Marco Gramaglia

Oscar Trullols-Cruces

Diala Naboulsi

Marco Fiore

Maria Calderon

Connectivity of Highway Vehicular Networks

Keywords: vehicle-to-vehicle communications, vehicular networks, mobility modeling, connectivity analysis

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

1I n t r o d u c t i o n

Despite the fact that technologies based on Dedicated Short-Range Communication (DSRC) and the associated services will soon hit the market, the vast majority of applications, protocols and architectures for upcoming vehicular networks is still evaluated via simulative studies. Within such a context, the level of realism of the simulation is a paramount aspect to account for, and the way the mobility of individual vehicles is represented is often the single feature that introduces the largest bias in the results.

During the past decade, there have been efforts aiming at gathering real-world road traffic data,developing tools for the simulation of vehicular movement.and generating synthetic mobility datasets.This notwithstanding, there is still a noticeable scarcity of road traffic datasets featuring the level of realism and of spatiotemporal granularity required for network simulation.The result of the lack of a set of realistic, publicly shared, heterogeneous scenarios is that simulationso fv e h i c u l a rn e t w o r k sa r eo f t e nu n r e l i a b l ea n d non-reproducible.

We contribute to the endeavor of enlarging the set of realistic datasets of vehicular mobility that are freely available to the research community. To that end, we present 16 original traces describing the road traffic on two highways around Madrid, Spain, at different times of multiple weekdays. The traces are derived from high-detail real-world traffic counts, and describe unidirectional free-flow traffic in quasistationary conditions on a road spanning over 10 km. We then provide a characterization of the vehicular network connectivity in the synthetic traces, investigating the impact of time (i.e., hour of the day, day of the week), highway diversity, and vehicular communication range. Our results highlight that : (i) the level of global vehicular connectivity can be primarily ascribed to the V2V communication range ; (ii) for short communication ranges, the vehicular connectivity is drivenbyamixtureofslowandfasttemporaldynamics that are regulated by the road traffic density and by V2V contact durations, respectively.

2D a t a c o l l e c t i o n

The empirical data used throughout this work comes from measurement locations in the region of Madrid, Spain. The data, kindly provided to us by the Spanish office forthetrafficmanagement(DirecciónGeneral de Tráfico, DGT), detail road traffic conditions on two arterial highways.

The M40 is part of the middle layer of the Madrid city beltway system, which also comprises M-30 (innermost) and M-50 (outermost). This beltway, which has ana v eragedistanceof10.7kmfromthecity center, traverses both the most peripheral areas of the municipality and surrounding cities. The measurement point is placed at the 12.7-km milepost, where the M40 traverses the suburb of San Blas and the town of Coslada. The measures cover the internal carriageway (southbound) that includes 3 lanes with a speed limit of 100 km/h. The M40 is indicated as A in Fig. 1(a), while a photographic view of the exact measurement location is provided in Fig. 1

(b).

The Autovía A6 is a motorway that connects the city of A Coruña to the city of Madrid. This road enters into the urban area from the northwest and collects the trafficdemandoftheconurbationthatwasbuiltalong it. The data collection point is placed around the 11-km milepost (Madrid direction), where the A6 features three lanes with a speed limit of 120 km/h. We remark, however,thatthislocationisrightafterapopular entrance ramp that joins the rightmost lane and significantlyslowsdowntheroadtraffic.Thehighwaymaps to B in Fig. 1(a). A photographic view of the collection point is in Fig. 1(c).

The sensors that DGT deployed on M40 and A6 are induction loops. They consist of loops of wires buried under the concrete layer that create a magnetic field. When a vehicle passes on the vertical axis of the loop, it creates a variation in the magnetic field, which iscons ideredasane wtrans it.Iftw oloopsare placed close to each other, also the vehicular speed (and possibly other metrics, such as the vehicle length) is obtained. A picture of the buried induction loops used to collect measurements on the M40 is shown in Fig. 1(d).

