim u→1- F -1 (u) ∈ (-∞, +∞].
• We respectively denote by x ∧ y and x ∨ y the minimum and the maximum of two real numbers x and y.

• We denote by x (resp. x ) the integer j such that j ≤ x < j + 1 (resp. j -1 < x ≤ j) and by {x} = x -x the fractional part of x ∈ R.

α

Necessary condition

Sufficient condition

α = 1 ˆR 1 {f (x)>0} f ρ-1 (x) dx < +∞ f (x) > 0 dx a.e. on {x ∈ R : 0 < F (x) < 1} (Thm. 2.4 [3]) and ˆR 1 {f (x)>0}
f ρ-1 (x) dx < +∞ (Thm. 5.15 [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF])

α ∈ 1 ρ , 1
F -1 continuous (Remark 5.22 (ii) [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF]) related to the modulus of continuity of F -1 when ρ > 1 and µ with bounded support (Prop. 2.1) (Example 1.8)

α = 1 ρ ∃λ > 0, ∀x ≥ 0, F (-x) + 1 -F (x) ≤ e -λx λ
µ with bounded support (Thm. 5.21 (ii) [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF]) (Prop. 2.7)

For ρ > 1, Example 2.6 with unbounded supp.

α ∈ 0, 1 ρ sup x≥0 x ρ 1-αρ F (-x) + 1 -F (x) < +∞ sup x≥0 x ρ 1-αρ F (-x) + 1 -F (x) < +∞ (Thm. 2.2) (Thm. 2.2)
Table 1: Conditions for the convergence of e N (µ, ρ) with order α : sup

N ≥1
N α e N (µ, ρ) < +∞.

• For two sequences (a N ) N ≥1 and

(b N ) N ≥1 of real numbers with b N > 0 for N ≥ 2 we denote a N b N when 0 < inf N ≥2 a N b N and sup N ≥2 a N b N < +∞.
Acknowledgement : We thank the referees for their useful comments that helped us to improve the presentation of the manuscript.

Preliminary results

In the present section, we state preliminary results which will prove useful for the study of the order of convergence of e N (µ, ρ) to 0 when N → 0 in the next section devoted to the case when the support of µ is not bounded. We first recall that the infimum over (x 1 , . . . , x N ) is attained in (0.1) and bound the first (resp. last) coordinate of the minimizer from below (resp. above). Next, we give estimations from below and above not involving x N i of each contribution in the sum in the right-hand side of (0.3). We also rewrite these contributions in terms of the cumulative distribution function F in place of the quantile function F -1 . We recall that the finiteness of the ρ-th order moment of µ is a necessary and sufficient condition for the error e N (µ, ρ) to go to 0 as N → ∞ before checking that when µ has a bounded support and a continuous quantile function, then lim N →∞ N 1/ρ e N (µ, ρ) = 0 with an order of convergence arbitrarily low as examplified by the beta distributions. We finally derive a non-asymptotic lower bound for N e N (µ, ρ) + (N + 1)e N +1 (µ, ρ).

1.1 Infimum is attained in (0.1)

When ρ = 1 (resp. ρ = 2), R y → N ˆi N i-1 N y -F -1 (u)
ρ du is minimal for y belonging to the set

F -1 2i-1 2N , F -1 2i-1 2N + of medians (resp. equal to the mean N ˆi N i-1 N F -1 (u) du) of the image of the uni- form law on i-1 N , i N by F -1 . For general ρ > 1, the function R y → ˆi N i-1 N y -F -1 (u)
ρ du is strictly convex and continuously differentiable with derivative

(1.1) ρ ˆi N i-1 N 1 {y≥F -1 (u)} y -F -1 (u) ρ-1 -1 {y<F -1 (u)} F -1 (u) -y ρ-1 du
non-positive for y = F -1 i-1 N + when either i = 1 and F -1 (0+) > -∞ or i ≥ 2 and non-negative for y = F -1 i N when either i ≤ N -1 or i = N and F -1 (1) < +∞. Since the derivative has a positive limit as y → +∞ and a negative limit as y → -∞, we deduce that R y →

ˆi N i-1 N y -F -1 (u) ρ du admits a unique minimizer x N i ∈ F -1 i-1 N + , F -1 i N
∩ R, as already stated in Corollary 4.4 [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF] (to keep notations simple, we do not explicit the dependence of x N i on ρ). Therefore (1.2)

e ρ N (µ, ρ) = N i=1 ˆi N i-1 N x N i -F -1 (u) ρ du with F -1 i -1 N + , F -1 i N x N i =              F -1 2i -1 2N if ρ = 1, N ˆi N i-1 N F -1 (u) du if ρ = 2,
not explicit otherwise.

We will see that when the support of µ is unbounded, the contributions with i ∈ {1, N } in the right-hand side of (0.3) are dominant. To prove our main result, Theorem 2.2 below which characterizes the convergence of e N (µ, ρ) to 0 with order α ∈ (0, 1 ρ ), we will need the following estimates on x N 1 and x N N .

Lemma 1.1. Let ρ ≥ 1 and α ∈ 0, 1 ρ .There is a finite constant C only depending on ρ and α such that the two extremal points in the optimal sequence (x N i ) 1≤i≤N for e N (µ, ρ) satisfy

∀N ≥ 1, x N 1 ≥ CN 1 ρ -α inf u∈(0, 1 N ) u 1 ρ -α F -1 (u) and x N N ≤ CN 1 ρ -α sup u∈(0, 1 N ) u 1 ρ -α F -1 (1 -u). If sup u∈(0,1/2] u 1 ρ -α F -1 (1 -u) -F -1 (u) < +∞, then sup N ≥1 N α-1 ρ x N N ∨ -x N 1 < +∞.
The proof which relies on the expression (1.1) of the derivative when ρ / ∈ {1, 2} is postponed to the appendix.

Bounds on e N (µ, ρ)

To circumvent the lack of explicit formula for the optimal point x N i (unless ρ ∈ {1, 2}) when studying the order of convergence of e N (µ, ρ) to 0, we are now going to derive bounds where x N i does not appear for the i-th contribution in the right-hand side of (0.3).

When needing to bound e N (µ, ρ) from above (see in particular the derivation of the order of convergence of e N 1 {x>0} βx β-1 exp -x β dx, ρ for β > 0 in Example 2.6 below), we may replace the optimal point x N i by F -1 2i-1 2N :

(1.3) ∀i ∈ {1, . . . , N }, ˆi N i-1 N F -1 (u) -x N i ρ du ≤ ˆi N i-1 N F -1 (u) -F -1 2i -1 2N ρ du,
a simple choice particularly appropriate when linearization is possible since

i -1 N , i N v → ˆi N i-1 N |u -v| ρ du is minimal for v = 2i-1 2N .
To bound e N (µ, ρ) from below, we can use that, by Jensen's inequality and the minimality of

F -1 2i-1 2N for ρ = 1, ˆi N i-1 N F -1 (u) -x N i ρ du ≥ N ρ-1 ˆi N i-1 N F -1 (u) -x N i du ρ ≥ N ρ-1 ˆi N i-1 N F -1 (u) -F -1 2i -1 2N du ρ ≥ N ρ-1 1 4N F -1 2i -1 2N -F -1 4i -3 4N + F -1 4i -1 4N -F -1 2i -1 2N ρ ≥ 1 4 ρ N F -1 4i -1 4N -F -1 4i -3 4N ρ . (1.4)
This estimate will be used in the proof that the order of convergence of e N (µ, ρ) cannot exceed 1/ρ when the support of µ is not bounded (Proposition 2.1 below), in the derivation of the order of convergence of e N (β1 [0,1] (x)x β-1 dx, ρ) and e N 1 {x>0} βx β-1 exp -x β dx, ρ for β > 0 in Examples 1.8 and 2.6 below and in the derivation of the necessary condition for convergence with boundary order 1/ρ (Proposition 2.7 below).

Alternative formula in terms of the cumulative distribution function

In the proof of our main result (Theorem 2.2 below which characterizes the convergence of e N (µ, ρ) to 0 with order α ∈ (0, 1 ρ )) and in the derivation of our necessary condition for the convergence of e N (µ, ρ) with boundary order 1/ρ (Proposition 2.7 below), we will need to rewrite in terms of the cumulative distribution function F in place of the quantile function F -1 contributions in the decomposition over i ∈ {1, . . . , N } in the right-hand side of (0.3). This is possible thanks to the next lemma.

Lemma 1.2. Assume that ˆR |x| ρ µ(dx) < +∞ with ρ ≥ 1. For i ∈ {1, . . . , N } and x ∈ F -1 i-1 N , F -1 i N ∩ R (with convention F -1 (0) = F -1 (0+)), we have: ˆi N i-1 N x -F -1 (u) ρ du = ρ ˆx F -1 ( i-1 N ) (x -y) ρ-1 F (y) - i -1 N dy + ρ ˆF -1 ( i N ) x (y -x) ρ-1 i N -F (y) dy,
and the right-hand side is minimal for

x = x N i . Remark 1.3. Under the convention F -1 (0) = -∞, when, for some i ∈ {1, . . . , N }, F -1 i-1 N + > F -1 i-1 N , then F (y) = i-1 N for y ∈ F -1 i-1 N , F -1 i-1 N + and ˆF -1 ( i-1 N +) F -1 ( i-1 N ) (x -y) ρ-1 F (y) - i -1
N dy = 0 so that the lower integration limit in the first integral in the right-hand side of (1.5) may be replaced by F -1 i-1 N + . In a similar way, the upper integration limit in the second integral may be replaced by

F -1 i N + under the convention F -1 (1+) = +∞.
Plugging this equality written for x = x N i in the right-hand side of (0.3), we immediately deduce the following alternative formulation of e N (µ, ρ) in terms of the cumulative distribution function F :

Proposition 1.4. (1.5) e ρ N (µ, ρ) = ρ N i=1 ˆxN i F -1 ( i-1 N +) x N i -y ρ-1 F (y) - i -1 N dy + ˆF -1 ( i N ) x N i y -x N i ρ-1 i N -F (y) dy .
Remark 1.5. When ρ = 1, the equality (1.5) follows from the interpretation of W 1 (ν, η) as the integral of the absolute difference between the cumulative distribution functions of ν and η (equal, as seen with a rotation with angle π 2 , to the integral of the absolute difference between their quantile functions) and the integral simplifies into:

(1.6) e N (µ, 1) = N i=1 ˆF -1 ( 2i-1 N ) F -1 ( i-1 N +) F (y) - i -1 N dy + ˆF -1 ( i N ) F -1 ( 2i-1 N ) i N -F (y) dy = 1 N ˆR min j∈N |N F (y) -j| dy.
For ρ > 1, this equality can be deduced from the general formula for W ρ ρ (ν, η) in terms of the cumulative distribution functions of µ and η (see for instance Lemma B.3 [START_REF] Jourdain | Propagation of chaos for rank-based interacting diffusions and long time behaviour of a scalar quasilinear parabolic equation[END_REF]).

