come

Unique continuation for many-body Schrodinger

operators and the Hohenberg-Kohn theorem

UNIQUE CONTINUATION FOR MANY-BODY SCHR ÖDINGER OPERATORS AND THE HOHENBERG-KOHN THEOREM*

LOUIS GARRIGUE

Abstract. We prove the strong unique continuation property for manybody Schrödinger operators with an external potential and an interaction potential both in L p loc (R d ), where p > max(2d/3, 2), independently of the number of particles. With the same assumptions, we obtain the Hohenberg-Kohn theorem, which is one of the most fundamental results in Density Functional Theory.

Density Functional Theory (DFT) is one of the most successful methods in quantum physics and chemistry to simulate matter at the microscopic scale [START_REF] Cancès | Computational quantum chemistry: a primer[END_REF][START_REF] Dreizler | Density functional theory[END_REF][START_REF] Engel | Density Functional Theory: An Advanced Course[END_REF]. It is a very active field of research, applied to very diverse physical situations, going from atoms and small molecules to condensed matter systems [START_REF] Jones | Density functional theory: Its origins, rise to prominence, and future[END_REF].

One of the basis of DFT is due to Hohenberg and Kohn in 1964 [START_REF] Hohenberg | Inhomogeneous electron gas[END_REF], who showed that in equilibrium, the knowledge of the ground state density alone is sufficient to characterize the system. In other words, all the information of a quantum system is contained in its ground state one-particle density. The Hohenberg-Kohn theorem was precised by Lieb in [START_REF] Lieb | Density functionals for Coulomb systems[END_REF], who emphasized that it relies on a unique continuation property (UCP) for the many-particle Hamiltonian.

A typical (strong) unique continuation result [START_REF] Tataru | Unique continuation problems for partial differential equations[END_REF] is that if a wavefunction Ψ vanishes sufficiently fast at one point and solves Schrödinger's equation HΨ = 0, then Ψ = 0. Unique continuation properties began to be developped by Carleman in [START_REF] Carleman | Sur un problème d'unicité pour les systèmes d'équations aux dérivées partielles à deux variables indépendantes[END_REF] and, today, a broad range of results exists when the operator is H = -∆ + V (x), with V in some L p loc space. A famous result of Jerison and Kenig [START_REF] Jerison | Unique continuation and absence of positive eigenvalues for Schrödinger operators[END_REF] covers the case p = d/2 in dimension d. It was later improved by Koch and Tataru in [START_REF] Koch | Carleman estimates and unique continuation for secondorder elliptic equations with nonsmooth coefficients[END_REF].

Unfortunately, these results are not well adapted to the situation of Schrödinger operators describing N particles, which are defined on R dN . In order to apply the existing results, one would need assumptions on the potentials depending on N . To the best of our knowledge, two works, due to Georgescu [START_REF] Georgescu | On the unique continuation property for Schrödinger Hamiltonians[END_REF] and Schechter-Simon [START_REF] Schechter | Unique continuation for Schrödinger operators with unbounded potentials[END_REF], provide a unique continuation property for many-particle Hamiltonians with an assumption on the potentials independent of N . However, they require the wavefunction to vanish on an open set (weak UCP), and for the Hohenberg-Kohn theorem strong UCP is needed.

Recently, Laestadius, Benedicks and Penz [START_REF] Laestadius | Unique Continuation for the Magnetic Schrödinger Equation[END_REF] have proved the first strong UCP result for many-body operators using ideas of Kurata [START_REF] Kurata | A unique continuation theorem for the Schrödinger equation with singular magnetic field[END_REF] and Regbaoui [START_REF] Regbaoui | Unique continuation from sets of positive measure[END_REF], but they need extra assumptions on the negative part of 2V + x • ∇V , which naturally appears in the Virial identity. In [START_REF]A mathematical aspect of Hohenberg-Kohn theorem[END_REF], Zhou used the result of Schechter and Simon to state a weak form of the Hohenberg-Kohn theorem, but this was already implicit in the work of Lieb [START_REF] Lieb | Density functionals for Coulomb systems[END_REF]. We refer to [START_REF] Cancès | Computational quantum chemistry: a primer[END_REF][START_REF] Englisch | Hohenberg-Kohn theorem and non-V-representable densities[END_REF][START_REF] Kryachko | On the original proof by reductio ad absurdum of the Hohenberg-Kohn theorem for many-electron coulomb systems[END_REF][START_REF] Lammert | In search of the Hohenberg-Kohn theorem[END_REF][START_REF] Levy | On the simple constrained-search reformulation of the Hohenberg-Kohn theorem to include degeneracies and more (1964-1979[END_REF][START_REF] Lieb | Density functionals for Coulomb systems[END_REF][START_REF] Pino | A re-statement of the Hohenberg-Kohn theorem and its extension to finite subspaces[END_REF][START_REF] Zhou | Hohenberg-Kohn theorem for Coulomb type systems and its generalization[END_REF] for a discussion on the importance of the unique continuation principle for the Hohenberg-Kohn theorem.

In this article, we provide the first strong UCP for many-body operators in L p spaces and deduce the first complete proof of the Hohenberg-Kohn theorem in these spaces. Our proof mainly uses the method of Georgescu [START_REF] Georgescu | On the unique continuation property for Schrödinger Hamiltonians[END_REF], together with a Carleman estimate proved in [START_REF] Garrigue | Unique continuation for many-body Schrödinger operators and the Hohenberg-Kohn theorem. II. The Pauli Hamiltonian[END_REF]. We also use ideas from Figueiredo-Gossez [START_REF] De Figueiredo | Strict monotonicity of eigenvalues and unique continuation[END_REF] to pass from the vanishing of Ψ on a set of positive measure to the vanishing to infinite order at one point. In short, we can handle any number N of particles living in R d , with potentials in L p loc (R d ) with p > max(2d/3, 2). We deduce the Hohenberg-Kohn theorem with similar assumptions.

Acknowledgement. I warmly thank Mathieu Lewin, my PhD advisor, for having supervised me during this work. This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement MDFT No 725528).

Main results

1.1. Strong unique continuation property. We denote by B R the ball of radius R centered at the origin. Our main result is the following.

Theorem 1.1 (Strong UCP). Let V ∈ L 2 loc (R n
) such that for some δ > 0 and for every R > 0, there exists c R 0 such that for any u ∈ H 2 (R n ),

ˆBR |V | 2 |u| 2 ǫ δ,n ˆRn (-∆) 3 4 -δ u 2 + c R ˆRn |u| 2 , (1)
where ǫ δ,n is a constant depending only on δ and on the dimension n. Let Ψ ∈ H 2 loc (R n ) be a solution to -∆Ψ + V Ψ = 0. If Ψ vanishes on a set of positive measure or if it vanishes to infinite order at a point, then Ψ = 0.

The constant ǫ δ,n depends on the best constant of the Carleman inequality (8) which we are going to use later. We recall that Ψ vanishes to infinite order at x 0 ∈ R n when for all k 1, there is a c k such that

ˆ|x-x 0 |<ǫ |Ψ| 2 dx < c k ǫ k , for every ǫ < 1.
The assumption (1) can be rewritten in the sense of operators, in the form

|V | 2 1 B R ǫ δ,n (-∆) 3 2 -2δ + c R . This is satified if V ∈ L p loc (R n ) with p > max(2n/3, 2) (see the proof of Corollary 1.
2). However, the condition (1) has a better behavior with respect to the dimension than a condition in L p spaces. It is more appropriate to deal with N -body operators for which n = dN , as we will see. We denote by B R (x) the ball of radius R and centered on x ∈ R n . Assumption ( 1) is equivalent to saying that for any x ∈ R n , there exists c x such that

|V | 2 1 B 1 (x) ǫ ′ δ,n (-∆) 3 2 -2δ + c x in R n . (2)
Indeed (2) follows from (1) by taking R = |x| + 1 whereas the converse statement can be obtained by (fractional) localization, e.g. as in [START_REF] Lenzmann | Minimizers for the Hartree-Fock-Bogoliubov theory of neutron stars and white dwarfs[END_REF]Lemma A.1]. We have stated our main result in R n for simplicity, but there is a similar statement in a connected domain Ω. One should then replace ( 1) by ( 2) with small balls B R (x) ⊂ Ω. Our proof is really local in space. Following Simon in [29, section C.9], we conjecture that Theorem 1.1 holds under the weaker condition

|V | 1 B R ǫ n (-∆) + c R , with Ψ in H 1 loc (R n ).
A weak UCP was proved by Schechter and Simon in [28] using estimates from Protter [START_REF] Protter | Unique continuation for elliptic equations[END_REF], but with the stronger hypothesis

|V | 2 1 B R ǫ n (-∆) + c R .
Our Theorem 1.1 improves the weak UCP of Georgescu in [START_REF] Georgescu | On the unique continuation property for Schrödinger Hamiltonians[END_REF], which has an assumption similar to [START_REF] Cancès | Computational quantum chemistry: a primer[END_REF]. He used the estimate from Theorem 8.3.1 of [START_REF] Hörmander | Linear partial differential operators[END_REF], due to Hörmander and we instead use a new Carleman estimate that we proved in [START_REF] Garrigue | Unique continuation for many-body Schrödinger operators and the Hohenberg-Kohn theorem. II. The Pauli Hamiltonian[END_REF].

1.2. Application to N -body operators. We consider N particles in R d , submitted to an external potential v and interacting with an even two-body potential w. The corresponding N -body Hamiltonian takes the form

H N (v) = - N i=1 ∆ x i + N i=1 v(x i ) + 1 i<j N w(x i -x j ), (3)
on L 2 (R dN ). In order to ensure that the total potential

V (x 1 , . . . , x N ) := N i=1 v(x i ) + 1 i<j N w(x i -x j ), (4)
satisfies the assumption (1) in R dN , it is sufficient that v and w satisfy (1) in R d , but with an ǫ that can be taken as small as we want.

Corollary 1.2 (UCP for many-body Schrödinger operators). Assume that the potentials satisfy

|v| 2 1 B R + |w| 2 1 B R ǫ δ,d,N (-∆) 3 2 -2δ + c R in R d , (5)
for some δ > 0 and for all R > 0, where ǫ δ,d,N is a small constant depending only on δ, d and N . For instance v, w ∈ L p loc (R d ) with p > max(2d/3). Let Ψ ∈ H 2 loc (R dN ) be a solution to H N (v)Ψ = 0. If Ψ vanishes on a set of positive measure or if it vanishes to infinite order at a point, then Ψ = 0. 1.3. Hohenberg-Kohn theorem. The one-particle density of a wavefunction Ψ is defined as

ρ Ψ (x) := N i=1 ˆ|Ψ(x 1 , ..., x i-1 , x, x i+1 , ..., x N )| 2 dx 1 • • • dx i-1 dx i+1 • • • dx N .
From Corollary 1.2, we can deduce the following version of the Hohenberg-Kohn theorem.

Theorem 1.3 (Hohenberg-Kohn). Let w, v 1 , v 2 ∈ (L p + L ∞ )(R d , R), with p > max(2d/3
). If there are two normalised eigenfunctions Ψ 1 and Ψ 2 of H N (v 1 ) and H N (v 2 ), corresponding to the first eigenvalues, and such that ρ Ψ 1 = ρ Ψ 2 , then there exists a constant c such that

v 1 = v 2 + c.
The exact same theorem is valid if we take spin into account and assume that Ψ 1 , Ψ 2 are the first eigenfunctions of H N (v 1 ), H N (v 2 ) in any subspace invariant by the two operators. In particular the theorem applies to bosons and fermions. Our result covers the physical case of Coulomb potentials as in [START_REF] Zhou | Hohenberg-Kohn theorem for Coulomb type systems and its generalization[END_REF]. However, in this situation, eigenfunctions are real analytic on an open set of full measure, and the argument is much easier.

We recall the proof from [START_REF] Hohenberg | Inhomogeneous electron gas[END_REF][START_REF] Lieb | Density functionals for Coulomb systems[END_REF] for the convenience of the reader.

Proof. We denote by ρ := ρ Ψ 1 = ρ Ψ 2 the common density. Since Ψ 1 is the ground state for v 1 , then

E 1 := Ψ 1 , H N (v 1 )Ψ 1 Ψ 2 , H N (v 1 )Ψ 2 = Ψ 2 , H N (v 2 )Ψ 2 + ˆRd ρ(v 1 -v 2 ).
We also have

E 2 := Ψ 2 , H N (v 2 )Ψ 2 Ψ 1 , H N (v 1 )Ψ 1 + ˆRd ρ(v 2 -v 1 )
.