Usually, these devices are programmed to supply coarse-grained data, as public transportation authorities are generally interested in aggregate measures, (e.g.,thenumberofvehiclestransitingonaroad,their average speed, or the percentage of heavy vehicles) so as to detect exceptional alterations in the road capacity.The loops installed by DGT were configured to supply dataaveragedover60-secondintervals,buttheir configuration was changed specifically for our study : the real-world traffic counts we collected provide fine-grained time, speed and lane information on each single transiting vehicle. To the best of our knowledge, this is the most precise traffic count dataset recordedt od a t e ,a n dr e p r e s e n t sa ni d e a li n p u tt oo u r microscopic simulation of highway traffic.

The timing of the data collection is a very important variabletoaccountfor ,sincevehiculartrafficpresents, as many other human activities, important temporal variability : for instance, rush hours yield diverse traffic conditions than off-peak hours, especially on main arterial roads like those we consider. The dataset analyzed in this work was collected on multiple days of May 2010, namely Friday the 7th, Monday the 10th, Tuesday the 11th, and Wednesday the 12th. For each measurement location and each day, two traces are available : one capturing the traffic peak (from 8 :30 a.m. to 9 :00 a.m.), and one during off-peak hours (from 11 :30 a.m. to 12 :00 p.m.). As a result, our empirical dataset provides a heterogeneous view of traffic conditions, allowing us to generate synthetic mobility traces that are representative of different highway congestion levels.

3D a t a s e t g e n e r a t i o n

Our objective is to generate road traffic traces that are representative of unidirectional free flow highway traffic in a quasi-stationary state, i.e., such that comparable traffic conditions are present in between the in-flow and out-flow boundaries of the simulated highways. Thedatasetsshouldcoverageographicaldistance that is sufficient for networking studies (e.g., for warning message dissemination or floating car data upload). In the following, we simulate a 10-km road segment, but we remark that our approach can easily accommodate shorter or longer highway stretches.

In order to achieve the goal above, we fed the real-world traffic count data presented in Sec. 2 to a microscopic vehicular mobility simulator based on well-knownm o d e l so fc a r -f o l l o w i n ga n dl a n e -c h a n g i n g behaviors of individual drivers. Specifically, we employed the Intelligent Driver Model (IDM) [START_REF] Treiber | Congested traffic states in empirical observations and microscopic simulations[END_REF] to characterize the instantaneous acceleration of each vehicle, and the Minimizing Overall Braking Induced by Lane-changes model (MOBIL) [START_REF] Treiber | Realistische mikrosimulation von strassenverkehr mit einem einfachen modell[END_REF] to represent inter-lane movement decisions.

In order to obtain quasi-stationary traffic conditions over the simulated highway segment, some calibration of the IDM and MOBIL parameters proved necessary. Specifically, for the acceleration a,deceleration b,politenessfactorp and minimum safety distance ∆x sa f e we could use the default values suggested in [START_REF] Treiber | Congested traffic states in empirical observations and microscopic simulations[END_REF][START_REF] Treiber | Realistische mikrosimulation von strassenverkehr mit einem einfachen modell[END_REF]. The other parameters had instead to be adapted to the specificities of each road scenario. The final IDM and MOBIL settings are summarized in Tab. 3, where f V (v) is a Gaussian-shaped distribution of maximum desired speed, which is calibrated from empirical per-lane free-flow ingress velocities. Therefore, the exact parameters of f V (v) depend on the considered scenario. As an example, distributions for the three lanes of A6 at 8 :30 a.m. are shown in Fig. 3 : there, the histograms portray empirical per-lane free-flow ingress speed distributions, while the curves map to the fitted f V (v) normal distributions.

We present some results of the connectivity analysis we carried out on the resulting synthetic mobility traces. Specifically, in the following we present results fordifferentvaluesofthevehicle-to-vehiclecommunication range R,from50to300meters.Thesevaluesrepresentthelowerandupper bounds for acceptable packet delivery ratios, according to experimental studies [START_REF] Martelli | A measurement-based study of beaconing performance in IEEE 802.11p vehicular networks[END_REF][START_REF] Bai | Toward understanding characteristics of dedicated short range communications (DSRC) from a perspective of vehicular network engineers[END_REF]. We focus on the distributions of the number of components C and of the size of the largest component S max .Indeed,C is a measure of how fractioned the network is, while S max is the maximum number of nodes that can be reached via multi-hop communication. Clearly, the lower is C and the larger is S max ,thebetterconnectedisthevehicularnetwork.

