On asymptotic normality of nonparametric estimate for a stationary pairwise interaction point process

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On asymptotic normality of nonparametric estimate for a stationary pairwise interaction point process

Nadia Morsli

To cite this version:

Nadia Morsli. On asymptotic normality of nonparametric estimate for a stationary pairwise interaction point process. 2015. �hal-01121114v3�

On asymptotic normality of nonparametric estimate for a stationary pairwise interaction

point process

Nadia Morsli [email protected]

Laboratory Jean Kuntzmann, Grenoble University, France.

Abstract

We prove the asymptotic normality of nonparametric estimator of pair- wise interaction function for a stationary pairwise interaction point process characterized by the Papangelou conditional intensity and observed in a bounded window of a sequence of cubes growing up toR^d. Formula for the variance of the resulting estimator can be obtained using Papangelou condi- tional intensity of the point process. This is a random function satisfying the counterpart of the Georgii-Nguyen-Zessin formula. The proof of the asymp- totic normality of the resulting estimator is based on them_n-approximation method in the setting of dependent random fields indexed byZ^dwheredis a positive integer.

Keywords: Georgii-Nguyen-Zessin formula, nonparametric estimation, pairwise interaction point process, Papangelou conditional intensity, asymptotic normality.

1 Introduction

A special type of Gibbs processes that are noteworthy both for their abundant use in statistical physics are the pairwise interaction point processes. Several methods of estimation are available for parametric families of pairwise interac- tion point processes: approximate maximum likelihood ([29], [30]), maximum pseudo-likelihood ([4]), Monte Carlo likelihood ([14]).

In this paper we are concerned with nonparametric statistics for stationary pairwise interaction point processes. They have been introduced by [34], [7] and [13]. These provide a large variety of complex patterns starting from simple po- tential functions (or pairwise interaction function) which are easily interpretable

as attractive or repulsive forces acting among points and are of practical impor- tance because of their ability to model a wide variety of spatial point patterns, especially those displaying some degree of spatial regularity. A great deal is un- derstood about pairwise interaction models because they are very natural with respect to the derivation of Papangelou conditional intensity ([25]).

The goal of the present paper is to establish the asymptotic normality of the nonparametric estimator of the pairwise interaction function for a stationary pair- wise interaction point process characterized by the Papangelou conditional inten- sity. The asymptotic normality in statistics is based on central limit theorems for the sequences of random variables. These classical limit theorems have been extended to the setting of spatial processes. Some results on the central limit the- orem and its functional versions are [2], [3], [9], [10] and [22]. Our proof is based on the m-approximation method (m_n→∞asn→∞) in the setting of dependent random fields indexed by Z^d where d is a positive integer. We apply a central limit theorem for triangular arrays of stationarym-dependent random fields with unbounded m established by [39]. This result improves a central limit theorem established by [15]. The notion of m-dependence was first introduced by [17].

Central limit theorems for m-dependent random fields has already been consid- ered by many researchers. For example [6], [38], [35] [24].

The paper is organized as follows. Section 2 sets up the basic tools of the Gibbs point processes in R^d. Section 3 contains the main asymptotic results on proposed estimators of stationary pairwise interaction point process. This con- cerns firstly the asymptotic behaviour of variance and covariance of the kernel- type estimator under certain smoothness conditions. The latter is based on the knowledge of Papangelou conditional intensity and the iterated Georgii-Nguyen- Zessin formula. Secondly, we are able to study asymptotic Gaussianity of pro- posed estimators and prove it in Section4.

2 Basic tools

LetB^d be the Borelσ-algebra (generated by open sets) inR^d(thed-dimensional space) andB_O^d ⊆B^dbe the system of all bounded Borel sets. A point processXin R^dis a locally finite random subset ofR^d, i.e. the number of pointsN(Λ) =n(X_Λ) of the restriction of XtoΛ is a finite random variable whenever Λ is a bounded Borel set ofR^d(see [7]). We define the space of locally finite point configurations inR^d asN_{l f} ={x⊆R^d;n(x_Λ)<∞,∀Λ∈B₀^d}, wherex_Λ=x∩Λ. We equipN_{l f} withσ-algebraNl f =σ{{x∈N_{l f} :n(x_Λ) =m},m∈N0,Λ∈B^d₀}, whereN0= N∪ {0}={0,1,2,3, . . .}. The volume of a bounded Borel setΛofR^d is denoted by |Λ|. For any finite subset I of Z^d, we denote |I| the number of elements in I,Ja,bK≡ {a,a+1, . . . ,b}fora,b∈Zand the sum∑⁶⁼signifies summation over

distinct pairs. For any Gibbs point processes in a bounded window, the conditional intensity at a location u given the configuration x is related to the probability density f by

λ(u,x) = f(x∪ {u}) f(x)

(foru∈/x), the ratio of the probability densities for the configuration xwith and without the pointuadded. The Papangelou conditional intensity can be interpreted as follows: for any u∈R^d andx∈N_{l f},λ(u,x)ducorresponds to the conditional probability of observing a point in a ball of volume duaroundugiven the rest of the point process isx.