Proof of Lemma 1.2. Let i ∈ {1, . . . , N } and x ∈ F -1 i-1 N , F -1 i N ∩ R. We have i-1 N ≤ F (x) and F (x-) ≤ i N . Since F -1 (u) ≤ x ⇔ u ≤ F (x) and F -1 (u) = x for u ∈ (F (x-), F (x)], we have: ˆi N i-1 N x -F -1 (u) ρ du = ˆF (x) i-1 N x -F -1 (u) ρ du + ˆi N F (x) F -1 (u) -x ρ du.
Using the well-known fact that the image of

1 [0,1] (v) dvµ(dz) by (v, z) → F (z-) + vµ({z}) is the Lebesgue measure on [0, 1] and that 1 [0,1] (v) dvµ(dz) a.e., F -1 (F (z-) + vµ({z})) = z , we obtain that: ˆF (x) i-1 N x -F -1 (u) ρ du = ˆ1 v=0 ˆz∈R 1 { i-1 N ≤F (z-)+vµ({z})≤F (x)} (x -z) ρ µ(dz) dv = ˆ1 v=0 ˆz∈R 1 { i-1 N ≤F (z-)+vµ({z})≤F (x)} ˆρ(x -y) ρ-1 1 {z≤y≤x} dyµ(dz) dv = ρ ˆx y=-∞ (x -y) ρ-1 ˆ1 v=0 ˆz∈R 1 { i-1 N ≤F (z-)+vµ({z})} 1 {z≤y} µ(dz) dv dy. (1.7) For v > 0, {z ∈ R : F (z-) + vµ({z}) ≤ F (y)} = (-∞, y] ∪ {z ∈ R : z > y and F (z) = F (y)} with µ ({z ∈ R : z > y and F (z) = F (y)}) = 0 and therefore ˆz∈R 1 { i-1 N ≤F (z-)+vµ({z})} 1 {z≤y} µ(dz) = ˆz∈R 1 { i-1 N ≤F (z-)+vµ({z})≤F (y)} µ(dz).
Plugging this equality in (1.7), using again the image of

1 [0,1] (v) dvµ(dz) by (v, z) → F (z-) + vµ({z}) and the equivalence i-1 N ≤ F (y) ⇔ F -1 i-1 N
≤ y , we deduce that:

ˆF (x) i-1 N x -F -1 (u) ρ du = ρ ˆx y=-∞ (x -y) ρ-1 ˆ1 u=0 1 { i-1 N ≤u≤F (y)} du dy = ρ ˆx F -1 ( i-1 N ) (x -y) ρ-1 F (y) - i -1 N dy.
In a similar way, we check that:

ˆi N F (x) F -1 (u) -x ρ du = ρ ˆF -1 ( i N ) x (y -x) ρ-1 i N -F (y) dy,
which concludes the proof.

Convergence of the error to 0

According to Corollary 5.12 [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF], the finiteness of the moment ´R |x| ρ µ(dx) with order ρ implies that the error e N (µ, ρ) goes to 0 as N → ∞. Since, by the inverse transform sampling, the moment clearly has to be finite for e N (µ, ρ) to be finite for some N ≥ 1, the following equivalence holds.

Proposition 1.6.

For each ρ ≥ 1, we have ˆR |x| ρ µ(dx) < +∞ ⇔ lim N →+∞ e N (µ, ρ) = 0.
The direct implication can also be deduced from the inequality e N (µ, ρ) ≤ W ρ δ Xi , µ for (X i ) i≥1 i.i.d. according to µ deduced from the strong law of large numbers and stated for instance in Theorem 2.13 [START_REF] Bobkov | One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances[END_REF]. For the sake of completeness, we give an alternative simple argument based on (1.5).

Proof. According to the introduction, the finiteness of e N (µ, ρ) for some N ≥ 1 implies that ˆR |x| ρ µ(dx) < +∞. So it is enough to check the zero limit property under the finite moment condition. When respectively

F -1 i N ≤ 0, F -1 i-1 N + < 0 < F -1 i N or F -1 i-1
N + ≥ 0 , then, by Lemma 1.2, the term with index i in (1.5) is respectively bounded from above by

ˆF -1 ( i N ) F -1 ( i-1 N +) F -1 i N -y ρ-1 F (y) - i -1 N dy ≤ ˆF -1 ( i N ) F -1 ( i-1 N +) (-y) ρ-1 1 N ∧ F (y) dy, ˆ0 F -1 ( i-1 N +) (-y) ρ-1 1 N ∧ F (y) dy + ˆF -1 ( i N ) 0 y ρ-1 1 N ∧ (1 -F (y)) dy, ˆF -1 ( i N ) F -1 ( i-1 N +) y -F -1 i -1 N + ρ-1 i N -F (y) dy ≤ ˆF -1 ( i N ) F -1 ( i-1 N +) y ρ-1 1 N ∧ (1 -F (y)) dy.
After summation, we deduce that:

e ρ N (µ, ρ) ≤ ρ ˆ0 -∞ (-y) ρ-1 1 N ∧ F (y) dy + ρ ˆ+∞ 0 y ρ-1 1 N ∧ (1 -F (y)) dy.
Since, by Fubini's theorem, ρ

ˆ0 -∞ (-y) ρ-1 F (y) dy+ρ ˆ+∞ 0 y ρ-1 (1-F (y)) dy = ˆR |x| ρ µ(dx) < +∞, Lebesgue's
theorem ensures that the right-hand side and therefore e N (µ, ρ) go to 0 as N → +∞.

According to Theorem 5.21 (ii) [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF], when the support of µ is bounded, then sup N ≥1 N 1 ρ e N (µ, ρ) < +∞, while Remark 5.22 (ii) [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF] ensures that lim sup N →+∞ N 1 ρ e N (µ, ρ) > 0 when F -1 is discontinuous. We complement these results by the following lemma.

Lemma 1.7. If F -1 is continuous and the support of µ bounded, then for each ρ > 1, lim

N →+∞ N 1/ρ e N (µ, ρ) = 0. Proof. By (1.3), e ρ N (µ, ρ) ≤ 1 2N N i=1 F -1 2i -1 2N -F -1 i -1 N + ρ + F -1 i N -F -1 2i -1 2N ρ ≤ 1 2N F -1 (1) -F -1 (0+) max 1≤j≤2N F -1 j 2N -F -1 j -1 2N ρ-1
, where we use the convention

F -1 (0) = F -1 (0+) in max 1≤j≤2N F -1 j 2N -F -1 j-1 2N ρ-1 . When the support of µ is bounded then F -1 (1) -F -1 (0+)
< ∞ and when moreover F -1 is continuous, then this function is uniformly continuous on (0, 1) and the conclusion follows.

The next example shows that when µ is compactly supported with F -1 continuous then, for each ρ > 1, the rate of convergence of N 1/ρ e N (µ, ρ) to 0 as N → +∞ may be arbitrarily small.

Example 1.8. Let µ β (dx) = β1 [0,1] (x)x β-1 dx with β > 0. Then F -1 (u) = u 1/β . Let us suppose that ρ > 1 and β ≥ ρ ρ-1 .
Using (1.4) with i = 1 for the second inequality, we obtain that

e ρ N (µ β , ρ) ≥ ˆ1 N 0 F -1 (u) -x N 1 ρ du ≥ 1 4 ρ N 1+ ρ β 3 4 1 β - 1 4
1 β ρ .
On the other hand, under the convention F -1 (0) = 0:

e ρ N (µ β , ρ) ≤ W ρ ρ 1 N N i=1 δ F -1 ( i-1 N ) , µ β = N i=1 ˆi N i-1 N u 1 β - i -1 N 1 β ρ du ≤ ˆ1 N 0 u ρ β du + 1 N 1+ ρ β N i=2 i 1 β -(i -1) 1 β ρ ≤ β β + ρ × 1 N 1+ ρ β + 1 β ρ N 1+ ρ β N i=2 (i -1) ρ β -ρ . (1.8) When β > ρ ρ-1 , the last sum is smaller than j∈N * j -(ρ-ρ β ) which is finite since ρ -ρ β > 1 and e N (µ β , ρ) N -1 ρ -1
β . Notice that according Theorem 5.15 [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF],

∀β > 0, ∀ρ ≥ 1, lim N →+∞ N e N (µ β , ρ) = 1 2β(ρ + 1) 1/ρ ˆ1 0 u ρ β -ρ du 1/ρ
, with the right-hand side finite if and only if ρ = 1 or ρ > 1 and β < ρ ρ-1 and then equal to

1 2β(ρ+1) 1/ρ β ρ+β-ρβ 1/ρ . When ρ > 1, for the limiting value β = ρ ρ-1 , one has 1 ρ + 1 β = 1 and, by (1.8), e ρ N (µ β , ρ) ≤ 1 ρ + 1 β ρ N ρ N i=2 1 i -1 ∼ ln N β ρ N ρ as N → +∞.
On the other hand, according to (1.4),

e ρ N (µ β , ρ) ≥ 1 4 2ρ-1 N ρ N i=1 (4i -1) 1 β -(4i -3) 1 β ρ ≥ 2 ρ 4 2ρ-1 β ρ N ρ N i=1 1 4i -1 ≥ 2 ρ 4 2ρ β ρ N ρ N i=1 1 i ∼ 2 ρ ln N 4 2ρ β ρ N ρ so that e N (µ β , ρ) N -1 (ln N ) 1 ρ .
According to Corollary 6.15 [START_REF] Bobkov | One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances[END_REF], for

(X i ) i≥1 i.i.d. according to µ β , E 1/ρ W ρ ρ 1 N N i=1 δ Xi , µ β N -1 ρ -1 β if ρ > 2 and β > 2ρ ρ-2 and E 1/ρ W ρ ρ 1 N N i=1 δ Xi , µ β N -1/2 if ρ ≤ 2 and β ≥ 1 or ρ > 2 and β ∈ [1, 2ρ ρ-2
). Note that apart from the restriction β ≥ 1 made in [START_REF] Bobkov | One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances[END_REF] to ensure that the distribution is log-concave, the results concerning the optimal deterministic choice and the random choice share the same structure with different maximal orders of convergence 1 and 1/2. When ρ > 2 and β > 2ρ ρ-2 , the deterministic and random orders of convergence are both equal to 1 ρ + 1 β .

Non-asymptotic lower bound for the error

According to Theorem 5.20 [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF] and its proof, lim sup

N →∞ N e N (µ, ρ) ≥ 1 2 ´1 2 0 (F -1 (u + 1 2 ) -F -1 (u)
)du with the right-hand side positive apart when µ is a Dirac mass and e N (µ, ρ) vanishes for all N, ρ ≥ 1. This result may be complemented by the following non-asymptotic bound.

Lemma 1.9. ∀ρ ≥ 1, ∀N ≥ 1, N e N (µ, ρ) + (N + 1)e N +1 (µ, ρ) ≥ 1 2 ˆR F (x) ∧ (1 -F (x)) dx.
Remark 1.10.

• One has ˆR F (x)∧(1-F (x)) dx = ˆF -1 ( 1 2 ) -∞ F (x)dx+ ˆ+∞ F -1 ( 1 2 ) 1 2 -F (x) - 1 2 dx = ˆ1 2 0 F -1 u + 1 2 -F -1 (u) du
as easily seen since the sum and the last integral correspond to the area of the points at the right to

(F (x)) -∞≤x≤F -1 ( 1 2 ) , at the left to (F (x) -1 2 ) F -1 ( 1
2 )<x<+∞ , above 0 and below 1 2 respectively computed by integration with respect to the abscissa and to the ordinate.