In Fig. 3, we present the distributions of C (top plots) and S max (bottom plots), for various values of the communication range R (on different columns). In every plot, each candlestick refers to one mobility trace, and is obtained by aggregating the metrics computed in all instantaneous graphs observed at every second during the 30-minute timespan of that trace. The lowest and highest values in the candlesticks (in grey) are the minimum and maximum values that the metric attains. The inner error bars (in red) depict the first and ninth deciles, while the box highlights the first and thirdquartiles(inred)aroundthemedianvalue(in black). Also, the step function in the S max plots is the maximum number of vehicles N max observed in all instances of the respective trace. It thus represents the upper bound to S max :thecloserisS max to N max ,the closer the vehicular network is to a fully connected single component.

Let us now focus on the case of R =5 0m ,i nF i g .3 ( a )a n dF i g .3 ( e ) .W eo b s e r v et h a tt h ec o n n e c t i vity metrics are very similar on different days of the week, forming four clear clusters in both plots. This implies that we can expect the network to have consistent topologicalpropertiesfromMondaytoFriday.Theaforementioned clusters are instead separated by highway (A6 or M40) and hour (8 :30 or 11 :30). Specifically, the network appears more fragmented on A6 than on M40, and, in both cases, the connectivity is worse at 11 :30 a.m. than at 8 :30 a.m. Intuitively, we can ascribe such a variability to the different road traffic conditions encountered on the two highways at different times of the day.

The observations above still hold when considering different communication ranges separately. Nonetheless, when comparing the plots for varying R,itisevidentthatincreasingthecommunicationrangecan dramatically improve the connectivity of the network. Indeed, for R =50m,theree xistona verage30-50 disconnected components, the largest of which only comprises around one tenth of the vehicles. As R grows, however, the network fragmentation is reduced, and more nodes join the largest component. When R =300 malmostallvehiclesbelongallthetimetoonesinglecomponent, as in Fig. 3(d) and Fig. 3(h). We conclude that the communication range is the foremost parameter controlling the vehicular network connectivity,as it can induce variations in C and S max that are much more significant than those imputable to road traffic conditions. Indeed, R =300mguaranteesawellconnectednetw orkindependentlyofthe traffic scenario, but reducing that value causes the topology to rapidly break apart.

FIGURE 1 :

 1 FIGURE 1: Geographical location of the two highways near Madrid, Spain(a).Close-byviewofmeasurementpoints (b,c), and of the induction loop on M40 (d).

FIGURE 3 :

 3 FIGURE 3: Candlestick istributions of C (top), and S max (bottom), for each mobility trace under different values of R.

TABLE 1 :

 1 IDM and MOBIL parameter settings.

	Empirical Probability Density Function	0.4 0.0 0.4 0.2 0.2			0.0 0.4 0.2
		0.0
		40	80	120	160
			Speed (km/h)
	FIGURE 2: Per-lane f V (v) on A6, 8 :30 a.m.

† The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union's Seventh Framework Programme (FP7/2007-2013) underREAgrantagreementn.630211.

We also carried out additional topological analyses in presence of communication ranges below 200 m, which are not reported here due to space limitations. Such analyses revealed that the network connectivity undergoes temporal variations that are a combination of slowdynamics,controlledbytheroadtrafficdensity, and fast dynamics, due to the predominant presence of short-lived contacts. Interestingly, we found such properties to be invariant throughout multiple weekdays and on different highways.

R éf érences


Mohamed El Machkouri
email: mohamed.elmachkouri@univ-rouen.fr

Kernel density estimation for stationary random fields

Keywords: Kernel density estimation for stationary random fields. 2014. hal-00622861v5 AMS Subject Classifications (2000): 60F05, 60G60, 62G07, 62G20 Central limit theorem, spatial processes, m-dependent random fields, physical dependence measure, nonparametric estimation, kernel density estimator, rate of convergence Kernel density estimation for random fields

come

Introduction and main results

Let (X i ) i∈Z be a stationary sequence of real random variables defined on a probability space (Ω, F, P) with an unknown marginal density f . The kernel density estimator f n of f introduced by Rosenblatt [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF] and Parzen [START_REF] Parzen | On the estimation of a probability density and the mode[END_REF] is defined for all positive integer n and any real x by