Pairwise interaction point processes in R^d can be defined and characterized through the Papangelou conditional intensity (see [25]) which is a function λ : R^d×N_{l f} →R+. The Georgii-Nguyen-Zessin (GNZ) formula (see [31], [40], [12], [28]) states that for any nonnegative measurable functionhonR^d×N_{l f}

E

∑

u∈X

h(u,X\u) =E Z

R^d

h(u,X)λ(u,X)du. (2.1) Using induction we obtain the iterated GNZ-formula: for nonnegative func- tionsh:(R^d)ⁿ×N_{l f} −→R

E

6=

∑

u1,...,un∈X

h(u₁, . . . ,u_n,X\ {u₁, . . . ,u_n})

= Z

. . . Z

Eh(u₁, . . . ,u_n,X)λ(u₁, . . . ,u_n,X)du₁. . .du_n (2.2) whereλ(u₁, . . . ,un,x)is Papangelou conditional intensity and is defined (not uniquely) byλ(u₁, . . . ,u_n,x) =λ(u₁,x)λ(u₂,x∪ {u₁}). . .λ(u_n,x∪ {u₁, . . . ,u_n−1}).Exam- ples of Gibbs point process models and their conditional intensities are presented in [1], [25] and [26]. The formula (2.2) will also be used extensively later in this document.

3 Assumptions and the main results

For the general pairwise interaction process the conditional intensity is λ(u,x) =g₀(u)exp

−

∑

v∈x\u

g₀(u,v)

and note thatg0(u,v) =g0(v,u)(i.e. symmetric pairwise interaction).

Alternatively, if g₀(u) is a constant and g₀(u,v) =g(v−u) is translation in- variant, then a pairwise interaction point process is called stationary.

Throughout this paper, we define a stationary pairwise interaction point process via the Papangelou conditional intensity at a locationugiven by

λ_β^?(u,x) =β^?exp

−

∑

v∈x\u

g(v−u)

(3.1) whereβ^?is the true value of the Poisson intensity parameter,grepresents a pair- wise interaction potential. Specifically, g is nonnegative measurable function;

the practical significance of this assumption is that pairwise interaction processes are a wide and useful class of models for spatial point processes with inhibition between points; for discussion see [11] and [36]. We denote G=exp(−g) the pairwise interaction function. The basic assumption throughout this paper is the Papangelou conditional intensity (3.1) has a finite rangeR, i.e.

λ_β^?(u,x) =λ_β^?(u,x∩B(u,R)), (3.2) for anyu∈R^d, x∈N_{l f}, whereB(u,R)is the closed ball inR^d with centeruand radiusR.

Suppose that a single realization x of a point process X is observed in a bounded window Λ_n∈B₀^d where (Λ_n)_n≥1 is a sequence of cubes growing up to R^d. We face a missing data problem, which in the spatial point process literature is referred to as a problem of edge effects, we can avoid this problem by reducing the window by introducing the 2R-interior of the cubesΛ_n, i.e. Λ_n,R ={u∈Λ_n: B(u,2R)⊂Λ_n}and assume this has non-zero area. The choice ofΛn,R is induced by the fact that whenuis on the edge ofΛn,Randv∈B(u,R), then we can observe the pointv. Various edge corrections have been suggested by [33], [5], [20], [25]

and [37]). We assume that the support of the interaction functionG=exp(−g)is T ={t∈R^d;g(t)>0, for||t||<R}. Throughout in this paper,his a nonnegative measurable function defined for allw∈R^d,x∈N_{l f}, by

h(w,x) =11(x∩B(w,R) = /0), (3.3) and we also introduce the following function

F(o,¯ w) =E[h(o,X)h(w,X)] =P(X∩B(o,R) = /0,X∩B(w,R) =/0). (3.4) To estimate the function β^?F¯(o,t), we propose empiric estimator Fb¯n(t)defined fort∈T by

b¯

Fn(t) = 1

|Λn,R|

∑

u∈X_Λn,R

h(u,X\ {u})h(t+u,X\ {u}). (3.5)

To estimate the function β^?2G(t)F¯(o,t), we propose kernel-type estimatorHb_n(t) defined fort∈T by

Hb_n(t) = 1 b^d_n|Λ_n,R|

6=

u,v∈X

∑

v−u∈B(o,R)

11_Λn,R(u)h(u,X\ {u,v})h(v,X\ {u,v})K

v−u−t b_n

, (3.6) where6=over the summation sign means that the sum runs over all pairwise dif- ferent pointsu;vinXandK:R^d→Rdenotes a smoothing kernel function asso- ciated with a sequence (b_n)_n≥1 of bandwidths. Plugging in the above estimators (3.5) and (3.6) and with the conventionc/0=0 for all real c, we suggest a new edge-corrected nonparametric estimatorGb_n(t)forβ^?G(t)fort∈T by

Gbn(t) =Hb_n(t) b¯ F_n(t)

. (3.7)

Moreover, we have to impose certain natural restrictions on the kernel function K and the sequence(b_n)_n≥1.

ConditionK(d).

The sequence of bandwidths (b_n)_n≥1, is a sequence of positive real numbers sat- isfying:

n→∞limb_n=0 and lim

n→∞b^d_n|Λ_n,R|=∞.

The kernel functionK :R^d−→Ris nonnegative and bounded with bounded support and satisfies:

Z

R^d

K(z)dz=1, Z

R^d

K²(z)dz<∞.