Hence E 1 -E 2 = ´Rd ρ(v 1 -v 2 ) and Ψ 2 , H N (v 1 )Ψ 2 = E 1 , so Ψ 2 is a ground state for H N (v 1 )
, and

H N (v 1 )Ψ 2 = E 1 Ψ 2 . Together with H N (v 2 )Ψ 2 = E 2 Ψ 2 ,
this gives

E 1 -E 2 + N i=1 (v 2 -v 1 )(x i ) Ψ 2 = 0.
Since, by Corollary 1.2, the normalised function Ψ 2 cannot vanish on a set of positive measure, we get

E 1 -E 2 + N i=1 (v 2 -v 1 )(x i ) = 0
almost everywhere. Integrating this relation over x 2 , . . . , x N in a bounded domain we conclude, as wanted, that v 1 -v 2 = c. Eventually, using again

E 1 -E 2 = ´Rd ρ(v 1 -v 2 ) yields c = (E 1 -E 2 )/N .
The rest of the paper is devoted to the proof of our main results.

Proof of Theorem 1.1

Step 1. Vanishing on a set of positive measure implies vanishing to infinite order at one point. We will need to pass from Ψ vanishing on a set of positive measure, which is the needed hypothesis for the Hohenberg-Kohn theorem, to Ψ vanishing to infinite order at a point, which is the usual hypothesis for strong unique continuation. We reformulate here Proposition 3 of [START_REF] De Figueiredo | Strict monotonicity of eigenvalues and unique continuation[END_REF] with slightly weaker assumptions.

Proposition 2.1 (Figueiredo-Gossez [3]). Let V ∈ L 1 loc (R n ) such that for every R > 0, there exist a R < 1 and c R > 0 such that |V | 1 B R a R (-∆) + c R . (6)
If Ψ ∈ H 1 loc (R n ) vanishes on a set of positive measure and if -∆Ψ + V Ψ = 0 weakly, then Ψ has a zero of infinite order.

The proof is written in [START_REF] De Figueiredo | Strict monotonicity of eigenvalues and unique continuation[END_REF] under the assumption that V ∈ L n/2 loc (R n ) but after inspection, one realises that it only relies on [START_REF] Englisch | Hohenberg-Kohn theorem and non-V-representable densities[END_REF]. We remark that our assumption (1) is stronger than [START_REF] Englisch | Hohenberg-Kohn theorem and non-V-representable densities[END_REF]. This is because the square root is operator monotone, and therefore

|V | 1 B R ǫ n (-∆) 3 2 -2δ + c R ǫ 2 3 n (-∆) + c ′ R .
For this reason, we will assume for the rest of the proof that Ψ vanishes to infinite order at one point, which can be taken to be the origin without loss of generality.

Step 2. ∇Ψ and ∆Ψ vanish to infinite order as well. First we remark that if Ψ ∈ L 2 (R n ), then vanishing to infinite order at the origin is equivalent to ´B1 |x| -τ |Ψ| 2 dx being finite for every τ 0. Indeed, if Ψ vanishes to infinite order at the origin, that is ´Bǫ |Ψ| 2 c k ǫ k , then we get, after integrating over ǫ,

c k ˆ1 0 ´Bǫ |Ψ| 2 ǫ k dǫ = ˆB1 ˆ1 0 |Ψ(x)| 2 1 |x| ǫ ǫ k dǫ dx = 1 k -1 ˆB1 |Ψ(x)| 2 1 |x| k-1 -1 dx. Conversely, if ´|x| 1 |x| -τ |Ψ| 2 is finite for every τ 0, then ǫ -k ˆBǫ |Ψ| 2 ˆBǫ |x| -k |Ψ(x)| 2 dx ˆB1 |x| -k |Ψ(x)| 2 dx.
The finiteness of these integrals will play an important role later.

Lemma 2.2 (Finiteness of weighted norms)

. i) Let V ∈ L 1 loc (R n ) such that there exist a < 1 and c > 0 such that |V | 1 B 1 a(-∆) + c.
Let Ψ ∈ H 1 loc (R n ) satisfying -∆Ψ+V Ψ = 0 weakly. If Ψ vanishes to infinite order at the origin, then ∇Ψ as well.

ii) Let V ∈ L 2 loc (R n ) such that there exist a < 1 and c > 0 such that

|V | 2 1 B 1 a(-∆) 2 + c. Let Ψ ∈ H 2 loc (R n ) satisfying -∆Ψ + V Ψ = 0.
If Ψ vanishes to infinite order at the origin, then ∇Ψ and ∆Ψ as well.

Proof. i) We take ǫ ∈ (0, 1/2] and define a smooth localisation function η with support in B 2ǫ , equal to 1 in B ǫ , and such that |∇η| c/ǫ and |∆η| c/ǫ 2 . Multiplying the equation by η 2 Ψ and taking the real parts yields

-Re ˆV |ηΨ| 2 = -Re ˆΨη 2 ∆Ψ = Re ˆ∇Ψ • ∇ η 2 Ψ = ˆ|η∇Ψ| 2 + Re ˆΨ∇Ψ • ∇η 2 = ˆ|η∇Ψ| 2 + 1 2 ˆ∇ |Ψ| 2 • ∇η 2 = ˆ|η∇Ψ| 2 - 1 2 ˆ|Ψ| 2 ∆η 2 .
So by the assumption on V ,

ˆ|η∇Ψ| 2 a ˆ|∇ (ηΨ)| 2 + c ˆ|ηΨ| 2 + 1 2 ˆ|Ψ| 2 ∆η 2 = a ˆ|η∇Ψ| 2 + a ˆ|Ψ∇η| 2 + 1 -a 2 ˆ|Ψ| 2 ∆η 2 + c ˆ|ηΨ| 2 .
So, since a < 1, we get

ˆBǫ |∇Ψ| 2 ˆ|η∇Ψ| 2 c a ǫ -2 ˆB2ǫ |Ψ| 2 < c a c k 2 k ǫ k-2 ,
for any k 0, where we used that Ψ vanishes to infinite order. This proves the result.

ii) By i), we know that ∇Ψ vanishes to infinite order at the origin. We take the same function η as in i) and use the Schrödinger equation pointwise to get

ˆ|η∆Ψ| 2 = ˆ|V ηΨ| 2 a ˆ|∆ (ηΨ)| 2 + c ˆ|ηΨ| 2 a(1 + α) ˆ|η∆Ψ| 2 + 2 1 + 1 α ˆ|Ψ∆η| 2 + 4 1 + 1 α ˆ|∇Ψ • ∇η| 2 + c ˆ|ηΨ| 2 ,
for any α > 0. We take α such that a(1 + α) < 1 and thus

ˆBǫ |∆Ψ| 2 ˆ|η∆Ψ| 2 c a ǫ -4 ˆB2ǫ |Ψ| 2 + |∇Ψ| 2 < c a c k 2 k ǫ k-4 ,
which proves the result.

Step 3. Carleman estimate. One common tool for unique continuation results is the Carleman estimate. We use here a new Carleman inequality with singular weights which we have recently proved in [8, Corollary 1.2]. We define the weight function

φ(x) := -ln |x| + (-ln |x|) -1/2 . (7)
In dimension n, for any ξ > 0 there exist constants κ ξ,n and τ 0 1 such that for any s ∈ [0, 1], any τ τ 0 and any u

∈ C ∞ c (B 1 \ {0} , C), we have 1 τ 3 2 -2s (-∆) (1-ξ)s e τ φ u L 2 (R n ) κ ξ,n e τ φ ∆u L 2 (B 1 ) . (8)
Step 4. Proof that Ψ = 0. We consider some number τ 0 (large), and we call c any constant that does not depend on τ . We take a smooth localisation function η, equal to 1 in B 1/2 ⊂ R n , supported in B 1 , and such that 0 η 1. We take the weight function φ as in [START_REF] Garrigue | Unique continuation for many-body Schrödinger operators and the Hohenberg-Kohn theorem. II. The Pauli Hamiltonian[END_REF]. It verifies e φ(x) + |∇φ| c |x| -1 and |∆φ| c |x| -2 for c sufficiently large.

In step 1, we have shown that Ψ vanishes to infinite order at the origin and in step 2 we have deduced the same property for ∇Ψ and ∆Ψ. Moreover,

ˆB1 |Ψ(x)| 2 |x| τ dx + ˆB1 |∇Ψ(x)| 2 |x| τ dx + ˆB1 |∆Ψ(x)| 2 |x| τ dx < +∞,
for all τ 0. All the integrals with e τ φ are finite as well and the following calculations are valid. In addition, from the Carleman inequality (8), we know that e τ φ Ψ belongs to H 3/2 loc (R n ) for all τ . By the assumption (1) on V , we have

e τ φ V ηΨ L 2 (B 1 ) √ ǫ δ,n (-∆) 3 4 -δ e τ φ ηΨ L 2 (R n ) + c e τ φ ηΨ L 2 (B 1 )
.

Applying the Carleman estimate ( 8) with s = 3/4 and ξ = 4δ/3, we get

(-∆) 3 4 -δ e τ φ ηΨ L 2 (R n ) κ 4δ/3,n e τ φ ∆(ηΨ) L 2 (B 1 )
, and hence e τ φ V ηΨ

L 2 (B 1 )
κ 4δ/3,n √ ǫ δ,n e τ φ ∆(ηΨ)

L 2 (B 1 )
+ c e τ φ ηΨ

L 2 (B 1 )
.

Now we estimate

e τ φ ∆(ηΨ)

L 2 (B 1 )
e τ φ η∆Ψ

L 2 (B 1 )
+ 2 e τ φ ∇η • ∇Ψ

L 2 (B 1 ) + e τ φ Ψ∆η L 2 (B 1 )
e τ φ V ηΨ

L 2 (B 1 )
+ c e τ φ ∇Ψ

L 2 (B 1 \B 1/2 ) + c e τ φ Ψ L 2 (B 1 \B 1/2 ) κ 4δ/3,n √ ǫ δ,n e τ φ ∆ (ηΨ) L 2 (B 1 )
+ c e τ φ ηΨ

L 2 (B 1 )
+ ce τ φ( 1 2 ) .

We take ǫ δ,n = 1 4κ 2 4δ/3,n , and move the term e τ φ ∆(ηΨ) L 2 (B 1 ) to the left side of the inequality, which yields e τ φ ∆(ηΨ)

L 2 (B 1 ) c e τ φ ηΨ L 2 (B 1 )
+ ce τ φ( 1 2 ) . 1 In the published version of the present article [START_REF] Garrigue | Unique continuation for many-body Schrödinger operators and the Hohenberg-Kohn theorem[END_REF], we used another Carleman estimate with the different weight φ(x) = -ln |x| + λ |x| 2 , taken from [START_REF] Tataru | Unique continuation problems for partial differential equations[END_REF]Theorem 12]. After publication, we realized that the estimate on ∆ e τ φ u was not known for this weight.

The estimate with the weight [START_REF] Garrigue | Unique continuation for many-body Schrödinger operators and the Hohenberg-Kohn theorem[END_REF], proved in [START_REF] Garrigue | Unique continuation for many-body Schrödinger operators and the Hohenberg-Kohn theorem. II. The Pauli Hamiltonian[END_REF], is of remedies.

But by the Carleman inequality (8) applied with s = 0, we have

e τ φ ηΨ L 2 (B 1 ) cτ -3 2 e τ φ ∆(ηΨ) L 2 (B 1 )
, so for τ big enough so that cτ -3 2 < 1/2, we find

||ηΨ|| L 2 (B 1/2 ) e τ (φ(•)-φ( 1 2 )) ηΨ L 2 (B 1/2 ) cτ -3 2 .
Eventually, letting τ → +∞, we get Ψ = 0 almost everywhere in B 1/2 . We can then propagate this small region B 1/2 , where Ψ vanishes, to the whole space, as explained for instance in [START_REF] Reed | Methods of Modern Mathematical Physics. IV. Analysis of operators[END_REF].