f n (x) = 1 nb n n i=1 K x -X i b n
where K is a probability kernel and the bandwidth b n is a parameter which converges slowly to zero such that nb n goes to infinity. The literature dealing with the asymptotic properties of f n when the observations (X i ) i∈Z are independent is very extensive (see Silverman [START_REF] Silverman | Density Estimation for Statistics and Data Analysis[END_REF]). Parzen [START_REF] Parzen | On the estimation of a probability density and the mode[END_REF] proved that when (X i ) i∈Z are independent and identically distribut (i.i.d) and the bandwidth b n goes to zero such that nb n goes to infinity then (nb n ) 1/2 (f n (x 0 ) -Ef n (x 0 )) converges in distribution to the normal law with zero mean and variance f (x 0 ) R K 2 (t)dt. Under the same conditions on the bandwidth, this result was extended by Wu an Mielniczuk [START_REF] Wu | Kernel density estimation for linear processes[END_REF] for causal linear processes with i.i.d.

innovations and by Dedecker and Merlevède [START_REF] Dedecker | Necessary and sufficient conditions for the conditional central limit theorem[END_REF] for strongly mixing sequences.

In this paper, we are interested by the kernel density estimation problem in the setting of dependent random fields indexed by Z d where d is a positive integer. The question is not trivial since Z d does not have a natural ordering for d ≥ 2. In recent years, there is a growing interest in asymptotic properties of kernel density estimators for random fields. One can refer for example to Carbon et al. ( [START_REF] Carbon | Kernel density estimation for random fields: the l 1 theory[END_REF], [START_REF] Carbon | Kernel density estimation for random fields[END_REF]), Cheng et al. [START_REF] Cheng | A note on asymptotic normality of kernel estimation for linear random fields on Z 2[END_REF], El

Machkouri [START_REF] Machkouri | Asymptotic normality for the parzen-rosenblatt density estimator for strongly mixing random fields[END_REF], Hallin et al. [START_REF] Hallin | Density estimation for spatial linear processes[END_REF], Tran [START_REF] Tran | Kernel density estimation on random fields[END_REF] and Wang and Woodroofe [START_REF] Wang | On the asymptotic normality of kernel density estimators for causal linear random fields[END_REF]. In [START_REF] Tran | Kernel density estimation on random fields[END_REF], the asymptotic normality of the kernel density estimator for strongly mixing random fields was obtained using the Bernstein's blocking technique and coupling arguments. Using the same method, the case of linear random fields with i.i.d. innovations was handled in [START_REF] Hallin | Density estimation for spatial linear processes[END_REF]. In [START_REF] Machkouri | Asymptotic normality for the parzen-rosenblatt density estimator for strongly mixing random fields[END_REF], the central limit theorem for the Parzen-Rosenblatt estimator given in [START_REF] Tran | Kernel density estimation on random fields[END_REF] was improved using the Lindeberg's method (see [START_REF] Lindeberg | Eine neue Herleitung des Exponentialgezetzes in der Wahrscheinlichkeitsrechnung[END_REF]) which seems to be better than the Bernstein's blocking technique approach. In particular, a simple criterion on the strong mixing coefficients is provided and the only condition imposed on the bandwith is n d b n → ∞ which is similar to the usual condition imposed in the independent case (see Parzen [START_REF] Parzen | On the estimation of a probability density and the mode[END_REF]). In [START_REF] Machkouri | Asymptotic normality for the parzen-rosenblatt density estimator for strongly mixing random fields[END_REF], the regions where the random field is observed are reduced to squares but a carrefull reading of the proof allows us to state that the main result in [START_REF] Machkouri | Asymptotic normality for the parzen-rosenblatt density estimator for strongly mixing random fields[END_REF] still holds for very general regions Λ n , namely those which the cardinality |Λ n | goes to infinity such that |Λ n |b n goes to zero as n goes to infinity (see Assumption (A3) below). In [START_REF] Cheng | A note on asymptotic normality of kernel estimation for linear random fields on Z 2[END_REF], Cheng et al. investigated the asymptotic normality of the kernel density estimator for linear random fields with i.i.d. innovations using a martingale approximation method (initiated by Cheng and Ho [START_REF] Cheng | Central limit theorems for instantaneous filters of linear random fields on Z 2[END_REF]) but it seems that there is a mistake in their proof (see Remark 6 in [START_REF] Wang | On the asymptotic normality of kernel density estimators for causal linear random fields[END_REF]). Since the mixing property is often unverifiable and might be too restrictive, it is important to provide limit theorems for nonmixing and possibly nonlinear random fields. We consider in this work a field (X i ) i∈Z d of identically distributed real random variables with an unknown marginal density f such that

X i = g ε i-s ; s ∈ Z d , i ∈ Z d , (1)
where (ε j ) j∈Z d are i.i.d. random variables and g is a measurable function defined on R Z d . In the one-dimensional case (d = 1), the class (1) includes linear as well as many widely used nonlinear time series models as special cases. More importantly, it provides a very general framework for asymptotic theory for statistics of stationary time series (see e.g. [START_REF] Wu | Nonlinear system theory: another look at dependence[END_REF] and the review paper [START_REF] Wu | Asymptotic theory for stationary processes[END_REF]).

We introduce the physical dependence measure first introduced by Wu [START_REF] Wu | Nonlinear system theory: another look at dependence[END_REF]. Let (ε ′ j ) j∈Z d be an i.i.d. copy of (ε j ) j∈Z d and consider for all positive integer n the coupled version

X * i of X i defined by X * i = g ε * i-s ; s ∈ Z d where ε * j = ε j 1 1 {j =0} + ε ′ 0 1 1 {j=0} for all j in Z d .
In other words, we obtain X * i from X i by just replacing ε 0 by its copy ε ′ 0 . Let i in Z d and p > 0 be fixed. If X i belongs to L p (that is, E|X i | p is finite), we define the physical dependence measure δ i,p = X i -X * i p where . p is the usual L p -norm and we say that the random field

(X i ) i∈Z d is p-stable if i∈Z d δ i,p < ∞. For d ≥ 2
, the reader should keep in mind the following two examples already given in [START_REF] Machkouri | A central limit theorem for stationary random fields[END_REF] :

Linear random fields: Let (ε i ) i∈Z d be i.i.d random variables with ε i in L p , p ≥ 2.
The linear random field X defined for all i in Z d by

X i = s∈Z d a s ε i-s with (a s ) s∈Z d in R Z d such that i∈Z d a 2 i < ∞ is of the form (1) with a linear functional g. For all i in Z d , δ i,p = |a i | ε 0 -ε ′ 0 p . So, X is p-stable if i∈Z d |a i | < ∞. Clearly, if H is a Lipschitz continuous function, under the above condition, the subordinated process Y i = H(X i ) is also p-stable since δ i,p = O(|a i |).
Volterra field : Another class of nonlinear random field is the Volterra process which plays an important role in the nonlinear system theory (Casti [4], Rugh [START_REF] Rugh | Nonlinear system theory[END_REF]): consider the second order Volterra process

X i = s 1 ,s 2 ∈Z d a s 1 ,s 2 ε i-s 1 ε i-s 2 ,
where a s 1 ,s 2 are real coefficients with a s 1 ,s 2 = 0 if s 1 = s 2 and

(ε i ) i∈Z d are i.i.d. random variables with ε i in L p , p ≥ 2. Let A i = s 1 ,s 2 ∈Z d (a 2 s 1 ,i + a 2 i,s 2 ) and B i = s 1 ,s 2 ∈Z d (|a s 1 ,i | p + |a i,s 2 | p ).
By the Rosenthal inequality, there exists a constant C p > 0 such that

δ i,p = X i -X * i p ≤ C p A 1/2 i ε 0 2 ε 0 p + C p B 1/p i ε 0 2 p .
From now on, for all finite subset Λ of Z d , we denote |Λ| the number of elements in Λ and we observe (X i ) i∈Z d on a sequence (Λ n ) n≥1 of finite subsets of Z d which only satisfies |Λ n | goes to infinity as n goes to infinity. It is important to note that we do not impose any condition on the boundary of the regions Λ n . The density estimator f n of f is defined for all positive integer n and any real x by

f n (x) = 1 |Λ n |b n i∈Λn K x -X i b n
where b n is the bandwidth parameter and K is a probability kernel. Our aim is to provide sufficient conditions for the L 1 -distance between f n and f to converge to zero (Theorem 1) and for (|Λ

n |b n ) 1/2 (f n (x i ) -Ef n (x i )) 1≤i≤k , (x i ) 1≤i≤k ∈ R k , k ∈ N\{0},
to converge in law to a multivariate normal distribution (Theorem 2) under minimal conditions on the bandwidth parameter. We give also a Berry-Esseen's type central limit theorem for the considered estimator (Theorem 3). In the sequel, we denote |i| = max 1≤k≤d |i k | for all i = (i 1 , ..., i d ) ∈ Z d and we denote also δ i for δ i,2 . The following assumptions are required.

(A1) The marginal density function f of each X k is Lipschitz.

(A2) K is Lipschitz, R K(u) du = 1, R u 2 |K(u)| du < ∞ and R K 2 (u) du < ∞. (A3) b n → 0 and |Λ n | → ∞ such that |Λ n |b n → ∞. (A4) i∈Z d |i| 5d 2 δ i < ∞.
Theorem 1 If (A1), (A2), (A3) and (A4) hold, then there exists κ > 0 such that for all integer n ≥ 1,

E R |f n (x) -f (x)| dx ≤ κ b n + 1 |Λ n |b n 2 3
.

(

) 2
Remark 1. One can optimize the inequality (2) by taking

b n = |Λ n | -1 3 . Then, we obtain E R |f n (x) -f (x)| dx = O |Λ n | -2 9 .
Remark 2. The convergence in probability of R |f n (x)f (x)| dx to 0 was obtained (without rate) by Hallin et al. ([15], Theorem 2.1) for rectangular region Λ n . The authors defined the so-called stability coefficients

(v(m)) m≥1 by v(m) = X 0 -X 0 2 2
where X 0 = E (X 0 |H m ) and H m = σ (ε s , |s| ≤ m). Under minimal conditions on the bandwidth b n , with our notations, their result holds as soon as v(m) = o(m -4d ). Arguing as in the proof of Lemma 5 below, one can relate the stability coefficients with the physical dependence measure ones by the inequality v(m)

≤ C |i|>m δ 2 i , m ≥ 1, C > 0.
In the sequel, we consider the sequence (m n ) n≥1 defined by

m n = max      v n ,      1 b 3 n |i|>vn |i| 5d 2 δ i   1 3d    + 1      (3)
where v n = b

m n → ∞, m d n b n → 0 and 1 (m d n b n ) 3/2 |i|>mn |i| 5d 2 δ i → 0.
For all z in R and all i in Z d , we denote

K i (z) = K z -X i b n and K i (z) = E (K i (z)|F n,i ) (4)
where

F n,i = σ (ε i-s ; |s| ≤ m n ). So, denoting M n = 2m n + 1, (K i (z)) i∈Z d is an M n - dependent random field (i.e. K i (z) and K j (z) are independent as soon as |i -j| ≥ M n ).
Lemma 2 For all p > 1, all x in R, all positive integer n and all

(a i ) i∈Z d in R Z d , i∈Λn a i K i (x) -K i (x) p ≤ 8m d n b n p i∈Λn a 2 i 1/2 |i|>mn δ i,p .
In order to establish the asymptotic normality of f n , we need additional assumptions:

(B1) The marginal density function of each X k is positive, continuous and bounded.

(B2) K is Lipschitz, R K(u) du = 1, R |K(u)| du < ∞ and R K 2 (u) du < ∞. (B3) There exists κ > 0 such that sup (x,y)∈R 2 i∈Z d \{0} f 0,i (x, y) ≤ κ where f 0,i is the joint density of (X 0 , X i ).
Theorem 2 Assume that (A3), (A4), (B1), (B2) and (B3) hold. For all positive integer k and any distinct points x 1 , ..., x k in R,

(|Λ n |b n ) 1/2    f n (x 1 ) -Ef n (x 1 ) . . . f n (x k ) -Ef n (x k )    Law -----→ n→∞ N (0, Γ) (5)
where Γ is a diagonal matrix with diagonal elements 5) is a classical problem in density estimation theory. Let s ≥ 2 be a positive integer and κ > 0. If the sth

γ ii = f (x i ) R K 2 (u)du. Remark 3. A replacement of Ef n (x i ) by f (x i ) for all 1 ≤ i ≤ k in (
derivative f (s) of f exists such that |f (s) | ≤ κ and the kernel K satisfies R u r K(u)du = 0 for r = 1, 2, ..., s -1 and 0 < R |u| s |K(u)|du < ∞ then |Ef n (x i ) -f (x i )| = O(b s n
) and thus the centering Ef n (x i ) may be changed to f (x i ) without affecting the above result provided that

|Λ n |b 2s+1 n converges to zero. Remark 4. If (X i ) i∈Z d is a linear random field of the form X i = j∈Z d a j ε i-j where (a j ) j∈Z d are real numbers such that j∈Z d a 2 j < ∞ and (ε j ) j∈Z d are i.i.d.

real random variables with zero mean and finite variance then

δ i = |a i | ε 0 -ε ′ 0 2 and Theorem 2 holds provided that i∈Z d |i| 5d 2 |a i | < ∞. For Λ n rectangular, Hallin et al. [14] obtained the same result when |a j | = O (|j| -γ ) with γ > max{d + 3, 2d + 0.5} and |Λ n |b (2γ-1+6d)/(2γ-1-4d) n
goes to infinity. So, in the particular case of linear random fields, our assumption (A4) is more restrictive than the condition obtained by Hallin et al. [START_REF] Hallin | Density estimation for spatial linear processes[END_REF] but our result is valid for a larger class of random fields and under only minimal conditions on the bandwidth (see Assumption (A3)). Finally, for causal linear random fields, Wang and Woodroofe [START_REF] Wang | On the asymptotic normality of kernel density estimators for causal linear random fields[END_REF] obtained also a sufficient condition on the coefficients (a j ) j∈N d for the kernel density estimator to be asymptotically normal. Their condition is less restrictive than the condition

i∈Z d |i| 5d 2 |a i | < ∞ but they assumed also E(|ε 0 | p ) < ∞ for some p > 2.
Now, we are going to investigate the rate of convergence in [START_REF] Chen | Normal approximation under local dependence[END_REF]. For all positive integer n and all x in R, we denote D n (x) = sup t∈R |P (U n (x) ≤ t) -Φ(t)| where Φ is the distribution function of the standard normal law and

U n (x) = |Λ n |b n (f n (x) -Ef n (x)) f (x) R K 2 (t)dt . Theorem 3 Let n in N\{0} and x in R be fixed. Assume that R |K(t)| τ dt < ∞ for some 2 < τ ≤ 3. If there exist α > 1 and p ≥ 2 such that i∈Z d |i| dα δ i,p < ∞ then there exists a constant κ > 0 such that D n (x) ≤ κ|Λ n | -θ where θ = θ(α, τ, p) = 1 2 - 1 τ 3p(1 -τ ) + 2p(α -1) (τ -1)(p + 1) + p(α -1)
.

Remark 5. If τ = 3, p = 2 and i∈Z d |i| dα δ i < ∞ for some α > 4 then

D n (x) ≤ κ|Λ n | -θ(α) where θ(α) = 2α -8 3(4 + 2α) -----→ α→∞ 1 3 .

Numerical illustration

In this section, we give some simulations with a view to illustrate the results given in this paper. We assume d = 2 and we consider the autoregressive random field

(X i,j ) (i,j)∈Z 2 defined by X i,j = αX i-1,j + βX i,j-1 + ε i,j (6)
where α = 0.2, β = 0.7 and (ε i,j ) (i,j)∈Z 2 are iid random variables uniformly distributed over the intervalle [-5, 5]. Since |α|+|β| < 1, the equation (6) has a stationary solution X i,j (see [START_REF] Kulkarni | Estimation of parameters of a two-dimensional spatial autoregressive model with regression[END_REF]) defined by

X i,j = k 1 ≥0 k 2 ≥0 k 1 + k 2 k 1 α k 1 β k 2 ε i-k 1 ,j-k 2 (7)
and each X i,j is uniformly distributed over the intervalle [-5γ, 5γ] with

γ = k 1 ≥0 k 2 ≥0 k 1 + k 2 k 1 α k 1 β k 2 = 1 1 -(α + β) = 10.
We simulate the ε i,j 's over the rectangular grid [0, 2t] 2 ∩ Z 2 where t is a positive integer and the data X i,j over the grid Λ t = [t + 1, 2t] 2 ∩ Z 2 following (7). We take the data X i,j for (i, j) in the region Λ t as our data set and we calculate from this data set the kernel density estimator

ft (x) = 1 t 2 × b t (i,j)∈Λt K x -X i,j b t ( 8
)
where x is fixed in R, b t is the bandwith parameter and K is the Epanachnikov kernel defined by K(s) and we obtain the following histogram (see figure 1) which seems to fit well to the target distribution, that is the standard normal law N (0, 1). In the simulation given in Figure 1, we fixed the bandwith b 20 = 0.7 arbitrarily since we do not investigate in this work any procedure for a data-driven choice of the bandwith parameter. Such a study is an important task and will be done in a forthcoming paper.

= 3 4 (1 -s 2 ) if s ∈] -1, 1[ and K(s) = 0 if s / ∈] -

Proofs

The proof of all lemmas of this section are postponed to the appendix. In the sequel, the letter κ denotes a positive constant which the value is not important.

Proof of Theorem 1

For all positive integer n, denote

J n = R |f n (x) -f (x)| dx. For all real A ≥ 1, we have J n = J n,1 (A) + J n,2 (A) where J n,1 (A) = |x|>A |f n (x) -f (x)| dx and J n,2 (A) = |x|≤A |f n (x) -f (x)| dx. Moreover EJ n,1 (A) ≤ |x|>A E|f n (x)|dx + 1 A 2 R x 2 f (x)dx and |x|>A E|f n (x)|dx ≤ |x|>A R |K(t)|f (x -b n t)dtdx = |t|> A 2 |K(t)| |x|>A f (x -b n t)dxdt + |t|≤ A 2 |K(t)| |x|>A f (x -b n t)dxdt ≤ |t|> A 2 |K(t)| |y+bnt|>A f (y)dydt + |t|≤ A 2 |K(t)| |y|>A(1-bn 2 )
f (y)dydt

≤ 4 A 2 R t 2 |K(t)|dt + 4 A 2 R |K(t)|dt R y 2 f (y)dy.
Consequently, we obtain

EJ n,1 (A) ≤ κ A 2 . (9)
Now, J n,2 (A) ≤ J

n,2 (A) + J

n,2 (A) where

J (1) n,2 (A) = |x|≤A |f n (x) -Ef n (x)| dx and J (2) n,2 (A) = |x|≤A |Ef n (x) -f (x)| dx. Since |Ef n (x) -f (x)| = R K(t) (f (x -b n t) -f (x)) dt ≤ R |K(t)| |f (x -b n t) -f (x)| dt ≤ κb n R |t||K(t)|dt, we obtain J (2) n,2 (A) ≤ κAb n . ( 10
)
Keeping in mind the notation (4) and denoting

f n (x) = 1 |Λn|bn i∈Λn K i (x), we have J (1) n,2 (A) ≤ I n,1 (A) + I n,2 (A) where I n,1 (A) = |x|≤A |f n (x) -f n (x)| dx and I n,2 (A) = |x|≤A |f n (x) -Ef n (x)| dx.
By Lemma 2, we have

f n (x) -f n (x) 2 ≤ κ |i|>mn |i| 5d 2 δ i |Λ n |b n (m d n b n ) 3/2 .
Applying Lemma 1, we obtain

EI n,1 (A) ≤ κA |Λ n |b n . (11)
Now,

f n (x) -Ef n (x) 2 2 equals to 1 |Λ n | 2 b n     |Λ n |E Z 2 0 (x) + j∈Z d \{0} |j|<Mn |Λ n ∩ (Λ n -j)|E Z 0 (x)Z j (x)     (12)
where we recall that

Z i (x) = 1 √ bn K i (x) -EK i (x) and M n = 2m n + 1. Lemma 3 Let x, s and t be fixed in R. