Christophe Ange

Napoléon Biscio

Frédéric Lavancier

Contrast estimation for parametric stationary determinantal point processes

Keywords: Ripley's K function, pair correlation function, Brillinger mixing, central limit theorem. disjoint subsets D 1, . .

published or not. The documents may come

Contrast estimation for parametric stationary determinantal point processes

Introduction

Determinantal point processes (DPPs) are models for repulsive (or regular, or inhibitive) point processes data. They have been introduced by O. Macchi in [START_REF] Macchi | The coincidence approach to stochastic point processes[END_REF] to model the position of fermions, which are particles that repel each others. Their probabilistic aspects have been studied thoroughly, in particular in [START_REF] Soshnikov | Determinantal random point fields[END_REF], [START_REF] Shirai | Random point fields associated with certain Fredholm determinants. I. Fermion, Poisson and boson point processes[END_REF] and [START_REF] Hough | Zeros of Gaussian Analytic Functions and Determinantal Point Processes[END_REF]. Recently, DPPs have been studied and applied from a statistical perspective. A description of their main statistical aspects is conducted in [START_REF] Lavancier | Determinantal point process models and statistical inference[END_REF] and they actually turn out to be a well-adapted statistical model in domains as statistical learning [START_REF] Kulesza | Determinantal point processes for machine learning[END_REF], telecommunications [START_REF] Deng | The Ginibre point process as a model for wireless networks with repulsion[END_REF][START_REF] Miyoshi | A cellular network model with ginibre configurated base stations[END_REF], biology and ecology (see the examples in [START_REF] Lavancier | Determinantal point process models and statistical inference[END_REF] and [START_REF] Lavancier | Determinantal point process models and statistical inference : Extended version[END_REF]).

A DPP is defined through a kernel C, basically a covariance function. Assuming a parametric form for C, several estimation procedures are considered in [START_REF] Lavancier | Determinantal point process models and statistical inference[END_REF], specifically the maximum likelihood method and minimum contrast procedures based on the Ripley's K function or the pair correlation g. These methods are implemented in the spatstat library [START_REF] Baddeley | Spatial Point Patterns: Methodology and Applications with R[END_REF][START_REF] Baddeley | spatstat: An R package for analyzing spatial point patterns[END_REF] of R [START_REF] Core | R: A Language and Environment for Statistical Computing[END_REF]. From the simulation study conducted in [START_REF] Lavancier | Determinantal point process models and statistical inference[END_REF] and [START_REF] Lavancier | Determinantal point process models and statistical inference : Extended version[END_REF], see also Section 2.2, the maximum likelihood procedure seems to be the best method in terms of quadratic loss. However, the expression of the likelihood relies in theory on a spectral representation of C, which is rarely known in practice, and some Fourier approximations are introduced in [START_REF] Lavancier | Determinantal point process models and statistical inference[END_REF]. The likelihood also involves the determinant of a n × n matrix, where n is the number of observed points, which is prohibitively time consuming to compute when n is large. In contrast, the estimation procedures based on K or g do not require the knowledge of any spectral representation of C and are faster to compute in presence of large datasets, which explain their importance in practice.

From a theoretical point of view, neither the likelihood method nor the minimum contrast methods for DPPs have been studied thoroughly, even in assuming that a spectral method for C is known. In this work, we focus on parametric stationary DPPs and we prove the strong consistency and the asymptotic normality of the minimum contrast estimators based on K and g. These questions are in connection with the general investigation of Y. Guan and M. Sherman [START_REF] Guan | On least squares fitting for stationary spatial point processes[END_REF], who study the asymptotic properties of the latter estimators for stationary point processes. However the setting in [START_REF] Guan | On least squares fitting for stationary spatial point processes[END_REF] has a clear view to Cox processes and the assumptions involve both α-mixing and Brillinger mixing conditions, which are indeed satisfied for a large class of Cox processes. Unfortunately these results do no apply straightforwardly to DPPs. We consider instead more general versions of the asymptotic theorems in [START_REF] Guan | On least squares fitting for stationary spatial point processes[END_REF] and we prove that they apply nicely to DPPs. Our main ingredient then becomes the Brillinger mixing property of stationary DPPs, recently proved in [START_REF] Biscio | Brillinger mixing of determinantal points processes and statistical applications[END_REF], and we do not need any α-mixing condition. Our asymptotic results finally gather a very large class of stationary DPPs, where the main assumptions are quite standard and only concern the regularity of the kernel C with respect to the parameters. As an extension to the results in [START_REF] Guan | On least squares fitting for stationary spatial point processes[END_REF], it is worth mentioning the study of [START_REF] Waagepetersen | Two-step estimation for inhomogeneous spatial point processes[END_REF] dealing with constrast estimation for some inhomogeneous spatial point processes, still under a crucial α-mixing condition. We do not address this generalization for DPPs in the present work.

The remainder of this paper is organized as follows. In Section 2, we recall the definition of stationary DPPs, some of their basic properties and we discuss parametric estimation of DPPs. Our main results are presented in Section 3, namely the asymptotic properties of the minimum contrast estimators of a DPP based on the K or the g function. Section 4 gathers the proofs of our main results. In the appendix, we finally present our general asymptotic result for minimum contrast estimators and some auxiliary materials.

Stationary DPPs and parametric estimation 2.1 Stationary DPPs

We refer to [START_REF] Daley | An Introduction to the Theory of Point Processes[END_REF][START_REF] Daley | An Introduction to the Theory of Point Processes[END_REF] for a general presentation on point processes. Let X be a simple point process on R d . For a bounded set D ⊂ R d , denote by X(D) the number of points of X in D and let E be the expectation over the distribution of X. If there exists a function ρ (k) : (R d ) k → R + , for k ≥ 1, such that for any family of mutually

F (h)(t) = R d h(x)e 2iπx•t dx, ∀t ∈ R d
and we consider its extension to L 2 (R d ) by Plancherel's theorem, see [START_REF] Stein | Introduction to Fourier Analysis on Euclidean Spaces[END_REF].

Condition K(ρ). A kernel C is said to verify condition K(ρ) if C is a symmetric continuous real-valued function in L 2 (R d ) with C(0) = ρ and 0 ≤ F (C) ≤ 1.

Proposition 2.2 ([28, 19]). Assume C satisfies K(ρ). Then DP P (C) exists and is unique if and only if

0 ≤ F (C) ≤ 1.
In short, DP P (C) exists whenever C is a continuous covariance function in L 2 (R d ) with F (C) ≤ 1. This makes easy the construction of parametric families of DPPs, simply considering parametric families of covariance functions where the condition F (C) ≤ 1 appears as a constraint on the parameters. Some examples are given in [START_REF] Lavancier | Determinantal point process models and statistical inference[END_REF], [START_REF] Biscio | Quantifying repulsiveness of determinantal point processes[END_REF] and in the next section.

By definition, all moments of a DPP are known, in particular the pair correlation (pcf) and the Ripley's K-function can explicitly be expressed in terms of the kernel. For C satisfying K(ρ), let R(x) = C(x)/C(0) be the correlation function associated to C. The pcf, defined in the stationary case for all x ∈ R d by g(x) = ρ (2)

(0, x)/ρ 2 , writes g(x) = 1 -R 2 (x).
The Ripley's K-function is in turn given for all t ≥ 0 by

K(t) = B(0,t) g(x)dx = B(0,t) (1 -R 2 (x))dx (2.1)
where B(0, t) is the Euclidean ball centred at 0 with radius t. For later purposes, we denote by c red [k] the density of the reduced factorial cumulant moment measures of order k of X. We refer to [START_REF] Biscio | Brillinger mixing of determinantal points processes and statistical applications[END_REF] for the definition and further details, where the following particular cases are derived. Assuming that the kernel C of X satisfies K(ρ), we have for all (u, v, w

) ∈ R 3d c red [2] (u) = -C 2 (u), (2.2
)

c red [3] (u, v) = 2 C(u)C(v)C(v -u), (2.3) c red [4] (u, v, w) = -2 C(u)C(v)C(u -w)C(v -w) + C(u)C(w)C(u -v)C(v -w) + C(v)C(w)C(u -v)C(u -w) .
(2.4)

Parametric estimation of DPPs

We consider a parametric family of DPPs with kernel C ρ,θ where ρ = C ρ,θ (0) > 0 and θ belongs to a subset Θ ρ of R p , for a given p ≥ 1. To ensure the existence of DP P (C ρ,θ ), we assume that for all ρ > 0 and any θ ∈ Θ ρ , the kernel C ρ,θ verifies K(ρ), which explains the indexation of Θ ρ by ρ. We assume further that for a given ρ 0 > 0 and θ 0 in the interior of Θ ρ 0 (provided this interior is non-empty) we observe the point process X ∼ DP P (C ρ 0 ,θ 0 ) on a bounded domain

D n ⊂ R d .
The standard estimator of the intensity ρ 0 is

ρ n = 1 |D n | x∈X 1 {x∈Dn} (2.5)
where |D n | denotes the Lebesgue volume of D n . Since a stationary DPP is ergodic, see [START_REF] Soshnikov | Determinantal random point fields[END_REF], this estimator is strongly consistent by the ergodic theorem, and it is asymptotically normal, cf [START_REF] Soshnikov | Gaussian limit for determinantal random point fields[END_REF] and [START_REF] Biscio | Brillinger mixing of determinantal points processes and statistical applications[END_REF]. In the following, we focus our attention on the estimation of θ 0 . As explained in [START_REF] Lavancier | Determinantal point process models and statistical inference : Extended version[END_REF], likelihood inference is in theory feasible if we know a spectral representation of C ρ,θ on D n . Unfortunately no spectral representations are known in the general case and some Fourier approximations are introduced in [START_REF] Lavancier | Determinantal point process models and statistical inference : Extended version[END_REF]. Another option is to consider minimum contrast estimators (MCE) as described below.

For ρ > 0 and θ ∈ Θ ρ , let J(., θ) be a function from R d into R + which is a summary statistic of DP P (C ρ,θ ) that does not depend on ρ. In the DPP's case, the most important and natural examples are the K-function and the pcf g, that we study in detail in the following. Consider J n an estimator of J from the observation of X on D n . Further, let c ∈ R, c = 0, be a parameter such that J n (t) c and J(t, θ) c are well defined for all t ∈ R and θ ∈ Θ ρ 0 . Finally, define for 0 ≤ r min < r max , the discrepancy measure

U n (θ) = rmax r min w(t) J n (t) c -J(t, θ) c 2 dt (2.6)
where w is a smooth weight function. The MCE of θ 0 is

θ n = arg min θ∈Θ ρn U n (θ). (2.7)
For example, let us consider the parametric family of DPPs with Gaussian kernels

C(x) = ρe -| x α | 2 , x ∈ R d , ( 2.8)
where |.| denote the Euclidean norm on R d , ρ > 0 and α ≤ 1/( √ πρ 1/d ), the latter constraint on the parameter space being a consequence of the existence condition F (C) ≤ 1 in K(ρ). Some realizations are shown in Figure 1. For comparison, we have estimated the parameter α of this model with the MCE (2.7) when J corresponds to K or g, and with the maximum likelihood method (using the Fourier approximation of the spectral representation of C introduced in [START_REF] Lavancier | Determinantal point process models and statistical inference[END_REF]). The estimators of K and g, in place of J n in (2.7), are standard and recalled in Sections 3.2-3.3, see also [START_REF] Møller | Statistical Inference and Simulation for Spatial Point Processes[END_REF]Chapter 4]. For the tuning parameters, we followed the standard choice w(t) = 1, r min = 0.01, r max as one quarter of the side length of the window and c = 0.5 as recommended in [START_REF] Diggle | Statistical Analysis of Spatial Point Patterns[END_REF] for repulsive point processes. This simulation study has been carried out with the functions implemented in the spatstat library. Table 1 reports the mean squared errors of the three mentioned methods over 500 realisations of DP P (C) with ρ = 100 and α = 0.01, 0.03, For all methods considered in Table 1, the estimators seem consistent and the precision, in the sense of the mean squared errors, increases with the size of the observation window. From these results, the maximum likelihood method seems to be the best method in terms of quadradic loss, which agrees with the observations made in [START_REF] Lavancier | Determinantal point process models and statistical inference[END_REF]. However, MCEs, especially the one based on g, seem to perform reasonably well. Moreover, their computation is faster than the maximum likelihood method and do not rely on an approximated spectral representation of C. For instance, with a regular laptop, the estimation of α for 500 realizations on [0, 3] 2 took about 30 minutes for the MCEs based on K and g against more than 7 hours by the maximum likelihood method. Finally, it seems that each estimator has an asymptotic Gaussian behaviour, as illustrated in Figure 2 where we have represented the histograms obtained from the estimations of α = 0.03 over 500 realizations on [0, 1] 2 as in Table 1. The remainder of this paper is dedicated to proving the asymptotic normality of the MCE (2.7) when J = K or J = g and X is a stationary DPP. The asymptotic properties of the maximum likelihood estimator remain an open problem. Note finally that a solution to improve the efficiency of the MCEs, still avoiding the computation of the likelihood, is to construct an optimal linear combination of the MCE based on K and the MCE based on g, see [START_REF] Lavancier | A general procedure to combine estimators[END_REF] for a general presentation of the procedure and [START_REF] Lavancier | Modelling aggregation on the large scale and regularity on the small scale in spatial point pattern datasets[END_REF] for an example in spatial statistics. 3 Asymptotic properties of minimum contrast estimators based on K and g

1 10 √ π , observed on [0, 1] 2 , [0, 2] 2 and [0, 3] 2 .
[0, 1] 2 [0, 2] 2 [0, 3] 2 K g ML K g ML K g ML α = 0

Setting

In the next sections we study the asymptotic properties of (2.7) when J = K and J = g, respectively. The asymptotic is to be understood in the following way. We assume to observe one realization of X on D n and we let D n to expand to R d as detailed below. We denote by ∂D n the boundary of D n .

Definition 3.1. A sequence of subsets {D n } n∈N of R d is called regular if for all n ∈ N, D n ⊂ D n+1 , D n is compact, convex
and there exist constants α 1 and α 2 such that

α 1 n d ≤ |D n | ≤ α 2 n d , α 1 n d-1 ≤ H d-1 (∂D n ) ≤ α 2 n d-1
where

H d-1 is the (d -1)-dimensional Hausdorff measure.
Henceforth, we consider the estimator (2.7) under the setting of Section 2.2 where {D n } n∈N is a sequence of regular subsets of R d . Moreover, for any ρ > 0 and θ ∈ Θ ρ , we assume that the correlation function associated to C ρ,θ , denoted by R θ , does not depend on ρ but only on θ, i.e. R θ = C ρ,θ /ρ. Note that this is the case for all parametric families considered in [START_REF] Lavancier | Determinantal point process models and statistical inference[END_REF] and [START_REF] Biscio | Quantifying repulsiveness of determinantal point processes[END_REF], including the Whittle-Matèrn, the generalized Cauchy and the generalized Bessel families.

For r > 0, we denote by Θ ⊕r ρ 0 := Θ ρ 0 + B(0, r) the r-dilation of Θ ρ 0 , where B(0, r) denotes the closed ball centred at 0 with radius r. Further, for all x ∈ R d , denote R

(1) θ (x) and R

(2) θ (x), the gradient, respectively the Hessian matrix, of R θ (x) with respect to θ. We make the following assumptions. Specific additional hypotheses in the case J = K and J = g are described in the respective sections.

(H1) For all ρ > 0, Θ ρ is a compact convex set with non-empty interior and the mapping ρ → Θ ρ is continuous with respect to the Haussdorff distance on the compact sets.

(H2) For all θ ∈ Θ ρ 0 , C ρ 0 ,θ verifies the condition K(ρ 0 ) and there exists ǫ > 0 such that for all

θ ∈ Θ ⊕ǫ ρ 0 , C ρ 0 ,θ ∈ L 2 (R d ) and F (C ρ 0 ,θ ) ≥ 0.
(H3) There exists ǫ > 0 such that for all x ∈ B(0, r max ), the function

θ → R θ (x) is of class C 2 on Θ ⊕ǫ ρ 0 . Further, for i ∈ {1, 2}, there exists M > 0 such that for all x ∈ B(0, r max ) and θ ∈ Θ ⊕ǫ ρ 0 , R (i) θ (x) ≤ M.
The first assumption is needed to handle the fact that the minimisation (2.7) is done over the random set Θ ρn in place of Θ ρ 0 . The two other assumptions deal with the regularity of the kernel with respect to the parameters.

MCE based on K

Since for any ρ > 0 and θ ∈ Θ ρ R θ = C ρ,θ /ρ is assumed to not depend on ρ, the K-function (2.1) of DP P (C ρ,θ ) does not depend on ρ. Consequently we denote it by K(., θ). For all t ≥ 0 and n ∈ N, we consider the estimator of the K-function, see [START_REF] Møller | Statistical Inference and Simulation for Spatial Point Processes[END_REF]Chapter 4],

K n (t) := 1 ρ n 2 = (x,y)∈X 2 1 {x∈Dn} 1 {y∈D ⊖t n } 1 {|x-y|≤t} |D ⊖t n | (3.1)
where ρ n is as in (2.5) and for t ≥ 0,

D ⊖t n := {x ∈ D n , B(x, t) ∈ D n }. For all t ∈ [
r min , r max ], denote by K (1) (t, θ) and K (2) (t, θ) the gradient and the Hessian matrix of K(t, θ) with respect to θ. We consider the following assumptions.

(H K 1) w is a positive and integrable function in [r min , r max ]. (H K 2) If r min = 0, then c ≥ 2. (H K 3) For θ 1 = θ 2 , there exists a set A ∈ [r min , r max ] of positive Lebesgue measure such that x∈B(0,t) R θ 1 (x) 2 dx = x∈B(0,t) R θ 2 (x) 2 dx, ∀t ∈ A.
(H K 4) The matrix rmax r min w(t)K(t, θ 0 ) 2c-2 K (1) (t, θ 0 )K (1) (t, θ 0 ) T dt is invertible.

Assumption (H K 1) is not restrictive. The constraint on c implied by (H K 2) in the case r min = 0 tends to confirm the practice, which consists in the choice r min > 0. (H K 3) is an identifiability assumption and (H K 4) turns out to be the main technical assumption. Define for all t ∈ [r min , r max ],

j K (t) := w(t)K(t, θ 0 ) 2c-2 K (1) (t, θ 0 ).
The following theorem states the strong consistency and the asymptotic normality of the MCE based on K for stationary DPPs. It is proved in Section 4.1. Theorem 3.2. Let X be a DPP with kernel C ρ 0 ,θ 0 = ρ 0 R θ 0 for a given ρ 0 > 0 and θ 0 an interior point of Θ ρ 0 . For all n ∈ N, let U n be defined as in (2.6) with J = K and J n = K n . Assume that (H1)-( H3) and (H K 1)-(H K 4) hold. Then, the minimum contrast estimator θ n defined by (2.7) exists and is strongly consistent for θ 0 . Moreover, it satisfies

|D n |( θ n -θ 0 ) distr. ----→ n→+∞ N 0, B -1 θ 0 Σ ρ 0 ,θ 0 {B -1 θ 0 } T with B θ 0 := rmax r min w(t)K(t, θ 0 ) 2c-2 K (1) (t, θ 0 )K (1) (t, θ 0 ) T dt (3.2)
and

Σ ρ 0 ,θ 0 = rmax r min rmax r min h ρ 0 ,θ 0 (t 1 , t 2 )j K (t 1 )j K (t 2 )dt 1 dt 2
where h ρ 0 ,θ 0 can be expressed in terms of C ρ 0 ,θ 0 . Specifically, for all

(t 1 , t 2 ) ∈ [r min , r max ] 2 , h ρ 0 ,θ 0 (t 1 , t 2 ) := 2 R d 1 {0<|x|≤t 1 } 1 {0<|x|≤t 2 } c red [2] (x) + ρ 2 0 dx + 4 R 2d 1 {0<|x|≤t 1 } 1 {0<|y-x|≤t 2 } c red [3] (x, y) + ρ 0 c red [2] (y) dxdy + 4ρ 0 R 2d 1 {0<|x|≤t 1 } 1 {0<|y|≤t 2 } 2c red [2] (y) + ρ 2 0 dxdy + R 3d 1 {0<|x|≤t 1 } 1 {0<|z-y|≤t 2 } c red [4] (x, y, z)dxdydz + 4ρ 0 R 3d 1 {0<|x|≤t 1 } 1 {0<|z-y|≤t 2 } c red [3] (y, z)dxdydz + 2 R 3d 1 {0<|x|≤t 1 } 1 {0<|x+z-y|≤t 2 } c red [2] (y)c red [2] (z)dxdydz + 4ρ 2 0 R 3d 1 {0<|x|≤t 1 } 1 {0<|z-y|≤t 2 } c red [2] (y)dxdydz -4ρ 0 R 2d 1 {0<|x|≤t 1 } K(t 2 , θ 0 ) c red [3] (x, y) + 2ρ 0 c red [2] (y) dxdy -8ρ 0 R d 1 {0<|x|≤t 1 } K(t 2 , θ 0 ) c red [2] (x) + ρ 2 0 dx + 4ρ 2 0 K(t 1 , θ 0 )K(t 2 , θ 0 ) ρ 0 - R d C ρ 0 ,θ 0 (x) 2 dx
where c red [START_REF] Baddeley | spatstat: An R package for analyzing spatial point patterns[END_REF] , c red [3] and c red [START_REF] Biscio | Brillinger mixing of determinantal points processes and statistical applications[END_REF] are given with respect to C ρ 0 ,θ 0 in (2.2)-(2.4). Let us notice that the finiteness of the integrals involved in the last expression follows from the Brillinger mixing property of the DPPs with kernel verifying the condition K(ρ 0 ), see [START_REF] Biscio | Brillinger mixing of determinantal points processes and statistical applications[END_REF].

MCE based on g

We assume in this section that all DPPs of the parametric family are isotropic, which is the usual practice when dealing with the pair correlation function. In this case, for all ρ > 0 and θ ∈ Θ ρ , there exists

R θ such that R θ (x) = R θ (|x|) for all x ∈ R d so that the pcf of DP P (C ρ,θ ) writes g(x, θ) = 1 -R θ (|x|) 2 =: g(|x|, θ) (3.3)
and does not depend on ρ. In the following, to alleviate the notation, we omit the symbol tilde and for all θ ∈ Θ ρ , we consider that the domain of definition of R θ (.) and g(., θ) is R + . Moreover, by symmetry we extend this domain to R. Denote, for all d ≥ 2, the surface area of the d-dimensional unit ball,

σ d := 2π d/2 Γ (d/2) .
For n ∈ N and t > 0, we consider the kernel estimator of g, see [23, Section 4.3.5],

g n (t) := 1 σ d t d-1 ρ n 2 = (x,y)∈X 2 1 {x∈Dn, y∈Dn} 1 b n |D n ∩ D x-y n | k t -|x -y| b n (3.4)
where for any

z ∈ R d D z n := {u, u + z ∈ D n }, ρ n is as in (2.5
) and b n and k are the bandwidth and the kernel to be chosen according to the assumptions below. For all t ∈ [r min , r max ], denote by g (1) (t, θ) and g (2) (t, θ) the gradient and the Hessian matrix of g with respect to θ. We consider the assumptions:

(H g 1) r min > 0.
(H g 2) w is a positive and continuous function on [r min , r max ].

(H g 3) The kernel k is positive, symmetric and bounded with compact support included in [-T, T ] for a given

T > 0. Further, R k(x)dx = 1. (H g 4) {b n } n∈N is a positive sequence, b n → 0, b n |D n | → +∞ and b 4 n |D n | → 0. (H g 5) There exists ǫ > 0 such that for all θ ∈ Θ ⊕ǫ ρ 0 , R θ (.) is of class C 2 on R \ {0}. (H g 6) For θ 1 = θ 2 , there exists a set A ∈ [r min , r max ] of positive Lebesgue measure such that |R θ 1 (t)| = |R θ 2 (t)| , ∀t ∈ A.
(H g 7) The matrix rmax r min w(t)g(t, θ 0 ) 2c-2 g (1) (t, θ 0 )g (1) (t, θ 0 ) T dt is invertible.

The first four assumptions are easy to satisfy by appropriate choices of r min , w, b n and k. (H g 5) is not restrictive and is satisfied by all parametric families considered in [START_REF] Lavancier | Determinantal point process models and statistical inference[END_REF] and [START_REF] Biscio | Quantifying repulsiveness of determinantal point processes[END_REF]. (H g 6) is an identifiability assumption and as in the previous section, the main technical assumption is in fact (H g 7). The proof of the following theorem is postponed to Section 4.2. Put

j g (t) := w(t)g(t, θ 0 ) 2c-2 g (1) (t, θ 0 ), t ∈ [r min , r max ].
Theorem 3.3. Let X be an isotropic DPP with kernel C ρ 0 ,θ 0 = ρ 0 R θ 0 for a given ρ 0 > 0 and θ 0 an interior point of Θ ρ 0 . For all n ∈ N, let U n be defined as in (2.6) with J = g and J n = g n . Assume that (H1)-( H3) and (H g 1)-(H g 7) hold. Assume further that for all θ ∈ Θ ρ 0 , R θ (.) is isotropic. Then, the minimum contrast estimator θ n defined by (2.7) exists and is consistent for θ 0 . Moreover, it satisfies (1) (t, θ 0 )g (1) (t, θ 0 ) T dt and

|D n |( θ n -θ 0 ) distr. ----→ n→+∞ N 0, B -1 θ 0 Σ ρ 0 ,θ 0 {B -1 θ 0 } T with B θ 0 := rmax r min w(t)g(t, θ 0 ) 2c-2 g
Σ ρ 0 ,θ 0 = 2 R d 1 {r min ≤|x|≤rmax} j g (|x|)j g (|x|) σ 2 d |x| 2(d-1) c red [2] (x) + ρ 2 0 dx + 4 R 2d 1 {r min ≤|x|,|y-x|≤rmax} j g (|x|)j g (|y -x|) σ 2 d |x| d-1 |y -x| d-1 c red [3] (x, y) + ρ 0 c red [2] (y) dxdy + 4ρ 0 R 2d 1 {r min ≤|x|,|y|≤rmax} j g (|x|)j g (|y|) σ 2 d |x| d-1 |y| d-1 2c red [2] (x) + ρ 2 0 dxdy + R 3d 1 {r min ≤|x|,|z-y|≤rmax} j g (|x|)j g (|z -y|) σ 2 d |x| d-1 |z -y| d-1 c red [4] (x, y, z) dxdydz + 4ρ 0 R 3d 1 {r min ≤|x|,|z-y|≤rmax} j g (|x|)j g (|z -y|) σ 2 d |x| d-1 |z -y| d-1 c red [3] (y, z) dxdydz + 2 R 3d 1 {r min ≤|x|,|z-y+x|≤rmax} j g (|x|)j g (|z -y + x|) σ 2 d |x| d-1 |z -y + x| d-1 c red [2] (y)c red [2] (z) dxdydz + 4ρ 2 0 R 3d 1 {r min ≤|x|,|z-y|≤rmax} j g (|x|)j g (|z -y|) σ 2 d |x| d-1 |z -y| d-1 c red [2] (y) dxdydz -4ρ 0 rmax r min g(t, θ 0 )j g (t)dt R 2d 1 {r min ≤|x|≤rmax} j g (|x|) σ d |x| d-1 c red [3] (x, y) + 2ρ 0 c red [2] (y) dxdy -8ρ 0 rmax r min g(t, θ 0 )j g (t)dt R d 1 {r min ≤|x|≤rmax} j g (|x|) σ d |x| d-1 c red [2] (x) + ρ 2 0 dx + 4ρ 2 0 rmax r min g(t, θ 0 )j g (t)dt 2 ρ 0 - R d C ρ 0 ,θ 0 (x)

Proofs

Proof of Theorem 3.2

Since C ρ 0 ,θ 0 verifies K(ρ 0 ), ρ n converges almost surely to ρ 0 , so by (H1), for all ǫ > 0, there exists N ∈ N such that for all n ≥ N, Θ ρn ⊂ Θ ⊕ǫ ρ 0 almost surely. Henceforth, without loss of generality, we let ǫ > 0 and assume that Θ ρn ⊂ Θ ⊕ǫ ρ 0 for all n ∈ N. We apply below the general Theorems 5.1-5.2 of the appendix to prove that the estimator θ n defined in (5.2) with Θ = Θ ⊕ǫ ρ 0 , J = K and J n = K n is consistent and asymptotically normal. As a consequence, almost surely, there exist r > 0 such that B(θ 0 , r) ⊂ Θ ρ 0 and N r ∈ N such that for all n ≥ N r , θ n ∈ B(θ 0 , r). From Lemma 5.4 in the appendix and (H1), we deduce that for n sufficiently large, B(θ 0 , r) ⊂ Θ ρn . Hence, almost surely, for n large enough, the minimum of U n is attained in Θ ρn ⊂ Θ ⊕ǫ ρ 0 so that θ n in (5.2) and θ n in (2.7) coincide. Let us now prove the strong consistency and asymptotic normality of θ n in (5.2) when Θ = Θ ⊕ǫ ρ 0 , J = K and J n = K n . To that end, we verify all the assumptions of Theorems 5.1-5.2. The general setting in Section 3.1, Assumptions (H1) and (H K 1) imply directly (A1)-(A2). For all θ ∈ Θ, we have

K(t, θ) = σ d t d - x∈B(0,t) R θ (x) 2 dx (4.1)
where F (R θ ) ≥ 0 by (H2). Further, by [START_REF] Sasvári | Multivariate Characteristic and Correlation Functions[END_REF]Corollary 1.4.13], for all θ ∈ Θ, if for a given x = 0, |R θ (x)| = 1, then R θ is invariant by translation of x. Since for all θ ∈ Θ, R θ (.) ∈ L 2 (R d ), this is impossible so, for all x = 0 and θ ∈ Θ, |R θ (x)| < 1. Hence, by (4.1), K(t, θ) > 0 on (r min , r max ] × Θ and K(., .) is continuous on [r min , r max ] × Θ. Consequently, K(., .) c is continuous for all c ∈ R if r min > 0 and for all c > 0 if r min = 0. Therefore, under (H1)-(H3) and (H K 2), (A3) holds. By the same arguments, K(., .) c-2 and K(., .) 2c-2 are continuous for all c ∈ R if r min > 0 and for all c ≥ 2 if r min = 0. Thus (A8) holds. For all t ∈ [r min , r max ], K n (t) is bounded by K n (r max ) and it follows from the ergodic theorem that K n (r max ) is almost surely finite as soon as n and so D n is large enough. Moreover, by Lemma 4.1, K n (t) is almost surely strictly positive for t > 0 and n large enough. Hence, under (H1)-(H3) and (H K 2), (A4) holds. We have for all θ ∈ Θ and t ∈ (0, r max )

K (1) (t, θ) = - ∂ ∂θ x∈B(0,t) R θ (x) 2 dx.
By (H3), the function (x, θ) → R

(1) θ (x) is continuous with respect to θ and bounded for all x ∈ B(0, r max ) and θ ∈ Θ. Thus by the dominated convergence theorem,

K (1) (t, θ) = -2 x∈B(0,t) R θ (x)R (1) θ (x)dx. (4.2)
We obtain similarly

K (2) (t, θ) = -2 x∈B(0,t) R (1) θ (x)R (1) θ (x) T + R (2) θ (x)R θ (x) dx.
By (H3), the terms inside the integral in the last equation are bounded uniformly with respect to (x, θ) ∈ B(0, r max ) × Θ. Therefore, K (1) (t, θ) and K (2)

K n (t) -K(t) ≤ sup t∈[r min ,rmax] ρ 2 n K n (t) -ρ 2 0 K(t) + K(r max ) sup t∈[r min ,rmax] ρ 2 n -ρ 2 0 .
Hence

H n (t)s T j K (t)dt = [r min ,rmax] 2 h(t 1 , t 2 )s T j K (t 1 )s T j K (t 2 )dt 1 dt 2
where h ρ 0 ,θ 0 is defined as in Theorem 3.2.

Proof. From (3.1), we have

rmax r min H n (t)s T j K (t)dt = (x,y)∈X 2 f n (x, y) - x∈X h n (x)
where for all n ∈ N,

f n (x, y) := 1 {x∈Dn} rmax r min 1 |D ⊖t n | 1 {y∈D ⊖t n } 1 {0<|x-y|≤t} s T j K (t)dt and h n (x) = 2ρ 0 |D n | 1 {x∈Dn} rmax r min K(t, θ 0 )s T j K (t)dt.
Notice that for all n ∈ N and x ∈ R d , f n (x, x) = 0. Thus, we have from the last equation,

Var rmax r min H n (t)s T j K (t)dt = Var   = (x,y)∈X 2 f n (x, y)   +Var x∈X h n (x) -2 Cov   = (x,y)∈X 2 f n (x, y), x∈X h n (x)   .
These terms are developed in Lemmas 7.1-7.3 of [4], whereby we deduce the limit by a long but straightforward calculus.

Lemma 4.4. If (H1)-(H3) and (H

K 1)-(H K 2) hold, then |D n | rmax r min K n (t) -K(t, θ 0 ) j K (t)dt distr.
----→ n→+∞ N (0, Σ ρ 0 ,θ 0 )

where Σ ρ 0 ,θ 0 is defined as in Theorem 3.2.

Proof. For all n ∈ N, we have

ρ 2 0 |D n | rmax r min K n (t) -K(t, θ 0 ) j K (t)dt = |D n | rmax r min ρ 2 0 -ρ 2 n K n (t)j K (t)dt + |D n | rmax r min ρ 2 n K n (t) -ρ 2 0 K(t, θ 0 ) j K (t)dt. (4.6)
Since X is ergodic by [START_REF] Soshnikov | Determinantal random point fields[END_REF]Theorem 7], ρ n converges almost surely to ρ 0 . Then, by Taylor expansion of the function x → x 2 at ρ 0 , we have almost surely

ρ 2 0 -ρ 2 n = 2ρ 0 [ρ 0 -ρ n ] + o (ρ 0 -ρ n ) . (4.7) Moreover, 2ρ 0 |D n | rmax r min [ρ 0 -ρ n ] K n (t)j K (t)dt = 2ρ 0 |D n | rmax r min [ρ 0 -ρ n ] K n (t) -K(t, θ 0 ) j K (t)dt + 2ρ 0 |D n | rmax r min [ρ 0 -ρ n ] K(t, θ 0 )j K (t)dt. (4.8)
Using the notation

A n = 2ρ 0 |D n | [ρ 0 -ρ n ] rmax r min K n (t)j K (t)dt, B n = 2ρ 0 |D n | [ρ 0 -ρ n ] rmax r min K n (t) -K(t, θ 0 ) j K (t)dt, C n = |D n | rmax r min [ρ 0 -ρ n ] 2ρ 0 K(t, θ 0 ) + ρ 2 n K n (t) -ρ 2 0 K(t, θ 0 ) j K (t)dt,
we have by (4.6)-(4.8),

ρ 2 0 |D n | rmax r min K n (t) -K(t, θ 0 ) j K (t)dt = B n + C n + o (A n ) . ( 4
K n (t) -K(t, θ 0 ) a.s. ----→ n→+∞ 0 so rmax r min K n (t)j K (t)dt a.s. ----→ n→+∞ rmax r min K(t, θ 0 )j K (t)dt. (4.10) Since K(., θ 0 ) is continuous on [r min , r max ], rmax r min K(t, θ 0 )j K (t)dt is finite by Lemma 4.
C n = |D n | rmax r min H n (t)j K (t)dt -- rmax r min ρ 2 0 K(t, θ 0 )j K (t)dt . (4.11)
We prove the convergence in distribution of C n by the Cramer-Wold device, see for instance [START_REF] Billingsley | Probability and measure[END_REF]Theorem 29.4]. For all t ∈ [r min , r max ] and s ∈ R p , we have

s T C n = |D n | rmax r min H n (t)s T j K (t)dt -- rmax r min ρ 2 0 K(t, θ 0 )s T j K (t)dt . By (3.1), we have rmax r min H n (t)s T j K (t)dt = (x,y)∈X 2 f Dn (x, y) (4.12)
where

f Dn (x, y) := 1 {x∈Dn} rmax r min 1 {y∈D ⊖t n } |D ⊖t n | 1 {0<|x-y|≤t} -2ρ 0 K(t, θ 0 ) |D n | 1 {x-y=0} s T j K (t)dt.
Notice that for t ∈ [r min , r max ], s T j K (t) ≤ |j K (t)||s| and K(t, θ 0 ) ≤ K(r max , θ 0 ) so we have

|f Dn (x, y)| ≤ |s| |D ⊖rmax n | 1 Dn (x) 1 {0<|x-y|≤rmax} + 1 {x-y=0} 2ρ 0 K(r max , θ 0 ) rmax r min |j K (t)| dt. (4.13)
The right-hand term in (4.13) is compactly supported and is bounded by Lemma 4.2. Moreover,

E rmax r min H n (t)s T j K (t) dt ≤ |s| E ρ 2 n K n (t) + 2ρ 0 K(r max , θ 0 )E (| ρ n |) rmax r min |j K (t)|dt.
Further, for n ∈ N and t ∈ [r min , r max ], ρ 2 n K n (t) and ρ n are positive and unbiased estimator of ρ 2 0 K(t, θ 0 ) and ρ 0 , respectively, see for instance [START_REF] Heinrich | Asymptotic methods in statistics of random point processes[END_REF]Section 4.2.2]. Thus,

E rmax r min H n (t)s T j K (t) dt ≤ 3|s|ρ 2 0 K(r max , θ 0 ) rmax r min |j K (t)|dt,
which is finite by Lemma 4.2. Then, by Fubini's theorem, (4.12) and the last equation, we have

E   (x,y)∈X 2 f Dn (x, y)   = - rmax r min ρ 2 0 K(t, θ 0 )s T j K (t)dt.
Moreover, by (4.12) and Lemma 4.3,

lim n→+∞ V ar   |D n | (x,y)∈X 2 f Dn (x, y)   = s T Σ ρ 0 ,θ 0 s.
Therefore, by (4.11)-(4.13), the last two equations and Theorem 5.5, we have

s T C n distr. ----→ n→+∞ N(0, s T Σ ρ 0 ,θ 0 s).
which proves that C n distr.

----→ n→+∞ N(0, Σ ρ 0 ,θ 0 ).

Proof of Theorem 3.3

As in the proof of Theorem 3.2, we consider without loss of generality ǫ > 0 such that Θ ρn ⊂ Θ ⊕ǫ ρ 0 , for all n ∈ N. We prove below the consistency and asymptotic normality of θ n defined in (5.2) with Θ = Θ ⊕ǫ ρ 0 , J = g and J n = g n . Then, for r ≥ 0 such that B(θ 0 , r) ⊂ Θ ρ 0 , we have

P ( θ n ∈ B(θ 0 , r)) ----→ n→+∞ 1.
Thus, by Lemma 5.3, with probability tending to one θ n ∈ Θ ρn so

P ( θ n = θ n ) ----→ n→+∞ 1.
Therefore, θ n has the same asymptotic behaviour than θ n .

Let us now determine the asymptotic properties of θ n by application of Theorems 5.1 and 5.2. The assumptions (A1), (A2), (A6), (A7) and (A9) are directly implied by (H1)-(H3), (H g 1), (H g 2), (H g 6) and (H g 7). Moreover, r min > 0 by (H g 1) so (A4) is directly implied by (3.4), (H g 3), (H g 4) and the ergodic theorem, see [START_REF] Nguyen | Ergodic theorems for spatial processes[END_REF] or [START_REF] Heinrich | Asymptotic methods in statistics of random point processes[END_REF]. By (H2), R θ 0 (.) is continuous on [r min , r max ] so is g. By [START_REF] Sasvári | Multivariate Characteristic and Correlation Functions[END_REF]Corollary 1.4.14], for all θ ∈ Θ, if for a given t > 0, |R θ (t)| = 1, then R θ is periodic of period t. This is incompatible with (H2) so, for all t > 0 and θ ∈ Θ, |R θ (t)| < 1. Consequently, by (3.3) and (H g 1), g(t, θ) is strictly positive for all (t, θ) ∈ [r min , r max ] × Θ. Thus, for all c ∈ R, g(., .) c is well defined and strictly positive on [r min , r max ]×Θ so (A3) holds. By the same arguments, it follows that (A8) holds. Finally, the assumptions (A5) and (T CL) are proved by Lemmas 4.5 and 4.9, respectively while the other lemmas are auxiliary results.

Lemma 4.5. If (H1)-(H3), (H g 1) and (H g 3)-(H g 4) hold then, for all

r max > r min > 0, there exists a set A verifying |[r min , r max ] \ A| = 0 such that sup t∈A | g n (t) -g(t, θ 0 )| P ----→ n→+∞ 0.
Proof. From (H2)-( H3) and (H g 3)-(H g 4) we can use Proposition 4.5 in [START_REF] Biscio | Brillinger mixing of determinantal points processes and statistical applications[END_REF] that gives

E rmax r min ρ 2 n g n (t) -ρ 2 0 g(t, θ 0 ) 2 dt = 2ρ 2 0 b n |D n | rmax r min g(t, θ 0 ) σ d t d-1 dt R k(x) 2 dx + O 1 |D n | + O(b 4 n ). (4.14)
By (H g 1), (H g 3) and (H3) we have rmax r min g(t,θ 0 )

σ d t d-1 dt R k(x) 2 dx < +∞.
Hence, with (H g 4), the right-hand term in (4.14) tends to 0 as n tends to infinity. Moreover, the term inside the expectation in (4.14) is positive so there exists a set A as in Lemma 4.5 such that

sup t∈A ρ 2 n g n (t) -ρ 2 0 g(t, θ 0 ) P ----→ n→+∞ 0. (4.15)
We have

ρ 2 n sup t∈A | g n (t) -g(t, θ 0 )| ≤ sup t∈A ρ 2 n g n (t) -ρ 2 0 g(t, θ 0 ) + sup t∈A g(t, θ 0 ) ρ 2 n -ρ 2 0 .
By (H1)-(H2), it follows from Corollary 5.6 that ρ n converges in probability to ρ 0 . Further, by (H3) and (3. Proof. By (3.3), we have for all t ∈ [r min , r max ]

|j g (t)| = 2 w(t) 1 -R θ 0 (t) 2 2c-2 R θ 0 (t)R (1)
θ 0 (t) .

By (H3), R θ 0 (.) and R

(1) θ 0 (.) are continuous on [r min , r max ]. Further, by (H g 1), r min > 0 and as noticed at the beginning of the proof of Theorem 3.3, for all t > 0, |R θ 0 (t)| < 1. Thus by (H3), the function t

→ (1 -R θ 0 (t) 2 )
2c-2 is well defined and continuous on [r min , r max ]. Finally, by (H g 2), w is continuous on [r min , r max ] so the lemma is proved.

To abbreviate, we define for all n ∈ N and t ∈ [r min , r max ],

H g n (t) := ρ 2 n g n (t) -2ρ 0 ρ n g(t,
H g n (t)s T j g (t)dt = (x,y)∈X 2 f n (x, y) - x∈X h n (x)
where for all n ∈ N,

f n (x, y) := 1 {x∈Dn} rmax r min k t-|x-y| bn 1 {|x-y|>0,y∈Dn} σ d t d-1 b n |D n ∩ D x-y n | s T j g (t)dt and h n (x) = 2ρ 0 |D n | 1 {x∈Dn} rmax r min g(t, θ 0 )s T j g (t)dt.
The result follows similarly as in the proof of Lemma 4.3 using Lemmas 7.1-7.3 in [START_REF] Biscio | Brillinger mixing of determinantal points processes and statistical applications[END_REF].

Lemma 4.8. Assume that (H1)-( H3) and (H g 1)-(H g 4) hold. For a given s ∈ R d and all n ∈ N, let f Dn be defined for any (x, y) ∈ R 2d by f Dn (x, y)

:= 1 {x∈Dn} rmax r min   k t-|x-y| bn 1 {|x-y|>0,y∈Dn} σ d t d-1 b n |D n ∩ D x-y n | - 2ρ 0 g(t, θ 0 ) |D n | 1 {x-y=0}   s T j g (t)dt.
Then, there exists M > 0 such that for all (x, y) ∈ R 2d ,

|f Dn (x, y)| ≤ |s|M1 {x∈Dn} |D ⊖rmax+T n | 1 σ d r d-1 min 1 {0<|x-y|≤rmax+T } + 2ρ 0 ||g|| ∞ 1 {x-y=0} . Proof.
σ d t d-1 b n |D n ∩ D x-y n | s T j g (t)dt ≤ 1 {x∈Dn} |s|M |D ⊖rmax+T n | 1 {0<|x-y|≤rmax+T } σ d r d-1 min b n rmax r min k t -|x -y| b n dt.
Finally, the result follows by the last inequality, (H2) and (H g 3).

Lemma 4.9. If (H1)-( H3) and (H g 1)-(H g 5) hold, then

|D n | rmax r min [ g n (t) -g(t, θ 0 )] j g (t)dt distr.
----→ n→+∞ N (0, Σ ρ 0 ,θ 0 )

with Σ ρ 0 ,θ 0 defined as in Theorem 3.3.

Proof. The arguments of this proof are similar the the ones of the proof of Lemma 4.4. Notice that

ρ 2 0 |D n | rmax r min [ g n (t) -g(t, θ 0 )] j g (t)dt = |D n | ρ 2 0 -ρ 2 n rmax r min g n (t)j g (t)dt + rmax r min ρ 2 n g n (t) -E ρ 2 n g n (t) j g (t)dt + rmax r min E ρ 2 n g n (t) -ρ 2 0 g(t, θ 0 ) j g (t)dt (4.17)
and Denote

|D n | [ρ 0 -ρ n ] rmax r min g n (t)j g (t)dt = |D n | [ρ 0 -ρ n ] rmax r min [ g n (t) -g(t, θ 0 )] j g (t)dt+ |D n | [ρ 0 -ρ n ]
T n = 2ρ 0 |D n | [ρ 0 -ρ n ] rmax r min g n (t)j g (t)dt U n = 2ρ 0 |D n | [ρ 0 -ρ n ] rmax r min [ g n (t) -g(t, θ 0 )] j g (t)dt V n = |D n | rmax r min E ρ 2 n g n (t) -ρ 2 0 g(t, θ 0 ) j g (t)dt W n = |D n | rmax r min ρ 2 n g n (t) -2ρ 0 ρ n g(t, θ 0 ) -E ρ 2 n g n (t) -2ρ 2 0 g(t, θ 0 ) j g (t)dt.
Using (4.7) in the proof of Lemma 4.4, (4.17) and (4.18), we get

ρ 2 0 |D n | rmax r min [ g n (t) -g(t, θ 0 )] j g (t)dt = U n + V n + W n + o (T n ) . ( 4
H g n (t) = ρ 2 n g n (t) -ρ 2 0 g(t, θ 0 ) + (ρ 0 -ρ n )ρ 0 g(t, θ 0 ) -ρ 0 ρ n g(t, θ 0 ),
we have

E rmax r min H g n (t)s T j g (t) dt ≤ |s|ME rmax r min ρ 2 n g n (t) -ρ 2 0 g(t, θ 0 ) dt + |s|M [E(|ρ 0 -ρ n |) + E( ρ n )]
+ |s|M(r max -r min )ρ 0 ||g|| ∞ (E(|ρ 0 -ρ n |) + E( ρ n )) . (4.21)
By the same arguments as in the proof of Lemma 4.5, we have

E rmax r min ρ 2 n g n (t) -ρ 2 0 g(t, θ 0 ) 2 dt = 2ρ 2 0 b n |D n | rmax r min g(t, θ 0 ) σ d t d-1 dt R k(x) 2 dx + O 1 |D n | + O(b 4 n ).
Thus by (H g 4), E rmax r min ( ρ 2 n g n (t)ρ 2 0 g(t, θ 0 )) 2 dt tends to 0. Moreover, as noticed in [START_REF] Heinrich | Minimum contrast estimates for parameters of spatial ergodic point processes[END_REF],

ρ n converge in L 1 to ρ 0 so E(|ρ 0 -ρ n |) + E( ρ n ) converges to ρ 0 .
s T W n = |D n | (X s n -E(X s n )) .
By (3.4), we have

X s n = (x,y)∈X 2 f Dn (x, y), (4.22)
where f Dn (x, y) is given in Lemma 4.

∈ R p |D n | (X s n -E(X s n ))
distr.

----→ n→+∞ N(0, s T Σ ρ 0 ,θ 0 s), which implies that W n distr.

----→ n→+∞ N(0, Σ ρ 0 ,θ 0 ).

Appendix

A general result for minimum contrast estimation

We present in this section two general theorems concerning the consistency and asymptotic normality of the estimator defined in (2.7). Contrary to the results in Sections 3.2-3.3, these theorems hold for an arbitrary stationary point process and an arbitrary statistic J, generalizing a study by [START_REF] Guan | On least squares fitting for stationary spatial point processes[END_REF]. The results of Sections 3.2-3.3 are in fact consequences in the particular case of a DPP and J = K or J = g, which simplifies the general assumptions below. Let X be a stationary point process belonging to a parametric family indexed by, among possibly other parameters, θ ∈ Θ where Θ ⊂ R p , for a given p ≥ 1. For any t ∈ [r min , r max ], let J(t, θ) be any real valued summary statistic of X that depends Theorem 5.1. Let X be a stationary point process with distribution ruled by a given θ 0 , assumed to be an interior point of Θ. For all n ∈ N, let U n be defined as in (2.6). Assume that(A1)-(A6) hold. Then, the minimum contrast estimator θ n defined by

θ n = arg min θ∈Θ U n (θ) (5.2)
exists almost surely, is consistent for θ 0 and strongly consistent if (A5) ′ holds.

Proof. For a sequence {θ m } m∈N belonging to Θ, we have for all n ∈ N,

|U n (θ m ) -U n (θ)| ≤ rmax r min |w(t)| 2 J n (t) c |J(t, θ m ) c -J(t, θ) c | + J(t, θ m ) 2c -J(t, θ) 2c dt. (5.3)
Denote A the intersection of the sets defined in (A3) and (A5). By (A3), J(., .) c is continuous on [r min , r max ] × Θ which is compact by (A1). We deduce that sup

t∈[r min ,rmax] |J(t, θ m ) c -J(t, θ) c | ≤ K.
By (A3)-(A4), for all θ ∈ Θ, J(t, θ) c and J n (t) c are almost surely bounded on [r min , r max ], for all n large enough. Further, by (A2), w is integrable on [r min , r max ] thus, by (5.3) and the dominated convergence theorem, we have the convergence

|U n (θ m ) -U n (θ)| a.s.
---→ θm→θ 0.

Therefore, for all n large enough, U n is almost surely continuous so the almost sure existence of θ n follows by (A1). Define for all θ ∈ Θ,

U * n (θ) = U n (θ) -U n (θ 0 ).
(5.4) By (2.6) and (5.4), ----→ n→+∞ N m, B(θ 0 ) -1 Σ B(θ 0 ) -1 T where B is defined as in (5.1) and Σ comes from (T CL).

U * n (θ) = 2 rmax r min w(t) J n (t) c -J(t, θ 0 ) c J(t, θ 0 ) c -J(t, θ) c dt + rmax r min w(t) J(t, θ 0 ) c -J(t, θ) c 2 dt. that from (5.4) U * n ( θ n ) ≤ U * n (θ 0 ) = 0, so rmax r min w(t) J(t,θ 0 ) c -J(t, θ n ) c 2 dt ≤ 2 rmax r min w(t) J n (t) c -J(t, θ 0 ) c J(t, θ 0 ) c -J(t, θ n ) c dt. ( 5
Proof. Denote by A the intersection of the sets defined in (A3) and (A5). Then, by (A3), (A7) and (A8), we see that U n is almost surely twice differentiable on Θ and that we can differentiate twice under the integral sign. Thus, by the mean value theorem, for all j = 1, . . . , p, there exists s ∈ (0, 1) and θ * j = θ 0 + s( θ nθ 0 ) such that

∂ j U n ( θ n ) -∂ j U n (θ 0 ) = ∂ 2 ij U n (θ * j )
i=1,...,p θ nθ 0 .

To shorten, denote by U (1) n the gradient of U n and by U (2) n (θ * n ) the matrix with entries ∂ 2 ij U n (θ * j ). Since U n is minimal at θ n , U (1) n ( θ n ) = 0 and the last equation becomes U (2) n (θ * n )( θ nθ 0 ) = -U (1) n (θ 0 ) = 2c rmax r min w(t) J n (t) c -J(t, θ 0 ) c J(t, θ 0 ) c-1 J (1) (t, θ 0 )dt. (5.6) Note that by (A3) and (A8), J(., θ 0 ) c-1 is bounded on A and strictly positive. Thus, by (A4), we can use the Taylor expansion of the function x → x c so, for all t ∈ A, J n (t) c -J(t, θ 0 ) c = cJ(t, θ 0 ) c-1 J n (t) -J(t, θ 0 ) + o J n (t) -J(t, θ 0 ) . Therefore, by (A5), (5.6) and the last equation, |D n | U (2) n (θ * n )( θ nθ 0 ) = 2c 2 A n (θ 0 ) + o(A n (θ 0 )) (5.7)

where

A n (θ 0 ) = |D n | A
J n (t) -J(t, θ 0 ) j(t)dt.

By (T CL), we have 2c 2 A n (θ 0 )

distr.

----→ n→+∞ 2c 2 N(m, Σ). Hence, by Slutsky's theorem and (5.7),

|D n | U (2) n (θ * n )( θ n -θ 0 )
distr.

----→ n→+∞ 2c 2 N(m, Σ).

(5.8)

Moreover, we have that

U (2) n (θ * n ) = 2c 2 B(θ * n ) -E n (5.9)
where B is as in (5.1) and Proof. Since B(x, r) belongs to the interior of Θ, there exists δ > 0 such that B(x, r + δ) ⊂ Θ.

E n := 2c(c -
(5.12)

Assume that the lemma is wrong, then for all N ∈ N, there exists n ≥ N such that B(x, r) Θ n . Denote y a point in B(x, r) that does not belong to Θ n . By Lemma 5.3, B(y, δ) Θ ⊕δ n . But by (5.12), B(y, δ) ⊂ Θ so Θ Θ ⊕δ n which contradicts the convergence of the sequence {Θ n } n∈N to Θ.

The following theorem appears in [START_REF] Jolivet | Central limit theorem and convergence of empirical processes for stationary point processes[END_REF] in a slightly less general framework, see also [START_REF] Krickeberg | Processus ponctuels en statistique[END_REF], and is proved in [START_REF] Biscio | Brillinger mixing of determinantal points processes and statistical applications[END_REF] in its present form. It is used in the proofs of our main results, Theorems 3.2 and 3.3. distr.

----→ n→+∞ N (0, σ 2 ) (5.16)

and the convergence of all moments to the corresponding moments of N (0, σ 2 ).

As a corollary when p = 1, we retrieve a theorem from [START_REF] Soshnikov | Gaussian limit for determinantal random point fields[END_REF] giving the asymptotic normality of the estimator of the intensity of a DPP.

Figure 1 :

 1 Figure 1: Realizations on [0, 1] 2 of DPPs with kernel (2.8) where ρ = 100 and from left to right α = 0.01, 0.03, 1 10 √ π .

Figure 2 :

 2 Figure 2: Histograms of the estimations of α = 0.03 from 500 realizations of DPPs with kernel (2.8) on [0, 1] 2 . From left to right : MCE (2.7) based on K, MCE (2.7) based on g and maximum likelihood estimator.

  2. Hence, by Corollary 5.6, (4.10) and Slutsky's theorem, we deduce that B n P to the term C n , notice that

Lemma 4 . 6 .

 46 3), g(., θ 0 ) is bounded on [r min , r max ]. Therefore, we have by (4.15) the convergence sup t∈A | g n (t)g(t, θ 0 )| P ----→ n→+∞ 0. If (H1)-(H3), (H g 1)-(H g 2) hold then j g (.) is continuous on [r min , r max ].

  , θ 0 )j g (t)dt.(4.18)

1 σ d r d- 1 min 1

 111 8 and satisfies |f Dn (x, y)| ≤ |s|M1 {x∈Dn} |D ⊖rmax+T n | {0<|x-y|≤rmax+T } + 2ρ 0 ||g|| ∞ 1 {x-y=0} . (4.23) The right-hand term in (4.23) is bounded and compactly supported. Therefore, by Lemma 4.7 and Theorem 5.5, we have for all s

Theorem 5 . 5 . 2 . ( 5 . 14 ) 2 , ( 5 . 15 )

 5525142515 Let {D n } n∈N and { D n } n∈N be two sequences of regular sets in the sense of Definition 3.1 such that | Dn| |Dn| n→+∞ ----→ κ for a given κ > 0. For all n ∈ N, let {f Dn } n∈N be a family of functions from R dp into R. Assume that there exists a bounded function F from R d(p-1) into R + with compact support such that for all n ∈ N and (x 1 , . . . , x p ) ∈ R dp , |f Dn (x 1 , . . . , x p )| ≤ 1 | D n | 1 {x 1 ∈Dn} F (x 2x 1 , . . . , x px 1 ). (5.13)Assume further that the point process X is ergodic, admits moment of any order and is Brillinger mixing. Then, for all k ≥ 2, we haveCum k |D n |N p (f Dn ) = O |D n | 1-k Moreover, if there exists σ > 0 such that Var |D n |N p (f Dn ) ----→ n→+∞ σthen we have the convergence|D n | [N p (f Dn ) -E (N p (f Dn ))]

  .9)We prove that B n + o(A n ) tends in probability to 0 and C n tends in distribution to a Gaussian variable. Then, the proof is concluded by Slutsky's theorem and (4.9).

	By Lemma 4.1,
	sup
	t∈[r min ,rmax]

  θ 0 ).

	Lemma 4.7. If (H1)-(H3) and (H lim n→+∞ |D n |Var rmax r min	H g n (t)s

g 1)-(H g 5) hold, we have for all s ∈ R d , T j g (t)dt = s T Σ ρ 0 ,θ 0 s with Σ ρ 0 ,θ 0 defined as in Theorem 3.3. Proof. Similarly to the proof of Lemma 4.3, we have by (3.4), rmax r min

  By (H g 3), for any t ∈ [r min , r max ] and (x, y) ∈ R 2d ,

		k	t -|x -y| b n	1 {|y-x|>0, y∈Dn} ≤ k	t -|x -y| b n	1 {0<|y-x|<t+T bn}
							≤ k	t -|x -y| b n	1 {0<|y-x|<t+T }
	whenever b n < 1 which, by (H g 4), we assume in the following without loss of gener-
	ality. Thus, for any t ∈ [r min , r max ] and (x, y) ∈ R 2d ,
	k	t -|x -y| b n	1 {|y-x|>0, y∈Dn} ≤ k	t -|x -y| b n	1 {0<|y-x|<rmax+T } .	(4.16)
	Further, by Lemma 4.6, j g is bounded on [r min , r max ] by a constant M so by (4.16)
	and Lemma 6.3 in [4], we have
	1 {x∈Dn}	r min rmax	k t-|x-y| bn	1 {|x-y|>0,y∈Dn}

  .[START_REF] Lavancier | Determinantal point process models and statistical inference[END_REF] We prove that U n + V n + o(T n ) tends in probability to 0 and we conclude by proving that W n tends in distribution to a Gaussian variable. From Corollary 5.6, Lemmas 4.5-4.6 and Slutsky's theorem, we have U n min , r max ] so bounded, we have byLemma 4.5 that g n is uniformly bounded in probability on [r min , r max ], see [31, Prohorov's theorem]. Thus, by Corollary 5.6 and Lemma 4.6, o(T n )We prove the convergence in distribution of W n by the Cramer-Wold device. To shorten, denote for all n ∈ N and s ∈ R p , By Lemma 4.6, j g is bounded on [r min , r max ] by a constant M. Then, since for all t ∈ [r min , r max ] and n ∈ N,

	P n→+∞ ----→ 0. Further, under (H1)-(H2) and (H g 1)-(H g 5), we 0. Further, since g is contin-deduce from Lemma 6.2 in [4] that sup t∈[r min ,rmax] (E [ ρ 2 uous on [r P ----→ n→+∞ 0 g(t, θ 0 )) < κb 2 n n g n (t)] -ρ 2 with κ > 0, which combined with (H g 4) and Lemma 4.6 proves that V n P ----→ n→+∞ 0.
	X s n :=	rmax r min	H g n (t)s T j g (t)dt.

  Under the same setting as in Theorem 5.1, if in addition (A7)-(A9) and (T CL) hold true, then|D n |( θ nθ 0 )

		.5)
	By (A3)-(A4), J(., .) c is continuous on [r min , r max ]×Θ and for n large enough, J n (.) c
	is almost surely bounded on [r min , r max ] so by (A5), the right-hand term in (5.5)
	tends in probability to 0. Hence, we have
	rmax r min	w(t) J(t,θ 0 ) c -J(t, θ n ) c 2 dt

P ----→ n→+∞ 0.

It follows by (A2) and (A6) that θ n converges in probability to θ 0 . Finally, by a similar argument, we prove by

(5.5

) that this last convergence is almost sure if (A5) ′ holds.

Theorem 5.2.

distr.

  For p ≥ 1, let Θ be a convex compact set in R p and {Θ n } n∈N be a sequence of convex compact sets in R p that converges to Θ with respect to the Hausdorff distance. Let r ≥ 0 and x be an interior point of Θ such that B(x, r) belongs to the interior of Θ. Then, there exists N ∈ N such that for all n ≥ N, B(x, r) ⊂ Θ n .

1) rmax r min w(t) J n (t) c -J(t, θ * n ) c J(t, θ * n ) c-2 J (1) (t, θ * n )J (1) (t, θ * n ) T dt + 2c rmax r min w(t) J n (t) c -J(t, θ * n ) c J(t, θ * n ) c-1 J (2) (t, θ * n )dt.

Lemma 5.4.

on θ (specific assumptions on J are listed below). For any t ∈ [r min , r max ], let J n (t) be an estimator of J(t, θ 0 ) where θ 0 is the true parameter ruling the distribution of X. We denote by J (1) (t, θ) and J (2) (t, θ) the gradient, respectively the Hessian matrix, of J(t, θ) with respect to θ. Define for all θ ∈ Θ, B(θ) := rmax r min w(t)J(t, θ) 2c-2 J (1) (t, θ)J (1) (t, θ) T dt, (5.1) and for all t ∈ [r min , r max ],

We consider the following assumptions.

(A1) Θ is a compact set with non-empty interior, 0 ≤ r min < r max , c = 0 and {D n } n∈N is a regular sequence of subsets of R d in the sense of Definition 3.1.

(A2) w is a positive and integrable function in [r min , r max ].

(A3) J(., .) and J(., .) c are well defined continuous functions on [r min , r max ] × Θ. Moreover, there exists a set A ∈ [r min , r max ] such that [r min , r max ] \ A is of Lebesgue measure null and for all t ∈ A, θ ∈ Θ, we have J(t, θ) > 0.

(A4) There exists n 0 ∈ N such that for all n ≥ n 0 , J n (.) and J n (. (A6) For θ 1 = θ 2 , there exists a set A of positive Lebesgue measure such that

(A7) For all t ∈ [r min , r max ], J (1) (t, θ) and J (2) (t, θ) exist, are continuous with respect to θ and uniformly bounded with respect to t ∈ [r min , r max ] and θ ∈ Θ.

(A8) There exists M > 0 such that for all (t, θ)

(A9) The matrix B(θ 0 ) is invertible.

(T CL) There exists m ∈ R and a covariance matrix Σ such that

Further, define (A5) ′ as the assumption (A5) with the convergence in probability replaced by the almost sure convergence.

By Theorem 5.1,

Then, by (A7)-(A8), E n tends in probability to 0. Note that by continuity of J(., θ) for all θ ∈ Θ, the integrability on [r min , r max ] of w(.)J(., θ) c-2 implies the one of w(.)J(., θ) c-1 . Further, we deduce by (A3), (A7) and (A8) that (t, θ) → J(t, θ) 2c-2 J (1) (t, θ)J (1) (t, θ) T is continuous with respect to θ ∈ Θ and uniformly bounded for t ∈ [r min , r max ]. Thus, by the dominated convergence theorem,

By (A9), B(θ 0 ) is invertible so by (5.9)

(5.10)

Since the group of invertible matrix is an open set, it follows from the last convergence that for n large enough, U (2) n (θ * n ) is invertible so we can write

By (5.8)-(5.10) and Slutsky's theorem, we get

and the conclusion of the theorem follows.

Auxiliary results

The two following lemmas are of topological nature and useful for the proofs of Theorems 3.2-3.3.

Lemma 5.3. For all p ≥ 1, let Ξ be a compact convex set in R p . Then, for all y ∈ R p \ Ξ and δ ≥ 0, B(y, δ) Ξ ⊕δ .

Proof. Since Ξ is a closed convex set, the projection of y onto Ξ, denoted by p Ξ (y), is the unique element belonging to Ξ that, for all u ∈ Ξ, verifies (yp Ξ (y)).(up Ξ (y)) ≤ 0.

(5.11)

For all δ ≥ 0, the line (y, p Ξ (y)) intersects ∂B(y, δ) at two points, one inside the segment [y, p Ξ (y)] and the other, that we denote by v, outside the segment. Thus, there exists t > 1 such that v = p Ξ (y) + t(yp Ξ (y)). Notice that for all u ∈ Ξ,

Thus, as t > 1, we deduce from (5.11) and the last equation that p Ξ (y) is the projection of v onto Ξ. It follows that d(v, Ξ) = d(y, Ξ) + δ and as y / ∈ Ξ and Ξ is closed, d(v, Ξ) > δ. Therefore, v / ∈ Ξ ⊕δ but v ∈ ∂B(y, δ) by construction so B(y, δ) Ξ ⊕δ . Corollary 5.6. Let X be a DPP with kernel C verifying the condition K(ρ) for a given ρ > 0 and {D n } n∈N be a family of regular sets. Define for all n ∈ N,

We have the convergence

distr.

----→ n→+∞ N(0, σ 2 )

where σ 2 = lim n→+∞ V ar


Metod Saniga
email: msaniga@astro.sk

Michel Planat
email: michel.planat@femto-st.fr

Multiple Qubits as Symplectic Polar Spaces of Order Two

Keywords: 51Exx, 81R99 PACS Numbers: 02.10.Ox, 02.40.Dr, 03.65.Ca Symplectic Polar Spaces of Order Two -N-Qubits

It is surmised that the algebra of the Pauli operators on the Hilbert space of N -qubits is embodied in the geometry of the symplectic polar space of rank N and order two, W 2N -1 (2). The operators (discarding the identity) answer to the points of W 2N -1 (2), their partitionings into maximally commuting subsets correspond to spreads of the space, a maximally commuting subset has its representative in a maximal totally isotropic subspace of W 2N -1 (2) and, finally, "commuting" translates into "collinear" (or "perpendicular").

It is well known that a complete basis of operators in the Hilbert space of N -qubits, N ≥ 2, can be given in terms of the Pauli operators -tensor products of classical 2 × 2 Pauli matrices. Although the Hilbert space in question is 2 N -dimensional, the operators' space is of dimension 4 N . Excluding the identity matrix, the set of 4 N -1 Pauli operators can be partitioned into 2 N + 1 subsets, each comprising 2 N -1 mutually commuting elements [START_REF] Lawrence | Mutually unbiased binary observable sets on N qubits[END_REF]. The purpose of this note is to put together several important facts supporting the view that this operators' space can be identified with W 2N -1 (q = 2), the symplectic polar space of rank N and order two.

A (finite-dimensional) classical polar space (see [START_REF] Ball | The geometry of finite fields[END_REF][START_REF] Cameron | Projective and polar spaces[END_REF][START_REF] De Clerck | Ovoids and spreads of polar spaces and generalized polygons," a lecture given at an intensive course on[END_REF][START_REF] Hirschfeld | General Galois Geometries[END_REF][START_REF] Payne | Finite Generalized Quadrangles[END_REF] for more details) describes the geometry of a d-dimensional vector space over the Galois field GF (q), V (d, q), carrying a non-degenerate reflexive sesquilinear form σ. The polar space is called symplectic, and usually denoted as W d-1 (q), if this form is bilinear and alternating, i.e., if σ(x, x) = 0 for all x ∈ V (d, q); such a space exists only if d = 2N , where N is called its rank. A subspace of V (d, q) is called totally isotropic if σ vanishes identically on it. W 2N -1 (q) can then be regarded as the space of totally isotropic subspaces of P G(2N -1, q), the ordinary (2N -1)-dimensional projective space over GF (q), with respect to a symplectic form (also known as a null polarity), with its maximal totally isotropic subspaces, also called generators G, having dimension N -1. For q = 2 this polar space contains

|W 2N -1 (2)| = |P G(2N -1, 2)| = 2 2N -1 = 4 N -1 (1)
points and

|Σ(W 2N -1 (2))| = (2 + 1)(2 2 + 1) . . . (2 N + 1) (2)
generators [START_REF] Ball | The geometry of finite fields[END_REF][START_REF] Cameron | Projective and polar spaces[END_REF][START_REF] De Clerck | Ovoids and spreads of polar spaces and generalized polygons," a lecture given at an intensive course on[END_REF]. An important object associated with any polar space is its spread, i. e., a set of generators partitioning its points. A spread S of W 2N -1 (q) is an (N -1)-spread of its ambient projective space P G(2N -1, q) [4, 5, 7], i. e., a set of (N -1)-dimensional subspaces of P G(2N -1, q) partitioning its points. The cardinalities of a spread and a generator of W 2N -1 (2) thus read

|S| = 2 N + 1 (3)
and

|G| = 2 N -1, (4)
respectively [START_REF] Ball | The geometry of finite fields[END_REF][START_REF] Cameron | Projective and polar spaces[END_REF]. Finally, it needs to be mentioned that two distinct points of W 2N -1 (q) are called perpendicular if they are "isotropically" collinear, i. e., joined by a totally isotropic line of W 2N -1 (q); for q = 2 there are

# ∆ = 2 2N -1 (5)
points that are not perpendicular to a given point of W 2N -1 (2) [START_REF] Ball | The geometry of finite fields[END_REF][START_REF] Cameron | Projective and polar spaces[END_REF]. Now, in light of Eq. ( 1), we can identify the Pauli operators with the points of W 2N -1 (2). If, further, we identify the operational concept "commuting" with the geometrical one "perpendicular," from Eqs. ( 3) and ( 4) we readily see that the points lying on generators of W 2N -1 (2) correspond to maximally commuting subsets (MCSs) of operators and a spread of W 2N -1 (2) is nothing but a partitioning of the whole set of operators into MCSs. From Eq. ( 2) we then infer that the operators' space possesses (2 + 1)(22 + 1) . . . (2 N + 1) MCSs and, finally, Eq. ( 5) tells us that there are 2 2N -1 operators that do not commute with a given operator; the last two statements are, for N > 2, still conjectures to be rigorously proven. However, the case of two-qubits (N = 2) is recovered in full generality [START_REF] Lawrence | Mutually unbiased binary observable sets on N qubits[END_REF][START_REF] Wootters | Picturing qubits in phase space[END_REF][START_REF] Saniga | Projective ring line encompassing two-qubits[END_REF], with the geometry behind being that of the generalized quadrangle of order two [START_REF] Saniga | Projective ring line encompassing two-qubits[END_REF] -the simplest nontrivial symplectic polar space. 1

This object can also be recognized as the projective line over the Jordan system of the full

× 2 matrix ring with coefficients in GF (2)[START_REF] Saniga | Projective ring line encompassing two-qubits[END_REF].

Acknowledgements

This work was partially supported by the Science and Technology Assistance Agency under the contract # APVT-51-012704, the VEGA project # 2/6070/26 (both from Slovak Republic) and the trans-national ECO-NET project # 12651NJ "Geometries Over Finite Rings and the Properties of Mutually Unbiased Bases" (France).