ConditionK(d,m).Letz= (z₁, . . . ,z_d)⁰,zi∈R, Z

R^d

zⁱ₁¹. . .zⁱ_d^dK(z₁, . . . ,z_d)dz₁. . .dz_d=0, for 0<

d

∑

i_j<m,

and _Z

R^d

|z|^mK(z)dz<∞.

3.1 Asymptotic behaviour of the variance and the covariance of the kernel-type estimator

In this section, we investigate the asymptotic behaviour of the variance and the covariance of the kernel-type estimator (3.6) estimatingβ^?2G(t)F(o,¯ t). The latter is based on the knowledge of the iterated GNZ-formula (2.2).

Theorem 3.1. Consider a stationary pairwise interaction point processXinR^d with Papangelou conditional intensity(3.1)satisfying condition(3.2). Further let the kernel function K satisfy Condition K(d). We have

n→∞limb^d_n|Λ_n,R|Var Hbn(t)

=β^?2G(t)F¯(o,t) Z

R^d

K²(z)dz at any continuity point t∈T\{o}of GF.¯

The following theorem presents an asymptotic representation of the covariance of the kernel-type estimator (3.6) in two pointst₁,t₂inT.

Theorem 3.2. Consider a stationary pairwise interaction point processXinR^d with Papangelou conditional intensity(3.1)satisfying condition(3.2). Further let the kernel function K satisfy Condition K(d)and let t₁6=t₂∈T . We have

n→∞limb^d_n|Λ_n,R|Cov Hbn(t₁),Hbn(t₂)

=0.

The proof of Theorem3.2is similar to that of Theorem3.1, and is omitted.

3.2 Asymptotic normality of a nonparametric estimator

Asymptotic normality of a nonparametric estimate of the pairwise interaction functionGfor a stationary pairwise interaction point process is essentially based on a multivariate central limit theorem for the estimatorHb_n(t)ofβ^?2G(t)F(o,t¯ ).

The proving idea of the below Theorem 3.3 consists in approximating the se- quence Hb_n(t) by a triangular array of m_n-dependent random fields. The corre- sponding Lindeberg-type CLT for m_n-dependent random fields has been proved in [39].

Now, we assume that the domainΛ_n is divided into a fixed number of sub- domains as follows Λ_n =∪_i∈InΛ_i, following [32], [21], [18] we will describe a point process in R^d as lattice process by means of this decomposition Λi = {ξ ∈R^d;eq(i_j−¹₂)≤ξ_j ≤q(ie _j+¹₂),j=1, . . . ,d}for a fixed numberqe>0, i= (i₁, . . . ,i_d), and settingX_i=X_Λi, i∈Z^d, this becomes a Gibbs lattice field. We will consider estimation of β^?2G(t)F¯(o,t) from Hbn(t), where the process is ob- served in Λ_n,R =∪_i∈eI

nΛ_i, where eI_n={i∈I_n;|i−j| ≤1, for all j ∈I_n}, and the norm is |j|= max{|j₁|, . . . ,|j_d|} and assume that I_n increases towards Z^d and write

Hbn(t) = 1 b^d_n|Λ_n,R|

∑

i∈eIn

6=

u∈X

∑

h(u,X\ {u,v})h(v,X\ {u,v})K

v−u−t b_n

.

We consider in this work the field{X_i}_i∈Zd (d∈N)in form of

X_i=F(εe _i−k,k∈Z^d),i∈Z^d. (3.8) For eachm∈N,i∈Z^d, we writeX_i,m=F(εe _i−k,k∈J0,m−1K

d),where{ε_i}_i∈Zd

are i.i.d. random variables andFeis measurable function. Let{ε_i⁰}_i∈Zd be an i.i.d.

copy of {ε_i}_i∈Zd and consider for all positive integer n the coupled version X^?_i defined byX^?_i =Fe(ε_i−k^? ,k∈Z^d), where

ε^?_j =

ε_j if j6=0 ε₀⁰ if j=0.

Let i∈Z^d and p>0 be fixed. We define δi,p=kX_i−X^?_ik_p, wherek · k_p is the usualL^p-norm. Thus, [23] obtained this sufficient condition:

∑

i∈Z^d

|i|^5d/2δi,p<∞ (3.9)

for a random field of the form (3.8) for the kernel density estimator to be asymp- totically normal.

The desired sequence(m_n)_n≥1can be chosen as m_n=max

( v_n,

1 b³_n

∑

|i|>vn

|i|^5d2 δi,2

_3d¹ +1

)

wherev_n= [b⁻

1

n2d]and[·]denotes the integer part function.

In order to establish the asymptotic normality of Hbn, we need additional as- sumptions:

ConditionW. There exists a sequence of integers(m_n)_n≥1such thatm_n→∞ asn→∞, and the following limits hold

m^d_nb^d_n→0 and 1

(m_nbn)^3d/2

∑

|i|>mn

|i|^5d2δ_i,2→0 if (3.9) holds. (3.10) m^2d_n

b^d_nn^d →0. (3.11)

Theorem 3.3. Consider a stationary pairwise interaction point processXinR^d with Papangelou conditional intensity(3.1)satisfying condition(3.2). Further let the kernel function K satisfy Condition K(d)and assume that ConditionW holds.

Then, for any positive integer s and any distinct points t1, . . . ,tsin T\{o}, we have q

b^d_n|Λn,R|

Hb_n(t_i)−EHb_n(t_i) s

i=1

−→d N (0,σ_H)

where the covariance matrixσ_H = [ri,j]i,j=1,...,s is given by ri,i=β^?2G(t_i)F(o,¯ ti)

Z

R^d

K²(z)dz, for i=1, . . . ,s and ri,j=0 if i6= j.

Corollary 1. Let, in addition to the assumptions of Theorem3.3 and the kernel function K satisfy Condition K(d,m) and let b^d+2m_n |Λ_n,R| →0 as n→+∞. Fur- thermore if G(t)F(o,¯ t)has bounded and continuous partial derivatives of order m in some neighborhood of the points t1, . . . ,t_s. Then

q

b^d_n|Λ_n,R|

Hbn(t_i)−β^?2G(t_i)F(o,¯ ti) s

i=1

−→d N (0,σH)

where the covariance matrixσ_H = [r_i,j]i,j=1,...,s is given by

ri,i=β^?2G(t_i)F(o,¯ t_i) Z

R^d

K²(z)dz, for i=1, . . . ,s and ri,j=0 if i6= j.

The estimator (3.5) turns out to be unbiased estimator ofβ^?F¯(o,t)and strongly consistent (the uniform strong consistency) as n tends infinity, since a classical ergodic theorem for spatial point processes obtained in [27]. This implies the following:

Theorem 3.4. Consider a stationary pairwise interaction point processXinR^d with Papangelou conditional intensity(3.1)satisfying condition(3.2). Further let the kernel function K satisfy Condition K(d), Condition K(d,m)and assume that ConditionW and b^d+2m_n |Λ_n,R| →0as n→∞hold. Then for any positive integer s and any distinct points t₁, . . . ,t_sin T\{o},

q

b^d_n|Λ_n,R|

Gb_n(t_i)−β^?G(t_i) s

i=1

−→d N (0,σ_G)

where the covariance matrixσ_G= [σi,j]i,j=1,...,s is given by σi,i= G(t_i)

F¯(o,t_i) Z

R^d

K²(z)dz for i=1, . . . ,s and σi,j=0 if i6= j.

4 Proofs

4.1 Proof of Theorem 3.1

Proof. We determine the asymptotic behaviour of the variance of the estimator Hb_n(t). For this we use the following result.

Lemma 1. Consider any Gibbs point process X in R^d with Papangelou condi- tional intensityλ. Let f :R^d×R^d×N_{l f} −→Rbe a nonnegative and measurable function. Then

Var 6=

u,v∈X

∑

f(u,v,X\{u,v})

=E Z

R^2d

f(u,v,X)

f(u,v,X) + f(v,u,X)

λ(u,v,X)dudv +E

Z

R^3d

f(u,v,X)

f(v,w,X) +f(w,v,X)]λ(u,v,w,X)dudvdw +E

Z

R^3d

f(u,v,X)

f(u,w,X) + f(w,u,X)

λ(u,v,w,X)dudvdw +E

Z

R^4d

f(u,v,X)f(w,y,X)λ(u,v,w,y,X)dudvdwdy

− Z

R^4d

E

f(u,v,X)λ(u,v,X) E

f(w,y,X)λ(w,y,X)

dudvdwdy.

Proof. After a simple calculation, but prolix (see [19] and [16]), we obtain the following decomposition:

6=

u,v∈X

∑

f(u,v,X\{u,v}) 2

=

6=

u,v∈X

∑

f²(u,v,X\{u,v})

+

6=

u,v∈X

∑

f(u,v,X\{u,v})f(v,u,X\{u,v})

+

6=

u,v,w∈X

∑

f(u,v,X\{u,v,w})f(v,w,X\{u,v,w})

+

6=

u,v,w∈X

∑

f(u,v,X\{u,v,w})f(w,v,X\{u,v,w})

+

6=

u,v,w∈X

∑

f(u,v,X\{u,v,w})f(u,w,X\{u,v,w})

+

6=

u,v,w∈X

∑

f(u,v,X\{u,v,w})f(w,u,X\{u,v,w})

+

6=

u,v,w,y∈X

∑

f(u,v,X\{u,v,w,y})f(w,y,X\{u,v,w,y}).

(4.1)

On the other hand using standard definition of variance, we have Var

6=

u,v∈X

∑

f(u,v,X\{u,v}) =E 6=

u,v∈X

∑

f(u,v,X\{u,v}) 2

−

E

6=

u,v∈X

∑

f(u,v,X\{u,v}) 2

. (4.2) After rearrangement with the formula of GNZ (2.2) and formulas (4.1) and (4.2), we obtain the desired result.

In the sequel, we denote

H(u₁,u2, . . . ,us,X) =h(u₁,X)h(u₂,X). . .h(u_s,X), (4.3) wherehis given by (3.3),

I_R(u₁,u₂, . . . ,u_s) =11 u₁∈B(o,R),u₂∈B(o,R), . . . ,u_s∈B(o,R)

(4.4) and we keep in mind the following function

F¯(u₁,u₂, . . . ,u_s) =E[H(u₁,u₂, . . . ,u_s,X)] =E[h(u₁,X)h(u₂,X). . .h(u_s,X)].

(4.5) Substitute

f(u,v,x) =11_Λn,R(u)I_R(v−u)H(u,v,x)K

v−u−t b_n

the variance term ofHb_n(t)is now expanded using Lemma1, therefore we find that VarHb_n(t) =T₁+T₂+T₃+T₄+T₅+T₆+T₇+T₈,

with

T₁= 1 b^2d_n |Λ_n,R|²E

Z

R^2d

11_Λn,R(u)I_R(v−u)H(u,v,X)K²

v−u−t b_n

λ_β^?(u,v,X)dudv, T2= 1

b^2d_n |Λn,R|² E

Z

Λ²_n,R

I_R(v−u,u−v)H(u,v,X)K

v−u−t bn

K

u−v−t bn

λ_β^?(u,v,X)dudv,

T₃= 1 b^2d_n |Λ_n,R|²E

Z

R^d

Z

Λ²_n,R

I_R(v−u,w−v)H(u,v,w,X)K

v−u−t b_n

K

w−v−t b_n

×λ_β^?(u,v,w,X)dudvdw, T₄= 1

b^2d_n |Λ_n,R|²E Z

R^d

Z

Λ²_n,R

I_R(v−u,v−w)H(u,v,w,X)K

v−u−t bn

K

v−w−t bn

×λ_β^?(u,v,w,X)dudvdw, T₅= 1

b^2d_n |Λ_n,R|²E Z

R^2d

Z

Λn,R

IR(v−u,w−u)H(u,v,w,X)K

v−u−t b_n

K

w−u−t b_n

×λ_β^?(u,v,w,X)dudvdw, T₆= 1

b^2d_n |Λ_n,R|²E Z

R^d

Z

Λ²_n,R

I_R(v−u,u−w)H(u,v,w,X)K

v−u−t b_n

K

u−w−t b_n

×λ_β^?(u,v,w,X)dudvdw, T7= 1

b^2d_n |Λn,R|²E Z

R^4d

11_Λn,R(u)11_Λn,R(w)I_R(v−u,y−w)H(u,v,w,y,X)K

v−u−t b_n

×K

y−w−t bn

λ_β^?(u,v,w,y,X)dudvdwdy and

T₈=− 1 b^2d_n |Λ_n,R|²

Z

R^4d

11_Λn,R(u)11_Λn,R(w)I_R(v−u,y−w)K

v−u−t b_n

K

y−w−t b_n

×E

H(u,v,X)λ_β^?(u,v,X) E

H(w,y,X)λ_β^?(w,y,X)

dudvdwdy.

The asymptotic behaviour of the leading term T₁ is obtained by applying the second order Papangelou conditional intensity given by:

λ_β^?(u,v,x) =λ_β^?(u,x)λ_β^?(v,x∪ {u}) for anyu,v∈R^d andx∈N_{l f}. And using the finite range property (3.2) for each functionλ_β^?(u,x)andλ_β^?(v,x∪ {u}). We recall whenx= /0, this implies that

λ_β^?(u,/0) =β^? and λ_β^?(v,/0∪ {u}) =β^?G(v−u) for all u,v∈R^d.

By stationarity ofX, it results T₁= 1

b^2d_n |Λ_n,R|²E Z

R^2d

11_Λn,R(u)I_R(v−u)H(u,v,X)K²

v−u−t bn

λ_β^?(u,v,X)dudv

= β^?2 b^2d_n |Λ_n,R|²

Z

R^2d

11_Λn,R(u)I_R(v−u)F¯(u,v)K²

v−u−t b_n

G(v−u)dudv

= β^?2 b^2d_n |Λ_n,R|

Z

R^d

I_R(s)F(o,¯ s)K² s−t

bn

G(s)ds

= β^?2 b^d_n|Λn,R|

Z

R^d

IR(b_nz+t)F¯(o,bnz+t)K²(z)G(b_nz+t)dz.

The continuity ofGF¯ int and the boundedness conditions on the kernel function yield the desired result by dominated convergence theorem, i.e.

lim_n→∞b^d_n|Λ_n,R|T₁=β^?2G(t)F(o,¯ t)^R_RdK²(z)dz.

The following lemma will justify the applicability of dominated convergence theorem in the proofs of the next asymptotic results.

Lemma 2. ([8], Lemma A.2) Let (Λ_n)_n∈N be a sequence of convex sets in R^d growing up to R^d and let y,z∈R^d. Let (b_n)_n∈N be a sequence of positive real numbers with bn−→

n→∞0. Then we have

n→∞lim

Λ_n∩(Λ_n−b_ny−z)

|Λ_n| =1.

We will now show that all the other terms of the variance of the estimator Hbn(t)vanish. With similar arguments to those used to obtain the results above, we

treat the termT₂. T₂= 1

b^2d_n |Λ_n,R|²E Z

R^2d

11_Λn,R(u)11_Λn,R(v)I_R(v−u,u−v)H(u,v,X)K

v−u−t bn

×K

u−v−t b_n

λ_β^?(u,v,X)dudv

= β^?2 b^2d_n |Λn,R|²

Z

R^2d

11_Λn,R(u)11_Λn,R(v)I_R(v−u,u−v)F¯(o,v−u)K

v−u−t b_n

×K

u−v−t bn

G(v−u)dudv

= β^?2 b^2d_n |Λ_n,R| Z

R^d

|Λ_n,R∩(Λ_n,R−s)|

|Λn,R| IR(s,−s)F¯(o,s)K s−t

b_n

K

−s−t b_n

G(s)ds

= β^?2 b^d_n|Λ_n,R|

Z

R^d

|Λ_n,R∩(Λ_n,R−(b_nz+t))|

|Λ_n,R| I_R(b_nz+t,−b_nz−t)F(o,¯ b_nz+t)K(z)

×K(−z−2t

b_n)G(b_nz+t)dz.

The boundedness conditions on the kernel and under condition t 6=o imply that b^d_n|Λ_n,R|T₂−→0 asn→∞, by dominated convergence theorem and the geometric properties ofΛn,R(Lemma2).

For the asymptotic behaviour of the third termT₃, we remember the third order Papangelou conditional intensity by

λ_β^?(u,v,w,x) =λ_β^?(u,x)λ_β?(v,x∪ {u})λ_β?(w,x∪ {u,v})

for any u,v,w∈R^d and x∈N_{l f}. Using the finite range property (3.2) for each functionλ_β^?(u,x),λ_β^?(v,x∪ {u})andλ_β^?(w,x∪ {u,v}), i.e.

λ_β^?(u,X) =λ_β^?(u,X∩B(u,R))

=β^? when X∩B(u,R) = /0, λ_β^?(v,X∪ {u}) =λ_β^?(v,(X∪ {u})∩B(v,R))

=β^?G(v−u) when X∩B(v,R) =/0,v−u∈B(o,R), and whenX∩B(w,R) = /0 andw−v∈B(o,R), we have

λ_β^?(w,X∪ {u,v}) =λ_β^?(w,(X∪ {u,v})∩B(w,R))

=λ_β^?(w,v,{u} ∩B(w,R)).

Since X is a point process to interact in pairs, the interaction terms due to triplets or higher order are equal to one, i.e.G(y) =1 whenn(y)>2, for /06=y⊆x.

Then,

λ_β^?(w,v,{u} ∩B(w,R)) =

β^?G(w−v) if u∈/B(w,R), β^? otherwise and

λ_β^?(u,v,w,/0) =

β^?3G(v−u)G(w−v) if u∈/B(w,R)

β^?3G(v−u) otherwise. (4.6) Which ensures that λ_β^?(u,v,w,/0) is a function that depends only variables v− u,w−v, denoted byG3(v−u,w−v).In this way from the definition of H (resp.

I_R) given by (4.3) (resp.(4.4)) and by stationarity ofX, it follows that T3= 1

b^2d_n |Λn,R| Z

R^2d

|Λ_n,R∩(Λ_n,R−v)|

|Λ_n,R| G¯3(v,w−v)K v−t

b_n

K

w−v−t b_n

dvdw

= 1

|Λ_n,R| Z

R^2d

|Λ_n,R∩(Λ_n,R−(b_nz+t))|

|Λ_n,R| G¯₃(b_nz+t,b_nz⁰+t)K(z)K(z⁰)dzdz⁰, where ¯G₃(v,w−v) =I_R(v,w−v)F(−v,¯ w−v)G₃(v,w−v), where ¯F is given by (4.5). By dominated convergence theorem and using the Lemma2, we getb^d_n|Λ_n,R|T₃−→

0 asn→∞. Analogously, one may show the other termsT₄,T₅andT₆.

For the asymptotic behaviour of the leading termT₇, letu,v,w,yany points in R^d andx∈N_{l f}, we define Papangelou conditional intensity of the fourth order by λ_β^?(u,v,w,y,x) =λ_β^?(u,x)λ_β^?(v,x∪ {u})λ_β^?(w,x∪ {u,v})λ_β^?(y,x∪ {u,v,w}).

Next we introduce the finite range property (3.2) and reasoning analogous with the foregoing onλ_β^?(u,v,w,/0)is defined by (4.6), which ensures thatλ_β^?(u,v,w,y,/0) is a function that depends only variables v−u,y−w,w−u,w−v, denoted by

G₄(v−u,y−w,w−u,w−v).Further by the stationarity ofX, this implies that T7= 1

b^2d_n |Λn,R|²E Z

R^4d

11_Λn,R(u)11_Λn,R(w)I_R(v−u,y−w)H(u,v,w,y,X)K

v−u−t b_n

×K

y−w−t bn

λ_β^?(u,v,w,y,X)dudvdwdy

= 1

b^2d_n |Λ_n,R| Z

R^3d

|Λ_n,R∩(Λ_n,R−w)|

|Λ_n,R| G¯₄(v,y−w,w,w−v)K v−t

bn

K

y−w−t bn

dvdwdy

= 1

|Λ_n,R| Z

R^3d

|Λ_n,R∩(Λn,R−w)|

|Λ_n,R| G¯₄(b_nz+t,b_nz⁰+t,w,w−b_nz−t)K(z)K(z⁰)dzdz⁰dw, where ¯G4(v,y−w,w,w−v) =IR(v,y−w)F(v,¯ w,y)G₄(v,y−w,w,w−v). Similar arguments show thatb^d_n|Λ_n,R|T₇−→0 asn→∞.

We now finish the proof of the asymptotic behaviour of the termT₈. Under the conditions imposed in Theorem3.1, we have

T₈=− 1 b^2d_n |Λ_n,R|²

Z

R^4d

11_Λn,R(u)11_Λn,R(w)I_R(v−u,y−w)K

v−u−t bn

×K

y−w−t b_n

E

H(u,v,X)λ_β^?(u,v,X) E

H(w,y,X)λ_β^?(w,y,X)

dudvdwdy

=− 1

b^2d_n |Λ_n,R|² Z

R^4d

11_Λn,R(u)11_Λn,R(w)K

v−u−t b_n

K

y−w−t b_n

×G¯₅(u,v,w,y,v−u,y−w)dudvdwdy

=− 1

b^2d_n |Λ_n,R|

× Z

R^3d

|Λ_n,R∩(Λ_n,R−w)|

|Λ_n,R| G¯₅(o,v,w,y,v,y−w)K v−t

bn

K

y−w−t bn

dvdwdy

=− 1

|Λn,R| Z

R^3d

|Λ_n,R∩(Λ_n,R−w)|

|Λn,R| G¯₅(o,bnz+t,w,bnz⁰+t+w,bnz+t,bnz⁰+t)

×K(z)K(z⁰)dwdzdz⁰, where

G¯₅(u,v,w,y,v−u,y−w) =I_R(v−u,y−w)F(u,¯ v)F(w,¯ y)G(v−u)G(y−w), where ¯F is given by (4.5). By dominated convergence theorem, we get the desired result.

4.2 Proof of Theorem 3.3

Proof. To prove the Theorem 3.3, we need (using the well-known Cram´er-Wold (1936)). Without loss of generality, we consider only the cases=2. Letλ1and λ₂ be two constants with λ₁+λ₂6=0 and let the pointst₁,t₂∈T\{o}be fixed, t₁6=t₂and define the triangular array of random variables

1

|Λ_n,R|^1/2

∑

i∈eIn

Y¯n,i=

λ₁(b^d_n|Λ_n,R|)^1/2

Hb_n(t₁)−E[Hb_n(t₁)]

+λ₂(b^d_n|Λ_n,R|)^1/2

Hb_n(t₂)−E[Hb_n(t₂)]

where

Y¯n,i=λ1

Yi(t₁)−E[Y_i(t₁)]

+λ2

Yi(t₂)−E[Y_i(t₂)]

and for all j=1,2, we consider Yi(t_j) = (b^d_n)^−1/2

6=

u∈X

∑

h(u,X\ {u,v})h(v,X\ {u,v})K

v−u−t_j b_n

.

The proving idea the below Theorem3.3consists in approximating the sequence Y¯n,iby a triangular array ofm-dependent random fields. The corresponding Lindeberg- type CLT m-dependent random fields has been proved by [39]. Therefore, it proves convenient to switch notation from the text and to define

Yen,i=λ1

Yei(t₁)−E[eYi(t₁)]

+λ2

Yei(t₂)−E[eYi(t₂)]

and for all j=1,2, we consider

Yei(t_j) = (b^d_n)^−1/2

6=

u∈Xi,

∑

h(u,X\ {u,v})h(v,X\ {u,v})K

v−u−t_j b_n

.

By construction,{eYn,i}_i∈Zd aremn-dependent, in the sense thatYen,iandYen,jare in- dependent if|i−j|_∞≥mn, where|j|_∞=max{|j1|, . . . ,|j_d|}. We will use{eYn,i;i∈ Z^d}_n∈Nto approximate{Y¯_n,i;i∈Z^d}_n∈N. We decompose

1

|Λ_n,R|^1/2

∑

i∈eIn

Y¯n,i= 1

|Λ_n,R|^1/2

∑

i∈eIn

Yen,i+ 1

|Λ_n,R|^1/2

∑

i∈eIn

Y¯n,i−Yen,i .

SinceKandhare bounded and applying Proposition 1 obtained [24] and under (3.10) of ConditionW we know that

1

|Λ_n|^1/2

∑

i∈eIn

Y¯n,i−Yen,i 2

≤c|λ₁|+|λ₂| (m_nb_n)^3d/2

∑

|i|>mn

|i|^5d2 δi,2=o(1).

Then, it suffices to establish the following result.

Proposition 1. Under(3.11)of ConditionW . As n→∞, we have 1

|Λ_n,R|^1/2

∑

i∈eIn

Yen,i

−→d N 0,σ² ,

where σ²=

β^?2λ₁²F(o,¯ t₁)G(t₁) +β^?2λ₂²F(o,¯ t₂)G(t₂) _Z

R^d

K²(z)dz. (4.7)

Proof of Proposition 1

Proof. Taking into account of{eY_n,i}_i∈Zd arem_n-dependent random fields, we ap- ply limit theorem for stationary triangular arrays of m-dependent random fields (Theorem 2 established by [39]). We need to verify that the two fundamental con- ditions (4.9) and (4.10), i.e. it suffices to show, for some sequence of increasing integers{l_n}_n∈Nsuch that

m_n/l_n→0 and l_n/n→0 as n→∞, (4.8) we have

n→∞lim 1 l_n^dE

∑

i∈J1,lnK^d

Yen,i

2

=σ² (4.9)

and

n→∞lim 1 l_n^dE

∑

i∈J1,lnK

d

Ye_n,i 2

11

∑

i∈J1,lnK

d

Ye_n,i

>εn^d/2

=0 for all ε>0 (4.10) whereJa,bK≡ {a,a+1, . . . ,b}fora,b∈Z.

Taking into account the definition of∑_i∈

J1,lnK

dYe_n,iwe get

n→∞lim 1 l_n^dVar

∑

i∈J1,lnK

d

Yen,i

= lim

n→∞b^d_nl^d_n

λ₁²Var(Hb_ln(t₁)) +λ₂²Var(Hb_ln(t₂)) +2λ₁λ₂Cov(Hb_ln(t₁),Hb_ln(t₂))

.

Using Theorem3.1entails replacingHb_n(t_j)byHb_ln(t_j), we have

n→∞limb^d_nl^d_nVar(Hb_ln(t_j)) =β^?2G(t_j)F¯(o,t_j) Z

R^d

K²(u)du.

Using Theorem3.2, we have

n→∞limb^d_nl_n^dCov

Hb_ln(t₁),Hb_ln(t₂)

=0 then, it follows that

n→∞lim 1 l_n^dVar

∑

i∈J1,lnK

d

Ye_n,i

=σ² (4.11)

whereσ²is defined by (4.7).

To prove (4.10), we writeξ_n=∑_i∈

J1,lnK^dYen,i, therefore we get E

ξ_n²11{|ξ_n|>εn^d/2}

≤ ||ξ_n||²_p P(|ξ_n|>n^d/2ε)(p−2)/p

≤ ||ξ_n||²_p

||ξ_n||²₂ n^dε²

(p−2)/p

.

Note that (4.11) yieldskξ_nk₂≤Cl_n^d/2for alln∈N. Forp>2, observe that, since K andhare bounded,

kξ_nk^p_p=E|ξ_n|^p≤E|ξ_n|² C

b^d/2_n l_n^d

p−2

.

By the above inequalities, we have obtained 1

l_n^dE

ξ_n²11{|ξ_n|>εn^d/2}

≤C l_n^2d

b^d_nn^d

(p−2)/p

.

Now, (3.11) of ConditionW entails thatl_ncan be chosen so that in addition to (4.8),l_n^2d/(n^db^d_n)→0 asn→∞; hence (4.10) follows.

This completes the proof of Theorem3.3.

4.3 Proof of Corollary 1

Proof. Let t ∈ {t₁, . . . ,ts} be fixed. Using the second-order Georgii-Nguyen- Zessin formula (2.2), and using the finite range property (3.2) and by stationarity

ofX, we obtain EHbn(t) = β^?2

b^d_n|Λn,R|E Z

R^2d

11_Λn,R(u)I_R(v−u)H(u,v)K

v−u−t b_n

G(v−u)dudv

= β^?2 b^d_n|Λ_n,R|

Z

R^2d

11_Λn,R(u)I_R(v−u)F¯(o,v−u)K

v−u−t b_n

G(v−u)dudv

=β^?2 Z

R^d

I_R(b_nz+t)F¯(o,b_nz+t)K(z)G(b_nz+t)dz.

By Taylor expansion of the integrand in neighborhood oft and making use of ConditionK(d,m), ConditionK(d)and the functionG(t)F¯(o,t)has bounded and continuous partial derivatives of order mof in some neighborhood of the pointt, we get the following rate of convergence

EHb_n(t) =β^?2G(t)F(o,¯ t) +O(b^m_n), as n→∞.

Under the conditionb^d+2m_n |Λ_n,R| −→0, we get q

b^d_n|Λ_n,R| EHb_n(t)−β^?2G(t)F(o,¯ t)

=Oq

b^d+2m_n |Λ_n,R|

. (4.12) By the application of Theorem3.3we conclude the result announced.

4.4 Proof of Theorem 3.4

Proof. Lett∈ {t₁, . . . ,t_s}be fixed. We may split the differenceGbn(t)−β^?G(t)as follows:

Gbn(t)−β^?G(t) = 1 b¯ F_n(t)

Hbn(t)−EHbn(t)

+ EHbn(t)−β^?G(t)Fb¯n(t)

= 1

b¯ Fn(t)

A⁽¹⁾_n +A⁽²⁾_n .

Hence, from [27] ergodic theorem, and additionally assume that P is ergodic, b¯

Fn(t) converges almost surely to β^?F¯(o,t). Note that there exists at least one stationary Gibbs measure. If this measure is unique, it is ergodic. Otherwise, it can be represented as a mixture of ergodic measures (see [13], Theorem 14.10).

Therefore, we can assume for this proof, that Pis ergodic. By Theorem 3.3, we have that as n→ ∞, p

b^d_n|Λ_n,R|A⁽¹⁾_n tends to a Gaussian distribution with zero mean. From strong consistency of the estimatorFb¯_n(t), and by inserting (4.12) and sinceb^d+2m_n |Λ_n,R| →0 asn→∞, we have that asn→∞,

q

b^d_n|Λ_n,R|A⁽²⁾_n −→^P 0.

which completes the proof.

Acknowledgements: The research was supported by laboratory Jean Kuntz- mann, Grenoble University, France.

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