Jean-François Coeurjolly

Rémy Drouilhet

Keywords: Stationary Gibbs point processes, maximum pseudo-likelihood estimator, Lennard-Jones model

Asymptotic properties of the maximum pseudo-likelihood estimator for stationary Gibbs point processes including the Lennard-Jones model

Jean-François Coeurjolly, Rémy

Asymptotic properties of the maximum pseudo-likelihood estimator for stationary Gibbs point processes including the Lennard-Jones model 1 Introduction

These last years, much attention has been paid to spatial point pattern data, and especially to models and methodologies for fitting them, see [START_REF] Møller | Parametric methods for spatial point processes[END_REF] for a recent overview of this topic and [START_REF] Daley | An introduction to the Theory of Point Processes[END_REF], [START_REF] Stoyan | Stochastic geometry and its applications[END_REF] [START_REF] Møller | Statistical Inference and Simulation for Spatial Point Processes[END_REF] or [START_REF] Illian | Statistical analysis and modelling of spatial point patterns[END_REF] for more general information. For spatial point pattern data, the reference model is the Poisson point process modelling a random configuration of points with no interaction between points. In particular, this leads to the independence of any two random sub-configurations lying in two non-overlapping domains. A way to introduce dependence is to consider the class of Gibbs models. In a bounded domain, a Gibbs point process is defined through its probability measure having a Radon-Nykodym derivative with respect to a Poisson point process measure proportional to e -V (ϕ) where V (ϕ) corresponds to the energy function (i.e. a cost function expressed in terms of interactions) of the configuration of points ϕ. The definition of Gibbs models in R d is essential when dealing with asymptotic properties of estimators based on a point process observed in a domain aimed at converging towards R d . The extension of this definition is not so straightforward. The probability measure of a Gibbs point process in R d has to be defined by specifying its conditional density (indirectly expressed in terms of the energy function V (ϕ)), see e.g. [START_REF] Preston | Random fields[END_REF] or Section 2 for more details. The class of Gibbs point processes is extremely rich. The energy function can penalize points, pairs or triplets of points (see e.g. [START_REF] Baddeley | Practical maximum pseudolikelihood for spatial point patterns (with discussion)[END_REF]). More sophisticated models can also be obtained by considering interactions based on the Delaunay or the k-nearest neighbor graphs (Bertin et al. (1999b,c)), Voronoï tessellations [START_REF] Dereudre | Campbell equilibrium equation and pseudo-likelihood estimation for non-hereditary Gibbs point processes[END_REF]) or random sets [START_REF] Kendall | Quermass-interaction processes conditions for stability[END_REF], [START_REF] Dereudre | The existence of quermass-interaction processes for nonlocally stable interaction and nonbounded convex grains[END_REF]).

Following the definition of a parametric Gibbs point process, the natural question of efficiently estimating the parameters arises. Many proposals have tried to estimate the energy function from an available point pattern data. The most well-known method is the use of the likelihood function, see e.g. [START_REF] Møller | Statistical Inference and Simulation for Spatial Point Processes[END_REF] and the references therein. The main drawback of this approach is that the likelihood function contains an unknown scaling factor whose value depends on the parameters. This parametric normalizing constant is difficult to calculate from a practical point of view. From a theoretical one, it also makes asymptotic results more complicated to obtain. An alternative approach relies on the use of the pseudo-likelihood function. The idea originated from [START_REF] Besag | Spatial interaction and the statistical analysis of lattice system[END_REF] in the study of lattice processes. [START_REF] Besag | Point process limits of lattice processes[END_REF] further considered this method for pairwise interaction point processes, and [START_REF] Jensen | Pseudolikelihood for exponential family models of spatial point processes[END_REF] extended the definition of the pseudo-likelihood function to the general class of marked Gibbs point processes. The construction of the pseudo-likelihood function is based on the conditional densities which spare the computation of the scaling factor.

Our paper deals with asymptotic properties of the maximum pseudo-likelihood estimator. In order to underline our theoretical improvements, let us discuss the two main different papers discussing this topic:

• In [START_REF] Billiot | Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes[END_REF], we obtain consistency and asymptotic normality for exponential family models of Gibbs point processes, that is, on models with energy functions that are linear in terms of the parameters. Moreover, we concentrate on models such that the local energy function is local and stable. The locality of the local energy expresses that the energy to insert a point x into ϕ, that is, V (x|ϕ) = V (ϕ ∪ x) -V (ϕ), depends only on the points of ϕ falling into some ball with a fixed radius whereas the stability of the local energy (property referred as the local stability) asserts that V (x|ϕ) is bounded from below by a finite negative constant. The paper [START_REF] Billiot | Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes[END_REF] extends several papers [START_REF] Jensen | Pseudolikelihood for exponential family models of spatial point processes[END_REF], [START_REF] Jensen | On asymptotic normality of pseudo likelihood estimates of pairwise interaction processes[END_REF]) and includes a large class of examples of practical interest: area-interaction point process, Multi-Strauss marked point process based on the complete graph or the k-nearest-neighbors graph, or the Geyer's triplet point process to name a few.

• Another work has been undertaken by Mase. The consistency for non necessarily stable local energy functions (actually for superstable and lower regular ones introduced by Ruelle (1970)) is obtained in [START_REF] Mase | Consistency of maximum pseudo-likelihood estimator of continuous state space gibbsian process[END_REF] for specific models with only two parameters -the chemical potential and the inverse temperature-which can be viewed as particular exponential family models. [START_REF] Mase | Marked gibbs processes and asymptotic normality of maximum pseudo-likelihhod estimators[END_REF] extended his work to the context of marked point processes and provided asymptotic normality by adding the assumption of finite range.

Based on this literature, the main goal of this paper is to derive asymptotic properties similar to the ones presented before (consistency and asymptotic normality) but in a more general framework. We provide asymptotic results for general Gibbs point processes with non (necessarily) linear and non (necessarily) stable local energy functions. The characteristic example we have in mind is the Lennard-Jones model. This model, from statistical physics, is a stationary pairwise interaction Gibbs point process where the local energy to insert a point x into a configuration ϕ is parameterized as follows: for

θ = (θ 1 , θ 2 , θ 3 ) ∈ R 3 with θ 2 , θ 3 > 0 V LJ (x|ϕ; θ) := θ 1 + 4θ 2 y∈ϕ θ 3 y -x 12 - θ 3 y -x 6 .
Let us notice that [START_REF] Mase | Consistency of maximum pseudo-likelihood estimator of continuous state space gibbsian process[END_REF] could only propose the estimation of θ 1 and θ 2 with known θ 3 . The Lennard-Jones model is of great interest from several points of view. From a physical point of view, this model arises when theoretically modelling a pair of neutral atoms or molecules subject to two distinct forces in the limit of large separation and small separation: an attractive force at long ranges (van der Waals force, or dispersion force) and a repulsive force at short ranges (the result of overlapping electron orbitals, referred to as a Pauli repulsion from the Pauli exclusion principle). In this literature, the parameters θ 2 and θ 3 are often referred to as the depth potential and the (finite) distance at which the interparticle potential is zero. From a probabilistic point of view, this model constitutes the main example of superstable, regular and lower regular energies studied in [START_REF] Ruelle | Superstable interactions in classical statistical mechanics[END_REF] where the author proves the existence of ergodic measures for such models. Finally, from a statistical point of view, this model has been considered by several authors, see e.g. [START_REF] Ogata | Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure[END_REF], [START_REF] Goulard | Parameter estimation for marked Gibbs point processes through the maximum pseudo-likelihood method[END_REF] for fitting spatial point patterns arising in forestry. In particular, let us note that, in [START_REF] Goulard | Parameter estimation for marked Gibbs point processes through the maximum pseudo-likelihood method[END_REF], the model is fitted by using the maximum pseudo-likelihood method. As the authors do not endeavour to justify the theoretical performances of the procedure, the result proposed in Section 4 of this paper fills this gap.

The rest of the paper is organized as follows. Section 2 introduces some background and notation on Gibbs point processes (general definitions, examples). The maximum pseudo-likelihood method and asymptotic results of the derived estimator are proposed in Section 3. For general Gibbs point processes, sufficient conditions, expressed in terms of the local energy function to establish strong consistency and asymptotic normality results of this estimator are presented. While no general condition on the model is assumed to obtain the consistency, the characteristic finite range of the local energy function is required to establish the asymptotic normality. For the sake of simplicity, Section 3 (and the resulting proofs) would concentrate on non-marked Gibbs point processes. However, as we have shown in our seminal paper [START_REF] Billiot | Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes[END_REF], no real mathematical difficulty occurs with the introduction of marks. Section 4 focuses on the Lennard-Jones model. We show that the general assumptions described in Section 3 are fulfilled for this model. Proofs have been postponed until Section 5.

Background and notation

For the sake of simplicity, we consider Gibbs point processes in dimension d = 2.

General notation, configuration space

Subregions of R 2 will typically be denoted by Λ or ∆ and will always be assumed to be Borel with positive Lebesgue measure. We write Λ ⋐ R 2 if Λ is bounded. Λ c denotes the complementary set of Λ inside R 2 . The notation |.| will be used without ambiguity for different kind of objects. For a countable set J , |J | represents the number of elements belonging to J ; For Λ ⋐ R 2 , |Λ| is the volume of Λ; For a vector x ∈ R 2 , |x| corresponds to its uniform norm while x is simply its euclidean norm. For all x ∈ R 2 , ρ > 0 and i ∈ Z 2 , let B(x, ρ) := {y ∈ R 2 , |y -x| < ρ} and (i, ρ)

:= B(i, ρ) ∩ Z 2 .
A configuration is a subset ϕ of R 2 which is locally finite in that ϕ Λ := ϕ ∩ Λ has finite cardinality N Λ (ϕ) := |ϕ Λ | for all Λ ⋐ R 2 . The space Ω of all configurations is equipped with the σ-algebra F that is generated by the counting variables N Λ (ϕ) with Λ ⋐ R 2 . Finally, let T = (τ x ) x∈R 2 be the shift group, where τ x : Ω → Ω is the translation by the vector -x ∈ R 2 .

Gibbs point processes

Our results will be expressed for general stationary Gibbs point processes. Since we are interested in asymptotic properties, we have to consider these point processes acting on the infinite volume R 2 . Let us briefly recall their definition.

A point process Φ is a Ω-valued random variable, with probability distribution P on (Ω, F ). The most prominent point process is the (homogeneous) Poisson process with intensity z > 0. Recall that its probability measure π z is the unique probability measure on (Ω, F ) such that the following holds for Λ ⋐ R 2 : (i) N Λ is Poisson distributed with parameter z|Λ|, and (ii) conditionally to N Λ = n, the n points in Λ are independent with uniform distribution on Λ, for each interger n ≥ 1. For Λ ⋐ R 2 , let us denote by π z Λ the marginal probability measure in Λ of the Poisson process with intensity z.

Let θ ∈ R p (for some p ≥ 1). For any Λ ⋐ R 2 , let us consider the parametric function V Λ (.; θ) from Ω into R ∪ {+∞}. From a physical point of view, V Λ (ϕ; θ) is the energy of ϕ Λ in Λ given the outside configuration ϕ Λ c .

In this article, we focus on stationary point processes on R 2 , i.e. with T -invariant probability measure. For any Λ ⋐ R 2 , we therefore consider V Λ (.; θ)) to be T -invariant, i.e. V Λ (τ x ϕ; θ) = V Λ (ϕ; θ) for any x ∈ R 2 . Furthermore, we assume that the family of energies is hereditary, which means that for any Λ ⋐ R 2 , ϕ ∈ Ω, and x ∈ Λ:

V Λ (ϕ; θ)) = +∞ ⇒ V Λ (ϕ ∪ {x}; θ)) = +∞.
In such a context, a Gibbs measure is usually defined as follows (see [START_REF] Preston | Random fields[END_REF]).

Definition 1 A probability measure P θ on Ω is a Gibbs measure for the family of energies (V Λ (.; θ)) Λ⋐R 2 if for every Λ ⋐ R 2 , for P θ -almost every outside configuration ϕ Λ c , the law of P θ given ϕ Λ c admits the following density with respect to π z Λ : (ϕ;θ) , where Z Λ (ϕ Λ c ; θ) := ΩΛ e -VΛ(ϕΛ∪ϕ Λ c ;θ) π z Λ (dϕ Λ ) is called the partition function.

f Λ (ϕ Λ |ϕ Λ c ; θ) = 1 Z Λ (ϕ Λ c ; θ) e -VΛ
Without loss of generality, the intensity of the Poisson process, z is fixed to 1 and we simply write π and π Λ in place of π 1 and π 1 Λ . In the previous definition, we implicitly assume the consistency of the family (f Λ (.|.; θ))

Λ⋐R 2 : for any ∆ ⊂ Λ ⋐ R 2 f ∆ (ϕ ∆ |ϕ ∆ c ; θ) = f Λ (ϕ ∆ ∪ ϕ Λ\∆ |ϕ Λ c ; θ) f Λ (ϕ Λ\∆ |ϕ Λ c ; θ) = f Λ (ϕ ∆ ∪ ϕ Λ\∆ |ϕ Λ c ; θ) Ω∆ f Λ (ψ ∆ ∪ ϕ Λ\∆ |ϕ Λ c ; θ)π ∆ (dψ ∆ )
.

A sufficient condition to directly fulfill this basic ingredient is to assume the compatibility of the family (V Λ (.))

Λ⋐R 2 : for every ∆ ⊂ Λ ⋐ R 2 , the function ϕ → V Λ (ϕ; θ) -V ∆ (ϕ; θ) from Ω into R ∪ {+∞} is measurable and only depends on ϕ Λ c .
The existence of a Gibbs measure on Ω which satisfies these conditional specifications is a difficult issue. We refer the interested reader to [START_REF] Ruelle | Statistical Mechanics[END_REF]; [START_REF] Preston | Random fields[END_REF]; Bertin et al. (1999a); [START_REF] Dereudre | Gibbs Delaunay tessellations with geometric hardcore condition[END_REF]; [START_REF] Dereudre | Existence of Gibbsian point processes with geometry-dependent interactions[END_REF] for the technical and mathematical development of the existence problem. The minimal assumption of our paper is then:

[Mod-E]: Our data consist in the realization of a point process Φ with Gibbs measure P θ ⋆ , where θ ⋆ ∈ Θ, Θ is a compact subset of R p and, for any θ ∈ Θ, there exists a stationary Gibbs measure P θ for the family (V Λ (.; θ)) Λ⋐R 2 .

In the rest of this paper, the reader has mainly to keep in mind the concept of local energy defined as the energy required to insert a point x into the configuration ϕ and expressed for any Λ ∋ x by

V (x|ϕ; θ) := V Λ (ϕ ∪ {x}) -V Λ (ϕ).
From the compatibility of the family of energies, the local energy does not depend on Λ.

Our asymptotic normality result will require the following locality property assumption.

[Mod-L]: There exists D ≥ 0 such that for all ϕ ∈ Ω

V (0|ϕ; θ) = V 0|ϕ B(0,D) ; θ .

Example : Lennard-Jones models

Let us present the main example studied in this paper. We call LJ-type model the stationary pairwise interaction point process defined for some D ∈]0, +∞] by

V LJ Λ (ϕ; θ) := θ 1 |ϕ Λ | + H LJ Λ (ϕ; θ) with H LJ Λ (ϕ; θ) := x1∈ϕΛ x2∈ϕ Λ c g LJ (||x 1 -x 2 ||; θ) and g LJ (r; θ) := 4θ 2 θ 3 r 12 - θ 3 r 6 1 [0,D] (r).
As a direct consequence, the local energy function is expressed as

V LJ (x|ϕ; θ) := θ 1 + H LJ (x|ϕ; θ) with H LJ (x|ϕ; θ) := y∈ϕ g LJ (||x -y||; θ).
where θ = (θ 1 , θ 2 , θ 3 ) ∈ R × (R + ) 2 . The cases D = +∞ and D < +∞ respectively correpond to the Lennard-Jones model (briefly presented in the introduction) and the Lennard-Jones model with finite range. [START_REF] Ruelle | Superstable interactions in classical statistical mechanics[END_REF] has proved the existence of an ergodic measure for superstable, regular and lower regular potentials. The Lennard-Jones model (including the finite range one) is known to be the characteristic example of such a family of models for which Ruelle managed to prove the existence of ergodic measures for any θ ∈ R × (R + ) 2 . In order to ensure [Mod-E], it is required to assume that θ ⋆ 2 , θ ⋆ 3 > 0. Finally, [Mod-L] is satisfied for the LJ-type model with D < +∞ since the parameter D corresponds for pairwise interaction point processes to the range of the Gibbs point process.

3 Asymptotic results of the Maximum pseudo-likelihood estimator

Maximum pseudo-likelihood method

The idea of maximum pseudo-likelihood is due to [START_REF] Besag | Spatial interaction and the statistical analysis of lattice system[END_REF] who first introduced the concept for Markov random fields in order to avoid the normalizing constant. This work was then widely extended and Jensen and Møller (1991) (Theorem 2.2) obtained a general expression for Gibbs point processes. Using our notation and up to a scalar factor, the pseudo-likelihood defined for a configuration ϕ and a domain of observation Λ is denoted by P L Λ (ϕ; θ) and given by

P L Λ (ϕ; θ) = exp - Λ e -V (x|ϕ;θ) dx x∈ϕΛ e -V (x|ϕ\x;θ) . (1)
It is more convenient to define and work with the log-pseudo-likelihood, denoted by LP L Λ (ϕ; θ)

LP L Λ (ϕ; θ) = - Λ e -V (x|ϕ;θ) dx - x∈ϕΛ V (x|ϕ \ x; θ) . (2)
The point process is assumed to be observed in a domain Λ n ⊕ D = ∪ x∈Λn B(x, D) for some D < +∞. For the asymptotic normality result, it is also assumed that D ≥ D and that Λ n ⊂ R 2 can be decomposed into ∪ i∈In ∆ i where

I n = (0, n) and for i ∈ Z 2 , ∆ i = ∆ i ( D) is the square centered at i with side-length D. As a consequence, as n → +∞, Λ n → R 2 such that |Λ n | → +∞ and |∂Λ n | |Λ n | → 0.
Define for any configuration ϕ, U n (ϕ; θ) = -1 |Λn| LP L Λn (ϕ; θ). The maximum pseudolikelihood estimate (MPLE), denoted by θ n (ϕ), is then defined by

θ n (ϕ) = arg max θ∈Θ LP L Λn (ϕ; θ) = arg min θ∈Θ U n (ϕ; θ) .
The following basic notation are introduced: for j, k = 1, . . . , p and Λ ⋐ R 2

• Gradient vector of U n : U (1) n (ϕ; θ) := -|Λ n | -1 LP L (1)
Λn (ϕ; θ) where

LP L

(1)

Λ (ϕ; θ) j = Λ ∂V ∂θ j (x|ϕ; θ) e -V (x|ϕ;θ) dx - x∈ϕΛ ∂V ∂θ j (x|ϕ \ x; θ) . • Hessian matrix of U n : U (2) n (ϕ; θ) := -|Λ n | -1 LP L (2) Λn (ϕ; θ) LP L (2) Λ (ϕ; θ) j,k = Λ ∂ 2 V ∂θ j ∂θ k (x|ϕ; θ) - ∂V ∂θ j (x|ϕ; θ) ∂V ∂θ k (x|ϕ; θ) e -V (x|ϕ;θ) dx + x∈ϕΛ ∂V ∂θ j (x|ϕ \ x; θ) ∂V ∂θ k (x|ϕ \ x; θ) .
Finally, note that from the decomposition of the observation domain Λ n , one has

U (1) n (ϕ; θ) = -|Λ n | -1 i∈In LP L (1) ∆i (ϕ; θ) and U (2) n (ϕ; θ) = -|Λ n | -1 i∈In LP L (2) ∆i (ϕ; θ) .

Consistency of the MPLE

The assumption [C] gathers the following four assumptions:

[C1] For all θ ∈ Θ, E e -V (0|Φ;θ) < +∞ and E |V (0|Φ; θ)| e -V (0|Φ;θ ⋆ ) < +∞.
[C2] Identifiability condition : there exists A 1 , . . . , A ℓ , ℓ ≥ p events in Ω such that:

the ℓ events A i are disjoint and satisfy

P θ ⋆ (B i ) > 0 -for all (ϕ 1 , . . . , ϕ ℓ ) ∈ A 1 × • • • × A ℓ D(0|ϕ i ; θ) = 0 i = 1 . . . , ℓ ⇒ θ = θ ⋆ where D(0|ϕ i ; θ) := V (0|ϕ i ; θ) -V (0|ϕ i ; θ ⋆ ) [C3] The function U n (ϕ; •) is continuous for P θ ⋆ -a.e. ϕ.
[C4] For all ϕ ∈ Ω, V (0|ϕ; θ) is continuously differentiable in θ and for all j = 1, . . . , p

E max θ∈Θ ∂V ∂θ j (0|Φ; θ) e -V (0|Φ;θ) 2 < +∞.
Theorem 1 Under the assumptions [Mod-E] and [C], for P θ ⋆ -almost every ϕ, the maximum pseudo-likelihood estimate θ n (ϕ) converges towards θ ⋆ as n tends to infinity.

Asymptotic normality of the MPLE

For establishing the asymptotic normality of the MPLE we need to assume the four additional following assumptions:

[N1] For all ϕ ∈ Ω, V (0|ϕ; θ) is differentiable in θ = θ ⋆ . For all k = 1, . . . , 3 and for all λ 1 , . . . , λ k , k positive integers such that k i=1 λ i = 3 and for ∆ ⋐ R 2 E ∆ k k i=1 ∂V ∂θ j 0 M |Φ; θ ⋆ λi e -V ({x1,...,xk}|Φ;θ ⋆ ) dx 1 . . . dx k < +∞.
[N2] There exists a neighbourhood V(θ ⋆ ) of θ ⋆ such that for all ϕ ∈ Ω, V (0|ϕ; θ) is twice continuously differentiable in θ ∈ V and, for all j, k = 1, . . . , p and θ ∈ V(θ ⋆ ),

E ∂ 2 V ∂θ j ∂θ k (0|Φ; θ) e -V (0|Φ;θ) < +∞, E ∂ 2 V ∂θ j ∂θ k (0|Φ; θ) e -V (0|Φ;θ ⋆ ) < +∞,
and

E ∂V ∂θ j (0|Φ; θ) e -V (0|Φ;θ) 2 < +∞.
[N3] There exists A 1 , . . . , A ℓ , ℓ ≥ p events in Ω such that:

the ℓ events A i are disjoint and satisfy

P θ ⋆ (A i ) > 0 -for all (ϕ 1 , . . . , ϕ ℓ ) ∈ A 1 × • • • × A ℓ the (ℓ, p) matrix with entries ∂V ∂θj (0|ϕ i ; θ ⋆ ) is injective.
[N4] There exists A 0 , . . . , A ℓ , ℓ ≥ p disjoint sub-events of Ω := {ϕ ∈ Ω :

ϕ ∆i = ∅, 1 ≤ |i| ≤ 2}
such that -for j = 0, . . . , ℓ, P θ ⋆ (A j ) > 0.

for all (ϕ 0 , . . . , ϕ ℓ )

∈ A 0 × • • • × A ℓ the (ℓ, p) matrix with entries LP L (1) Λ (ϕ i ; θ ⋆ ) j - LP L (1) Λ (ϕ 0 ; θ ⋆ ) j is injective, with Λ := ∪ i∈ (0,1) .
The assumptions [N3] and [N4] will ensure (see Section 5 for more details) that the matrices U (2) (θ ⋆ ) and Σ( D, θ ⋆ ) respectively defined by

U (2) (θ ⋆ ) j,k := E ∂V ∂θ j (0|Φ; θ ⋆ ) ∂V ∂θ k (0|Φ; θ ⋆ ) e -V (0|Φ;θ ⋆ ) (3)
and

Σ( D, θ ⋆ ) = D -2 i∈ (0,1) E LP L (1) ∆0 (Φ; θ ⋆ ) LP L (1) ∆i (Φ; θ ⋆ ) T , (4)
are definite positive.

Observe that, when the energy function is linear, the expressions of the assumptions [N1] and [N2] are clearly simpler (see [START_REF] Billiot | Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes[END_REF]) and that [C2] and [N3] are similar.

Theorem 2 Under the assumptions [Mod], [C], [N1], [N2] and [N3], we have the following convergence in distribution as

n → +∞ |Λ n | 1/2 U (2) (θ ⋆ ) θ n (Φ) -θ ⋆ → N 0, Σ( D, θ ⋆ ) , (5)
where Σ( D, θ ⋆ ) is defined by (4). In addition under the assumption [N4]

|Λ n | 1/2 Σ n (Φ; θ n (Φ)) -1/2 U (2) n (Φ; θ n (Φ)) θ n (Φ) -θ ⋆ → N 0, I p , (6)
where for some θ and any configuration ϕ, the matrix Σ n (ϕ; θ) is defined by

Σ n (ϕ; θ) = |Λ n | -1
i∈In j∈ (i,1)∩In

LP L

(1) ∆i (ϕ; θ) LP L Note that [C3] is obvious since g LJ (r, •) is continuous. For the integrability type assumptions, the following Lemma will be widely used.

Lemma 4 Let Φ be a stationary pairwise interaction Gibbs point process assumed to be superstable, regular and lower regular. For i = 1, 2, define H i (x|ϕ) = y∈ϕ g i (||x -y||) with g i a continuous function. Assume that there exists ε > 0 such that there exists a positive and decreasing function g(•) such that g ε (r) := g 2 (r)-ε|g 1 (r)| ≥ -g(r) for all r > 0 and +∞ 0 rg(r)dr < +∞. Then for all k ≥ 0,

E |H 1 (0|Φ)| k e -H2(0|Φ) < +∞.
Proof. For all finite configuration ϕ Hε(0|ϕ) ,

|H 1 (0|ϕ)| k e -H2(0|ϕ) = |H 1 (0|ϕ)| k e -ε|H1(0|ϕ)| e -(H2(0|ϕ)-εH1(0|ϕ)) ≤ c(ε, k)e -(H2(0|ϕ)-εH1(0|ϕ)) , with c(ε, k) = k εe k ≤ c(ε, k)e -
where

H ε (0|ϕ) := x∈ϕ g ε (||x||).
Now, the assumptions ensure that g ε is lower regular in the Ruelle sense. We may now apply the same argument as in Lemma 3 of [START_REF] Mase | Consistency of maximum pseudo-likelihood estimator of continuous state space gibbsian process[END_REF] to prove the integrability of the random variable e -Hε(0|Φ) . Before verifying the different assumptions, let us denote by

θ inf i := inf θ∈Θ θ i , θ sup i := sup θ∈Θ θ i , θ inf := min(θ inf 2 , θ inf 3
) and θ sup := max(θ sup 2 , θ sup 3 ).

Since Θ is a compact set of R × (]0, +∞[) 2 , then θ inf > 0 and θ sup < +∞.

Assumptions [C]

Assumption [C1]

The first part is a direct application of Lemma 4. For the second part, one has to prove that for all

θ ∈ Θ E |H LJ (0|Φ; θ) |e -H LJ (0|Φ;θ ⋆ ) < +∞ Let g ε (r) = g LJ (r; θ ⋆ ) -ε|g LJ (r; θ)|. We have g ε (r) :=        4θ ⋆ 2 (θ ⋆ 3 ) 12 -ε θ 2 θ ⋆ 2 θ 12 3 r 12 - (θ ⋆ 3 ) 6 -ε θ 2 θ ⋆ 2 θ 6 3 r 6 if r ≤ θ 3 4θ ⋆ 2 (θ ⋆ 3 ) 12 +ε θ 2 θ ⋆ 2 θ 12 3 r 12 - (θ ⋆ 3 ) 6 +ε θ 2 θ ⋆ 2 θ 6 3 r 6 if r ≥ θ 3
which satisfies the assumptions of Lemma 4 as soon as ε <

θ ⋆ 3 θ3 12 θ ⋆ 2 θ2
, that is, as soon as

ε < θ inf θ sup 13 .

Assumption [C2]

Let us denote for n ≥ 1, C n = B(0, n) \ B(0, n -1) and define for m, n ≥ 1 the following configuration sets

U m,n = {ϕ ∈ Ω : |ϕ Cn | ≤ m|C n |} U m = ∩ n≥1 U m,n .
In order to prove [C2], we need the following Lemma.

Lemma 5 Let R ∈ R + , θ ∈ Θ and ϕ ∈ U m , let us denote by Z(ϕ, R; θ) := x∈ϕ B(0,R) c g LJ (||x||; θ),
then for all δ > 0 there exists R 0 such that for all

R ≥ R 0 , |Z(ϕ, R; θ)| ≤ δ.
Proof.

Z(ϕ, R; θ) = x∈ϕ B(0,R) c g LJ (||x||; θ) ≤ n≥⌈R⌉ x∈ϕC n g LJ (||x||; θ) ≤ n≥⌈R⌉ |ϕ Cn | × sup x∈Cn g LJ (||x||; θ ⋆ ) .
There exists a constant k = k(R) such that for all n ≥ ⌈R⌉, sup x∈Cn g LJ (||x||; θ ⋆ ) ≤ kn -6 . Therefore,

x∈ϕ B(0,R) c g LJ (||x||; θ) ≤ km n≥⌈R⌉ |C n | × n -6 = O   n≥⌈R⌉ n -5   ,
which leads to the result since the previous series is convergent. Let θ ∈ Θ \ θ ⋆ and consider the following configuration sets defined for k ≥ 1 and for η small enough by

A 0 = {ϕ ∈ Ω : |ϕ ∩ B(0, D)| = 0} (8) A k (η) = ϕ ∈ Ω : |ϕ ∩ B(0, D)| = |ϕ ∩ B((0, Dk -1/12 ), η)| = 1 , (9)
where D is any positive real for the Lennard-Jones model and corresponds to the range of the function g LJ (•) for the finite range Lennard-Jones model. There exists m ≥ 1 such that for all η > 0 and for k = 2, 4

P θ ⋆ (A 0 ∩ U m ) > 0 and P θ ⋆ (A k (η) ∩ U m ) > 0. Now, let ϕ 0 ∈ A 0 ∩ U m , ϕ 2 ∈ A 2 (η) ∩ U m and ϕ 4 ∈ A 4 (η) ∩ U m . First, D(0|ϕ 0 ; θ) = θ 1 -θ ⋆ 1 + Z(ϕ 0 , D; θ) -Z(ϕ 0 , D; θ ⋆ ) = 0.
For the Lennard-Jones model, according to Lemma 5 one has, for D large enough,

|Z(ϕ 0 , D; θ) -Z(ϕ 0 , D; θ ⋆ )| ≤ 1 2 |θ 1 -θ ⋆ 1 | . 10
Hence for η small enough, and for both models

0 = |D(0|ϕ 0 ; θ)| ≥ |θ 1 -θ ⋆ 1 | -|Z(ϕ 0 , D; θ) -Z(ϕ 0 , D; θ ⋆ )| ≥ 1 2 |θ 1 -θ ⋆ 1 |, which leads to θ 1 = θ ⋆ 1 . Moreover, D(0|ϕ 2 ; θ) = 4θ 2 2 θ 3 D 12 - √ 2 θ 3 D 6 -4θ ⋆ 2 2 θ ⋆ 3 D 12 - √ 2 θ ⋆ 3 D 6 +f 2 (ϕ 2 ) + Z (ϕ 2 , D; θ) -Z (ϕ 2 , D; θ ⋆ ) D(0|ϕ 4 ; θ) = 4θ 2 4 θ 3 D 12 -2 θ 3 D 6 -4θ ⋆ 2 4 θ ⋆ 3 D 12 -2 θ ⋆ 3 D 6 +f 4 (ϕ 4 ) + Z (ϕ 4 , D; θ) -Z (ϕ 4 , D; θ ⋆ ) ,
where for any

ϕ k ∈ A k (η) (k = 2, 4), there exists a positive function f k (η) converging towards zero as η → 0 such that |f k (ϕ k )| is bounded by f k (η). Now, we have 2D(0|ϕ 2 ; θ) -D(0|ϕ 4 ; θ) = 4(2 -2 √ 2) D 6 θ 2 θ 6 3 -θ ⋆ 2 θ ⋆ 3 6 + 2f (ϕ 2 ) -f 4 (ϕ 4 ) + Z ′ (ϕ 2 , ϕ 4 , D; θ, θ ⋆ ) = 0 with Z ′ (ϕ 2 , ϕ 4 , D; θ, θ ⋆ ) := 2 (Z(ϕ 2 , D; θ) -Z(ϕ 2 , D; θ ⋆ )) -(Z(ϕ 4 , D; θ) -Z(ϕ 4 , D; θ ⋆ )) .
For η small enough, we have, for any

ϕ k ∈ A k (η) (k = 2, 4), |2f (ϕ 2 ) -f 4 (ϕ 4 )| ≤ 2 f 2 (η) + f 4 (η) ≤ 1 4 4(2 -2 √ 2) D 6 |θ 2 θ 6 3 -θ ⋆ 2 θ ⋆ 3 6 |.
For the finite range Lennard-Jones model, Z ′ (ϕ 2 , ϕ 4 , D; θ, θ ⋆ ) = 0. For the Lennard-Jones model, according to Lemma 5, one has for D large enough

|Z ′ (ϕ 2 , ϕ 4 , D; θ, θ ⋆ )| ≤ 1 4 4(2 -2 √ 2) D 6 |θ 2 θ 6 3 -θ ⋆ 2 θ ⋆ 3 6 |.
Hence for η small enough, and for both models

0 = 4(2 -2 √ 2) D 6 θ 2 θ 6 3 -θ ⋆ 2 θ ⋆ 3 6 + 2f (ϕ 2 ) -f 4 (ϕ 4 ) + Z ′ (ϕ 2 , ϕ 4 , D; θ, θ ⋆ ) ≥ 4(2 -2 √ 2) D 6 |θ 2 θ 6 3 -θ ⋆ 2 θ ⋆ 3 6 | -|2f (ϕ 2 ) -f 4 (ϕ 4 )| -|Z ′ (ϕ 2 , ϕ 4 , D; θ, θ ⋆ )| ≥ 1 2 4(2 -2 √ 2) D 6 |θ 2 θ 6 3 -θ ⋆ 2 θ ⋆ 3 6 | leading to θ 2 θ 6 3 = θ ⋆ 2 θ ⋆ 3 6
. By considering the combination √ 2D(0|ϕ 2 ; θ) -D(0|ϕ 4 ; θ) and using similar arguments as previously, one obtains:

θ 2 θ 12 3 = θ ⋆ 2 θ ⋆ 3 12
. By computing the ratio of the two last equations, one obtains θ 3 = θ ⋆ 3 and then θ 2 = θ ⋆ 2 .

Assumption [C4]

For all ϕ ∈ Ω and for any θ ∈ Θ, V LJ (0|ϕ; θ) is clearly differentiable in θ. First, note that [C4] is trivial for j = 1. For j = 2, 3, let us define:

X j (ϕ; θ) := ∂V LJ ∂θj (0|ϕ; θ) e -V LJ (0|ϕ;θ) .

Our aim will be to prove that for j = 2, 3 and for all k > 0

E max θ∈Θ X j (Φ; θ) k < +∞. ( 10
)
In particular, the Assumption [C4] corresponds to (10) with k = 2. Let us notice that for all ϕ ∈ Ω and for all θ ∈ Θ

V LJ (0|ϕ; θ) ≥ V inf (0|ϕ) := θ inf + x∈ϕ g inf (||x||),
with for some r > 0, g inf (r

) := 4θ inf (θ inf ) 12 r 12 - (θ sup ) 6 r 6
. Let us also underline that for j = 2, 3

∂g LJ ∂θ j (r; θ) ≥ g inf j (r) with g inf j (r) :=        4 (θ inf ) 12 r 12 - (θ sup ) 6 r 6 if j = 2, 4m 12(θ inf ) 11 r 12 - 6(θ sup ) 5 r 6 if j = 3.
Therefore, by defining V inf j (0|ϕ) := x∈ϕ g inf j (||x||), the result (10) will be ensured by proving

E V inf j (0|Φ)e -V inf (0|Φ) < +∞.
According to Lemma 4, in order to prove this, let us denote by g j,ε (•) the function defined for j = 2, 3, for some ε > 0 and for r > 0 by g j,ε (r) = g inf j (r)ε g inf (r) . On the one hand, one has

g 2,ε (r) =        4 (θ inf ) 13 -ε(θ inf ) 12 r 12 - θ inf (θ sup ) 6 -ε(θ sup ) 6 r 6 if r ≤ (θ inf ) 2 θ sup , 4 (θ inf ) 13 +ε(θ inf ) 12 r 12 - θ inf (θ sup ) 6 +ε(θ sup ) 6 r 6 if r ≥ (θ inf ) 2 θ sup ,
which satisfies the assumptions of Lemma 4 as soon as ε < θ inf . On the other hand

g 3,ε (r) =          4θ inf (θ inf ) 12 -12ε(θ inf ) 11 r 12 - (θ sup ) 6 -6ε(θ sup ) 5 r 6 if r ≤ 2 (θ inf ) 11 (θ sup ) 5 1/6 4θ inf (θ inf ) 12 +12ε(θ inf ) 11 r 12 - (θ sup ) 6 +6ε(θ sup ) 5 r 6 if r ≥ 2 (θ inf ) 11 (θ sup ) 5 1/6
, which satisfies the assumptions of Lemma 4 as soon as ε < θ inf /12, which ends the proof.

Assumptions [N]

Assumption [N1]

Let us present two auxiliary lemmas.

Lemma 6 Let ϕ be the realization of a stationary pairwise interaction point process with local energy function defined by

V (x|ϕ; θ) = θ 1 + H (x|ϕ; θ) with H (x|ϕ; θ) = y∈ϕ g(||y -x||; θ).
Let K < +∞ and let x 1 , . . . , x K ∈ R 2 \ ϕ, x i = x j for i, j = 1, . . . , K (where K < +∞), then

H ({x 1 , . . . , x K }|ϕ; θ) = K k=1 H (x k |ϕ; θ) + H ({x 1 , . . . , x K }; θ) V ({x 1 , . . . , x K }|ϕ; θ) = K k=1 V (x k |ϕ; θ) + H ({x 1 , . . . , x K }; θ)
This result comes from the definition of the local energy.

Lemma 7 Using the same notation and under the same assumptions of Lemma 6, assume that there exists g min such that for all r > 0 and any θ ∈ Θ, g(r; θ) ≥ g min , then

e -V ({x1,...,xK}|ϕ;θ) ≤ c K K k=1 e -V (xk|ϕ;θ) with c K = e -K(K-1) 2 gmin
Proof. The proof is immediate since

H ({x 1 , . . . , x K }; θ) = i<j g(||x i -x j ||; θ) ≥ K(K -1) 2 g min .
Let k = 1, . . . , 3 and let λ 1 , . . . , λ k , k positive integers such that k i=1 λ i = 3 and define the random variable

A(Φ) := ∆ k k i=1 ∂V ∂θ j (x i |Φ; θ ⋆ ) λi e -V ({x1,...,xk}|Φ;θ ⋆ ) dx i .
From Lemma 7, we have

E (A(Φ)) ≤ E c k ∆ k k i=1 ∂V ∂θ j (x i |Φ; θ ⋆ ) λi e -V (xi|Φ;θ ⋆ ) dx i = c k ∆ k E k i=1 ∂V ∂θ j (x i |Φ; θ ⋆ ) λi e -V (xi|Φ;θ ⋆ ) dx 1 . . . dx k ≤ c k ∆ k k i=1 E ∂V ∂θ j (x i |Φ; θ ⋆ ) k e -k λ i V (xi|Φ;θ ⋆ ) 1/k dx 1 . . . dx k = c k k i=1 ∆ E ∂V ∂θ j (x i |Φ; θ ⋆ ) k e -k λ i V (xi|Φ;θ ⋆ ) 1/k dx i = c k |∆| k k i=1 E ∂V ∂θ j (0|Φ; θ ⋆ ) k e -k λ i V (0|Φ;θ ⋆ ) 1/k
by using Hölder's inequality and the stationarity of the process. The result is then a simple consequence of ( 10) and Lemma 4.

Assumption [N2]

For all ϕ ∈ Ω, it is clear that for all θ ∈ Θ, V (0|ϕ; θ) is twice continuously differentiable in θ.

According to Lemma 4 and the fact that [N1] is satisfied, it is sufficient to prove that for all j, k = 1, 2, 3

E ∂ 2 V LJ ∂θ j ∂θ k (0|Φ; θ) e -V LJ (0|Φ;θ) < +∞.
This is obvious when either j or k equals 1 and when j = k = 2 (since ∂ 2 g LJ (∂θ2) 2 (r; •) = 0). Now, for the other cases, define for θ ∈ Θ g j,k,ε (r) := g LJ (r; θ)ε ∂ 2 g LJ ∂θj ∂θ k (r; θ) . We have

g 2,3,ε (r) = g 3,2,ε (r) =    4 θ2θ 12 3 -12εθ 11 3 r 12 - θ 6 3 -6εθ 5 3 r 6 if r ≤ 2 1/6 4 θ2θ 12 3 +12εθ 11 3 r 12 - θ 6 3 +6εθ 5 3 r 6
otherwise which satisfies the assumptions of Lemma 4 as soon as ε < θ2θ3 12 , that is, as soon as ε < (θ inf ) 2 12 . Finally,

g 3,3,ε (r) =    4 θ2θ 12 3 -132εθ 10 3 r 12 - θ2θ 6 3 -30εθ 4 3 r 6 if r ≤ 132 30 1/6 θ 3 4 θ2θ 12 3 +132εθ 10 3 r 12 - θ2θ 6 3 +30εθ 4 3 r 6
otherwise which satisfies the assumptions of Lemma 4 as soon as ε <

θ2θ 2 3
132 , that is, as soon as ε < (θ inf ) 3 132 .

Assumption [N3]

Let y = (y 1 , y 2 , y 3 ) ∈ R 3 and g(y, ϕ) := y T V

(1)

LJ (0|ϕ; θ ⋆ ). Let ϕ 0 ∈ A 0 and ϕ k (η) ∈ A k (η) (k = 2, 4
) where A 0 and A k (η) are defined by ( 8) and ( 9). Assume g(y, ϕ k ) = 0 for k = 0, 2, 4.

Since, g(y, ϕ 0 ) = y 1 , we have y 1 = 0. Now,

g(y, ϕ 2 ) = 4y 2 2 θ ⋆ 3 D 12 - √ 2 θ ⋆ 3 D 6 + 4y 3 θ ⋆ 2 2 12θ ⋆ 3 11 D 12 - √ 2 6θ ⋆ 3 5 D 6 + f 2 (y, ϕ 2 ) g(y, ϕ 4 ) = 4y 2 4 θ ⋆ 3 D 12 -2 θ ⋆ 3 D 6 + 4y 3 θ ⋆ 2 4 12θ ⋆ 3 11 D 12 -2 6θ ⋆ 3 5 D 6 + f 4 (y, ϕ 4 ),
where for any ϕ k ∈ A k (η) (k = 2, 4), there exists a positive function f k (y, η) converging towards zero as

η → 0 such that |f k (y, ϕ k )| is bounded by f k (y, η). Now, we have 2g(y, ϕ 2 ) -g(y, ϕ 4 ) = 4(2 -2 √ 2)) θ ⋆ 3 5 D 6 (θ ⋆ 3 y 2 + 6θ ⋆ 2 y 3 ) + 2f 2 (y, ϕ 2 ) -f 4 (y, ϕ 4 ) = 0.
For η small enough, we have, for any

ϕ k ∈ A k (η) (k = 2, 4), |2f (y, ϕ 2 ) -f 4 (y, ϕ 4 )| ≤ 2| f 2 (y, η)| + | f 4 (y, η)| ≤ 1 2 4(2 -2 √ 2) θ ⋆ 3 5 D 6 (θ ⋆ 3 y 2 + 6θ ⋆ 2 y 3 ) .
Hence for η small enough,

0 = |2g(y, ϕ 2 ) -g(y, ϕ 4 )| ≥ 1 2 4(2 -2 √ 2) θ ⋆ 3 5 D 6 (θ ⋆ 3 y 2 + 6θ ⋆ 2 y 3 ) ,
leading to the equation θ ⋆ 3 y 2 + 6θ ⋆ 2 y 3 = 0. By considering the linear combination √ 2g(y, ϕ 2 )g(y, ϕ 4 ), we may obtain the equation θ ⋆ 3 y 2 + 12θ ⋆ 2 y 3 = 0 with similar arguments. Both equations lead to y 2 = y 3 = 0.

Assumption [N4]

The assumption [N4] may be rewritten for all k = 1, • • • , ℓ and for all ϕ k ∈ A k and ϕ 0 ∈ A 0 :

∀y ∈ R 3 , y T LP L (1) Λ (ϕ k ; θ ⋆ ) -LP L (1) Λ (ϕ 0 ; θ ⋆ ) = y T (L(ϕ k ; θ ⋆ ) -R(ϕ k ; θ ⋆ )) = 0 =⇒ y = 0.
where for any configuration ϕ ∈ Ω and

ϕ 0 ∈ A 0 L(ϕ; θ ⋆ ) := Λ V (1) LJ (x|ϕ; θ ⋆ ) e -V LJ (x|ϕ;θ ⋆ ) dx - Λ V (1) LJ (x|ϕ 0 ; θ ⋆ ) e -V LJ (x|ϕ0;θ ⋆ ) dx R(ϕ; θ ⋆ ) := x∈ϕ∩Λ V (1) LJ (x|ϕ \ x; θ ⋆ ) - x∈ϕ0∩Λ V (1) LJ (x|ϕ 0 \ x; θ ⋆ ) .
Concerning this assumption, we choose

ϕ 0 ∈ A 0 = ϕ ∈ Ω : ϕ ∆0 = ∅ . Let y ∈ R 3 then Λ y T V (1) LJ (x|ϕ 0 ; θ ⋆ ) e -V LJ (x|ϕ0;θ ⋆ ) dx = y 1 e -θ ⋆ 1 Λ and x∈ϕ0∩Λ y T V (1) LJ (x|ϕ 0 \ x; θ ⋆ ) = 0.
Consider the following configuration set, defined for η, ε > 0, by

A 2 (η, ε) = ϕ ∈ Ω : ϕ ∆0 = {z 1 , z 2 } where z 1 ∈ B(0, η), z 2 ∈ B((0, 2η + ε), η) . Note that for z 1 ∈ B(0, η), z 2 ∈ B((0, 2η + ε), η), ε ≤ ||z 2 -z 1 || ≤ ε + 4η. Let ϕ 2 ∈ A 2 (η, ε) and
x ∈ Λ, then one may prove that for j = 2, 3

V LJ (x|ϕ 2 ; θ ⋆ ) = θ ⋆ 1 + 2g LJ (||x||; θ ⋆ ) + f (x, η, ε) ∂V LJ ∂θ j (x|ϕ 2 ; θ ⋆ ) = 2 ∂g LJ ∂θ j (||x||; θ ⋆ ) + f j (x, η, ε)
where f (x, η, ε) and f j (x, η, ε) are such that lim

(η,ε)→(0,0) f (x, η, ε) = lim (η,ε)→(0,0) f j (x, η, ε) = 0.
On the one hand, one may prove that there exists a function f L (y, η, ε) such that lim (η,ε)→(0,0) f L (y, η, ε) = 0 and such that

y T L(ϕ 2 ; θ ⋆ ) = y T I -y 1 e -θ ⋆ 1 |Λ| + f L (y, η, ε)
where

I := Λ h(||x||; θ ⋆ )e -θ ⋆ 1 -2g LJ (||x||;θ ⋆ ) dx and h(r; θ ⋆ ) := 1, 2 ∂g LJ ∂θ 2 (r; θ ⋆ ), 2 ∂g LJ ∂θ 3 (r; θ ⋆ ) T .
On the other hand, there exists a function f R (y, η, ε) such that lim η→0 f R (y, η, ε) = 0

y T R(ϕ 2 ; θ ⋆ ) = 2y 1 + 2y 2 4 θ ⋆ 3 ε 12 - θ ⋆ 3 ε 6 + 2y 3 4θ ⋆ 2 12θ ⋆ 3 11 ε 12 - 6θ ⋆ 3 5 ε 6 + f R (y, η, ε).
Since

ε 12 y T (L(ϕ 2 ; θ ⋆ ) -R(ϕ 2 ; θ ⋆ )) = ε 12 y T I -y 1 e -θ ⋆ 1 |Λ| + f L (y, η, ε) -f R (y, η, ε) -ε 6 2y 2 4θ ⋆ 3 6 + 2y 3 4θ ⋆ 2 6θ ⋆ 3 5 + 2y 2 4θ ⋆ 3 12 + 2y 3 4θ ⋆ 2 12θ ⋆ 3 11 .
For η and ε chosen small enough, one may prove that

0 = ε 12 y T (L(ϕ 2 ; θ ⋆ ) -R(ϕ 2 ; θ ⋆ )) ≥ 1 2 2y 2 4θ ⋆ 3 12 + 2y 3 4θ ⋆ 2 12θ ⋆ 3 11 leading to 2y 2 4θ ⋆ 3 12 + 2y 3 4θ ⋆ 2 12θ ⋆ 3 11 = 0 ⇔ θ ⋆ 3 y 2 + 12θ ⋆ 2 y 3 = 0. (11)
This means that

y T R(ϕ 2 ; θ ⋆ ) = 2y 1 - 1 ε 6 2y 2 4θ ⋆ 3 6 + 2y 3 4θ ⋆ 2 6θ ⋆ 3 5 + f R (y, η, ε).
With similar arguments, we obtain that

2y 2 4θ ⋆ 3 6 + 2y 3 4θ ⋆ 2 6θ ⋆ 3 5 = 0 ⇔ θ ⋆ 3 y 2 + 6θ ⋆ 2 y 3 = 0. (12)
Equations ( 11) and ( 12) lead to y 2 = y 3 = 0. Now consider the following configuration set defined for some k ≥ 1 and η > 0

A k (η) = ϕ ∈ Ω : ϕ ∆0 = |ϕ ∩ B(0, η)| = k and let ϕ k ∈ A k (η).
Then, one may prove that there exists a function f L (y, η) such that lim η→0 f L (y, η) = 0 and such that

y T (L(ϕ k ; θ ⋆ ) -R(ϕ k ; θ ⋆ )) = y 1 Λ e -θ ⋆ 1 e -kg LJ (||x||;θ ⋆ ) -1 dx -ky 1 + f L (y, η) = 0.
Let us denote by Λ 1 := B(0, min(θ ⋆ 3 , D)) and Λ 2 := B(0, D) \ Λ 1 Now let us consider two cases. Case 1: θ ⋆ 3 ≤ D. First note that for all x ∈ Λ, g LJ (||x||; θ ⋆ ) ≥ 0. Then, for k large enough and for η small enough, we have

1 k Λ1 e -θ ⋆ 1 e -kg LJ (||x||;θ ⋆ ) -1 dx ≤ |Λ 1 | k e -θ ⋆ 1 ≤ 1 4 and 1 k f L (y, η) ≤ |y 1 | 4 .
Hence for k large enough and for η small enough, we may obtain

0 = 1 k y T (L(ϕ k ; θ ⋆ ) -R(ϕ k ; θ ⋆ )) ≥ |y 1 | -y 1 1 k Λ1 e -θ ⋆ 1 e -kg LJ (||x||;θ ⋆ ) -1 dx + 1 k f L (y, η) ≥ |y 1 | - |y 1 | 4 - |y 1 | 4 = |y 1 | 2 , which leads to y 1 = 0. Case 2: θ ⋆ 3 ≥ D. First note that for all x ∈ Λ 2 , g LJ (||x||; θ ⋆ ) ≤ g m := g LJ (D; θ ⋆ ) = 4θ ⋆ 2 θ ⋆ 3 D 12 - θ ⋆ 3 D 6 < 0.
On the one hand, for k large enough and for η small enough, we may have

1 k y 1 Λ1 e -θ ⋆ 1 e -kg LJ (||x||;θ ⋆ ) -1 dx + 1 k f L (y, η) -y 1 ≤ |y 1 | 2 + |y 1 | ≤ 3 2 |y 1 |.
On the other hand, we have for k large enough

1 k y 1 Λ2 e -θ ⋆ 1 e -kg LJ (||x||;θ ⋆ ) -1 dx = |y 1 | k Λ2 e -θ ⋆ 1 e -kg LJ (||x||;θ ⋆ ) -1 dx ≥ |y 1 | k e -θ ⋆ 1 |Λ 2 | e -kgm -1 = |y 1 |e -θ ⋆ 1 e k|gm| -1 k ≥ 2|y 1 |.
Therefore for k large enough and for η small enough, we have

0 = 1 k y T (L(ϕ k ; θ ⋆ ) -R(ϕ k ; θ ⋆ )) ≥ 2|y 1 | - 3 2 |y 1 | = |y 1 | 2 ,
which leads to y 1 = 0.

5 Annex: proofs of Theorems 1 and 2

Let us start by presenting a particular case of the Campbell Theorem combined with the Glötz Theorem that is widely used in our future proofs.

Corollary 8 Assume that the point process Φ with probability measure P is stationary. Let Λ ⋐ R 2 , ϕ ∈ Ω and let g be a function satisfying g(x, ϕ) = g(0, τ x ϕ) for all x ∈ R 2 . Define f (ϕ) = g(0, ϕ)e -V (0|ϕ) and assume that f ∈ L 1 (P ). Then,

E x∈ΦΛ g(x, Φ \ x) = E Λ g(x, Φ)e -V (x|Φ) dx = |Λ| E g (0, Φ) e -V (0|Φ) (13)
Proof. see Corollary 3 of [START_REF] Billiot | Maximum pseudolikelihood estimator for exponential family models of marked Gibbs point processes[END_REF] Let us now present a version of an ergodic theorem obtained by [START_REF] Nguyen | Ergodic theorems for Spatial Process[END_REF] and widely used in this paper. Let ∆ 0 be a fixed bounded domain Theorem 9 [START_REF] Nguyen | Ergodic theorems for Spatial Process[END_REF]) Let {H G , G ∈ B b } be a family of random variables, which is covariant, that for all x ∈ R 2 , H τxG (τ x ϕ) = H G (ϕ), for a.e. ϕ and additive, that is for every disjoint G 1 , G 2 ∈ B b ,

H G1∪G2 = H G1 + H G2 , a.s.
Let I be the sub-σ-algebra of F consisting of translation invariant (with probability 1) sets. Assume there exists a nonnegative and integrable random variable

Y such that |H G | ≤ Y a.s. for every convex G ⊂ ∆ 0 . Then, lim n→+∞ 1 |G n | H Gn = 1 |∆ 0 | E(H ∆0 |I), a.s.
for each regular sequence G n → R 2 .

Proof of Theorem 1

Due to the decomposition of stationary measures as a mixture of ergodic measures (see [START_REF] Preston | Random fields[END_REF]), one only needs to prove Theorem 1 by assuming that P θ ⋆ is ergodic. From now on, P θ ⋆ is assumed to be ergodic. The tool used to obtain the almost sure convergence is a convergence theorem for minimum contrast estimators established by [START_REF] Guyon | Champs aléatoires sur un réseau[END_REF]. We proceed in three stages.

Step 1. Convergence of U n (Φ; θ).

Decompose U n (ϕ; θ) = 1 |Λn| (H 1,Λn (ϕ) + H 2,Λn (ϕ)) with H 1,Λn (ϕ) = Λn e -V (x|ϕ;θ) dx and H 2,Λn (ϕ) = x∈ΦΛ n V (x|ϕ \ x; θ) .
Under the assumption [C1], one can apply Theorem 9 [START_REF] Nguyen | Ergodic theorems for Spatial Process[END_REF]) to the process H 1,Λn . And from Corollary 8, we obtain P θ ⋆ -almost surely as n → +∞

1 |Λ n | H 1,Λn (Φ) → E e -V (0|Φ;θ) . (14)
Now, let G ⊂ ∆ 0 , we clearly have

|H 2,G (ϕ)| ≤ x∈ϕG |V (x|ϕ \ x; θ) | ≤ x∈ϕ∆ 0 |V (x|ϕ \ x; θ) |.
Under the assumption [Mod] and from Corollary 8, we have

E   x∈Φ∆ 0 |V (x|Φ \ x; θ) |   = |∆ 0 |E |V (0|Φ; θ) |e -V (0|Φ;θ ⋆ ) < +∞
This means that for all G ⊂ ∆ 0 , there exists a random variable Y ∈ L 1 (P θ ⋆ ) such that |H 2,G (Φ)| ≤ Y . Thus, under the assumption [C1] and from Theorem 9 [START_REF] Nguyen | Ergodic theorems for Spatial Process[END_REF]) and from Corollary 8, we have P θ ⋆ -almost surely

1 |Λ n | H 2,Λn (Φ) → 1 |∆ 0 | E x∈Φ∆ 0 V (x|Φ \ x; θ) = E V (0|Φ; θ) e -V (0|Φ;θ ⋆ ) . ( 15
)
We have the result by combining ( 14) and ( 15): P θ ⋆ -almost surely

U n (Φ; θ) → U (θ) = E e -V (0|Φ;θ) + V (0|Φ; θ) e -V (0|Φ;θ ⋆ ) ( 16
)
Step

= θ ⋆ ) such that P θ ⋆ -almost surely U n (Φ; θ)-U n (Φ; θ ⋆ ) → K(θ, θ ⋆ ). From Step 1, we have K(θ, θ ⋆ )=E e -V (0|Φ;θ ⋆ ) e V (0|Φ;θ)-V (0|Φ;θ ⋆ ) -1 + V (0|Φ; θ) -V (0|Φ; θ ⋆ ) . (17)
Since the function t → e t -(1 + t) is nonnegative and is equal to zero if and only if t = 0, K(θ, θ ⋆ ) ≥ 0 and

K(θ, θ ⋆ ) = 0 ⇔ e V (0|ϕ;θ)-V (0|ϕ;θ ⋆ ) -1 + V (0|ϕ; θ) -V (0|ϕ; θ ⋆ ) = 0 ⇔ D (0|ϕ; θ) := V (0|ϕ; θ) -V (0|ϕ; θ ⋆ ) = 0
for P θ ⋆a.e. ϕ. Let us consider the ℓ events A j (j = 1, . . . , ℓ) defined in Assumption [C2]. The previous equation is at least true for ϕ j ∈ A j , which leads under Assumption [C2] to θ = θ ⋆ . Therefore, K(θ, θ ⋆ ) = 0 ⇒ θ = θ ⋆ . The converse is trivial. Before ending this step, note that the assumption [C3] asserts that for any ϕ, U n (ϕ; •) and K(•, θ ⋆ ) are continuous functions.

Step 3. Modulus of continuity.

The modulus of continuity of the contrast process defined for all ϕ ∈ Ω and all η > 0 by

W n (ϕ, η) = sup U n (ϕ; θ) -U n (ϕ; θ ′ ) : θ, θ ′ ∈ Θ, ||θ -θ ′ || ≤ η
is such that there exists a sequence (ε k ) k≥1 , with ε k → 0 as k → +∞ such that for all k ≥ 1

P lim sup n→+∞ W n Φ, 1 k ≥ ε k = 0. ( 18
)
Let us start to write

W n ϕ, 1 k ≤ W 1,n ϕ, 1 k + W 2,n ϕ, 1 k with W 1,n ϕ, 1 k := sup W ′ 1,Λn (ϕ; θ, θ ′ ) : θ, θ ′ ∈ Θ, ||θ -θ ′ || ≤ 1 k W 2,n ϕ, 1 k := sup W ′ 2,Λn (ϕ; θ, θ ′ ) : θ, θ ′ ∈ Θ, ||θ -θ ′ || ≤ 1 k . and W ′ 1,Λn (ϕ; θ, θ ′ ) := 1 |Λ n | Λn e -V (x|ϕ;θ) -e -V (x|ϕ;θ ′ ) dx W ′ 2,Λn (ϕ; θ, θ ′ ) := 1 |Λ n | x∈ϕΛ n V (x|ϕ \ x; θ) -V x|ϕ \ x; θ ′ .
Let k ≥ 1 and let θ, θ ′ ∈ Θ such that ||θθ ′ || ≤ 1 k , then under the assumption [C1] and from Theorem 9 and Corollary 8, we have P θ ⋆ -almost surely as n → +∞

W ′ 1,Λn (Φ; θ, θ ′ ) -→ E e -V (0|Φ;θ) -e -V (0|Φ;θ ′ ) W ′ 2,Λn (Φ; θ, θ ′ ) -→ E V (0|Φ; θ) -V 0|Φ; θ ′ e -V (0|Φ;θ ⋆ )
Under Assumption [C4], one may apply the mean value theorem in R p as follows: there exist

ξ (1) , . . . , ξ (p) ∈ p j=1 min(θ j , θ ′ j ), max(θ j , θ ′ j ) such that for all ϕ ∈ Ω e -V (0|ϕ;θ) -e -V (0|ϕ;θ ′ ) = p j=1 θ j -θ ′ j ∂V ∂θ j 0|ϕ; ξ (j) e -V 0|ϕ;ξ (j)
.

This leads, under Assumption [C4], to the following inequality

E e -V (0|Φ;θ) -e -V (0|Φ;θ ′ ) 2 ≤ E e -V (0|Φ;θ) -e -V (0|Φ;θ ′ ) 2 ≤ E   ||θ -θ ′ || 2 p j=1 ∂V ∂θ j 0|Φ; ξ (j) e -V 0|Φ;ξ (j) 2 .   ≤ 1 k 2 γ 2 1 ,
with γ 1 := E p j=1 max θ∈Θ ∂V ∂θj (0|Φ; θ) e -V (0|Φ;θ) 2 < +∞. In such a way, one may also prove that

E V (0|Φ; θ) -V 0|Φ; θ ′ e -V (0|Φ;θ ⋆ ) 2 ≤ 1 k 2 γ 2 2 , with γ 2 := E p j=1 max θ∈Θ ∂V ∂θj (0|Φ; θ) e -V (0|Φ;θ ⋆ ) 2 .
Hence, for all k ≥ 1 and for all θ, θ ′ ∈ Θ such that ||θθ ′ || ≤ 1 k there exists n 0 (k) ≥ 1 such that for all n ≥ n 0 (k), we have

W ′ 1,Λn ϕ; θ, θ ′ ≤ 2 k γ 1 and W ′ 2,Λn ϕ; θ, θ ′ ≤ 2 k γ 2 , for P θ ⋆ -a.e. ϕ.
Since γ 1 and γ 2 are independent of θ and θ ′ , we have for all n ≥ n 0 (k)

W n ϕ, 1 k ≤ W 1,n ϕ, 1 k + W 2,n ϕ, 1 k ≤ 2 k (γ 1 + γ 2 ) := c k , for P θ ⋆ -a.e. ϕ. Finally, since lim sup n→+∞ W n ϕ, 1 k ≥ c k = m∈N n≥m W n ϕ, 1 k ≥ c k ⊂ n≥n0(k) W n ϕ, 1 k ≥ c k
for P θ ⋆ -a.e. ϕ, the expected result ( 18) is proved.

Conclusion step.

The Steps 1, 2 and 3 ensure the fact that we can apply Property 3.6 of [START_REF] Guyon | Champs aléatoires sur un réseau[END_REF] which asserts the almost sure convergence for minimum contrast estimators.

Proof of Theorem 2

Step 1. Asymptotic normality of U (1) n (Φ; θ ⋆ ) The aim is to prove the following convergence in distribution as n → +∞

|Λ n | 1/2 U (1) n (Φ; θ ⋆ ) → N 0, Σ( D, θ ⋆ ) ( 19
)
where the matrix Σ( D, θ ⋆ ) is defined by ( 4).

The idea is to apply to U (1) n (Φ; θ ⋆ ) a central limit theorem obtained by [START_REF] Jensen | On asymptotic normality of pseudo likelihood estimates of pairwise interaction processes[END_REF], Theorem 2.1. The following conditions have to be fulfilled to apply this result. For all j = 1, . . . , p

(i) For all i ∈ Z 2 , E LP L (1) ∆i (Φ; θ ⋆ ) j |Φ ∆ c i = 0. (ii) For all i ∈ Z 2 , E LP L (1) ∆i (Φ; θ ⋆ ) j 3 < +∞. (iii) The matrix Var |Λ n | 1/2 U (1) n (Φ; θ ⋆ ) converges to the matrix Σ( D, θ ⋆ ).
Condition (i) : From the stationarity of the process, it is sufficient to prove that

E LP L (1) ∆0 (Φ; θ ⋆ ) j |Φ ∆ c 0 = 0.
Recall that for any configuration ϕ LP L

(1)

∆0 (ϕ; θ ⋆ ) j = - ∆0 ∂V ∂θ j (x|ϕ; θ ⋆ ) e -V (x|ϕ;θ ⋆ ) dx + ∆0 ∂V ∂θ j (x|ϕ \ x; θ ⋆ ) ϕ(dx). ( 20
)
Denote respectively by G 1 (ϕ) and G 2 (ϕ) the first and the second right-hand term of (20) and by

E i = E G i (Φ)|Φ ∆ c 0 = ϕ ∆ c 0 .
Let us define for any ϕ, the measure µ ϕ := x∈ϕ δ x . From the definition of Gibbs point processes,

E 2 = 1 Z ∆0 (ϕ ∆ c 0 ) Ω∆ 0 π ∆0 (dϕ ∆0 ) R 2 µ ϕ∆ 0 (dx)1 ∆0 (x) ∂V ∂θ j (x|ϕ \ x; θ ⋆ ) e -V∆ 0 (ϕ;θ ⋆ ) .
Since π is a Poisson process,

Ω∆ 0 π ∆0 (dϕ ∆0 )f (ϕ) = Ω∆ 0 π ∆0 (dϕ ∆0 ) Ω ∆ c 0 π ∆ c 0 (dϕ ′ ∆ c 0 )f (ϕ)
and therefore, by introducing ψ :

= ϕ ∆0 ∪ ϕ ′ ∆ c 0 E 2 = 1 Z ∆0 (ϕ ∆ c 0 ) Ω π(dψ) R 2 µ ψ (dx)1 ∆0 (x) ∂V ∂θ j x|ψ ∆0 ∪ ϕ ∆ c 0 \ x; θ ⋆ e -V∆ 0 ψ∆ 0 ∪ϕ ∆ c 0 ;θ ⋆ . Now, from Campbell Theorem (applied to the Poisson measure π) E 2 = 1 Z ∆0 (ϕ ∆ c 0 ) ∆0 dx Ω π ! x (dψ) ∂V ∂θ j x|ψ ∆0 ∪ ϕ ∆ c 0 ; θ ⋆ e -V∆ 0 x∪ψ∆ 0 ∪ϕ ∆ c 0 ;θ ⋆ ,
where π ! x stands for the reduced Palm distribution of the Poisson point process. Since from Slivnyak-Mecke Theorem (see e.g. [START_REF] Møller | Statistical Inference and Simulation for Spatial Point Processes[END_REF]), π = π !

x , one can obtain

E 2 = 1 Z ∆0 (ϕ ∆ c 0 ) Ω π(dψ) ∆0 dx ∂V ∂θ j x|ψ ∆0 ∪ ϕ ∆ c 0 ; θ ⋆ e -V∆ 0 x∪ψ∆ 0 ∪ϕ ∆ c 0 ;θ ⋆ = 1 Z ∆0 (ϕ ∆ c 0 ) Ω∆ 0 π ∆0 (dϕ ∆0 ) ∆0 dx ∂V ∂θ j (x|ϕ; θ ⋆ ) e -V (x|ϕ;θ ⋆ ) e -V∆ 0 (ϕ;θ ⋆ ) = -E 1
Condition (ii) : For any bounded domain ∆ one may write for j = 1, . . . , p LP L Let j ∈ I n ∩ (i, 1) c , since LP L

(1) ∆i (ϕ; θ ⋆ ) is a measurable function of ϕ ∆ c i , we have by using condition (i):

E LP L

(1) ∆i (Φ; θ ⋆ ) LP L Step 2. Domination of U (2) n (Φ; θ) in a neighborhood of θ ⋆ and convergence of U (2) n (Φ; θ ⋆ ) Let j, k = 1, . . . , p, recall that U (2) n (ϕ; θ)

j,k
is defined in a neighborhood V(θ ⋆ ) of θ ⋆ for any configuration ϕ by Note that from Theorem 9 [START_REF] Nguyen | Ergodic theorems for Spatial Process[END_REF]), U (2) n (•; θ ⋆ ) converges almost surely as n → +∞ towards U (2) (θ ⋆ ) defined by (3). Note that U (2) (θ ⋆ ) is a symmetric positive matrix since for all y ∈ R p y T U (2) (θ ⋆ )y = E y T V (1) (0|Φ; θ ⋆ ) e -V (0|Φ;θ ⋆ ) ≥ 0, where for j = 1, . . . , p, ϕ ∈ Ω and for θ ∈ V(θ ⋆ ) V (1) (x|ϕ; θ ⋆ ) j := ∂V ∂θj (x|ϕ; θ) and it is a definite matrix under the assumption [N3].

U (2) n (ϕ; θ) j,k = - 1 |Λ n | Λn ∂ 2 V

Conclusion

Step Under the assumptions [Mod] and [Ident], and using Steps 1 and 2, one can apply a classical result concerning asymptotic normality for minimum contrast estimators e.g. Proposition 3.7 of [START_REF] Guyon | Champs aléatoires sur un réseau[END_REF] in order to obtain (5).

It remains to prove (6). This may de done in two different steps. The first one consists in verifying the positive definiteness of the matrix Σ( D, θ ⋆ ). The proof is strictly similar to the one of Billiot et al. (2008) (p. 261) except that the assumption [SDP] is now simply replaced by the more general one assumption [N4]. Now, the convergence in probability of Σ n (Φ; θ n (Φ)) towards Σ( D, θ ⋆ ) is obtained by applying Proposition 9 of [START_REF] Coeurjolly | Residuals for stationary marked Gibbs point processes. submitted for publication[END_REF].

  the following the assumption [N] will stand for the assumptions [N1], [N2], [N3] and [N4]. 4 Applications to the LJ-type model This section focuses on the LJ-type model presented in Section 2.3 and aims at proving the following result. Proposition 3 (i) Theorem 1 holds for the LJ-type model (with D ∈]0, +∞]), that is for the Lennard-Jones and the finite-range Lennard-Jones model. (ii) Theorem 2 holds only for the finite-range Lennard-Jones model. The proof of Proposition 3 consists in verifying Assumptions [C] for the LJ-type model and [N] only for the finite range Lennard-Jones model. In the following, we will deal with two types of assumptions: • Integrabilility type assumptions, i.e. Assumptions [C1], [C4], [N1] and [N2]. • Identifiability type assumptions, i.e. Assumptions [C2], [N3] and [N4].

..

  θ ⋆ ) e -V (x|Φ;θ ⋆ ) x|Φ \ x; θ ⋆ ) 3The assumption [N1] ensures the integrability of the first right-hand term. For the second one, |ϕ \x 1 ; θ ⋆ ) ∂V ∂θ j (x 2 |ϕ \ x 2 ; θ ⋆ ) ∂V ∂θ j (x 3 |ϕ \ x 3 ; θ ⋆ ) +3 x1,x2∈ϕ∆,x1 =x2 ∂V ∂θ j (x 1 |ϕ \ x 1 ; θ ⋆ ) 2 ∂V ∂θ j (x 2 |ϕ \ x 2 ; θ ⋆ ) + x1∈ϕ∆ ∂V ∂θ j (x 2 |ϕ \ x 1 ; θ ⋆ ) 3The result is obtained by using the assumption [N1] and iterated versions of Corollary 8.Condition (iii): let us start by noting that from the assumption [Mod-L], the vector LP L(1) ∆i (ϕ; θ ⋆ ) depends only on ϕ ∆j for j ∈ (i, 1). Let E i,j := E LP L



  by I n the following setI n = I n ∩ (∪ i∈∂In (i, 1)) .We now obtainVar |Λ n | 1/2 U (1) n (Φ; θ ⋆ ) = |Λ n | -1 i∈In j∈In∩ (i,1) E i,j = |Λ n | -1   i∈In\ In j∈In∩ (i,1) E i,j + i∈ In j∈In∩ (i,1) E i,jUsing the stationarity and the definition of the domain Λ n , one obtains|Λ n | -1 i∈In\ In j∈In∩ (i,1) E i,j = |Λ n | -1 |I n \ I n | j∈ (0,1) E 0,j → Σ( D, θ ⋆ ) as n → +∞and|Λ n | -1 i∈ In j∈In∩ (i,⌈ D D ⌉) E i,j ≤ |Λ n | -1 | I n | j∈ (0,1)|E 0,j | → 0 as n → +∞.Hence as n → +∞Var |Λ n | 1/2 U (1) n (Φ; θ ⋆ ) = |Λ n | -1i∈In j∈In∩ (i,1) k = Σ( D, θ ⋆ ).(21)

  2. U n (•; θ) is a contrast function Recall that U n (•; θ) is a contrast function if there exists a function K(•, θ ⋆ ) (i.e. nonnegative function equal to zero if and only if θ

  ∂θ j ∂θ k (x|ϕ; θ) exp (-V (x|ϕ; θ)) dxUnder the assumption [N1] and [N2], from Theorem 9[START_REF] Nguyen | Ergodic theorems for Spatial Process[END_REF]) and from Corollary 8, there exists n 0 ∈ N such that for all n ≥ n 0

	U (2) n (ϕ; θ)	j,k	≤ 2E		∂ 2 V ∂θ j ∂θ k	(0|Φ; θ) +	∂V ∂θ j	(0|Φ; θ)	∂V ∂θ k	(0|Φ; θ) e -V (0|Φ;θ)
			+2 × E	∂ 2 V ∂θ j ∂θ k	(0|Φ; θ) e -V (0|Φ;θ
			+ +	1 |Λ n | Λn 1 |Λ n | x∈ϕΛ n ∂V ∂θ j ∂θ j ∂θ k (x|ϕ; θ) ∂ 2 V (x|ϕ \ x; θ) . ∂V ∂θ k (x|ϕ; θ) exp (-V (x|ϕ; θ)) dx	(22)

⋆ )

Acknowledgements

We are grateful to the referee and the associate editor for their comments which helped us in improving a previous version.


Tahar Gherbi
email: tahar_gherbi@yahoo.com

Isabelle Borne
email: isabelle.borne@univ-ubs.fr

Djamel Meslati
email: meslati_djamel@yahoo.com

MDE and Mobile agents: another reflection on the agent migration

Keywords: MDE, PIM, PSM, mobile agent, agent factory, agent migration I

Model Driven Engineering (MDE) is a software development approach family based on the use of models in the software construction. It allows the exploitation of models to simulate, estimate, understand, communicate and produce code. Mobile agents are a very interesting technology to develop applications for mobile and distributed environments. A mobile agent is essentially a computer program that acts autonomously on behalf of a user and travels through a network of heterogeneous computers. By moving to remote computers, the mobile agent processes data there and hence saves remote communications and local computation. Many mobile-agent platforms have been developed. While some of them have been abandoned, others continue to release new versions that fix bugs detected or offer new interesting features.

In this paper, we present a reflection on how one can realize the migration operation when the mobile-agent applications are built using an MDE approach.

INTRODUCTION

Model Driven Engineering (MDE) is gaining more and more interest. Indeed, new technologies (design patterns, middle-wares, component programming, service approach, etc.) appear continuously and rapidly; then, it becomes increasingly difficult for companies to remain competitive. Obviously these technologies simplify the programming task and bring gains to the productivity but enormously increase the software complexity; consequently, companies are very embarrassed: adopt a new technology and pay for the expensive migration cost or do not adopt it and take the risk of seeing competitors which make the reverse choice become more competitive.

MDE comes to improve this situation: its principal aim is "Model once, generate anywhere [START_REF] Bézivin | Les nouveaux défis des systèmes complexes et la réponse MDA de l'OMG[END_REF]".

Several researchers and industrials agree to think that MDE is the future of applications and are actively involved in its evolution. It is thus reasonable to expect that the coming years will know a rich set of operational tools and working platforms supporting MDE.

In this paper we consider the mobile-agent platforms and more precisely the "migration" operation which moves an agent from one computer to another. We present a reflection on how this operation can be realized if the mobile-agent applications are built using an MDE approach.

The rest of the paper is organized as follows: Section 2 introduces MDE. Section 3 presents the mobile-agent technology, describes the migration operation and examines an approach (the generative migration using an agent factory [START_REF] Bézivin | Les nouveaux défis des systèmes complexes et la réponse MDA de l'OMG[END_REF]) supporting the agent migration over heterogeneous mobile-agent platforms. Section 4 discusses a reflection for an MDE based vision to realize the migration operation. Section 5 presents our conclusions.

II. MODEL DRIVEN ENGINEERING

Bill Gates said: "Modeling is the future, so every company that's working on this I think it's great" [START_REF]What Is Bill Gates Thinking?[END_REF].

The main idea of MDE is to consider that everything is a model, as in the object approach we considered that everything is an object.

Models (such as analysis and design ones) are mainly used to document software systems. Indeed, often analysts and designers produce models and provide them to programmers as materials of inspiration to facilitate and guide the software production. MDE let us pass to a new situation where production tools will be themselves directed by models; in other words, models can now be directly used to supply automatic software production tools.

Several alternatives exist in MDE; they do not necessarily adopt the same standards, but they share three basic principles: [START_REF] Booch | An MDA Manifesto[END_REF] • Direct representation, by offering families of DSLs, allowing the representation of each situation (aspect) of the system as well as co-operative communities. Let us see MDE principles in practice through the presentation of the OMG's MDA which was the first relevant proposal as an MDE approach.

A. MDA Architecture

MDA is an initiative of the OMG, published on November 2000. It can be defined as the realization of MDE principles around a set of standards, such as [START_REF] Bézivin | Sur les principes de base de l'ingénierie des modèles[END_REF]: Meta Object Facility (MOF), XML Metadata Interchange (XMI), Common Warehouse Meta-model (CWM), Unified Modeling Language (UML), Software Process Engineering Meta-model (SPEM), Query/View/Transformation (QVT), Object Constraint Language (OCL), etc.

Being an MDE alternative, MDA would cover all concerns which can appear during the realization or the maintenance of software: functional aspects (also called business descriptions) and non-functional aspects as: quality of service, reliability, safety, etc. However for the OMG, which was a priority in MDA was the separation between platform dependant aspects and platform independent aspects (neutral business descriptions).

MDA uses models in various steps of software development cycle. It recommends the elaboration of:

• Requirement models: Computation Independent Models (CIM), • Analysis and design models: Platform Independent Models (PIM) ,and • Code models: Platform Specific Models (PSM). Models to cover other steps of software development such as checking and testing, supervision, organization, etc., have not been integrated.

Experiments undertaken with MDA have allowed, at least in particular contexts, the validation of the principle of a model driven approach. Then MDA has been enlarged to the wider approach which is MDE [START_REF] Bézivin | Rapport de Synthèse de l'AS CNRS sur le MDA (Model Driven Architecture)[END_REF]. Indeed, MDA has been generalized, under the term MDE, to all model types (not only MOF/UML models, to not limit it to OMG technological advances) in conformity with the "Direct representation" principle of MDE. 2) Meta-modeling and Meta-Meta-modeling: CIM, PIM, PSM, transformation models and any other model which can appear in MDA are expressed, each one, in a particular formalism called meta-model.

MDA recommends the modeling of meta-models as well and proposes the MOF standard as the meta-meta-model. Fig.

3 [START_REF] Blanc | MDA in action : Ingénierie logicielle guidée par les modèles[END_REF] summarizes the situation. MDA is an approach, not a method. In order to use it, we need to define a method. Thus each enterprise which has practiced MDA in its context has defined its own method. These methods respect the MDA approach by using models as productive elements and differ in their automating degree and the set of model operations they propose.

An example of development steps with MDA can be as illustrated by Fig. 3 [2]: Note that:

• If we present the CIM models in this figure, they will appear on top of PIM models. • In addition any model, not only PSM models, can be stored and accessed using MOF facilities. • Finally, it's better to model the platforms with PDM, rather than describing a transformation for each target platform. This allows the construction of generic transformations [START_REF] Thomas | Contribution à la modélisation explicite des plates-formes d'exécution pour l'IDM[END_REF].

III. MOBILE-AGENT TECHNOLOGY

A mobile agent is essentially a computer program that acts autonomously on behalf of a user and travels through a network of heterogeneous computers. It executes within a mobile-agent platform. This later must offer a set of agent services, such as [START_REF] Falou | Programmation répartie, optimisation par agent mobile[END_REF]: creation and designation, access to local resources, communications, migration between computers, agent localization and traceability, agent protection from visited computers and other agents, computer protection against visiting agents, fault tolerance, etc.

A. Mobile agent structure

As a mobile agent is a computer program, it has then an execution flow defining its tasks; and because it can migrate from one computer to another, it is important to know which elements must be transferred with it.

An instance of mobile agent contains three parts [START_REF] Falou | Programmation répartie, optimisation par agent mobile[END_REF]:

• An executable code representing the mobile-agent static comportment, • An execution context (register values, execution stack) which reflects the mobile-agent current execution state, and • Resources which the mobile agent uses. These resources are of two kinds [START_REF] Fuggetta | Understanding Code Mobility[END_REF]: Transferable resources which are the agent attribute values forming its global state, Non transferable resources which form the execution environment offered by the mobileagent platform (for example, opened files, socket connections, etc.).

B. Mobile agent migration

This operation, as illustrated in Fig. 4, realizes the transfer of an agent in execution from one computer to another through a network. When it migrates, the agent takes with it: its code, its execution context and/or its transferable resources.

C. Migration strategies

Two types of migration have been proposed in the existing mobile-agent platforms [START_REF] De Groot | Cross-Platform Generative Agent Migration[END_REF]: strong and weak migration.

1) Strong migration: The agent can migrate at any moment regardless its execution state. After its migration, the agent continues its execution from the point where it stopped last.

The strong migration needs a mechanism which instantly captures the agent execution state. Then the agent's code, its execution context and its transferable resources are serialized and transferred.

Mobile Object Reactive Agents (Moorea) is an example of an agent platform supporting the strong migration. [START_REF] Falou | Programmation répartie, optimisation par agent mobile[END_REF] 2) Weak migration: In the weak migration, the agent takes with it only its code and its transferable resources. On the destination computer, the agent re-executes from the beginning and its execution context is re-initialized.

The weak migration operation can either be called at any moment during the agent execution (but in this case we lost the agent work), or it can be called at privileged moments (stop-points) which need to be defined in the agent's code.

As an example of an agent platform supporting the weak migration, we have Jade [START_REF] Magnin | Agents Need to Become Welcome[END_REF].

D. Migration scenarios

The sites involved in a distributed mobile-agent application (for example, an application searching for information on many different sites on the Internet) may not contain the same agent platform or not have the same (virtual) machine. In addition, the executable code of an agent usually contains a part that provides the interface between the agent and the agent platform on which it runs. Hence, several migration scenarios, as summarized in Table 1, result: [START_REF] Brazier | Agent Factory: Generative Migration of Mobile Agents in Heterogeneous Environments[END_REF]

1) Homogeneous migration:

In this scenario, an agent migrates to another site without any change either to the format of its executable code or to its interface to the agent platform. In practice, this form of migration is most common.

A homogeneous migration can be either strong or weak. Very few agent platforms support the strong migration; it requires that all sites must have exactly the same operating system, the same machine architecture and the same (version of the) programming language. The most of popular agent platforms are Java-based and support the weak migration. [START_REF] De Groot | Cross-Platform Generative Agent Migration[END_REF] 2) Cross-platform migration: In this scenario, an agent migrates to another site having a different agent platform but running on the same (virtual) machine. This generally implies changes on the agent platform interface but not necessarily changes to the format of the agent's executable code. [START_REF] Brazier | Agent Factory: Generative Migration of Mobile Agents in Heterogeneous Environments[END_REF] Many solutions have been proposed, such as: [START_REF] Brazier | Agent Factory: Generative Migration of Mobile Agents in Heterogeneous Environments[END_REF] • Constructing wrapper interfaces which hide differences between source and target platforms, or • Constructing agent platforms which are compliant to a standard interface for interoperability. As such standards, we have the Mobile Agent System Interoperability Facility (MASIF) of the OMG and the Foundation for Intelligent Physical Agents (FIPA).

3) Agent-regeneration migration:

In this scenario, an agent migrates to a site having the same agent platform but running on a different (virtual) machine. This requires the regeneration of the agent, resulting in a different executable code.

4) Heterogeneous migration: An agent migrates to another site having a different agent platform and running on a different (virtual) machine. In this case, the agent regeneration is also necessary.

E. Generative migration using an agent factory

Interesting initiatives for the agent migration across different platforms exist, such as proposed in Guest [START_REF] Magnin | Agents Need to Become Welcome[END_REF], Monads1 and the generative migration using an agent factory [START_REF] Brazier | Agent Factory: Generative Migration of Mobile Agents in Heterogeneous Environments[END_REF]. Unfortunately, no source-code or libraries on Guest or Monads are available for the public or the research usage [START_REF] De Groot | Cross-Platform Generative Agent Migration[END_REF].

Concerning the generative migration using an agent factory, it aims to support all the migration scenarios presented in the section 3.4 and especially the heterogeneous migration.

The concept of an agent factory requires: [START_REF] Brazier | Agent Factory: Generative Migration of Mobile Agents in Heterogeneous Environments[END_REF] 1. The agent to have a compositional structure (the resulting specification is called blueprint), 2. One or more libraries of re-usable agent components, and 3. One or more ways to describe the functionality of these agent components. An agent factory is a service capable of: [START_REF] De Groot | Cross-Platform Generative Agent Migration[END_REF] 1. (Re-) design (i.e., create a new blueprint) and adapt an agent (that it can, for example, execute in another agent platform), 2. Finding appropriate building libraries, and 3. Assembling and regenerating an agent adapted to its new environment at its destination site. In short, an agent factory constructs new agents and/or modifies existing ones [START_REF] Brazier | Automated servicing of agents[END_REF]. The (re-)design of agents is fully automated with very limited interactions with outside parties [START_REF] Brazier | Agent Factory: Generative Migration of Mobile Agents in Heterogeneous Environments[END_REF].

The generative-migration approach using an agent factory, considers that an agent has a blueprint which describes its structure and functionality and a state which represents its execution. The agent's state includes its transferable resources and eventually its execution context (in the case the strong migration is supported and solicited).

The key idea of this approach is, rather than sending the agent's code, we send an agent description in an implementation (agent platform and machine) independent format. Both the agent's blueprint and its state are expressed in this format (as a candidate format, we have the eXtended Markup Language: XML).

It's to be expected that the transmission of an agent's blueprint and information on its current state will generally require less network resources than migrating an agent using more traditional approaches. [START_REF] Brazier | Agent Factory: Generative Migration of Mobile Agents in Heterogeneous Environments[END_REF] The blueprint and state descriptions are sent to the destination site, where an agent factory processes them. This later:

• Re-builds the agent from its received blueprint. The appropriate building blocks (libraries) must be available at the agent platform to enable the regeneration of the agent's code. These building blocks are retrieved from a local repository (for the reusable building blocks) and the agent is assembled.

• Initiates the regenerated agent with its state and launches it into the platform where it continues its execution.

IV. MDE FOR ANOTHER REFLECTION ON THE AGENT

MIGRATION

MDE aims to mask the heterogeneity of development and deployment environments during the software construction. It consists of elaborating platform independent models, on which a set of model transformations is operated until obtaining the executable code.

We distinguish two types of model transformations in MDE: [START_REF] Parrend | Introduction à MDA : Principe[END_REF] • Horizontal transformations: they preserve the abstraction level of the model and are used by concern of: Representation: It is the question of passing from a format to another. The target format being rich in handling and representation tools, or for which tools are simply available, Optimization, to improve the performance, Rebuilding, to improve the maintenance and the visibility of the code. • Vertical transformations: they change the abstraction level of the model. We distinguish: Refinement mapping: it is used for example to pass from an analysis/conceptual model to an implementation model (as from PIM to PSM). Abstraction mapping: it is used for the reverse engineering or for migrate an application from one platform to another. Several researchers and industrials agree to think that MDE is the future of applications and are involved actively in its evolution. If MDE successes to meet its promises ("modeling once, generating everywhere", hiding the software complexity and preserving their continuity and evolution from obsolete technologies, etc.), we will expect to see MDE supports on the most sites. Consequently, it will be very interesting and even recommended to construct mobile-agent applications with an MDE approach and to elaborate mobile-agent platforms which exploit the MDE supports. Indeed, the agent applications will benefit from the MDE's advantages and the agent platforms will easily support all the migration scenarios presented in the section 3.4.

Let us suppose a mobile-agent application built with an MDE approach, a practical example for an MDA/MDE alternative supporting the weak agent migration (as it is the most common migration type) may be as follows:

1. Any running agent has its code generated from its PIM. This later is saved in a repository using MOF facilities. 2. During the agent execution, if the migration operation is invoked, the agent's transferable resources are gathered and made with the agent's PIM in an independent format (such as XML, using XMI facilities) message. 3. The resulting message is then sent to the target site. 4. On the target site, the agent's PIM and its transferable resource are extracted from the received message. The agent code is rebuilt from its PIM using appropriate transformations (horizontal transformations may eventually apply, for example when vertical transformations do not exist for the received PIM's format). 5. The agent is then activated and its execution context is initialized. One of the biggest challenges of MDE is to offer tools for testing and debugging the software artifacts at the model level [START_REF] Hann | 8 Reasons Why Model-Driven Approaches (will) Fail[END_REF]. Otherwise, for the strong migration case, if one can elaborate a mechanism which associates to the execution pointer on the agent's code (pointing its actual instruction) another execution pointer on the agent's PIM, so that the two pointers evolve in a coordinated manner during the agent execution; the pointer on the agent's PIM can thus be sent (within the agent's execution context at model level) to the target site to help restoring the execution of the rebuilt agent at the same instruction at which it was before its migration.

The ideas expressed in the "generative migration using an agent factory" approach [START_REF] De Groot | Cross-Platform Generative Agent Migration[END_REF] can be made closer to those of MDE. Indeed, to "blueprints" can correspond "PIM models", "agent generation" can be replaced by "model transformations", "repository for saving blueprints" can be made MOF compliant, saving and transferring models in XML format can be based on XMI facilities. Hence, the experiences with the implementation of the "generative migration using an agent factory" can effectively help to elaborate PIMs for the mobile-agent application conception, a Platform Description Model (PDM) for each combination of mobile-agent platform and (virtual) machine, model transformations enriching the PIM with the PDM's details and translating the resulting model into PSM. In addition, the corresponding meta-models can be elaborated: the PIM's meta-models can be inspired from the blueprint's languages, the meta-models for PDM and the model transformations can be inspired from the agent factory algorithm. To resume, the ideas expressed in this approach and others dealing with the question of agent migration across different agent platforms may concur to elaborate an MDE alternative for constructing mobile-agent applications and to build agent platforms exploiting the MDE supports. This make the agent applications benefit from the MDE advantages and the agent platforms to support the agent migration by only exploiting the MDE supports existing on each site.

V. CONCLUSION MDE is gaining more and more interest. It is actually a big undertaken framework; therefore, we cannot confirm yet if it will really solve all the software development problems; however, all the opinions confirm that this direction is very promising. Moreover, several researchers and industrials agree to think that MDE is the future of applications and are actively involved in its evolution.

MDE is making changes in all the software production industrial practices. This paper has presented a reflection on how we can support the agent migration if the agent applications are built with an MDE approach and the agent platforms can exploit the MDE supports. This allows the agent applications to benefit from the MDE advantages and allows the agent platforms to support the agent migration by just exploiting the MDE supports existing on each site.

We have examined an interesting approach for the agent migration (the generative migration using an agent factory [START_REF] Brazier | Agent Factory: Generative Migration of Mobile Agents in Heterogeneous Environments[END_REF]) and concluded that the ideas expressed in this approach and others dealing with the question of agent migration across different platforms may concur to concretize our reflection.

1 )

 1 Global view of MDA: As shown in Fig. 1 [14], the development of a new application starts by making one or more CIM; then PIM are established from CIM; finally, PIM are transformed into PSM.

Figure 1 .

 1 Figure 1. Global view of MDA

Figure 2 .

 2 Figure 2. Models, meta-models and meta-meta-model in MDA B. Software Development with MDA

Figure 3 .

 3 Figure 3. An example of development steps with MDA

Figure 4 .

 4 Figure 4. Mobile agent migration The migration operation executes the following steps [9]: 1. Suspends the agent execution and serializes its necessary details: code, execution context and/or transferable resources. 2. Suspends all communications with other agents and the agent dies. 3. Sends the serialized agent details to the destination computer. 4. Restores the agent on the destination computer. 5. Redirects suspended agent communications towards the destination computer. Note that the agent was restored but is not yet activated. 6. Activates the agent. The migration operation ends and any saved information concerning it, can be definitely erased from the source computer.

TABLE 1 .

 1 MIGRATION SCENARIOS

		Mobile-agent	(virtual)
		platforms on the	machines on
		source and target	the source and
		sites	target sites
	Homogeneous migration	same	Same
	Cross-platform migration	different	Same
	Agent-regeneration migration	same	different
	Heterogeneous migration	different	different

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