Proof of Corollary 1.2

We take n = dN . Let R > 0 and Ψ ∈ H 3/2 (R dN ). We apply the inequality (5) to the function x i → Ψ(. . . , x i , . . . ) and then integrate over

x 1 , . . . , x i-1 , x i+1 , . . . , x N to get ˆBR |v(x i )| 2 |Ψ| 2 ǫ δ,d,N ˆRdN (-∆ x i ) 3 4 -δ Ψ 2 + c R ˆRdN |Ψ| 2 .
For j = i, applying (5) with a radius 2R, we have similarly

ˆBR |w(x i -x j )| 2 |Ψ| 2 ǫ δ,d,N ˆRdN (-∆ x i ) 3 4 -δ Ψ 2 + c R ˆRdN |Ψ| 2 .
We consider the many-body potential

V (x 1 , . . . , x N ) := N i=1 v(x i ) + 1 i<j N w(x i -x j ), (9)
for which To finish, we prove that the assumption that v, w ∈ L p loc (R d ) with p > max(2d/3, 2) implies [START_REF] Engel | Density Functional Theory: An Advanced Course[END_REF]. This is very classical [START_REF] Kato | Fundamental properties of Hamiltonian operators of Schrödinger type[END_REF]23] tends to 0 when M → +∞. We can do a similar treatment for w. Therefore, this proves that for s ∈ (0, d/2) and q 1, if v, w ∈ L qd 2s loc (R d ), then for any R > 0 and any ǫ > 0, there is c ǫ,R such that

|V | 2 1 B R = 1 B R N i=1 v(x i ) + 1 i<j N w(x i -x j ) 2 N (N + 1) 2   N i=1 1 B R |v(x i )| 2 + 1 i<j N 1 B R |w(x i -x j )| 2
|v| q 1 B R + |w| q 1 B R ǫ(-∆) s + c ǫ,R in R d .
For the case d ∈ {1, 2}, we need v, w ∈ L 2 loc (R d ) because we use |V | 2 . We have the Sobolev embedding H 3/2 (R d ) ֒→ L ∞ (R d ), and the argument is the same.

3 2

 3 -2δ + N c R ,where in the last inequality we have used that

.

  Thus we can takeǫ δ,d,N = 4ǫ δ,dN N (N + 1) 2 = 1 N (N + 1) 2 κ 2 4δ/3,dN ,and we obtain the result by applying Theorem 1.1.

2 L 2 + M ||u|| 2 L 2 ,

 222 First let s ∈ (0, d/2),let v ∈ L d 2s loc (R d ), R > 0 and u ∈ H s (R d ) supported in B R ⊂ R d . We have v = v1 |v|>M + v1 |v|<M , so ˆRd |v| |u| 2 ˆRd v1 {|v|>M }∩B R |u| 2 + Mwhere, in the last line, we have used the Sobolev inequality. By dominated convergence,v1 |v|>M L d 2s (B R )


Nabil Kazi-Tani
email: nabil.kazi-tani@univ-lyon1.fr

Didier Rullière
email: didier.rulliere@univ-lyon1.fr

On a construction of multivariate distributions given some multidimensional marginals. *

Keywords: Multidimensional marginals, Copulas, Elliptical Distributions AMS Subject Classification(2010): 60E05, 60E10, 62H05, 62H20

In this paper, we investigate the link between the joint law of a d-dimensional random vector and the law of some of its multivariate marginals. We introduce and focus on a class of distributions, that we call projective, for which we give detailed properties. This allows us to obtain necessary conditions for a given construction to be projective. We illustrate our results by proposing some theoretical projective distributions, as elliptical distributions or a new class of distribution having given bivariate margins. In the case where the data do not necessarily correspond to a projective distribution, we also explain how to build proper distributions while checking that the distance to the prescribed projections is small enough.

Introduction

Multivariate analysis is now a cornerstone in probability theory. Constructing multivariate distributions from given marginals is mathematically interesting on its own, but also has huge impact in practical problems.

Let d ≥ 1 be an integer and let X 1 , . . . , X d be continuous random variables with cumulative distribution functions (CDFs) F 1 , . . . , F d , respectively. Sklar's theorem [START_REF] Sklar | Fonctions de répartition à n dimensions et leurs marges[END_REF] states that the joint distribution H of (X 1 , . . . , X d ) can be written, for all (x 1 , . . . , x d ) ∈ R d in the form

H(x 1 , . . . , x d ) = P(X 1 ≤ x 1 , . . . , X d ≤ x d ) = C F 1 (x 1 ), . . . , F d (x d ) ,
where C is a copula function, i.e. the cumulative distribution function of a probability measure on R d whose marginals are uniform on [0, 1]. Copula models have shown their interest in particular because, as it can be seen from the previous equation, they separate the study of the margins and the study of the dependence structure.

In this paper, we want to investigate the link between the joint law of a d-dimensional random vector and the law of its multivariate marginals. For any subset K = (j 1 , . . . , j k ) of {1, . . . , d} with cardinal k, and any random vector X = (X 1 , . . . , X d ), we will write X K the random vector with values in R k given by (X j1 , . . . , X j k ) and F K the cumulative distribution function of X K .

We will abuse notations and call F K a probability distribution on R k . Let n ≤ 2 d be a positive integer and let K 1 , . . . , K n be n subsets of {1, . . . , d} with cardinals k 1 , . . . , k n .

A question that has been extensively studied in the literature is the following: given n probability measures P 1 , . . . , P n such that P i is a probability measure on R ki , is it possible to construct a probability measure F on R d such that F Ki = P i for each i = 1, . . . , n?

(1.1)

The existence of such a measure F is not guaranteed. In the case where the subsets {K i , i = 1, . . . , n} are disjoint, the product distribution guarantees its existence. One could also try to extend the notion of Copula function to the case of non overlapping multidimensional marginals: Genest & al. [START_REF] Genest | De l'impossibilité de construire des lois marges multidimensionnelles données partir de copules[END_REF] showed that this approach is useless, since it only allows to model the product distribution. More precisely, they proved that if H(x 1 , . . . , x m , y 1 , . . . , y n ) = C F (x 1 , . . . , x m ), G(y 1 , . . . , y m ) defines a proper (m + n)-dimensional distribution function (with m + n ≥ 3) for every F and G with respective dimensions m and n, then C(u, v) = u • v.

When the subsets {K i , i = 1, . . . , n} are not disjoint, an obvious necessary condition is that the prescribed measures {P i , i = 1, . . . , n} have the same marginals on common subspaces. But this condition is not sufficient : Kellerer [START_REF] Kellerer | Distribution functions with given marginal distributions (german: Verteilungsfunktionen mit gegebenen marginalverteilungen)[END_REF] gave a necessary and sufficient condition only involving the structure of the sets {K i , i = 1, . . . , n}. We refer to [START_REF] Joe | Multivariate Models and Multivariate Dependence Concepts[END_REF] (in particular, chapters 3 and 4, and section 3.4.3 to 3.7, for some compatibility conditions) and [START_REF] Dall'aglio | Advances in Probability Distributions with Given Marginals: Beyond the Copulas[END_REF] and the references therein for further details and related problems, in particular extremal distributions with fixed multidimensional marginals and related optimization problems.

The question of uniqueness is also tricky. In [START_REF] Gutmann | Existence of probability measures with given marginals[END_REF], it is proved that if µ is a probability measure on R d with a density f with respect to the Lebesgue measure, then there exists a subset U of R d such that all the lower dimensional projections of the uniform distribution on U coincide with the lower dimensional projections of µ. This shows, at least in the case where measures have a density, that there is in general no hope for uniqueness of a measure with prescribed projections.

Possible explicit constructions of distribution functions F satisfying (1.1) have been given in [START_REF] Cohen | Probability distributions with given multivariate marginals[END_REF], [START_REF] Rüschendorf | Construction of multivariate distributions with given marginals[END_REF], [START_REF] Sánchez Algarra | Construcción de distribuciones con marginales multivariantes dadas[END_REF], [START_REF] Cuadras | Probability distributions with given multivariate marginals and given dependence structure[END_REF], [START_REF] Joe | Parametric families of multivariate distributions with given margins[END_REF], [START_REF] Joe | Multivariate extreme-value distributions with applications to environmental data[END_REF], [START_REF] Joe | Families of m-variate distributions with given margins and m(m -1)/2 bivariate dependence parameters[END_REF], [START_REF] Marco | On the construction of multivariate distributions with given nonoverlapping multivariate marginals[END_REF], [START_REF] Li | Linkages: A tool for the construction of multivariate distributions with given nonoverlapping multivariate marginals[END_REF], [START_REF] Li | Dynamic linkages for multivariate distributions with given nonoverlapping multivariate marginals[END_REF], [START_REF] Dolati | A method for constructing multivariate distributions with given bivariate margins[END_REF] for overlapping or non-overlapping marginals.

All these constructions are very useful, in particular in statistics. Indeed, when the dimension d is large, one can first estimate all the bivariate marginals, since fitting a 2-dimensional copula is doable, and then construct a valid d-dimensional vector having the prescribed 2-dimensional margins. One problem of this approach is that it may not provide a unique d-dimensional distribution, but as pointed out in [START_REF] Joe | Multivariate Models and Multivariate Dependence Concepts[END_REF], one can then use entropy maximization techniques to choose a distribution among all those that have the prescribed marginals. By comparison, directly fitting the right copula in large dimension is however quite difficult and often makes use of recent research developments (nested, hierarchical or vine copulas, see [START_REF] Mcneil | Sampling nested archimedean copulas[END_REF], [START_REF] Brechmann | Hierarchical kendall copulas: Properties and inference[END_REF], [START_REF] Joe | Tail dependence functions and vine copulas[END_REF], including pair-copula constructions or copulas with prescribed bivariate projections [START_REF] Aas | Pair-copula constructions of multiple dependence[END_REF], [START_REF] Acar | Beyond simplified pair-copula constructions[END_REF], [START_REF] Di Bernardino | On an asymmetric extension of multivariate archimedean copulas based on quadratic form[END_REF]).

The previously mentioned constructions all make use of the joint CDF, or the joint density. There are however other functions characterizing the joint distribution, for instance the characteristic function. We will call any such function a characterizing function. In this paper, we assume that a characterizing function m is given, and that there is a linear explicit decomposition of the characterizing function of the d-dimensional vector with respect to the characterizing functions of certain of its multi-dimensional marginals (Definition 2.1). We will say that a probability distribution satisfying Definition 2.1 is in a projective class. The main result of the paper is a complete analysis of the coefficients appearing in a decomposition of a projective class. Indeed, the distributions satisfying our Definition 2.1 are stable by projection, in the sense that they are such that all their multidimensional marginals also satisfy the same Definition 2.1. This allows us to give precise, and simple necessary conditions for a sequence of coefficients to generate a probability distribution on R d having fixed multidimensional marginals and belonging to a projective class. In particular, a necessary condition is that a matrix containing these coefficients is idempotent (Proposition 2.3). Notice that the linear form of the decomposition in a projective class is not as restrictive as one would initially imagine, since one could first apply a bijective non linear transformation to a characterizing function, and then obtain a linear relation in the form of (2.2). Said otherwise, given a family of probability distributions, if some linearity can be found in the expression of one of its characterizing function, then our approach allows to exploit this linearity to construct multi-dimensional distributions. The case of Elliptical random vectors illustrates well this last point.

In Section 2 we define the projective class that we are going to work with, and in Section 3 we give and analyse examples of elements of this class.

Projective class of random vectors 2.1 Definitions

Let D = {1, . . . , d}, t ∈ Rd , and denote by P(D) the power set of D. Consider a random vector X = (X i ) i∈D ; for K ∈ P(D) a subset of D, we denote by X K = (X i ) i∈K the subector of X, and by t K = (t i ) i∈K the subvector of t.

We assume the existence of a link between the joint distribution of X and the joint distributions of its projections X K . In general this link could be derived from the characteristic function of X, from its CDF, or from some other quantity. We thus define a function m for which the link will be investigated.

We will denote by X d the space of R d -valued random variables that we will work with (with k ≤ d). In the rest of the paper, the quantities involved (and our constructions) will only depend on the distributions of the random variables, and not on the random variables themselves. Because it has no impact on our results, we will nonetheless use random variables. Assumption 2.1 (Projective characterizing function) We assume that there exists a function m : Rd × X d × P(D) → R such that for any non-empty K ∈ P(D):

1. {m(t, X, K)} t∈ Rd characterizes the joint distribution of X K , i.e. m(t, X, K) = m(t, Y, K) for all t ∈ Rd if and only if X K and Y K have the same distribution.

2. there exists a ∈ R such that for all t ∈ Rd , m(P K t, X, D) = m(t, X, K), where

(P K t) i := t i if i ∈ K a if i / ∈ K. (2.1)
In the rest of the paper, to simplify notations, we will denote m(t, X, K) by m(t, X K ).

Remark 2.1 Assumption 2.1 2. implies that P ∅ t = (a, . . . , a) =: a ∈ Rd , and that m(t, X, ∅) = m(a, X, ∅) for all t. For simplicity, we will write this quantity m 0 := m(a, X, ∅), or abuse notation and write m(t, X ∅ ) = m 0 .

Remark 2.2 Such a function m always exists. Typically, m can be given by m(t,

X K ) = E e it K X K , with a = 0 and m 0 = 1. Another example is m(t, X K ) = F K (t K ), the CDF of X K , with a = +∞ and m 0 = 1. A third example is m(t, X K ) = φ -1 • F K (t K ) where φ : R + → [0, 1]
is an invertible Archimedean generator, with a = +∞ and m 0 = 0. Direct transformations of these functions, as entropy or survival functions, are also suitable characterizing functions.

Remark 2.3 Assumption 2.1 2. says that to study the marginal distribution of X on the subset K, it is enough to study the distribution of X, with the characterizing function m restricted to P K t. However, notice that not every function characterizing the distribution of random variables satisfies this Assumption. Let us give the example of the potential function ( [START_REF] Baxter | Potentials of stopped distributions[END_REF]). Define the potential kernel v on R d as follows:

v(x) = -|x| , d = 1, v(x) = -log |x| , d = 2, v(x) = |x| -d+2 , d ≥ 3.
Then the potential U X of a random vector X on R d is defined by

U X (t) = E [v(X -t)]
when the expectation exists. We have that U X = U Y if and only if X and Y have the same distribution, but the potential function does not satisfy Assumption 2.1 2.

We aim at defining the whole distribution of X, using only some of its projections, i.e. using only the laws of X K for K ∈ S, where S is a given subset of P(D). For example, S can gather some subsets of cardinal 3, their subsets, and some singletons, or S can gather only subsets of cardinal 1, as in copula theory. We assume that S is decreasing, in the sense that for all K ⊂ J, J ∈ S implies K ∈ S: knowing the distribution of a projection easily allows to know the distribution of every sub-vector. In the algebraic topology terminology, S is a simplicial complex [START_REF] Spanier | Algebraic topology[END_REF]. Simplicial complexes can be represented using points, line segments, triangles, and simplices in higher dimensions, which may ease the understanding of the projections and the model (see Figure 1 for an illustration). 7

Figure 1: Illustration of a multivariate distribution in the dimension 7, knowing the marginal distribution {3}, the bivariate projection {4, 7}, and the trivariate projections {1, 2, 7} and {4, 5, 6}. All subsets of these projections also correspond to known marginal distributions.

Definition 2.1 (Projective class) Let S ∈ P(D) be decreasing. For a given characterizing function m, we say that a random vector X ∈ R d belongs to the projective class F D (S) if there exist some real coefficients α K,D , K ⊂ D, such that for all t ∈ Rd ,

m(t, X) = K∈S α K,D m(t, X K ) , (2.2)
Remark that if a random vector X belongs the projective class F D (S), then the set of suitable coefficients {α K,D , K ⊂ D} satisfying Equation (2.2) is not necessarily unique. Notice also that if D ∈ S, any d-dimensional random vector is in F D (S), using for example constant coefficients α K,D equal to 1 if K = D and equal to 0 otherwise. This is quite natural, since the class F D (S) intends to define multivariate distributions that can be fully determined by some of their projections. This is obviously the case when the initial joint distribution is already in S.

The projective class explicitly depends on the set S of given projections. It also depends implicitly on the choice of the characterizing function m, but for the sake of simplicity, it will not be indexed by m. A distribution (e.g. a centered Gaussian distribution) can be projective for a characterizing function m 1 (e.g. the logarithm of the characteristic function), but not for a characterizing function m 2 (e.g. the cumulative distribution function). At last, notice that a priori, the coefficients α K,D may also depend on the choice of the characterizing function m, but for the sake of simplicity they will not be indexed by m. We will show however that under some simple conditions, one can find coefficients that do not depend on of the choice of m (see further Remark 2.6).

It is not trivial to assess the compatibility conditions between arbitrary characterizing functions of the projections: given arbitrary m(t, X K ) in Equation (2.2), determining if the resulting m(t, X) corresponds indeed to a characterizing function is hard. As an example, if m is a characteristic function, the verification may rely on known criteria as Bochner's theorem or multivariate extensions of Pólya's theorem. If m is a transformation of a cumulative distribution function, the verification may rely on differentiation of this function using chain rule differentiation and multivariate extensions of Faa Di Bruno's formula. For known projective families (see Section 3), the compatibility is ensured, but in practice (see Section 4) it may rely on some numerical verifications. As in Section 1, we refer to [START_REF] Kellerer | Distribution functions with given marginal distributions (german: Verteilungsfunktionen mit gegebenen marginalverteilungen)[END_REF][START_REF] Joe | Multivariate Models and Multivariate Dependence Concepts[END_REF][START_REF] Dall'aglio | Advances in Probability Distributions with Given Marginals: Beyond the Copulas[END_REF] for supplementary material on the question of projections compatibility.

Properties

In this subsection, we discuss several properties of projective distributions regarding uniqueness, stability, and statistical inference. In particular, we provide explicit expressions for the coefficients appearing in Definition 2.1, in the case of fixed projections up to a given dimension.

Assuming that a distribution is projective, we give here a necessary and sufficient condition ensuring the uniqueness of the coefficients {α K,D } K∈S , which further implies that the distribution characterized by Equation 2.2 is unique. The condition relies on the given projections m(t, X K ), K ∈ S. Proposition 2.1 (Condition on the projections for uniqueness) Define S := S if m 0 = 0, and S := S \ {∅} if m 0 = 0. Consider a finite set of points T ⊂ Rd , and denote the matrix M S (T ) := (m(t, X K )) t∈T ,K∈S . Assume that the distribution of X is projective, and denote the vector of coefficients α := (α K,D ) K∈S , then the following results hold: (i) If there exists a set T with |T | = |S| such that the matrix M S (T ) is invertible, then α is unique.

(ii) If α is unique, then for every set T of distinct points with |T | = |S|, the matrix M S (T ) is invertible.

(iii) In particular when S = {K ⊂ D, |K| ≤ 2} and m 0 = 0, α is unique if there exists t ∈ Rd such that for all K ∈ S,

m * K (t) = m(t, X K ) -J⊂K,J =K m(t, X J ) > 0.
Remark 2.4 The combination of items (i) and (ii) above implies the following necessary and sufficient condition for uniqueness: α is unique if and only if there exists a set T with |T | = |S| such that the matrix M S (T ) is invertible.

Proof. It is clear that if m 0 = 0, then α ∅,D m(t, X ∅ ) = 0 whatever the value of α ∅,D . Multiplied by zero, this coefficient has no impact, we thus exclude it from the analysis by setting S = S \{∅} in the case when m 0 = 0. Let us show the item (i). Write the vector m X (T ) = (m(t, X)) t∈T .

The main linear equality of Equation (2.2) writes:

m X (T ) = M S (T ) • α (2.3) so that when M S (T ) is invertible, α = [M S (T )] -1 m X (T ) is uniquely determined.
To prove item (ii), consider a set T of distinct points with |T | = |S|. Again, equation (2.2) can be written as (2.3). This linear system of |S| equations admits either no solution, which is excluded by the assumption that the distribution of X is projective, either an infinite number of solutions, which is excluded by the assumption that α is unique, or a unique solution, which is the only possible case. This implies, by the Rouché-Capelli theorem, that the rank of the matrix M S (T ) is equal to |S|, or equivalently that this matrix is invertible.

For the last part of the proposition, item (iii), write S = {K 1 , . . . , K s }, set t ∈ Rd and set T = {t K , K ∈ S}. Write P the matrix with components P ij = 1 if K i ⊂ K j or 0 otherwise. One can check that the component i, j of the matrix M S (T ) writes:

(M S (T )) ij = m(t Ki , X Kj ) = m(t, X Ki∩Kj ) (2.4)
For any set L, K = L if and only if K is a subset of L but not a strict subset of L, thus

(M S (T )) ij = K∈S,K⊂Ki∩Kj m(t, X K ) - K∈S,K Ki∩Kj m(t, X K ) . (2.5)
By assumption, for all K ∈ S, |K| ≤ 2. The following equality holds for |L| = 1, since both the left and right hand terms are equal to 0 in this case. The equality also holds for |L| = 2, because in the right hand side, K ∈ S, J ⊂ K, J = K implies |J| = 1 and |K| = 2, and K ⊂ L with |K| = |L| = 2 implies K = L:

K∈S,K L m(t, X K ) = K∈S,K⊂L J⊂K,J =K m(t, X J ) . (2.6)
Finally, for h ∈ {1, . . . , s}, as K h ⊂ K i ∩ K j if and only if

P hi P hj = 1, (M S (T )) ij = K h ∈S,K h ⊂Ki∩Kj m * K h (t) = K h ∈S P hi • m * K h (t) • P hj (2.7)
so that M S (T ) = P DP , where D is the diagonal matrix with diagonal {m * K (t), K ∈ S}. When S = {K ⊂ D, |K| ≤ 2}, up to a rearrangement, P is an upper triangular matrix with ones on its diagonal. Thus det(P ) = 1 and finally det(M S (T )) = det(D). Under the assumption of (iii), det(D) > 0 and the result holds.

In the next Proposition, we prove the following projection stability property: if a random vector belongs to the class F D (S), then any subvector also belong to F D (S), and we compute the corresponding coefficients. Proposition 2.2 (Projection stability) Let X be a d-dimensional random vector in F D (S), with associated coefficients {α K,D , K ∈ S}. Then for any non-empty L ⊂ D, the subvector X L belongs to F L (S), where for any non-empty subset J of L, a suitable set of associated coefficients is given by α

J,L = K∈S, K∩L=J α K,D , (2.8)
which implies in particular that α J,L = 0 if J ⊂ L.

Let d 0 be an integer such that 1 ≤ d 0 ≤ d. When S = {K ⊂ D, |K| ≤ d 0 }, and when the coefficients {α J,L , J ⊂ L} only depend on the subsets cardinals, i.e. α J,L = α j, with j = |J| ∈ {1, . . . , d} and = |L| ∈ {0, . . . , d}, then a suitable set of associated coefficients is given by

α j, = min(d0,d-+j) k=j d - k -j α k,d if j ≤ min( , d 0 ), (2.9)
and α j, = 0 otherwise.

Proof. First remark that, due to the second assumption on m(t, X), we have m(t, X K∩L ) = m(P K∩L t, X) = m(P K P L t, X) = m(P L t, X K ). Thus, using (2.2), we have

m(t, X L ) = m(P L t, X) = K∈S α K,D m(P L t, X K ) = K∈S α K,D m(t, X K∩L ) = J⊂L K∈S, K∩L=J α K,D m(t, X J ) .
By assumption, K ∈ S implies that

J = K ∩ L ∈ S. Setting α J,L = K∈S, K∩L=J α K,D as in Equation (2.8) we finally get m(t, X L ) = J⊂L,J∈S α J,L m(t, X J ) , (2.10) hence X L belongs to F L (S). Eventually remark that K ∩ L = J if and only if J ⊂ L and J ⊂ K and K ∩ L ⊂ J. As a consequence, if J ⊂ L, the sum in (2.8
) is empty and α J,L = 0.

We prove now the second part of the Proposition. When S = {K ⊂ D, |K| ≤ d 0 }, we have using (2.8)

α J,L = K ⊂D\L,|K |≤d0-|J| α K ∪J,D . Now, let j = |J| and = |L|. If j > , it is clear that J ⊂ L and that α J,L = 0. As K ⊂ D \ L and |K | ≤ d 0 -|J|, we get 0 ≤ |K | ≤ min(d 0 -j, d -). Thus when J ⊂ L, α J,L = min(d0-j,d-) k=0 K ⊂D\L,|K |=k α K ∪J,D (2.11)
and if the coefficients α J,L do only depend on the cardinals,

α j, = min(d0-j,d-) k=0 K ⊂D\L,|K |=k α k+j,d = min(d0-j,d-) k=0 d - k α k+j,d . (2.12)
Hence the second result.

Remark 2.5 (case of ∅) Notice that ∅ necessarily belongs to S. In the case where m(t, X ∅ ) = 0, Equation (2.2) may involve a constant α ∅,D , and implies that K⊂D,K∈S α K,D = 1. In this case, it becomes useful to determine the coefficients α ∅,L . For any non-empty L, the Equation

(2.8) writes α ∅,L = K⊂D\L,K∈S α K,D .
(2.13)

When L = ∅, the Equation (2.2) remains valid for D = ∅ if one defines α ∅,∅ = 1.
As previously noticed, the coefficients α K,D and α ∅,L may depend on the choice of the characterizing function m. However, one can check that using m(t, X) = cm(t, X) for a positive constant c leads to m(t, X ∅ ) = cm(t, X ∅ ) and to unchanged coefficients αK,D = α K,D .

Corollary 2.1 (Given projections up to dimension d 0 ) Let X be a d-dimensional random vector in F D (S). Assume that all projections of X are given up to a dimension d 0 , so that S = {K ⊂ D, |K| ≤ d 0 }. Assume that the associated coefficients {α J,L , J ⊂ L} only depend on the subsets cardinals, i.e. α J,L = α j, with j = |J| ∈ {1, . . . , d} and = |L| ∈ {0, . . . , d}.

Assume furthermore that α k,k = 1 for all k ≤ d 0 . Then the coefficients α d0-z,d can be obtained recursively, using

α d0-z,d = 1 - z i=1 d -d 0 + z i α i+d0-z,d , (2.14)
for z = 1, . . . , d 0 , starting with α d0,d = 1. In particular for 2

≤ d 0 ≤ d, we get α d0,d = 1, α d0-1,d = -(d -d 0 ), α d0-2,d = 1 + d-d0+2 2 (d -d 0 -1).
When

d 0 ≥ 3, α d0-3,d = 1 -(d -d 0 + 3){1 -d-d0+2 2 1 -d-d0+1 3
}. For higher orders, one can check by induction that these coefficients do only depend on d -d 0 , but their expression is omitted here.

Proof. Under the assumption α k,k = 1 for all k ≤ d 0 , this follows directly from Proposition 2.2, by writing Equation (2.9) in the case where j = ≤ d 0 , and setting i = k -j, j = d 0 -z: we get α d0,d = 1 when z = 0, and Equation (2.14) when z ≥ 1.

Remark 2.6 (Explicit coefficients)

The fact that α k,k = 1 for all k ≤ d 0 means that if we are given a k-dimensional marginal, we do not try to retrieve it from the given lower dimensional marginals. Under this assumption, and under the assumption that the coefficients in the projective decomposition only depend on the subsets cardinals, the previous Corollary provides a set of suitable coefficients {α j, } which are explicitly given, independently of the choice of m.

The case where all bivariate projections are given is a very natural and interesting case: in practical applications, bivariate projections can be graphically visualized, and the estimation of the dependence structure among each pair of random variable is still tractable. The following Remark shows that in this case, under some simple conditions, the coefficients α J,L can be computed explicitly.

Remark 2.7 (Given bivariate projections) Consider the same assumptions as in Corollary 2.1 and assume that all bivariate projections of a multivariate distribution are given, so that d 0 = 2 and S = {J ⊂ D, |J| ≤ 2}. Then for all non-empty L ⊂ D, we can reformulate equation (2.2) as follows:

m(t, X L ) = α 0,|L| m 0 -(|L| -2) J⊂L,|J|=1 m(t, X J ) + J⊂L,|J|=2 m(t, X J ) , (2.15)
where α 0,0 = 1, α 0, = 1 + 1 2 ( -3), ≥ 1 and where

m 0 = m(t, X ∅ ) is defined in Remark 2.1.
Let X ∈ R d be a random vector in F D (S). Since D = {1, . . . , d} is a finite set, the set of subsets of D is also finite, and we can define the following matrix, indexed by the subsets of D:

A = (α J,L ) J⊂D,L⊂D .
(2.16)

We will write A .,L for the column vector relative to the subset L.

Proposition 2.3

The coefficients in the matrix A satisfy the following constraints:

1. If the set of associated coefficients {α J,L } is unique, then the matrix A defined in (2. [START_REF] Gutmann | Existence of probability measures with given marginals[END_REF])

is idempotent: A 2 = A.
2. If furthermore the coefficients do only depend on the subset's cardinal, i.e. α J,L = α j, with j = |J| and = |L|, we get for 0 ≤ j ≤ ≤ d :

α j, = min(d0, ) k=j -j k -j α j,k α k, . (2.17)
Now define s j, := -2 j-2 α j, when j ≥ 2. The previous equation writes, when j ≥ 2,

s j, = min(d0, ) k=j s j,k s k, . (2.18)
Proof. Let L ⊂ D, |L| ≥ 1. From Proposition 2.2, m(t, X L ) = K⊂L,K∈S α K,L m(t, X K )
From Proposition 2.2, we also have m(t, X K ) = J⊂K,J∈S α J,K m(t, X J ), so that finally

m(t, X L ) = K⊂L,K∈S J⊂K,J∈S α J,K α K,L m(t, X J ) = J⊂L,J∈S K⊂L,K∈S α J,K α K,L m(t, X J ) , as α J,K = 0 if J ⊂ K. Then for all t, m(t, X L ) = J⊂L,J∈S    K⊂L,K∈S α J,K α K,L    m(t, X J )
so that, using the uniqueness of the set of coefficients {α J,L },

α J,L = K⊂L,K∈S α J,K α K,L
and thus A is idempotent.

Let us now focus on the second part of the proposition. For a subset L ⊂ D with cardinal and k ≤ , define [L] k := {K ⊂ L such that |K| = k}. Assume that when K ⊂ L the coefficients α K,L do only depend on the cardinals k = |K| and = |L| of the considered sets, and assume

α K,L = 0 if K / ∈ S, i.e. α k, = 0 if k > d 0 . Then m(t, X L ) = K⊂L α K,L m(t, X K ) = min(d0, ) k=0 α k, K∈[L] k m(t, X K ) = min(d0, ) k=0 α k, K∈[L] k    min(d0,k) j=0 α j,k J∈[K] j m(t, X J )   
Notice that, by a simple combinatorial argument, for j ≤ k,

K∈[L] k J∈[K] j m(t, X J ) = -j k -j J∈[L] j m(t, X J ) , which entails that m(t, X L ) = min(d0, ) k=0 min(d0,k) j=0 α k, α j,k -j k -j J∈[L] j m(t, X J ) .
On the other hand we have

m(t, X L ) = min(d0, ) j=0 α j, J∈[L] j m(t, X J )
so that for all t, for all j ≤ k,

α j, = min(d0, ) k=j -j k -j α j,k α k,
and the second result holds.

Remark 2.8 Notice that due to the projection stability property of Proposition 2.2, any column of the matrix A can be deduced from the last one by multiplication by a matrix with values in {0, 1}: A .,L = P (L) A .,D , where the 2 d × 2 d matrix P (L) is defined by its components

P (L) J,K = 1 if J = K ∩ L and K ∈ S 0 otherwise. ,
for J, K, L subsets of D. Indeed, we have from Proposition 2.2:

α J,L = K∈S,K∩L=J α K,D = K∈S P (L) J,K α K,D . (2.19)
Let us finish this Section with a useful property of our construction for statistical inference. Assume that we have at our disposal an i.i.d. sequence (X 1 , . . . , X n ), where for each i, X i ∈ R d . The following Proposition highlights the fact that if we have estimators with good properties for the distribution of subvectors X K , then these properties are maintained for the estimator of the distribution of the whole vector X.

As in Definition 2.1, let S ∈ P(D) be decreasing. For any K in S, assume that we can construct an estimator m n (t, X K ) of m(t, X K ) using the sample (X 1 , . . . , X n ), for t ∈ R d . Following the expression (2.2), a natural expression for an estimator m n (t, X) of the distribution of the whole vector X is given by

m n (t, X) = K∈S α K,D m n (t, X K ).
(2.20)

Proposition 2.4 (Unbiasedness and Consistency) The natural estimator given by (2.20) preserves unbiasedness and consistency:

1. If for each K, m n (t, X K ) is unbiased, then m n (t, X) is also unbiased. 2. If for each K, m n (t, X K ) is consistent, then m n (t, X) is also consistent.
Proof. Property 1. above is obvious, by the linearity of (2.1) and the linearity of expectation.

To prove the second statement, first suppose that for each K, m n (t, X K ) converges in probability to m(t, X K ) as n goes to infinity. By Slutsky's theorem, the vector ( m n (t, X K )) K∈S converges in probability to (m(t, X K )) K∈S , and by the continuous mapping theorem, any linear combination of elements of ( m n (t, X K )) K∈S converges in probability to the same linear combination of elements of (m(t, X K )) K∈S , which is the desired result. Notice that we are able to use Slutsky's theorem, which is a statement about convergence in distribution, because all the limits are towards real constant values. 2

Remark 2.9 If a central limit theorem (CLT) is available for each m n (t, X K ), then further work is needed to obtain a CLT for m n (t, X) defined in (2.20). Indeed for K 1 , K 2 ∈ S, K 1 ∩ K 2 is not necessarily empty, which implies that the elements in the linear combination (2.20) are dependent. A CLT for m n (t, X) is thus determined by the strength of this dependence.

Examples

Elliptical random vectors

Recall [START_REF] Joe | Multivariate Models and Multivariate Dependence Concepts[END_REF] that d-dimensional elliptical distributions are characterized by the fact that their characteristic function can be written in the following form: for any

t ∈ R d , E e it (X-µ) = φ t Σt (3.1)
for a given function φ : R + → R which is called the generator of the distribution, and where µ is the mean of the vector X and Σ is a non-negative definite matrix. We assume here that the generator φ does not depend on the dimension d of the random vector, i.e. that the elliptical distribution is consistent in the sense of [START_REF] Kano | Consistency property of elliptic probability density functions[END_REF].

One interesting feature of families of elliptical distributions is that they allow heavy tails, while preserving some advantageous properties of multivariate Gaussian distributions. Indeed, an elliptical distribution has a stochastic representation as a product of two independent random elements: a univariate radial distribution and a uniform distribution on an ellipsoid (see Chapter 2 in [START_REF] Fang | Symmetric multivariate and related distributions[END_REF]). This representation makes possible the analysis of the density functions, the moments, the conditional distributions, the symmetries, and the infinite-divisibility properties of elliptical distributions ( [START_REF] Frahm | Generalized Elliptical Distributions: Theory and Applications[END_REF]). Other than multivariate Gaussians, elliptical distributions include the Student t-distributions, the symmetric generalized hyperbolic distribution, the power exponential distributions and the sub-Gaussian α-stable distributions among others. Families of elliptical distributions are widely applied in statistics, in particular in the area of robust statistics, as a starting point for the development of the M -estimates of multivariate location and scale (see Chapter 2 in [START_REF] Maronna | Robust Statistics: Theory and Methods[END_REF]). It is also a rather standard family of distributions in financial modelling: see [START_REF] Frahm | Elliptical copulas: applicability and limitations[END_REF] and the references therein. Thus, constructing a distribution with given multivariate elliptical marginals is a useful and interesting problem, this is what we do in this subsection.

Let us first remark that, for a given generator φ, when one considers a centered multivariate elliptical distribution, the distribution is fully characterized by all components σ ij of the matrix Σ, that is to say by all bivariate elliptical projections of the distribution (it does not mean that the multivariate elliptical distribution is the only one having those projections).

It is thus quite natural to analyse, in the case of elliptical distributions, the links between the matrix Σ and a given set of submatrices Σ K1 , . . . Σ Kn , for K 1 , . . . , K n subsets of D = {1, . . . , d}. This is easier to do using the matrix Σ rather than its inverse Σ -1 . It thus seems easier to work with characteristic functions or entropy (which are expressed using Σ) rather than densities or cumulative distribution functions (which are expressed using Σ -1 ).

We will try to express the quantity t Σt as a linear combination of products t K Σ K t K , where K belongs to known projections indexes in S. A sequence of coefficients α K,D , K ∈ S, is said to be S-admissible if for all matrix Σ, for all t,

t Σt = K⊂D,K∈S α K,D • t K Σ K t K , (3.2)
where Σ K is the submatrix of Σ with indices in K.

The following Lemma provides a characterization of such coefficients:

α 1,d = - d0 k=2 s k d -k k -1 and α k,d = s k / d -2 k -2 , k = 2, . . . , d 0 (3.3)
for some real values s 2 , . . . , s d0 such that s 2 + . . . + s d0 = 1.

In the particular case where the coefficients are deduced from only two given dimensions, i.e. if there exists k 0 ≥ 2 such that s i = 0 whenever i / ∈ {1, k 0 }, we get a particular S-admissible sequence

α 1,d = - d -k 0 k 0 -1 and (3.4)
α k,d = 1 d-2 k0-2 , if k = k 0 and α k,d = 0 otherwise, k = 2, . . . , d 0 . (3.5)
Furthermore, when d 0 = 2, the only S-admissible sequence is

α 1,d = -(d -2) and α 2,d = 1 (3.6)
Proof. Assume that α d is S-admissible, and that α K,D does only depend on |K| and |D|. Let (i 0 , j 0 ) ∈ D 2 , i 0 = j 0 . Isolating the coefficient t i0 Σ i0,j0 t j0 on both sides of Equation (3.2), we get

1 = d0 k=2 |{K : K ⊂ D, |K| = k, {i 0 , j 0 } ⊂ K}| α k,d = d0 k=2 d -2 k -2 α k,d Denoting s k = α k,d d-2
k-2 for all k, we get d0 k=2 s k = 1. Now considering the coefficient t i0 Σ i0,i0 t i0 on both sides of Equation (3.2), we get

1 = d0 k=1 α k,d |{K : K ⊂ D, |K| = k, i 0 ∈ K}| = α 1,d + d0 k=2 d -1 k -1 α k,d
Now as

s k = α k,d d-2 k-2 for all k, 1 = α 1,d + d0 k=2 s k d-1 k-1 d-2 k-2 = α 1,d + d0 k=2 s k d -1 k -1
Finally, using 1 = d0 k=2 s k , Equation (3.3) holds. The rest of the proposition are direct applications of this last equation.

A direct application of such S-admissible sequence is given in the following Proposition. As a consequence, the further defined distance to admissibility (see subsection 4.1) is always 0 for elliptical distributions.

φ(t Σt) = φ   d0 k=1 α k,d K∈[D] k t K Σ K t K   , (3.7)
where Σ K is the submatrix of Σ with indices in K. In other words, setting m(t, X) = φ -1 E e it (X-µ) ,

and

S = {K ⊂ D, |K| ≤ 2}, we have X ∈ F D (S) (3.8)
In particular, when d 0 = 2 (i.e. starting from all bivariate projections), the admissible sequence is

α 1,d = -(d -2) and α 2,d = 1.
Proof. By definition, for any S-admissible sequence, t Σt =

d0 k=1 α k,d K∈[D] k t K Σ K t K .
One can also check that the functions m are suitable characterizing functions satisfying Assumption 2.1 with a = 0 and m 0 = φ -1 (1) = 0. Hence the result.

α J,L =    0 if J ⊂ L , -(|L| -2) if |J| = 1 and J ⊂ L , 1 if |J| = 2 and J ⊂ L .
In particular, if d = 3 and D = {1, 2, 3}, the matrix A = (α J,L ) J⊂D,L⊂D is

A =          
1 0 0 0 0 0 -1 0 1 0 0 0 0 -1 0 0 1 0 0 0 -1 0 0 0 1 0 0 1 0 0 0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 0 0

          , (3.9)
where the seven raws and columns correspond to successive subsets of D:

{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}
As m 0 = 0 in the elliptical case, it is not necessary to compute the coefficients α J,L for J = ∅ or L = ∅ (see Remark 2.5). One easily check that we can apply Proposition 2.3 to deduce that A is idempotent, which can also be verified by hand in this example.

Consider a centered elliptical random vector with zero mean and covariance matrix Σ. As seen before, its distribution is thus projective, so that for all t ∈ R d ,

t Σt = K⊂D,K∈S α K,D • t K Σ K t K , (3.10)
Let us denote by D K the d × d diagonal matrix having ones only at indices in K. We can write

t Σt = t   K⊂D,K∈S α K,D (D K ) ΣD K   t (3.11)
so that this holds for all t if and only if

Σ = K⊂D,K∈S α K,D Ext D (Σ K ) , (3.12)
where Ext D (Σ K ) is the extension of the matrix Σ K to the dimension d × d, i.e. the matrix having components (Σ K ) ij for all i, j ∈ K and 0 when i ∈ K or j ∈ K. Now assume that for all K ∈ S, one has a given estimator ΣK of the covariance matrix of X K . From the previous equation, a natural estimator of the full covariance matrix Σ is defined as

Σ := K⊂D,K∈S α K,D Ext D Σ K . (3.13)
Now assume that classical estimators Σ K are also available for any K ⊂ D, then we get the following result: if for all K ∈ S, Σ K is the submatrix of Σ D for indices in K, then Σ = Σ D .

(3.14)

This can be checked using Equation (3.10), when choosing t the vector having 0 everywhere except for two given components i and j.

In particular, consider for example a centered Gaussian random vector, if Σ K are Maximum Likelihood Estimators, K ⊂ D, it is well known that they are directly proportional to the sample covariance matrix, so that all Σ K are submatrices of Σ D , and thus Σ = Σ D . This also holds for many shrinkage estimators. On the contrary, when Σ K are built by inverting an estimated precision matrix, Σ D and Σ may differ. The study of all possible estimators of Σ is however out of the scope of the present paper. When the generator φ is unknown, another interesting perspective is to use the underlying linearity of the projective class in order to build a non parametric estimator of φ.

Vectors built from bivariate distributions

Assume that a family (µ i,j ) (i,j)∈D 2 of probability measures on R 2 is given, for some non-negative parameters θ i , θ j and θ i,j specific to each couple (i, j) ∈ D 2 , and that each µ i,j satisfies:

µ i,j ((t, +∞), (u, +∞)) := ψ(θ i t + θ j u + θ i,j tu), (t, u) ∈ R 2 + , (3.15)
where ψ is a given appropriate function.

Thanks to our previous results, we can construct a random variable X, with values in R d + , such that the survival function of each subvector (X i , X j ) is given by the right hand side of equation (3.15).

Let us first analyze the copula function associated to (X i , X j ). To do so, assume that ψ is a decreasing bijection from R + to (0, 1], such that ψ(0) = 1 and that derivatives of ψ exist up to the order d. Denote by ψ -1 the inverse function of ψ. Let t = (t i , t j ) ∈ R 2 + and S ij (t) = ψ(θ i t i + θ j t j + θ ij t i t j ). The one-dimensional survival functions are then given by S i (t) = ψ(θ i t i ), from which we get θ i t i = ψ -1 (S i (t)), so that finally

S ij (t) = ψ ψ -1 (S i (t)) + ψ -1 (S j (t)) + θ ij θ i θ j ψ -1 (S i (t))ψ -1 (S j (t)) .
Therefore we can write

S ij (t) = C Sij (S i (t), S j (t)),
where the survival copula C Sij is given by

C Sij (u, v) = ψ ψ -1 (u) + ψ -1 (v) + θ ij θ i θ j ψ -1 (u)ψ -1 (v) , (u, v) ∈ [0, 1] 2 . ( 3
C Sij (u i , u j ) = u i u j exp - θ ij θ i θ j ln u i ln u j ,
which is the survival copula associated with Gumbel's bivariate exponential distribution (see [START_REF] Nelsen | An Introduction to Copulas[END_REF], copula (4.2.9)).

2. In the case where θ i,j = 0 for every (i, j) ∈ D 2 and if the generator ψ is d-monotone (see Definition 3.2), we obtain a survival copula which is Archimedean with generator ψ. It is clear that in this specific case, the multivariate distribution in higher dimension will still have an Archimedean survival copula with the same generator, as it will appear further in Equation (3.18).

3. In the case where θ i,j = 0 for every (i, j) ∈ D 2 and all the coefficients θ i are equal, we obtain the class of Schur constant vectors, studied for example in [START_REF] Nelsen | Some properties of schur-constant survival models and their copulas[END_REF]. In that case the function ψ correspond to the generator of the Schur constant vector, which has an Archimedean survival copula in the bivariate case, whose generator is given by ψ -1 .

Let us now explicitly construct a distribution on R d + , with the given bivariate marginals. Given the form (3.15) of these marginals, we choose the following characterizing function:

m(t, X) = ψ -1 (P(X i > t i , i ∈ D)) .
(3.17)

With the assumptions we made for ψ, it is clear that the function m satisfies Assumption 2.1, with constant a = +∞ and m 0 = m(t, ∅) = 0. The function m is thus a suitable characterizing function. Now consider the decreasing set S = {J ⊂ D, |J| ≤ 2}, and assume that m belongs to the class F D (S) in Definition 2.1: each multivariate distribution is assumed here to depend only on its bivariate projections. Assume furthermore that the associated coefficients α J,K in Definition 2.1, J ⊂ K, do only depend on cardinals j = |J| and k = |K|, so that α J,K = α j,k .

Due to Remark 2.7, if a valid multivariate distribution belongs to the class F D (S), then its survival function must take the form

FK (t) = ψ   -(k -2) J⊂K,|J|=1 m(t, X J ) + J⊂K,|J|=2 m(t, X J )   , Notice that m(t, X {i} ) = θ i t i and m(t, X {i,j} ) = θ i t i + θ j t j + θ i,j t i t j . Now using {i,j}⊂K (θ i t i + θ j t j ) = (k -1) {i}⊂K θ i t i we get FK (t) = ψ   {i}⊂K θ i t i + {i,j}⊂K θ i,j t i t j   .
(3.18) Proposition 3.2 below shows that under some sufficient conditions this expression is a proper multivariate survival function. This proposition makes use of the following definition of dmonotony, as given in [START_REF] Mcneil | Multivariate archimedean copulas, d-monotone functions and 1 -norm symmetric distributions[END_REF]:

Definition 3.2 (d-monotone function) A real function f is called d-monotone in (a, b),
where a, b ∈ R and d ≥ 2, if it is differentiable there up to the order d -2 and the derivatives satisfy

(-1) k f (k) (x) ≥ 0, k = 0, 1, . . . , d -2,
for any x ∈ (a, b) and further if (-1) d-2 f (d-2) is non-increasing and convex in (a, b). For d = 1, f is called 1-monotone in (a, b) if it is nonnegative and non-increasing over (a, b).

Proposition 3.2

The following three conditions ensure that, for any fixed subset K of size k := |K|, FK (t) is a proper multivariate survival function:

(i) ψ and its derivatives goes fast enough to zero: for every n ≤ k -1, lim

x→+∞ xψ (n) (x) = 0, (ii) ψ is k-monotone, (iii) for all distinct i, j in K, θ i,j ∈ [0, θ i θ j ρ ψ,k ],
where

ρ ψ,k = inf x∈R + , r≤k/2, r odd γ -1 k,r ψ (k+1-r) (x) ψ (k-r) (x)
and

γ k,r = 1 r k-2r+2 2
.

For example, if |K| = k = 3 and ψ(x) = exp(-x), then ψ is a k-monotone function satisfying condition (i) and (ii). It also satisfies (iii) with coefficient ρ ψ,k = 1 3 , and the function FK defined in Equation (3.18) is a valid multivariate survival function if θ i,j ≤ θ i θ j /3 for all i, j ∈ K. Proof. We have

FK (t) = ψ   {i}⊂K θ i t i + {i,j}⊂K θ i,j t i t j   = ψ(Q(t)) , with Q(t) = {i}⊂K θ i t i + {i,j}⊂K θ i,j t i t j .
Let us consider without loss of generality K = {1, . . . , k} and t = (t 1 , . . . , t k ). Let us define

f K (t) = (-1) k ∂ k ∂t 1 • • • ∂t k FK (t).
If f K is a non-negative function whose integral is one, then it will be the density of a random vector, and FK (t) will be a proper multivariate cdf.

Positivity Let us first establish conditions under which f K is a non-negative function. The multivariate Faa Di Bruno's formula gives

f K (t) = (-1) k π∈Π K ψ (|π|) (Q(t)) • B∈π ∂ |B| Q(t) j∈B ∂t j . (3.19)
where Π K is the set of all partitions of K. B ∈ π means that B runs through all non-empty blocks of a considered partition π. In the following, we will write

∂ 2 ∂t B Q(t) = ∂ ∂ti∂tj Q(t)
, and θ B = θ i,j , where B = {i, j}.

Notice that ∂ |B| Q(t) is 0 when |B| ≥ 3. Thus the only partitions π involved in the calculation contain blocks of 1 or 2 elements only. Hereafter, we denote by Π r K the partitions in Π K that contains exactly r distinct blocks of size 2. For a partition π ∈ Π r K , these r blocks will be denoted B π 1 , . . . , B π r . Such a partition π ∈ Π r K contains r blocks of size 2 and k -2r blocks of size 1, so that |π| = k -r. Thus we get (with the convention r) , and setting

0 i=1 = 1) f K (t) = (-1) k k/2 r=0 π∈Π r K ψ (k-r) (Q(t)) • r i=1 ∂ 2 Q(t) ∂t B π i j∈K\∪iB π i ∂Q(t) ∂t j . If ψ is k-monotone, ψ (k-r) = (-1) k-r ψ (k-
N k = {0, . . . , k/2 }, f K (t) = r∈N k π∈Π r K ψ (k-r) (Q(t)) • r i=1 - ∂ 2 Q(t) ∂t B π i j∈K\∪iB π i ∂Q(t) ∂t j .
One can write f K (t) = r∈N k ξ(r). Assume that all θ i ≥ 0 and θ i,j ≥ 0, i, j ∈ K. Under this assumption, one can check that, when r is even, ξ(r) ≥ 0. As a consequence,

r∈N k ξ(r) ≥ r∈N k ,r odd [ξ(r) + ξ(r -1)] .
Let us try to simplify ξ(r) + ξ(r -1). First remark that for r ≥ 1,

|Π r K | = k 2 • • • k -2(r -1) 2 /r! = 2 -r k!/(k -2r)!r! and |Π r K | = γ k,r |Π r-1 K |, with γ k,r = 1 r k-2r+2 2 . Let us write ξ(r) = π∈Π r K z(B 1 , . . . , B π r ).
The term ξ(r -1) can be written

π∈Π r-1 K z(B π 1 , . . . , B π r-1 ) = γ -1 k,r π∈Π r K z(B π 1 , . . . , B π r-1 ) .
and thus

ξ(r) + ξ(r -1) = π∈Π r K z(B π 1 , . . . , B π r ) -γ -1 k,r z(B π 1 , . . . , B π r-1 ) .
As a consequence, a sufficient condition to ensure that f K (t) ≥ 0 is that: for any B = {i, j}, for any odd r, for any t,

- ∂ 2 Q(t) ∂t Bi ψ (k-r) (Q(t)) + γ -1 k,r ψ (k+1-r) (Q(t)) ∂Q(t) ∂t i ∂Q(t) ∂t j ≥ 0
Thus the sufficient condition to ensure the positivity of f K (x).

Absolute continuity Let us now check if the integral of f K is summing to one. If so, FK would be a valid absolutely continuous distribution, without singular component. First assume that for all integers n ≤ k -1, lim

x→+∞ xψ (n) (x) = 0 . (3.20)
Recall that FK (u) = ψ(Q(u 1 , . . . , u k )). We now make use of the multivariate Faa Di Bruno's formula as is Equation (3.19). Seen as function of u n , the derivative of FK (u) with respect to u n+1 , . . . , u k writes as a sum of terms ψ (i) (au n + b) • P (u n ), where P is a polynomial of degree at most 1 and a, b some real values. Thus under assumption (i), for any

u ∈ R k , for all integer n ≤ k -1, lim un→+∞ ∂ k-n ∂u n+1 • • • ∂u k FK (u) = 0 . (3.21)
As a consequence, one can show by recursion that in this case

FK (t 1 , . . . , t k ) = +∞ t1 • • • +∞ t k ∂ k ∂u 1 • • • ∂u k FK (u 1 , . . . , u k )du k • • • du 1 . (3.22)
Using the fact that FK (0, . . . , 0) = 1, we conclude that the derivative function f K is non-negative and is integrating to one on the whole domain R k + . Under chosen assumptions, it thus defines a proper probability measure and FK is a valid multivariate survival function.

Practical implementation 4.1 Distance to admissibility

We have seen several examples of projective distributions that can be used in practice, as Elliptically contoured distributions or distributions satisfying Equation (3.18). In the following, we show that the method is also applicable in situations where the data does not necessarily come from a projective distribution.

In practice, it can naturally happen that the multivariate distribution of a considered data is not projective. In such a case, the resulting multivariate function F , obtained from the characterizing function and from multivariate projections by Equation (2.2), may not be a proper cdf. However, by construction, this function have exactly the prescribed multivariate margins.

It is quite a usual problem that estimators do not always satisfy all the required constraints: among many examples, an empirical copula is not a copula ( [START_REF] Erdely | Exact distribution under independence of the diagonal section of the empirical copula[END_REF]), some Kaplan-Meier estimated distributions are defective ( [START_REF] Portnoy | Censored regression quantiles[END_REF]), some estimated quantiles do not satisfy monotonicity ( [START_REF] Chernozhukov | Quantile and probability curves without crossing[END_REF]), some nested copulas do not satisfy C-measure positivity on all hyperrectangles ( [START_REF] Mcneil | Sampling nested archimedean copulas[END_REF][START_REF] Hofert | Densities of nested archimedean copulas[END_REF]), etc. Here the obtained F , even when it is increasing on each component, may not satisfy the positivity of its cross-derivatives. In this Section, we approximate the function F that we obtain by an admissible cdf.

We show hereafter that, even in the case where the resulting function F is not a cdf, one can find a proper cdf F + that is close to F in some sense, and such that its projections are close to the prescribed ones. Furthermore, we will see in numerical illustrations that the maximal distance between F and F + can be easily estimated, and that it is very small in some considered applications.

In the following we denote [-∞, x] = {(s 1 , . . . , s d ) :

s 1 ≤ x 1 , . . . , s d ≤ x d } for any vector x = (x 1 , . . . , x d ) ∈ R d .
+ : D → [0, 1] such that (i) F + : D → [0, 1] is a proper multivariate cdf, (ii) d KS (F, F + ) ≤ ∆ 1+∆ , (iii) for any K ⊂ {1, . . . , d}, d KS (F K , F + K ) ≤ ∆ 1+∆ where F K (x) = F (P K x), F + K (x) = F + (P K x) and P K x = (p 1 , . . . , p d ) with p i = x i if i ∈ K, p i = +∞ otherwise, as defined in (2.1). Here d KS (F, G) = sup x∈D |F (x) -G(x)|
denotes the Kolmogorov-Smirnov distance between two functions F and G. Such a function F + is given by

F + (x) = Dx f + (x)dx , where f + (x) = 1 1+∆ f (x) if x ∈ D \ D -and f + (x) = 0 otherwise. Proof. Let F + (x) = Dx f + (x)dx
, where f + is defined in the Proposition, and let D + := D \ D -for simplicity. First, f + is nonnegative and using D f (x)dx = 1, we get

D + f + (x)dx = 1, so that f + a proper pdf and item (i) holds. When ∆ > 0, define α(x) ∈ [0, 1] by α(x) := Dx∩D -|f (x)| dx D -|f (x)| dx .
and α(x) = 0 otherwise. Notice that α(x) is such that Dx∩D -f (x)dx = -α(x)∆, and

F + (x) = 1 1 + ∆ Dx∩D + f (x)dx = 1 1 + ∆ Dx f (x)dx - Dx∩D - f (x)dx .
From this, we get

F (x) -F + (x) = ∆ 1+∆ (F (x) -α(x)). As sup x∈D |F (x) -α(x)| ≤ 1 the result (ii) holds for this specific function F + . As (ii) holds, |F (x) -F + (x)| ≤ ∆
1+∆ for any x ∈ D, and in particular for any x = P K x, so that (iii) holds.

Notice that ∆ = D -|f (s)| ds = -D -f (s)ds = -(1 -D + f (s)ds) = D + f (s)ds -1, so that D + f (s)ds = 1 + ∆.
As a consequence, one can define the following failure ratio

R = D -|f (s)| ds D + |f (s)| ds = ∆ 1 + ∆ , (4.1)
which bounds the errors in Proposition 4.1. This ratio can easily be estimated by a discrete approximation of each integral, using classical techniques which avoid any normalization of the integrals.

In the case where m(t, X) = ϕ • F (t) where ϕ is a bijection from (0, 1] to R + , one gets

F L (t L ) = ϕ -1   J⊂L,J∈S α J,L • ϕ • F J (t J )   .
In this case, if ϕ, ϕ -1 and all given projections F K are differentiable up to a sufficient order, then applying chain differentiation rules and multivariate Faa Di Bruno's formula, one can show that F satisfies the assumptions of Proposition 4.1.

One can notice that in practice, depending on the application, and when ∆ is small, one may use F instead of F + , as the expression of F do not require to compute an integral. In particular, some sampling procedures like MCMC may be easily adapted to the function F instead of F + . In the latter case, the produced sample have by definition a valid multivariate cdf.

Numerical illustration on a real data set

In the following, we give an example on a real data set. First, we illustrate a natural fit procedure on a trivariate data, by adjusting each marginal distribution, and then each copula. Second, we show that the proposed construction allows to build a valid trivariate cumulative distribution with projections close enough to each of the prescribed ones.

Marginals and copulas fit

The purpose here is to provide reasonable univariate and bivariate fits using standard tools, without distorting the data, in order to illustrate the flexibility of our result and its applicability on some usual data. Surely, better fits can be proposed, but are out of the scope of the present paper.

We have obtained best univariate fits using classical existing tools in R software, and in particular the package fitdistrplus. Best copulas were obtained by the package VineCopula. q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0 5 10 15 20 (a) cdf margin F1 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 25 30 35 (b) cdf margin F2 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 1 2 3 4 1.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q (a) copula C12

-1.0 -0.5 0.0 0.5 1.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q (b) copula C13 -8 -6 -4 -2 0 2 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q (c) copula C23 We present here the results obtained for the data LifeCycleSavings from the standard R Software library datasets. We have used the first three columns of this data. We do not detail here the data, as the purpose here is just to build a parametric fit on a multivariate data, with given projections.

The marginal fits obtained for the considered data are gathered in Figure 2. One can see that, independently of our method, the fits with usual tools may be sensitive to multimodality of the data. The copula fit illustrations are presented in Figure 3. All univariate and bivariate fits are summarized in Table 1.

Concerning both the cdf and the copulas, they exhibit quite different shapes, as it usually happens on many datasets. It thus seems particularly challenging to propose a trivariate function having exactly those fitted univariate and bivariate margins. In the next paragraph we show how to build such a function, using the method proposed in this article.

m(t, X L ) = ϕ θ [F X L (t L )] . (4.2)
For the link function ϕ θ , we have chosen some parametric monotone functions such that ϕ θ (1) = 0, detailed hereafter. One can check that in this case, m 0 = m(t, X ∅ ) = ϕ θ (F X (+∞, . . . , +∞)) = 0 as defined in Remark 2.1. Using the proposed method and the result in Remark 2.7, when |L| = 3 and with t L ∈ R 3 , we obtain a fitted trivariate function, denoted F θ :

F θ (t L ) = ϕ -1 θ   -(|L| -2) J⊂L,|J|=1 ϕ θ (F X J (t J )) + J⊂L,|J|=2 ϕ θ (F X J (t J ))   . (4.3)
Notice that for some specific link functions and margins, theoretically valid distributions with this shape are given in Proposition 3.2.

As it was easier to use classical expressions with known parametric inverse functions, we have used specific link functions ϕ θ . We have considered strictly positive and decreasing functions ϕ θ : [0, 1] → R + of Table 4.1. in [START_REF] Nelsen | An Introduction to Copulas[END_REF], that are also known to be Archimedean generators. We have tried the six first strict generators of this table 4.1. (respectively Clayton, AMH, Gumbel, Frank, Joe and Hougaard generators). Then, we have selected the generator and its parameter that was minimizing the estimation of the failure ratio R = ∆ 1+∆ in Equation (4.1). The parameter θ is thus used to reduce the maximal distance between the obtained function F θ and a proper multivariate cdf F + θ , as detailed in Proposition 4.1. We also compared the trivariate function F θ and the empirical trivariate cdf F emp of the data, by computing the average absolute error δ = 1 n n i=1 |F θ (x i ) -F emp (x i )|, where x 1 , . . . , x n are the points in the dataset. The Figure 4 shows the different values of the estimated failure ratio R and the distance δ with the empirical cdf. The ratio R was estimated by replacing the integrals by sums over a regular grid of 1000 points between minimal and maximal values of each component of the dataset. Whereas the average absolute error δ is quite stable on this data, the failure ratio R is sensitive to the choice of the parameter θ. We verify hereafter that the results are not too sensitive to the grid used for the estimation.

Finally, the results for this dataset, using the Hougaard generator ϕ θ given by (4.2.9) in Table 4 For the estimation of R, using a more precise grid of 64000 points, in the extended domain [0.9m, 1.1M] where m and M are the componentwise minimum and maximum of the points in the data, we obtained quite similar results: an estimated R equal to 0.000984 at θ = 0.766. Notice that it is usual that some compatibility conditions can only be verified numerically over a grid of points, see [START_REF] Joe | Multivariate Models and Multivariate Dependence Concepts[END_REF], page 75.

One can check that the estimated admissibility of the fitted function F θ is very good, so that in practice, as one can see on Figures 5 and6, it may be unecessary to compute the proper cdf F + θ of Proposition 4.1. Indeed, the numerical approximations involved by numerical differentiation or integration may be greater to the maximal distance ∆ and the failure ratio R ≤ ∆.

Values of the fitted function on given bivariate projections correspond exactly, by construction, to the prescribed ones so that it is useless to draw these values. One can instead compute the values of the fitted function for a set of points belonging to a 1D or a 2D diagonal hyperplane. The considered data has 3 columns. For each column i in {1, 2, 3}, denote m i the minimal observed value in this column, over all observations, and M i the maximal observed value. Let m = (m 1 , m 2 , m 3 ) and M = (M 1 , M 2 , M 3 ) (so that the cube [m, M] contains all observations of the dataset). Let us also define the two points A = (M 1 , m 2 , M 3 ) and B = (m 1 , M 2 , M 3 ). Suppose that the true distribution of the data set is given by the law of a vector (X 1 , X 2 , X 3 ) and suppose that we are interested in estimating the cdf of the univariate random variable Z := max( X1-m1 M1-m1 , X2-m2 M2-m2 , X3-m3 M3-m3 ). In Figure 5, we have drawn the estimated cdf of Z coming from our construction, using P

[Z ≤ a] = P [X 1 ≤ m 1 + a(M 1 -m 1 ), ..., X 3 ≤ m 3 + a(M 3 -m 3 )].
More precisely, the figure shows the values of the function F θ that we constructed, evaluated at several points of a diagonal m + a(M -m), for different values of a. Despite that the fit is not perfect compared to the empirical values (the fits of the bivariate projections in Figure 2 were not perfect either), we observe that the estimated univariate cdf behaves normally (starting from 0 up to 1, and increasing).

Along the same lines, on the right panel of Figure 6, we have drawn values of the fitted function F θ for points belonging to a plane m + a(A -m) + b(B -m), for different values of a and b (left panel), the empirical counterpart is also drawn (right panel).

One can check that, again, the fitted function behaves normally and is increasing over each component: the admissibility problem can arise even for functions increasing on each components, with values in [0, 1], as this requirement is not sufficient to define a cdf having positive cross derivatives.

At last, we have conducted the same analysis on other datasets of the R Software library datasets. Results are gathered in the Table 2. For some data (stackloss, rock), the estimated values of R are not negligible and more investigations are needed: change in the bivariate projections, in the one-parameter link function ϕ θ or in the characterizing function m. For other datasets, we obtain very good numerical results, with sometimes estimated ratios R of order 10 -6 , and a data as MathAchieve also leads to a very good global fit of the trivariate distribution. In our experiments, we did not use the (tedious) parametric expressions of the bivariate projections density, but instead some numerical differentiation. As a result, the distance between the proper cdf F + in Proposition 4.1 and the obtained fitted function would probably be comparable to the numerical errors induced by the numerical differentiation, so that we did not build this proper cdf. We have presented here a detailed procedure for a specific one-parameter class of characterizing functions. Introducing more parameters would logically result in smaller ratios R.

The proposed method cannot guarantee that the final fitted multivariate function F does not correspond to a signed measure. However, as a conclusion of this numerical investigation, the following advantages are of practical utility: (i) The obtained fitted distribution has exactly the prescribed projections, by construction.

(ii) The Kolmogorov-Smirnov distance between the final fitted function and a proper cdf is bounded by a quantity R that can be estimated, and is small in our experiments. Thus, the fitted function can be directly used for many practical applications.

(iii) One can build a proper cdf from the final fitted function. In such a case, the projections of this proper cdf are at maximal distance R from the prescribed ones.

Conclusion

We have considered in this paper specific multivariate distributions, belonging to a class which was called projective. They rely on a linear link between some functional of the considered distribution and their multivariate margins. The choice of a linear link is is not as restrictive as one would initially imagine, since it covers a variety of classical distributions, from Elliptical ones to some natural survival models, as presented in Section 3.

In theory, for those distributions, the compatibility between given multivariate margins and multivariate distribution is automatically ensured, by definition, and the coefficients linking multivariate margins with the whole distribution are easily obtained using for example Proposition 2.2, Equation (2.14) and Equation (2.15).

In practice, when dealing with a given data, one possibility is to use a class projective distributions, as those described in Section 3 (Elliptical distributions, specific survival model, etc.), and estimate its parameters. This way, fitted projections are necessarily compatible with each other, and the admissibility is ensured for the resulting multivariate construction having prescribed projections. However, the fitted projections are then modeled by the same parametric family of functions.

Another possibility is to assume the validity of the linearity assumption for some characterizing functions belonging to a set of functions, as described in Section 4. In that case, given a data, fitted multivariate marginals, and a chosen characterizing function m, the coefficients and the expression of the candidate function for the whole multivariate distribution are easily obtained. This allows a huge variety of fitted projections. It remains to verify whether the fitted function with prescribed projections is a valid distribution, as done theoretically in the Section 3.2 (but it involves many chain rules differentiations), or numerically in Section 4.2. In the latter case, it is always possible to build a proper multivariate cdf while controlling the distance to the prescribed projections, as detailed in Section 4.1.

At last, other characterizing functions can be tried, eventually relying on several parameters. This way, one can build theoretically new classes of projective distributions, or try to satisfy, at least numerically, the validity of the fitted functions on some data.

Definition 3 . 1 (

 31 S-admissible sequence) Let D ⊂ N, and S be a decreasing subset of D.

Lemma 3 . 1 (

 31 Characterization of S-admissible sequence) Let d 0 ∈ N be such that 2 ≤ d 0 ≤ d, and assume that S = {K ⊂ D, |K| ≤ d 0 }. Assume furthermore that for any sets K, D, α K,D = α |K|,|D| . A sequence α d = (α k,d ) k≤d of coefficients is S-admissible if and only if it can be written

Proposition 3 . 1 (

 31 Elliptical distributions are projective) Consider a d-dimensional random vector X having elliptical distribution with mean µ, matrix Σ and invertible generator φ. Let D = {1, . . . , d} and [D] k = {K ⊂ D, |K| = k}. Consider that all projections are given up to a dimension d 0 , 2 ≤ d 0 ≤ d, so that S = {K ⊂ D, |K| ≤ d 0 }. Then for any S-admissible sequence α d = (α 1,d , . . . , α d0,d ), the following equality holds

Remark 3 . 1 (

 31 Matrix A in the elliptical case, d 0 = 2) Consider the elliptical case with d 0 = 2. Then we get by Lemma 3.1,

. 16 ) 3 . 1 1 .

 16311 Example Examples of survival functions in the form of Equation(3.15), or Equation(3.18) in the d-dimensional case, include the following particular cases: If ψ(x) = exp(-x), then (3.16) reduces to

Proposition 4 . 1 (

 41 Maximal admissibility distance) Let D ⊂ R d and denote D x = D ∩ [-∞, x]. Consider a function F : D → [0, 1], and assume that there exists a function f : R d → R such that for any x ∈ D, F (x) = Dx f (s)ds. Assume furthermore that D f (s)ds = 1. Denote D -= {s ∈ D, f (s) < 0} and ∆ = D -|f (s)| ds. Then there exists a function F

Figure 2 :

 2 Figure 2: Univariate margins fits: empirical cdf (discontinuous line, black) and fitted cdf (continuous line, red) obtained by R software package fitdistrplus.

Figure 3 :

 3 Figure 3: Copula fits: pseudo observations scatterplot and heatmap of the copula log-density, obtained by R software package VineCopula.

Figure 4 :

 4 Figure 4: Estimated failure ratio R (red solid line) and distance δ to empirical cdf (black dashed line), as a function of the parameter θ.

Figure 5 :

 5 Figure 5: Values of the fitted function F θ (blue solid line) and the empirical cdf F emp (black dotted line), on several points of a diagonal line m + a(M -m) in R 3 , as a function of the abscissa a.

Figure 6 :

 6 Figure 6: Level curves of the fitted function F θ (left panel) and the empirical cdf F emp (right panel), on several points of the diagonal plane m + a(A -m) + b(B -m), as a function of the abscissa a and the ordinate b (right panel).

Table 1 :

 1 Fits obtained for the considered dataset, obtained by R software packages fitdistrplus and VineCopula. Parametric expressions of the fits are those indicated in these packages.Trivariate fit with prescribed bivariate projections For more flexibility, we consider here a one-parameter class of characterizing functions, in the sense of Definition 2.1. As the given bivariate fits rely here on copulas, it is easier to deal with multivariate cdf or survival functions. The considered class is a parametric transformation of multivariate cdf:

	object	parametric fit
	F 1	norm(9.671,4.43537653418512)
	F 2	weibull(4.48356587156989,38.6038169621111)
	F 3	weibull(1.89863954400289,2.59468797211236)
	C 12	Rotated BB8 90 degrees(-6,-0.396616316431917)
	C 13	BB8(6,0.316859224873494)
	C 23	Rotated BB8 270 degrees(-6,-0.857389852442906)

  .1 of[START_REF] Nelsen | An Introduction to Copulas[END_REF], are gathered in the following table:

	estimated failure ratio R	0.00 0.01 0.02 0.03 0.04 0.05 0.00 0.01 0.02 0.03 0.04 0.05	estimated failure ratio R distance to empirical cdf		0.00 0.01 0.02 0.03 0.04 0.05	distance to empirical
		0.0 0.0		0.5 0.5	1.0 θ 1.0	1.5 1.5	2.0 2.0
	dataset (columns)	link function ϕ θ (parameter θ) estimation of R distance δ
	LifeCycleSavings(1,2,3)	(4.2.9) Hougaard (θ = 0.768)	0.000872	0.0332

Table 2 :

 2 Final estimated failure ratio R and distance δ to empirical, for different datasets or for different considered columns Results are sorted by estimated ratio R. Detail of each bivariate fits are omitted here. The horizontal dashed lines highlights the previously detailed case study. The horizontal plain line separate the cases with estimated failure ratio greater than 0.05.

	dataset (columns)	link function ϕ θ (parameter θ) estimation of R	distance δ
	MathAchieve(4,5,6)	(4.2.1) Clayton (θ = 1.10)	0.00000392	0.006324558
	LifeCycleSavings(1,3,5)	(4.2.5) Frank (θ = 2.15)	0.000136078	0.075148513
	LifeCycleSavings(2,3,5)	(4.2.9) Hougaard (θ = 0.66)	0.000243695	0.071234939
	LifeCycleSavings(1,2,5)	(4.2.9) Hougaard (θ = 0.96)	0.000365772	0.068639313
	LifeCycleSavings(1,2,3)	(4.2.9) Hougaard (θ = 0.768)	0.000872	0.0332
	airquality(1,2,4)	(4.2.1) Clayton (θ = 0.61)	0.003274217	0.026502691
	airquality(1,3,4)	(4.2.5) Frank (θ = -3.57)	0.022312892	0.010637131
	mtcars(1,5,6)	(4.2.5) Frank (θ = -5.55)	0.032730639	0.026896243
	mtcars(3,4,6)	(4.2.4) Gumbel (θ = 5.95)	0.033680629	0.034147215
	swiss(1,2,3)	(4.2.9) Hougaard (θ = 1.12)	0.056176129	0.013166825
	trees(1,2,3)	(4.2.1) Clayton (θ = 1.24)	0.058231572	0.021837067
	stackloss(2,3,4)	(4.2.5) Frank (θ = 3.14)	0.076542359	0.037345426
	rock(1,2,3)	(4.2.6) Joe (θ = 1.00)	0.099751509	0.028614845

Acknowledgment

This paper was written during an invited stay of the authors at the Vietnam Institute for Advanced Study in Mathematics (VIASM). The authors warmly thank the VIASM institute for his support. The authors are very grateful to the anonymous reviewers, the associate editor and the editor, for their careful reading and their many insightful comments and suggestions.