• The analalogous result in the random case is stated in Theorem 3.1 [START_REF] Bobkov | One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances[END_REF] : when

(X i ) i≥1 are i.i.d. according to µ, E W 1 1 N N i=1 δ Xi , µ ≥ 1 2 √ 2N E |X 1 -F -1 (1/2)| .
In other words, unless µ is a Dirac mass, the random rate cannot be quicker than the usual Monte Carlo rate 1 √ N .

Proof. Note that when ρ ≥ ρ ≥ 1 and ˆR |x| ρ µ(dx) < +∞, with (x N i ) 1≤i≤N denoting the optimal points for ρ ≥ 1,

(1.9) e N (µ, ρ) = W ρ 1 N N i=1 δ x N i , µ ≥ W ρ 1 N N i=1 δ x N i , µ ≥ e N (µ, ρ).
Hence ρ → e N (µ, ρ) is non-decreasing and it is enough to check the statement for ρ = 1. It follows by plugging into (1.6) the following inequality written for v = F (x) :

∀v ∈ [0, 1], ∀N ≥ 1, min j∈N |N v -j| ∨ min j∈N |(N + 1)v -j| ≥ v ∧ (1 -v) 2 ,
To prove this inequality, we remark that for v ∈ (0, 1), there are two possibilities

• Either N v ≤ N v < (N +1)v ≤ N v +1, which implies that (N v -N v )∨( N v +1-(N +1)v) ≥ 1-v 2 while N v + 1 -N v = N v + 1 -(N + 1)v + v ≥ v and (N + 1)v -N v = N v -N v + v ≥ v so that min j∈N |N v -j| ∨ min j∈N |(N + 1)v -j| ≥ v ∧ 1 -v 2 . • Or N v ≤ N v < N v +1 ≤ (N +1)v, which implies that ( N v +1-N v)∨((N + 1)v -( N v + 1)) ≥ v 2 while N v-N v = (N +1)v-( N v +1)+1-v ≥ 1-v and N v +2-(N +1)v = N v +1-N v+1-v > 1 -v so that min j∈N |N v -j| ∨ min j∈N |(N + 1)v -j| ≥ v 2 ∧ (1 -v).
Synthetising the two cases and remarking that the inequality still holds for v ∈ {0, 1}, we conclude.

2 The case when the support of µ is unbounded

According to Theorem 5.21 (ii) [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF], when the support of µ is bounded, sup N ≥1

N 1/ρ e N (µ, ρ) < +∞ and, when moreover F -1 is continuous, then lim N →∞ N 1/ρ e N (µ, ρ) = 0 for ρ > 1 by Lemma 1.7. In this section, we first show that in the unbounded support case, the order of convergence cannot exceed the minimal order 1/ρ in the bounded support case. Next we state our main result : a necessary and sufficient condition for convergence with order α ∈ (0, 1 ρ ) which precises the implication stated in Theorem 5.21 (i) [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF] :

ˆRd |x| ρ 1-αρ µ(dx) < +∞ ⇒ lim N →∞ N α e N (µ, ρ) = 0.
We finally address the boundary case α = 1/ρ where we only obtain a necessary condition and illustrate the possibility that, when ρ > 1, lim N →∞ N 1/ρ e N (µ, ρ) = 0 for some probability measure µ with unbounded support.

The order of convergence cannot exceed 1/ρ

According to the next result, the order of convergence of e N (µ, ρ) cannot exceed 1 ρ when the support of µ is not bounded.

Proposition 2.1. Let ρ > 1. Then ∃ α > 1 ρ , sup N ≥1 N α e N (µ, ρ) < +∞ =⇒ F -1 (1) -F -1 (0+) < +∞. Proof. Let ρ > 1 and α > 1 ρ be such that sup N ≥1 N α e N (µ, ρ) < +∞ so that, by (1.2), sup N ≥1 N αρ ˆ1 N 0 F -1 (u) -x N 1 ρ du + ˆ1 N -1 N F -1 (u) -x N N ρ du < +∞.
By (1.4) for i = 1 and N ≥ 1, we have:

ˆ1 N 0 F -1 (u) -x N 1 ρ du ≥ 1 4 ρ N F -1 1 2N -F -1 1 4N ρ . Therefore C := sup N ≥1 (2N ) α-1 ρ F -1 1 2N -F -1 1 4N < +∞. For k ∈ N * , we deduce that F -1 2 -(k+1) - F -1 2 -k ≥ -C2
1-αρ ρ k , and after summation that: 1) .

(2.1) ∀k ∈ N * , F -1 2 -k ≥ F -1 (1/2) - C 2 α-1 ρ -1 1 -2 1-αρ ρ (k-
When k → +∞, the right-hand side goes to

F -1 1 2 - C 2 α-1 ρ -1 > -∞ so that F -1 (0+) > -∞.
In a symmetric way, we check that F -1 (1) < +∞ so that µ is compactly supported.

2.2 Necessary and sufficient condition for convergence with order α ∈ (0, 1 ρ )

Our main result is the following necessary and sufficient condition for e N (µ, ρ) to go to 0 with order α ∈ 0, 1 ρ .

Theorem 2.2. Let ρ ≥ 1 and α ∈ 0, 1 ρ . We have

sup x≥0 x ρ 1-αρ F (-x) + 1 -F (x) < +∞ ⇔ sup N ≥1 N α e N (µ, ρ) < +∞ ⇔ sup N ≥2 sup x 2:N -1 N α W ρ (µ N (x 2:N -1 ), µ) < +∞ where µ N (x 2:N -1 ) = 1 N δ F -1 ( 1 N )∧(-N 1 ρ -α ) + N -1 i=2 δ xi + δ F -1 ( N -1 N )∨N 1 ρ -α
and sup x 2:N -1 means the supremum over the choice of

x i ∈ [F -1 ( i-1 N +), F -1 ( i N )] for 2 ≤ i ≤ N -1. Moreover, lim x→+∞ x ρ 1-αρ F (-x) + 1 -F (x) = 0 ⇔ lim N →+∞ N α e N (µ, ρ) = 0. Remark 2.3.
• Let us relate the order of convergence to the maximal integrable power β := sup{β ≥ 0 : ´R |x| β µ(dx) < ∞} of µ. For β ∈ (0, β), ´R |x| β µ(dx) < ∞ while when β < ∞, for β > β, ´R |x| β+β 2 µ(dx) = +∞, so that, by Lemma 2.4 below, sup x≥0 x β F (-x) + 1 -F (x) = +∞. Let ρ ≥ 1. If β > ρ, we deduce from Theorem 5.21 (i) [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF] that for each α ∈ (0, 1 ρ -1 β ), lim N →∞ N α e N (µ, ρ) = 0 and, when β < +∞, Theorem 2.2 ensures that for each α >

1 ρ -1 β , sup N ≥1 N α e N (µ, ρ) = +∞ since ρ 1-αρ > β.
In this sense, when ρ < β < +∞ the order of convergence of e N (µ, ρ) to 0 is 1 ρ -1 β . Moreover, the boundedness and the vanishing limit at infinity for the sequence (N

+ x → x β F (-x) + 1 -F (x) . Note that lim sup x→+∞ x β F (-x) + 1 -F (x) can be +∞, in which case, sup N ≥1 N 1 ρ -1 β e N (µ, ρ) = +∞.
• In the proof (see (2.4) below), we estimate sup

N ≥2 sup x 2:N -1 N αρ W ρ ρ (µ N (x 2:N -1 ), µ) in terms of C = sup x≥0 x ρ 1-αρ F (-x) + 1 -F (x) .
• According to Theorem 7.16 [START_REF] Bobkov | One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances[END_REF], for (X i ) i≥1 i.i.d. according to µ,

sup N ≥1 N 1 2ρ E 1/ρ W ρ ρ 1 N N i=1 δ Xi , µ ≤ ρ2 ρ-1 ˆR |x| ρ-1 F (x)(1 -F (x)) dx 1/ρ with ∃ ε > 0, ´R |x| 2ρ+ε µ(dx) < +∞ ⇒ ´R |x| ρ-1 F (x)(1 -F (x)) dx < +∞ ⇒ ´R |x| 2ρ µ(dx) < +∞
(the reverse implications fail) by the discussions just after this theorem and after Theorem 3.2 [START_REF] Bobkov | One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances[END_REF]. The

condition sup x≥0 x 2ρ F (-x) + 1 -F (x) < +∞ equivalent to sup N ≥1 N 1 2ρ
e N (µ, ρ) < +∞ is slightly weaker than ´R |x| 2ρ µ(dx) < +∞, according to Lemma 2.4 just below. Moreover, we address similarly any order of convergence α with α ∈ 0, 1 ρ for e N (µ, ρ), while the order 1 2ρ seems to play a special role for

E 1/ρ W ρ ρ 1 N N i=1
δ Xi , µ in the random case. When ρ = 1, the order of convergence α for α ∈ (0, 1/2) is addressed in the random case in Theorem 2.2 [START_REF] Del Barrio | Central limit theorems for the Wasserstein distance between the empirical and the true distributions[END_REF] where the finiteness of sup x≥0 x

1 1-α F (-x) + 1 -F (x)
is stated to be equivalent to the stochastic boundedness of the sequence

N α W 1 1 N N i=1 δ Xi , µ N ≥1
. When α = 1/2, the stochastic boundedness property is, according to Theorem 2.1 (b) [START_REF] Del Barrio | Central limit theorems for the Wasserstein distance between the empirical and the true distributions[END_REF], equivalent to

´R F (x)(1 -F (x)) dx < +∞.
The proof of Theorem 2.2 relies on the next lemma, the proof of which is postponed to the appendix.

Lemma 2.4. For β > 0, we have

ˆR |y| β µ(dy) < +∞ =⇒ lim x→+∞ x β F (-x) + 1 -F (x) = 0 =⇒ sup x≥0 x β F (-x) + 1 -F (x) < +∞ =⇒ ∀ε ∈ (0, β] , ˆR |y| β-ε µ(dy) < +∞ and sup x≥0 x β F (-x) + 1 -F (x) < +∞ ⇔ sup u∈(0,1/2] u 1 β F -1 (1 -u) -F -1 (u) < +∞ with (2.2) sup u∈(0,1/2] u 1 β F -1 (1 -u) -F -1 (u) ≤ sup x≥0 x β F (-x) 1 β + sup x≥0 x β (1 -F (x)) 1 β . Last, lim x→+∞ x β F (-x) + 1 -F (x) = 0 ⇔ lim u→0+ u 1 β F -1 (1 -u) -F -1 (u) = 0.
Proof of Theorem 2.2. Since by Lemma 2.4,

sup u∈(0,1/2] u 1 ρ -α F -1 (1 -u) -F -1 (u) < +∞ =⇒ sup x≥0 x ρ 1-αρ F (-x) + 1 -F (x) < +∞ and, by (1.2), e ρ N (µ, ρ) ≥ ˆ1 N 0 F -1 (u) -x N 1 ρ du + ˆ1 N -1 N F -1 (u) -x N N ρ du
to prove the equivalence, it is enough to check that

sup x≥0 x ρ 1-αρ F (-x) + 1 -F (x) < +∞ =⇒ sup N ≥1
N αρ e ρ N (µ, ρ) < +∞ and that

sup N ≥1 N αρ ˆ1 N 0 F -1 (u) -x N 1 ρ du + ˆ1 N -1 N F -1 (u) -x N N ρ du < +∞ =⇒ sup u∈(0,1/2] u 1 ρ -α F -1 (1 -u) -F -1 (u) < +∞.
We are now going to do so and thus prove that the four suprema in the two last implications are simultaneously finite or infinite.

Let us first suppose that

C := sup x≥0 x ρ 1-αρ F (-x) + 1 -F (x) < +∞ and set N ≥ 2. Let x i ∈ [F -1 ( i-1 N +), F -1 ( i N )] for 2 ≤ i ≤ N -1.
We have

e ρ N (µ, ρ) ≤ W ρ ρ (µ N (x 2:N -1 ), µ) = L N + M N + U N with L N = ´1 N 0 F -1 (u) -F -1 1 N ∧ (-N 1 ρ -α ) ρ du, U N = ´1 N -1 N F -1 (u) -F -1 N -1 N ∨ N 1 ρ -α ρ du and
M N = N -1 i=2 ˆi N i-1 N F -1 (u) -x i ρ du ≤ N -1 i=2 ˆi N i-1 N F -1 i N -F -1 i -1 N ρ du ≤ 1 N N i=1 F -1 N -1 N -F -1 1 N ρ-1 F -1 i N -F -1 i -1 N = 1 N F -1 N -1 N -F -1 1 N ρ (2.3) ≤ 2 ρ C 1-αρ N -αρ ,
where we used (2.2) applied with β = ρ 1-αρ for the last inequality. Let x + = 0 ∨ x denote the positive part of any real number x. Applying Lemma 1.2 with x = F -1 1 N ∧ -N 1 ρ -α , we obtain that

L N = ρ ˆF -1 ( 1 N )∧ -N 1 ρ -α -∞ F -1 1 N ∧ -N 1 ρ -α -y ρ-1 F (y) dy + ρ ˆF -1 ( 1 N ) F -1 ( 1 N )∧ -N 1 ρ -α y -F -1 1 N ∧ -N 1 ρ -α ρ-1 1 N -F (y) dy ≤ ρ ˆ+∞ N 1 ρ -α y ρ-1 F (-y) dy + 1 N N 1 ρ -α + F -1 1 N ρ + .
In a symmetric way, we check that

U N ≤ ρ ˆ+∞ N 1 ρ -α y ρ-1 (1 -F (y)) dy + 1 N N 1 ρ -α -F -1 N -1 N ρ + so that L N + U N ≤ ρC ˆ+∞ N 1 ρ -α y -1-αρ 2 1-αρ dy + 1 N N 1 ρ -α + F -1 (1/2) ρ + + N 1 ρ -α -F -1 (1/2) ρ + ≤ 1 -αρ αρ CN -αρ + 1 + 2 ρ-1 N -αρ + 2 ρ-1 F -1 (1/2) ρ N -1 .
Since N -1 ≤ 2 αρ-1 N -αρ , we conclude that with sup x 2:N -1 denoting the supremum over

x i ∈ [F -1 ( i-1 N +), F -1 ( i N )] for 2 ≤ i ≤ N -1, sup N ≥2 N αρ e ρ N (µ, ρ) ≤ sup N ≥2 sup x 2:N -1 N αρ W ρ ρ (µ N (x 2:N -1 ), µ) ≤ 2 ρ C 1-αρ + 1 -αρ αρ C + 1 + 2 ρ-1 + 2 ρ+αρ-2 F -1 (1/2) ρ .
(2.4)

We may replace sup N ≥2 N αρ e ρ N (µ, ρ) by sup N ≥1 N αρ e ρ N (µ, ρ) in the left-hand side, since, applying Lemma 1.2 with x = 0, then using that for y ≥ 0, F (-y) + 1 -F (y) = µ((-∞, -y] ∪ (y, +∞)) ≤ 1, we obtain that

e ρ 1 (µ, ρ) ≤ ρ ˆ+∞ 0 y ρ-1 F (-y) + 1 -F (y) dy ≤ ρ ˆ1 0 y ρ-1 dy + ρC ˆ+∞ 1 y -1-αρ 2 1-αρ dy = 1 + 1 -αρ αρ C.
Let us next suppose that sup

N ≥1 N αρ ˆ1 N 0 F -1 (u) -x N 1 ρ du + ˆ1 N -1 N F -1 (u) -x N N ρ du < +∞.
Like in the proof of Proposition 2.1, we deduce (2.1). With the monotonicity of F -1 , this inequality implies that

∃C < +∞, ∀u ∈ (0, 1/2], F -1 (u) ≥ F -1 (1/2) - C 1 -2 α-1 ρ u α-1 ρ -1 ,
and therefore that inf u∈(0,1/2] u

1 ρ -α F -1 (u) > -∞.
With a symmetric reasoning, we conclude that

sup u∈(0,1/2] u 1 ρ -α F -1 (1 -u) -F -1 (u) < +∞.
Let us now assume that lim sup x→+∞ x ρ 1-αρ

F (-x) + 1 -F (x) ∈ (0, +∞), which, in particular implies that sup x≥0 x ρ 1-αρ F (-x) + 1 -F (x)
< +∞ and check that lim sup N →∞ N α e N (µ, ρ) > 0. For x > 0, we have, on the one hand

x αρ 2 1-αρ ˆ+∞ x y ρ-1 F (-y) + 1 -F (y) dy ≥ x αρ 2 1-αρ ˆ2x x x ρ-1 F (-2x) + 1 -F (2x) dy = x ρ 1-αρ F (-2x) + 1 -F (2x) .
On the other hand, still for x > 0,

x αρ 2 1-αρ ˆ+∞ x y ρ-1 F (-y) + 1 -F (y) dy ≤ x αρ 2 1-αρ sup y≥x y ρ 1-αρ F (-y) + 1 -F (y) ˆ+∞ x y -αρ 2 1-αρ -1 dy = 1 -αρ αρ 2 sup y≥x y ρ 1-αρ F (-y) + 1 -F (y) (2.
:= sup N ≥1 N α-1 ρ x N N ∨ -x N 1 < +∞ (notice that since x N 1 ≤ x N N , κ ≥ 0). With (1.5), we deduce that: e ρ N (µ, ρ) ρ ≥ ˆxN 1 -∞ x N 1 -y ρ-1 F (y) dy + ˆ+∞ x N N y -x N N ρ-1 (1 -F (y)) dy ≥ ˆ-κN 1 ρ -α -∞ -κN 1 ρ -α -y ρ-1 F (y) dy + ˆ+∞ κN 1 ρ -α y -κN 1 ρ -α ρ-1 (1 -F (y)) dy ≥ 2 1-ρ ˆ+∞ 2κN 1 ρ -α y ρ-1 F (-y) + 1 -F (y) dy.
Applying (2.6) with y N = 2κN 1 ρ -α , we conclude that lim sup

N →+∞ N αρ e ρ N (µ, ρ) > 0. If x ρ 1-αρ F (-x) + 1 -F (x)
does not go to 0 as x → +∞ then either sup x≥0 x ρ 1-αρ

F (-x) + 1 -F (x) = +∞ = sup N ≥1 N α e N (µ, ρ) or lim sup x→+∞ x ρ 1-αρ F (-x) + 1 -F (x) ∈ (0,
+∞) and lim sup N →+∞ N α e N (µ, ρ) ∈ (0, +∞) so that, synthesizing the two cases, N α e N (µ, ρ) does not go to 0 as N → +∞. Therefore, to conclude the proof of the second statement, it is enough to suppose lim x→+∞ x ρ 1-αρ F (-x)+1-F (x) = 0 and deduce lim N →+∞ N α e N (µ, ρ) = 0, which we now do. By Lemma 2.4,

lim N →∞ N α-1 ρ F -1 N -1 N -F -1 1 N = 0.
Since, reasoning like in the above derivation of (2.3), we have

N -1 i=2 ´i N i-1 N |F -1 (u) -x N i | ρ du ≤ 1 N F -1 N -1 N -F -1 1 N ρ for N ≥ 3, we deduce that lim N →∞ N αρ N -1 i=2 ´i N i-1 N |F -1 (u) -x N i | ρ du = 0. Let S N = sup x≥N 1 2ρ -α 2 x ρ 1-αρ F (-x) + 1 -F (x) 1-αρ 2αρ 2 and y N = F -1 N -1 N ∨ N 1 2ρ -α 2 ∨ (S N N 1 ρ -α ).
Using Lemma 1.2 for the first inequality, (2.5) for the second one, then the definition of y N for the third, we obtain that

N αρ ˆ1 N -1 N |F -1 (u) -x N N | ρ du ≤ ρN αρ ˆyN F -1 ( N -1 N ) (y N -y) ρ-1 F (y) - N -1 N dy + ρN αρ ˆ+∞ y N (y -y N ) ρ-1 (1 -F (y))dy ≤ N αρ-1 ˆyN F -1 ( N -1 N ) ρ(y N -y) ρ-1 dy + 1 -αρ αρ 2 N αρ y -αρ 2 1-αρ N sup y≥y N y ρ 1-αρ F (-y) + 1 -F (y) ≤ N αρ-1 N 1 2ρ -α 2 ∨ (S N N 1 ρ -α ) -F -1 N -1 N ρ + + 1 -αρ αρ 2 N αρ S N N 1 ρ -α -αρ 2 1-αρ sup y≥N 1 2ρ -α 2 y ρ 1-αρ F (-y) + 1 -F (y) ≤ N α 2 -1 2ρ ∨ S N -N α-1 ρ F -1 N -1 N ρ + + 1 -αρ αρ 2 S αρ 2 1-αρ N . Since lim N →∞ N α-1 ρ F -1 N -1 N = 0 = lim N →∞ N α 2 -1 2ρ = lim N →+∞ S N , we deduce that lim N →∞ N αρ ´1 N -1 N |F -1 (u)- x N N | ρ du = 0. Dealing in a symmetric way with N αρ ´1 N 0 |F -1 (u)-x N N | ρ du, we conclude that lim N →∞ N αρ e ρ N (µ, ρ) = 0. Example 2.5. let µ β (dx) = f (x) dx with f (x) = β 1 {x≥1}
x β+1 be the Pareto distribution with parameter β > 0.

Then F (x) = 1 {x≥1} 1 -x -β and F -1 (u) = (1 -u) -1 β . To ensure that ´R |x| ρ µ(dx) < +∞, we suppose that β > ρ. Since ρ 1-ρ( 1 ρ -1 β ) = β we have lim x→+∞ x ρ 1-ρ ( 1 ρ - 1
β ) (F (-x) + 1 -F (x)) = 1.
Replacing lim sup by lim inf in the last step of the proof of Theorem 2.2, we check that lim inf N →+∞ N 1 ρ -1 β e N (µ β , ρ) > 0 and deduce with the statement of this theorem that e N (µ β , ρ) N -

1 ρ + 1 β sup x 2:N -1 W ρ (µ N (x 2:N -1 ), µ β ).

The boundary case α = 1/ρ

Before giving a necessary condition for convergence with the boundary order 1/ρ, we show in the next example that for β > 0, e N (1 {x>0} βx β-1 exp -x β dx, ρ) converges to 0 with this order up to some logarithmic factor. In particular, the case β > 1 illustrates the possibility that, when ρ > 1, lim N →+∞ N 1/ρ e N (µ, ρ) = 0 for some probability measures µ with unbounded support. Of course, F -1 is then continuous on (0, 1), since, by Remark 5.22 (ii) [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF], lim sup

N →+∞ N 1/ρ e N (µ, ρ) > 0 otherwise.
Example 2.6. For the Weibull distribution µ β (dx) = f (x) dx with f (x) = 1 {x>0} βx β-1 exp -x β with β > 0 (the exponential distribution case β = 1, was addressed in Example 5.17 and Remark 5.22 (i) [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF]), we have that

F (x) = 1 {x>0} 1 -exp(-x β ) , F -1 (u) = (-ln(1 -u)) 1 β and f F -1 (u) = β(1 -u)(-ln(1 -u)) 1-1 β .
The density f is decreasing on [x β , +∞) where x β = (β-1)∨0

β 1 β . Using (1.4), the equality F -1 (w)-F -1 (u) = ´w u dv f (F -1 (v))
valid for u, w ∈ (0, 1) and the monotonicity of the density, we obtain that for N large enough so that N F (x β ) ≤ N -1,

e ρ N (µ β , ρ) ≥ 1 4 ρ N N i= N F (x β ) +1 ˆ4i-1 4N 4i-3 4N du f (F -1 (u)) ρ ≥ 1 8 ρ N ρ+1 N i= N F (x β ) +1 1 f ρ F -1 4i-3 4N ≥ 1 (8N ) ρ N i= N F (x β ) +2 ˆi-1 N i-2 N du f ρ (F -1 (u)) = 1 (8N ) ρ ˆN-1 N N F (x β ) N du f ρ (F -1 (u)) . (2.7)
Using (1.3) for the first inequality, Hölder's inequality for the second inequality, then Fubini's theorem for the third, we obtain that

e ρ N (µ β , ρ) - ˆ1 N -1 N x N N -F -1 (u) ρ du ≤ N -1 i=1 ˆi N i-1 N ˆu 2i-1 2N dv f (F -1 (v)) ρ du ≤ N -1 i=1 ˆi N i-1 N u - 2i -1 2N ρ-1 ˆu 2i-1 2N dv f ρ (F -1 (v)) du ≤ 1 (2N ) ρ ρ ˆN-1 N 0 dv f ρ (F -1 (v))
.

We have F (x β ) < 1 and, when β > 1, F (x β ) > 0. By integration by parts, for ρ > 1, (ρ -1)

ˆN-1 N F (x β ) β ρ du f ρ (F -1 (u)) = ˆN-1 N F (x β ) (ρ -1)(1 -u) -ρ (-ln(1 -u)) ρ β -ρ du = (1 -u) 1-ρ (-ln(1 -u)) ρ β -ρ N -1 N F (x β ) + ρ β -ρ ˆN-1 N F (x β ) (1 -u) -ρ (-ln(1 -u)) ρ β -ρ-1 du = N ρ-1 (ln N ) ρ β -ρ + o ˆN-1 N F (x β ) (1 -u) -ρ (-ln(1 -u)) ρ β -ρ du ∼ N ρ-1 (ln N ) ρ β -ρ ,
as N → +∞. We obtain the same equivalent when replacing the lower integration limit F (x β ) in the left-hand side by

N F (x β ) N or 0 since lim N →+∞ ˆ N F (x β ) N F (x β ) du f ρ (F -1 (u))
= 0 and

ˆF (x β ) 0 du f ρ (F -1 (u))
< +∞. On the other hand,

ˆ1 N -1 N x N N -F -1 (u) ρ du ≤ ˆ1 N -1 N (-ln(1 -u)) 1 β -(ln N ) 1 β ρ du. When β < 1, for u ∈ N -1 N , 1 , (-ln(1 -u)) 1 β -(ln N ) 1 β ≤ 1 β (-ln(1 -u)) 1 β -1 (-ln(1 -u) -ln N ) so that ˆ1 N -1 N (-ln(1 -u)) 1 β -(ln N ) 1 β ρ du ≤ 1 β ρ ˆ1 N -1 N (-ln(1 -u)) ρ β -ρ (-ln(N (1 -u))) ρ du = 1 β ρ N ˆ1 0 (ln N -ln v) ρ β -ρ (-ln(v)) ρ dv ≤ 2 ( ρ β -ρ-1)∨0 β ρ N (ln N ) ρ β -ρ ˆ1 0 (-ln(v)) ρ dv + ˆ1 0 (-ln(v)) ρ β dv . (2.8) When β ≥ 1, for N ≥ 2 and u ∈ N -1 N , 1 , (-ln(1 -u)) 1 β -(ln N ) 1 β ≤ 1 β (ln N ) 1 β -1 (-ln(1 -u) -ln N ) so that ˆ1 N -1 N (-ln(1 -u)) 1 β -(ln N ) 1 β ρ du ≤ (ln N ) ρ β -ρ β ρ N ˆ1 0 (-ln(v)) ρ β dv. (2.9)
We conclude that for ρ > 1 and β > 0, e N (µ β , ρ) N -1 ρ (ln N )

1 β -1 W ρ ( 1 N ( N -1 i=1 δ F -1 ( 2i-1 2N ) +δ F -1 ( N -1 N ) ), µ β ).
In view of Theorem 5.20 [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF], this rate of convergence does not extend continuously to e N (µ β , 1), at least for β > 1. In fact, by Remark 2.2 [START_REF] Jourdain | Optimal convergence rate of the multitype sticky particle approximation of one-dimensional diagonal hyperbolic systems with monotonic initial data[END_REF], for β > 0, e N (µ β , 1) N -1 (ln N ) 1 β , which in view of (2.8) and (2.9), implies that

N -1 i=1 ´i N i-1 N x N i -F -1 (u) ρ du N -1 (ln N ) 1 β .
In the Gaussian tail case β = 2, e N (µ 2 , ρ) N -1 ρ (ln N ) -1 2 +1 {ρ=1} for ρ ≥ 1, like for the true Gaussian distribution, according to Example 5.18 [START_REF] Xu | Best finite constrained approximations of one-dimensional probabilities[END_REF]. This matches the rate obtained when ρ > 2 in Corollary 6.14 [START_REF] Bobkov | One-Dimensional Empirical Measures, Order Statistics, and Kantorovich Transport Distances[END_REF] for

E 1/ρ W ρ ρ 1 N N i=1 δ Xi , µ
where (X i ) i≥1 are i.i.d. with respect to some Gaussian distribution µ with positive variance. When ρ = 2, still according to this corollary the random rate is N -1/2 (ln ln N ) 1/2 (of course worse than the standard Monte Carlo rate N -1/2 ).

The next proposition gives a necessary condition for e N (µ, ρ) to go to 0 with order α = 1 ρ . Proposition 2.7. For ρ ≥ 1, sup

N ≥1 N 1/ρ e N (µ, ρ) < +∞ ⇒ sup N ≥1 N ˆ1 N 0 F -1 (u) -x N 1 ρ du + ˆ1 N -1 N F -1 (u) -x N N ρ du < +∞ ⇔ sup u∈(0,1/2] F -1 (1 -u/2) -F -1 (1 -u) + F -1 (u) -F -1 (u/2) < +∞ ⇒ sup u∈(0,1/2] F -1 (1 -u) -F -1 (u) ln(1/u) < +∞ ⇔∃λ ∈ (0, +∞), ∀x ≥ 0, F (-x) + 1 -F (x) ≤ e -λx /λ ⇒ sup N ≥2 sup x 2:N -1 N 1/ρ 1 + ln N W ρ (µ N,λ (x 2:N -1 ), µ) < +∞ ⇒ sup N ≥1 N 1/ρ 1 + ln N e N (µ, ρ) < +∞, where µ N,λ (x 2:N -1 ) = 1 N δ F -1 ( 1 N )∧(-ln N λ ) + N -1 i=2 δ xi + δ F -1 ( N -1 N )∨ ln N λ
and sup x 2:N -1 means the supremum over the choice of

x i ∈ [F -1 ( i-1 N +), F -1 ( i N )] for 2 ≤ i ≤ N -1. Remark 2.8.
• The first implication is not an equivalence for ρ = 1. Indeed, in Example 2.6, for β ≥ 1, lim

N →+∞ N e N (µ, 1) = +∞ while sup N ≥1 N ´1 N 0 F -1 (u) -x N 1 du + ´1 N -1 N F -1 (u) -x N N du < +∞.
• The second implication is not an equivalence as examplified by F -1 (u) = g(u) ln(u) where g(u)

= k∈N k k+1 1 (2 -(k+1) 3 ,2 -k 3 ] (u).
The function (0, 1) u → g(u) ln(u) is a quantile function since it is leftcontinuous and non-decreasing as the product of the left-continuous, non-increasing and non-negative function g by the continuous, non-decreasing and non-positive logarithm function. Moreover, since g is bounded by 1, one easily checks that sup u∈(0,1/2]

F -1 (1-u)-F -1 (u) ln(1/u)
< +∞. On the other hand,

F -1 2 1-(k+1) 3 -F -1 2 -(k+1) 3 = k k + 1 (1 -(k + 1) 3 ) ln 2 + k + 1 k + 2 (k + 1) 3 ln 2 = k k + 1 ln 2 + (k + 1) 2 k + 2 ln 2
goes to ∞ with k.

• The exponential tail condition ∃λ ∈ (0, +∞), ∀x ≥ 0, F (-x) + 1 -F (x) ≤ e -λx /λ is not equivalent to

sup N ≥1 N 1/ρ 1+ln N e N (µ, ρ) < +∞ when ρ > 1 since in Example 2.6, for β ∈ [1/2, 1), sup N ≥1 N 1/ρ
1+ln N e N (µ, ρ) < +∞ while the exponential tail condition fails.

Proof. The first implication is an immediate consequence of (1.2). To prove the first equivalence, we first suppose that:

(2.10) sup

N ≥1 N 1 ρ ˆ1 N 0 F -1 (u) -x N 1 ρ du + ˆ1 N -1 N F -1 (u) -x N N ρ du 1 ρ < +∞.
and denote by C the finite supremum in this equation. By (1.4)

for i = 1, ∀N ≥ 1, F -1 1 2N -F -1 1 4N ≤ 4C. For u ∈ (0, 1/2], there exists N ∈ N * such that u ∈ 1 2(N +1) , 1 2N
and, by monotonicity of F -1 and since 4N ≥ 2(N + 1), we get

F -1 (u) -F -1 (u/2) ≤ F -1 1 2N -F -1 1 4(N + 1) ≤ F -1 1 2N -F -1 1 4N + F -1 1 2(N + 1) -F -1 1 4(N + 1) ≤ 8C.
Dealing in a symmetric way with

F -1 (1 -u/2) -F -1 (1 -u), we obtain that sup u∈(0,1/2] F -1 (1 -u/2) -F -1 (1 -u) + F -1 (u) -F -1 (u/2) ≤ 16C.
On the other hand, for N ≥ 2, by Lemma 1.2 applied with x

= F -1 1 N , 1 ρ ˆ1 N 0 F -1 (u) -x N 1 ρ du ≤ k∈N ˆF -1 ( 1 2 k N ) F -1 ( 1 2 k+1 N ) F -1 1 N -y ρ-1 F (y) dy ≤ k∈N F -1 1 2 k N -F -1 1 2 k+1 N 2 k N   k j=0 F -1 1 2 j N -F -1 1 2 j+1 N   ρ-1 ≤ 1 N sup u∈(0,1/2] F -1 (u) -F -1 (u/2) ρ k∈N (k + 1) ρ-1 2 k ,
where the last sum is finite. Dealing in a symmetric way with ´1 N -1

N F -1 (u) -x N N ρ du, we conclude that (2.10) is equivalent to the finiteness of sup u∈(0,1/2] F -1 (1 -u/2) -F -1 (1 -u) + F -1 (u) -F -1 (u/2) . Under (2.10)
with C denoting the finite supremum, for k ∈ N * , F -1 2 -(k+1) -F -1 2 -k ≥ -4C and, after summation,

F -1 (2 -k ) ≥ F -1 (1/2) -4C(k -1).
With the monotonicity of F -1 , we deduce that:

∀u ∈ (0, 1/2], F -1 (u) ≥ F -1 (1/2) + 4C ln 2 ln u
and therefore that sup u∈(0,1/2] -F -1 (u) ln(1/u) < +∞. With the inequality F -1 (F (x)) ≤ x valid for x ∈ R, this implies that sup {x∈R:0<F (x)≤1/2} -x ln(1/F (x)) < +∞ and therefore that ∃ λ ∈ (0, +∞), ∀x ≤ 0, F (x) ≤ e λx /λ. Under the latter condition, since u ≤ F (F -1 (u)) and F -1 (u) ≤ 0 for u ∈ (0, F (0)], we have sup u∈(0,F (0)]

-F -1 (u) ln(1/u) < ∞ and even sup u∈(0,1/2] -F -1 (u) ln(1/u) < ∞ since when F (0) < 1 2 , sup u∈(F (0),1/2] -F -1 (u)
ln(1/u) ≤ 0. By a symmetric reasoning, we obtain the two equivalent tail properties sup u∈(0,1/2]

F -1 (1-u)-F -1 (u) ln(1/u) < +∞ and ∃ λ ∈ (0, +∞), ∀x ≥ 0, F (-x) + 1 -F (x) ≤ e -λx /λ.
Let us finally suppose these two tail properties and deduce that sup N ≥2 sup x 2:N -1

N 1/ρ 1+ln N W ρ (µ N,λ (x 2:N -1 ), µ) < +∞. We use the decomposition W ρ ρ (µ N,λ (x 2:N -1 ), µ) = L N + M N + U N introduced in the proof of Theorem 2.2 but with F -1 1 N ∧ -ln N λ and F -1 N -1 N ∨ ln N λ respectively replacing F -1 1 N ∧ (-N 1 ρ -α ) and F -1 N -1 N ∨ (N 1 ρ -α ) in L N and U N . By (2.3), we get: ∀N ≥ 3, M N ≤ 1 N F -1 N -1 N -F -1 1 N ρ ≤ sup u∈(0,1/2] F -1 (1 -u) -F -1 (u) ln(1/u) ρ (ln N ) ρ N .
Applying Lemma 1.2 with x = F -1 1 N ∧ -ln N λ then the estimation of the cumulative distribution function, we obtain that for N ≥ 2,

L N ≤ ρ ˆF -1 ( 1 N )∧(-ln N λ ) -∞ F -1 1 N ∧ - ln N λ -y ρ-1 F (y) dy + ρ ˆF -1 ( 1 N ) F -1 ( 1 N )∧(-ln N λ ) y -F -1 1 N ∧ - ln N λ ρ-1 1 N -F (y) dy ≤ ρ λ ˆ-ln N λ -∞ (-y) ρ-1 e λy dy + 1 N ln N λ + F -1 1 N ρ + ≤ ρ λ k≥1 ˆ(k+1) ln N λ k ln N λ (k + 1) ln N λ ρ-1 e -λy dy + 1 N ln N λ + F -1 1 2 ρ + ≤ ρ(ln N ) ρ λ ρ+1 N k≥1 (k + 1) ρ-1 2 k-1 + 1 N 1 λ ln N + F -1 1 2 ρ + ,
where we used that N k ≥ N 2 k-1 for the last inequality. Dealing in a symmetric way with U N , we conclude that sup

N ≥2 sup x 2:N -1 N W ρ ρ (µ N,λ (x 2:N -1 ),µ) 1+(ln N ) ρ < +∞.

Conclusion

In the present paper, we have characterized the convergence of e N (µ, ρ) to 0 with order α ∈ (0, 1 ρ ) and also studied the convergence with boundary order α = 1/ρ between the unbounded support case and the bounded support case. In view of Example 2.6, it would be nice to investigate whether the leading factor remains N -1/ρ for µ with unbounded support and superpolynomial but subexponential tails. Characterizing the order of convergence when the support of µ is bounded and its quantile function F -1 is continuous is another open question.

Of course, generalizing our results to higher dimension would be of great interest. This appears to be challenging since our approach heavily relies on one-dimensional tools like the cumulative distribution function and the quantile function.

(0, +∞). Since for fixed y ∈ R, R x → 1 {y≥x} (y -x) ρ-1 -1 {y<x} (x -y) ρ-1 is non-increasing, we deduce that ∀y ∈ R, f (y) ≥ ρS ρ-1 N g( y S N ) where

g(z) = ˆ1 N 0 1 z≥u α-1 ρ z -u α-1 ρ ρ-1 -1 z<u α-1 ρ u α-1 ρ -z ρ-1
du.

For z ≥ (4N ) . Hence for z > ((κ ∨ 4) N ) Clearly, this inequality remains valid when S N = +∞. It holds in full generality since S N ≥ 0 and S N = 0 ⇔ F -1 (1) ≤ 0 a condition under which x N N ≤ 0 since x N N ∈ F -1 N -1 N + , F -1 (1) ∩ R. By a symmetric reasoning, we check that x N 1 ≥ ((κ ∨ 4) N )

1 ρ -α inf u∈(0, 1 N ) u 1 ρ -α F -1 (u).
Proof of Lemma 2.4. Let β > 0. For x > 0, using the monotonicity of F for the first inequality then that for y ∈ [ For the necessary condition, we set u ∈ (0, 1/2]. Either F -1 (u) ≥ 0 or, since for all v ∈ (0, 1), F (F -1 (v)) ≥ v, we have -F -1 (u) β u ≤ sup x≥-F -1 (u) x β F (-x) and therefore F -1 (u) ≥ -sup x≥0 x β F (-x) 1 β u -1 β . Either F -1 (1 -u) ≤ 0 or, since for all v ∈ (0, 1), F (F -1 (v)-) ≤ v, we have (F -1 (1 -u)) β u ≤ sup x≥F -1 (1-u) x β (1 -F (x-)) and therefore F -1 (1 -u) ≤ sup x≥0 x β (1 -F (x))

1 β u -1
β . Hence (2.2) holds. For the sufficient condition, we remark that the finiteness of sup u∈(0,1/2] u 1 β (F -1 (1 -u) -F -1 (u)) and the inequality F -1 (1-u)-F -1 (u) ≥ F -1 (1/2) -F -1 (u) ∨ F -1 (1 -u) -F -1 (1/2) valid for u ∈ (0, 1/2] imply that inf u∈(0,1/2] u 1 β F -1 (u) > -∞ and sup u∈(0,1/2] u 1 β F -1 (1 -u) < +∞. With the inequality x ≥ F -1 (F (x)) valid for x ∈ R such that 0 < F (x) < 1, this implies that inf x∈R:F (x)≤1/2 (F (x)) 1 β x > -∞ and therefore that sup x≥0 x β F (-x) < +∞. With the inequality x ≤ F -1 (F (x)+) valid for x ∈ R such that 0 < F (x) < 1, we obtain, in a symmetric way sup x≥0 x β (1 -F (x)) < +∞.

Let us finally check that lim x→+∞ x β F (-x) + 1 -F (x) = 0 ⇔ lim u→0+ u

1 β F -1 (1 -u) -F -1 (u) = 0.
For the necessary condition, we remark that either F -1 (1) < +∞ and lim u→0+ u 1 β F -1 (1 -u) = 0 or F -1 (1 -u) goes to +∞ as u → 0+. For u small enough so that F -1 (1 -u) > 0, we have (F -1 (1 -u)) β u ≤ sup x≥F -1 (1-u) x β (1 -F (x-)), from which we deduce that lim u→0+ u(F -1 (1 -u)) β = 0. The fact that lim u→0+ u 1 β F -1 (u) = 0 is deduced by a symmetric reasoning. For the sufficient condition, we use that x(1 -F (x)) 1 β ≤ sup u≤1-F (x) u 1 β F -1 ((1 -u)+) and x (F (x))

1 β ≥ inf u≤F (x) u 1 β F -1 (u) for x ∈ R such that 0 < F (x) < 1.

N 1 d

 1 , and ζ(p/d)N -1 ρ with ζ denoting the zeta Riemann function when ρ < d, ρ = d and ρ > d.

δ

  Xi , µ and the almost sure convergence to 0 of W ρ 1 N N i=1

1 ρ - 1 β

 1 e N (µ, ρ)) N ≥1 are respectively equivalent to the same property for the function R

1 F 1 F

 11 (-y) + 1 -F (y) dy ∈ (0, +∞) and, by monotonicity of the integral, (-y) + 1 -F (y) dy ∈ (0, +∞) along any sequence (y N ) N ∈N of positive numbers increasing to +∞ and such that lim sup N →+∞ y N +1 y N < +∞. By Lemmas 2.4 and 1.1, we have κ

1 ρ 1 ρ -α ≥ 1 -2 α- 1 ρ 1 ρ 1 (

 111111 -α , we have zρ αρ-1 ≤ 1 4N and z -(2N )The right-hand side is positive for z > (κN ) -α with κ := 1+(ρ-1)αρ)

1 ρ 1 ρ 1 ρ

 111 -α , g(z) > 0 so that for y > ((κ ∨ 4) N ) -α S N , f (y) > 0 and thereforex N N ≤ ((κ ∨ 4) N ) -α S N .

x 2 , 2 β- 1 ∧β∨1 x β ˆ+∞ x/ 2 y β- 1 F 0 y β- 1 Fx 1 β

 22121011 x], y β-1 ≥ x x β-1 = x β-1 2 (β-1)∨0 , we obtain thatF (-x) + 1 -F (x) ≤ 2 x ˆx x/2 F (-y) + 1 -F (y) dy ≤ 2 (-y) + 1 -F (y) dy.Since ´+∞ (-y) + 1 -F (y) dy = 1 β ´R |y| β µ(dy), the finiteness of ´R |y| β µ(dy) implies by Lebesgue's theorem thatlim x→∞ x β (F (-x) + 1 -F (x)) = 0. Since x → x β (F (-x) + 1 -F (x)) is right-continuous with left-hand limits on [0, +∞), sup x≥0 x β F (-x) + 1 -F (x) < +∞ ⇔ lim sup x→∞ x β F (-x) + 1 -F (x) < +∞,with the latter property clearly implied by lim x→∞ x β (F (-x) + 1 -F (x)) = 0.For ε ∈ (0, β), using that for y ≥ 0, F (-y) + 1 -F (y) = µ((-∞, -y] ∪ (y, +∞)) ≤ 1, we obtain ˆR |x| β-ε µ(dx) = (β -ε) ˆ+∞ 0 y β-ε-1 (F (-y) + 1 -F (y)) dy ≤ (β -ε) ˆ1 0 y β-ε-1 dy + (β -ε) sup x≥0 x β F (-x) + 1 -F (x) β F (-x) + 1 -F (x) .Therefore supx≥0 x β F (-x) + 1 -F (x) < +∞ =⇒ ∀ε ∈ (0, β), ˆR |x| β-ε µ(dx) < +∞.Let us next check that (2.11) sup x≥0 x β F (-x) + (1 -F (x)) < +∞ ⇔ sup u∈(0,1/2]u (F -1 (1 -u) -F -1 (u)) < +∞.

Appendix

Proof of Lemma 1.1. Since the finiteness of sup u∈(0,1/2] u

, the second statement is a consequence of the first one, that we are now going to prove. When ρ = 1 (resp. ρ = 2), then the conclusion easily follows from the explicit form

In the general case ρ > 1, we are going to take advantage of the expression

Let us first suppose that S N := sup u∈(0, 1 N ) u 1 ρ -α F -1 (1 -u) ∈


Zakaria Moutaoukil

Cyril Ronco

Rachid Benhida

One-pot synthesis of dihydropyrimidines via eco-friendly phosphorus derivatives catalysis

A straightforward, cost effective and eco-friendly protocol for the Biginelli reaction relying on the use of readily available hypophosphorous acid is presented. The methodology developed displays improvements compared to existing methods, is high-yielding, robust and was applied to a panel of dihydropyrimidines and thio-derivatives with various substituents. Related urea derivatives such as guanidines, benzimidazoles and benzothiazoles also reacted efficiently to afford more complex scaffolds. Thus, this rapid and convenient catalysis allows access to a wide diversity of structures including original biologically relevant heterocycles.

Introduction

3,4-Dihydropyrimidin-2(1H)-one (DHPM) is a biologically relevant heterocycle with great interest thanks to its anticancer, [START_REF] Kapoor | Probing spindle assembly mechanisms with monastrol, a small molecule inhibitor of the mitotic kinesin, Eg5[END_REF] antifungal, 2 anti-hypertensive, [START_REF] Rovnyak | Dihydropyrimidine calcium channel blockers. 4. Basic 3substituted-4-aryl-1, 4-dihydropyrimidine-5-carboxylic acid esters. Potent antihypertensive agents[END_REF] antimalarial, [START_REF] October | Reversed Chloroquines Based on the 3, 4-Dihydropyrimidin-2 (1H)-one Scaffold: Synthesis and Evaluation for Antimalarial, β-Haematin Inhibition, and Cytotoxic Activity[END_REF] anti-HIV [START_REF] Fatima | One pot efficient diversity oriented synthesis of polyfunctional styryl thiazolopyrimidines and their bio-evaluation as antimalarial and anti-HIV agents[END_REF] and anti-tubercular activities [START_REF] Desai | Preparation, biological evaluation and molecular docking study of imidazolyl dihydropyrimidines as potential Mycobacterium tuberculosis dihydrofolate reductase inhibitors[END_REF] . It is found in several drug candidates such as monastrol, [START_REF] Kaan | Structural basis for inhibition of Eg5 by dihydropyrimidines: stereoselectivity of antimitotic inhibitors enastron, dimethylenastron and fluorastrol[END_REF] an effective non-tubulin-interacting mitosis inhibitor, SQ-32926, [START_REF] Atwal | Dihydropyrimidine calcium channel blockers. II. 3-Substituted-4-aryl-1, 4-dihydro-6-methyl-5-pyrimidinecarboxylic acid esters as potent mimics of dihydropyridines[END_REF] a calcium channel antagonist with antihypertensive activity, or Bay 41-4109, [START_REF] Wichmann | Structure-activity relationships of substituted 5H-thiazolo [3, 2-a] pyrimidines as group 2 metabotropic glutamate receptor antagonists[END_REF] a potent human hepatitis B virus (HBV) inhibitor with an IC50 of 53 nM (Figure 1). A straightforward way to produce the DHPM scaffold is the well-known Biginelli reaction. [START_REF] Biginelli | Ueber aldehyduramide des acetessigäthers[END_REF] This multicomponent reaction involves a one-pot cyclocondensation of ethyl acetoacetate, urea and an aromatic aldehyde. It follows the principles of diversity-oriented synthesis and is therefore useful to produce rapidly series of analogues with great structural diversity during drug discovery processes. The synthesis of DHPM derivatives is presently an active research area, efforts are currently dedicated to the development of new ecofriendly synthetic routes.

Usual

© 2021 published by Elsevier. This manuscript is made available under the CC BY NC user license https://creativecommons.org/licenses/by-nc/4.0/ Version of Record: https://www.sciencedirect.com/science/article/pii/S1319610321002039 conditions for the Biginelli reaction are based on prolonged heating of the ternary mixture of reactants under strong Brønsted acidic conditions. [START_REF] Biginelli | Derivati aldeiduredici degli eteri acetile dossal-acetico[END_REF][START_REF] Xiao | One-pot synthesis of 3,4-dihydropyrimidine-2(1H)-ones and their derivatives[END_REF] Other conditions have been reported for this reaction such as microwave-assisted transformations, [START_REF] Kappe | Microwave-assisted high-speed parallel synthesis of 4-aryl-3, 4dihydropyrimidin-2 (1H)-ones using a solventless Biginelli condensation protocol[END_REF][START_REF] Kuraitheerthakumaran | Microwave-assisted multicomponent reaction for the synthesis of 3, 4-dihydropyrimidin-2 (1H)-ones and their corresponding 2 (1H)-thiones using lanthanum oxide as a catalyst under solvent-free conditions[END_REF] Mechanochemical conditions, [START_REF] Bose | A simplified green chemistry approach to the Biginelli reaction using 'Grindstone Chemistry[END_REF][START_REF] Ould | Green high-yielding one-pot approach to Biginelli reaction under catalyst-free and solvent-free ball milling conditions[END_REF] catalysis using adequate bases such as K2CO3, [START_REF] Kamal Raj | A mechanistic investigation of Biginelli reaction under base catalysis[END_REF] Lewis acids such as BF3.Et2O, [START_REF] Chaltin | Pyrimidine derivatives as novel viral replication inhibitors and their preparation and use in the treatment of HIV infection[END_REF] lanthanum salts LaCl3, [START_REF] Lu | One-pot synthesis of 3, 4-dihydropyrimidin-2 (1H)-ones using lanthanum chloride as a catalyst[END_REF] ionic liquids, [START_REF] Freitas | Tuning the Biginelli reaction mechanism by the ionic liquid effect: the combined role of supported heteropolyacid derivatives and acidic strength[END_REF][START_REF] Zarchi | Synthesis of [P 4-VP]-Fe 3 O 4 supported Brønsted acid ionic liquid and its application as a highly efficient heterogeneous and reusable nanocatalyst in the Biginelli reaction under solvent-free conditions[END_REF] boric acid [START_REF] Meshram | Boric acid promoted an efficient and practical synthesis of fused pyrimidines in aqueous media[END_REF] or α-zirconium sulfophenylphosphonate. [START_REF] Rosati | α-zirconium sulfophenylphosphonate as a catalyst for the synthesis of 3, 4-dihydropyrimidin-2 (1H)-one derivatives under solvent free conditions[END_REF] Despite undeniable improvements, all these protocols still present drawbacks such as extended reaction times, high temperatures, use of hazardous chemicals or toxic solvents, high loadings of expensive heavy metal catalysts, or displaying low product yields. [START_REF] Kappe | 100 years of the biginelli dihydropyrimidine synthesis[END_REF] One the other hand, sequential multistep strategies have been proposed to produce DHPM compounds with somewhat higher overall yields but lack the attractiveness of the one-pot procedure. [START_REF] Bc | Synthesis of substituted 1, 2, 3, 4-tetrahydro-6-methyl-2-oxo-5pyrimidinecarboxylic acid esters: The Biginelli condensation revisited[END_REF] In this context, we found appealing to develop new conditions for the Biginelli reaction that overcome these drawbacks and will be closer to the precepts of green synthesis. Phosphorus derivatives particularly attracted our attention as they are widely available from phosphate, a natural resource almost inexhaustible (world resources of phosphate rock are more than 300 billion tons, with no imminent shortages according to USGS 2020). [START_REF] Bernhardt | Mineral commodity summaries[END_REF] Moreover, they have been recently proposed as efficient catalysts in several organic chemical processes such as the asymmetric hydrosilylation of alkenes, [START_REF] Feng | Application of Pd-Monodentate Phosphorus Catalysts in the Asymmetric Hydrosilylation Reactions of Alkenes[END_REF] or the production of 5hydroxymethylfurfural. [START_REF] Souleymanou | Production of 5-hydroxymethylfurfural in the presence of organic phosphorus catalysts[END_REF] Therefore, in continuation of our studies directed at the development of green processes, [START_REF] Tachallait | A straightforward and versatile FeCl 3 catalyzed Friedel-Crafts C-glycosylation process. Application to the synthesis of new functionalized C-nucleosides[END_REF][START_REF] Marzag | FeCl3-promoted and ultrasoundassisted synthesis of resveratrol O-derived glycoside analogs[END_REF][START_REF] Malnuit | Tandem azide-alkyne 1, 3-dipolar cycloaddition/electrophilic addition: A concise three-component route to 4, 5-disubstituted triazolylnucleosides[END_REF][START_REF] Alaoui | Ultrasound-assisted facile one-pot sequential synthesis of novel sulfonamide-isoxazoles using cerium (IV) ammonium nitrate (CAN) as an efficient oxidant in aqueous medium[END_REF][START_REF] Spadafora | Friedel-Crafts and modified Vorbrüggen ribosylation. A short synthesis of aryl and heteroaryl-C-nucleosides[END_REF][START_REF] Botubol-Ares | Methylene-Linked Bis-NHC Half-Sandwich Ruthenium Complexes: Binding of Small Molecules and Catalysis toward Ketone Transfer Hydrogenation[END_REF] we report here the screening of phosphorus derivatives for the catalysis of Biginelli reaction and their use to produce a panel of DHPM and related heterocycles under eco-friendly conditions.

Results and discussion

Our interest first focused on common readily accessible phosphorus derivatives to test the catalytic activity for the Biginelli reaction. These derivatives were selected for their availability, their safety and eco-friendly features.

They encompass synthesized organic derivatives, mineral acids, natural ores and recycled industrial waste (Table 1).

The catalytic activity was assayed on a model Biginelli reaction with benzaldehyde, ethyl acetoacetate and urea as substrates (Table 1). The catalyst loading was first set up at 20 mol% to evaluate the reactivity of various phosphorus derivates, compared to most efficient procedures described in the literature (Table 1, entries 1-7).

Hypophosphorous acid was found to be the best reagent to promote this three-component reaction with 99% conversion after 6h in ethanol at reflux, and was therefore selected for further optimization. Decreasing the reaction time from 6h to 4h still retained high conversion (97%) whereas decreasing the amount of catalyst led to lower yields (Table 1, entries 8-10). Thus, the ratio of hypophosphorous acid was maintained at 20 mol% and several solvents and temperatures were assayed. Aqueous conditions proved clearely detrimental because of solubility issues. Acetonitrile or solvent-free conditions gave interesting results, even if they presented lower conversions than ethanol ( From a mechanistic point of view, the urea derivative reacts with the aldehyde to form an hemiaminal product.

Hypophosphorous acid (pKa 1.2) is believed to protonate this hemiaminal, promoting its dehydration to an iminium intermediate, which subsequently reacts with ethyl acetoacetate. Final cyclodehydration provides the desired DHPM derivative (see below Figure 2). [START_REF] Nagarajaiah | Biginelli reaction: an overview[END_REF][START_REF] Kappe | A reexamination of the mechanism of the Biginelli dihydropyrimidine synthesis. Support for an N-Acyliminium ion intermediate1[END_REF] With these optimized conditions in hand (heating in ethanol at 80°C for 4h in the presence of 20% of hypophosphorous acid), the scope of the protocol was studied by reacting various (hetero)aromatic/aliphatic aldehydes, diketones and (thio)urea derivatives (Table 2). All compounds were recovered by simple filtration and washing after ice-cooling of the reaction mixture and were purified by recrystallization. Under these conditions, model dihydropyrimidinone 1a was obtained with 96% yield. 2, entries 6-12). A benzaldehyde derivative with an electron-accepting nitro or cyano group on the ring showed also a good reactivity to afford the products 1m-o in good isolated yields ranging from 82%-97% (Table2, entries 13-15). Benzaldehyde derivatives with halogen atoms at different positions on the aromatic ring (4-fluoro, 2,6-dichloro, and 3-bromo) also underwent reaction to form the corresponding products (1p-r) in good isolated yields that ranged from 91-94% (Table2, entries 16-18). Heteroaromatic aldehydes, such as 3-pyridinealdehyde, 2-furylaldehyde, thiophene-2-aldehyde and 4-bromo-thiophene-2-aldehyde led to the corresponding products with the same efficiency 78%-95% (Table 2, entries 19-24). Remarkably, the aliphatic phenethylaldehyde were also converted to their DHPM products with 92% yield (Table 2, entries 25).

H 3 C R 2 O O N H NH 2 X HN NH X R 1 O R 2 H 3 C 20% H 3 PO 2 R 1 H O + + R 3 Ethanol, 80°C Entry R1 R2 R3 X Product b Yield c (%)
In summary, the developed protocol proved robust upon variation of the substituents on the benzaldehyde both in terms of electronic nature and position. Moreover, these optimized reaction conditions are suitable for heteroaromatic aldehydes as well as for aliphatic phenethylaldehyde that reacted with the same efficiency. The scope of the reaction proved also broad towards replacement of β-ketoester by 1,3-diketone or using substituted urea and thiourea derivatives. In all cases, high yields were achieved after a straightforward isolation and purification procedure. All the synthesized compounds were fully characterized (see experimental part) and showed analytical data in accordance with the literature, except for 5, 20 and 23 that represent new compounds.

To extend the scope of the present methodology, the synthesis of fused-dihydropyrimidinones and more complex heterocycles was attempted by extrapolating the urea reagent. We first tried the reaction with 2aminobenzimidazole in view to obtain a 1,4-dihydrobenzo [4,5] 61% to 92% (Table 3). To further explore the efficiency of hypophosphorous acid to produce more complex scaffolds via the Biginelli reaction, the conditions developed were applied to other urea derivatives: 2-aminobenzothiazole, N 3 -phenyl-4H-1,2,4-triazole-3,5-diamine and N-amidinothiourea (Table 4). The reaction proceeded smoothly with 2aminobenzothiazole furnishing, after cyclocondensation, the desired 4H-benzo [4,5]thiazolo[3,2-a]pyrimidine derivative 3 with 84% yield. Interestingly, the obtained product 3 showed a tautomeric dearomatisation behavior on the benzothiazole core, as already reported for 2-aminobenzothiazole derivatives linked to electron-withdrawing groups. [START_REF] Baladi | Sulfonylguanidine Derivatives as Potential Antimelanoma Agents[END_REF] The reaction also proceeded with N 3 -phenyl-4H-1,2,4-triazole-3,5-diamine to provide 5,8dihydro- [1,2,4]triazolo [4,3-a]pyrimidine analog 4 with 43% yield. Finally, these conditions were attempted with N-amidinothiourea as substrate to study the regioselectivity of the cyclocondensation. Indeed, two reaction products 6 and 6' can be possibly formed (Figure 2). A full selectivity was obtained towards the cyclisation via the thiourea part, relative to the guanidine part. However, a second benzaldehyde condensation occurred on the terminal guanidine nitrogen of 6, leading to the corresponding imine derivate 5 in 72% yield. Beyond the establishment of efficient conditions based on the use of readily available hypophosphorous acid, the developed methodology is particularly interesting to open access to complex heterocycles with biologically relevant structures. Indeed, the 1,4-dihydrobenzo [4,5]imidazo[1,2-a]pyrimidine tricyclic scaffold presents a benzimidazole condensed ring system that has shown interesting anticancer activities against a panel of cancer cell lines and the 4H-benzo [4,5]thiazolo [3,2-a]pyrimidine core has demonstrated efficiency in antibacterial and antimycobacterial models. [START_REF] Bhoi | Synthesis, biological evaluation and computational study of novel isoniazid containing 4H-Pyrimido [2, 1-b] benzothiazoles derivatives[END_REF] Moreover, the new scaffolds 4 and 5 possess very interesting structural features to be evaluated for potential future applications in anticancer, inflammation and antibacterial models.

In summary, we developed an efficient hypophosphorous acid-based protocol for the Biginelli reaction,

showing effectivity improvements compared to the existing methods, and advantages in the context of green chemistry as it is catalyzed by a readily available simple chemical: H3PO2, and does not require the use of toxic metals or solvents. Moreover, it displays a broad scope in terms of substrates, exemplified through the synthesis of 33 compounds DHPM, thio derivatives, as well as new bi-and tricyclic structures such as benzo [4,5]imidazo[1,2-a]-pyrimidine, 4H-pyrimido[2,1-b]benzothiazole and 5,8-dihydro- [1,2,4]triazolo [4,3a]pyrimidine. Overall, the methodology presented allows a rapid, versatile and ecofriendly access to a great diversity of functionalized DHPMs and related structures with potential biological interest.

Figure 1 :

 1 Figure 1: Examples of pharmacologically active DHPMs.

a

  Reaction conditions: aldehyde (1 mmol), β-dicarbonyl derivative (1.2 mmol), urea derivative (1.2 mmol) for 6h at 80°C, b : All products were characterized by 1 H NMR, 13 C NMR and LC-MS compared with those reported in the literature, c : Isolated yields. Derivatives bearing electron-donating groups, such as 4-dimehylamino, 4-methoxy, 4-hydroxy, and 3-hydroxy, respectively, at different positions on the ring reacted in efficient way with both ethyl acetoacetate and acetylacetone as R2 and (thio)urea R3 to produce the corresponding DHPMs 1f-l in excellent isolated yields (92-98%) (Table

  imidazo[1,2-a]pyrimidine tricyclic scaffold. The reaction could occur by condensation of 2-aminobenzimidazole and the aldehyde under hypophosphorous acid catalysis. Subsequent reaction with a 1,3-dicarbonyl compound and cyclodehydration should provide the desired heterocycle. The reaction was performed with five different (hetero)aromatic aldehydes allowing the formation of the corresponding 1,4-dihydrobenzo[4,5]imidazo[1,2-a]pyrimidines 2a-2e with satisfactory yields, ranging from

Table 3 :

 3 Synthesis of 1,4-dihydrobenzo[4,5]imidazo[1,2-a]pyrimidines 2a-2e by Biginelli type tricomponent reaction in the presence of hypophosphorous acid. products were characterized by 1 H NMR,[START_REF] Kappe | Microwave-assisted high-speed parallel synthesis of 4-aryl-3, 4dihydropyrimidin-2 (1H)-ones using a solventless Biginelli condensation protocol[END_REF] C NMR and HR-MS / LC-MS, b : Isolated yields.

Table 4 :

 4 Synthesis of complex heterocyclic systems by Biginelli-like reaction using urea analogues products were characterized by 1 H NMR,[START_REF] Kappe | Microwave-assisted high-speed parallel synthesis of 4-aryl-3, 4dihydropyrimidin-2 (1H)-ones using a solventless Biginelli condensation protocol[END_REF] C NMR and HR-MS / LC-MS, b : Isolated yields.

Figure 2 :

 2 Figure 2: Reaction mechanism and possible regioisomer products for the three-component reaction with Namidinothiourea

Table 1

 1

, entries 11-14). Finally, decreasing the temperature to 50°C or 20°C led to a significant drop in the reaction rate (Table 1, entries 15-16).

Table 1 :

 1 Screening and optimization of reaction conditions for the synthesis of dihydropyrimidinone 1a

	Entry	Catalyst	Amount of catalyst	Solvent	Time	Temperature (°C)	Yields a (%)
	Reference 1 35	H3[PW12O40]	0.1g	ethanol	5h		95
	Reference 2 23	α-zirconium sulfophenylphosphonate	12%	---	18h		89
	1	---	---	ethanol	8h		13
	2	phosphoric acid	20%	ethanol	6h		91
		N-
	3	(phosphonomethyl)iminodiacetic	20%	ethanol	6h		80
		acid hydrate
	4	2-ethylhexylphosphate b	20%	ethanol	6h		96
	5	hypophosphorous acid	20%	ethanol	4h, 6h		97, 99
	6	phosphogypsum c	20%	ethanol	6h		12
	7	apatite d	20%	ethanol	6h		3
	8	hypophosphorous acid	10%	ethanol	6h		81
	9	hypophosphorous acid	5%	ethanol	6h		74
	10	hypophosphorous acid	1%	ethanol	6h		60
	11	hypophosphorous acid	20%	water	6h		28
	12	hypophosphorous acid	20%	ethanol/water, 50/50	6h		44
	13	hypophosphorous acid	20%	acetonitrile	6h		93
	14	hypophosphorous acid	20%	solvent-free	6h		85
	15	hypophosphorous acid	20%	ethanol	6h		32
	16	hypophosphorous acid	20%	ethanol	6h		77

a : Conversion based on LCMS analyses at 280 nm. b : 2-ethylhexylphosphate (MPE) was synthesized by a procedure adapted from the literature. 36 c : byproduct formed after the attack of phosphate rock with sulfuric acid (ore transformed in Jorf Al Asfar plant, Morocco). d : Natural phosphate from an ore extracted in Khouribga region (Morocco)

Table 2 :

 2 Scope and limitations of the proposed conditions. a Synthesis of 3,4-dihydropyrimidin-2(1H)-(thi)ones

	1a-y

Acknowledgements

The authors acknowledge Mohammed VI Polytechnique University (Morocco), for PhD grant to ZM, Université Côte d'Azur (IDEX UCA-JEDI innovation program) and CNRS for funding support.

Keywords: Phosphate derivatives catalysis, Biginelli reaction, dihydropyrimidines, one-pot protocol, fused heterocycles

Declaration of interests

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.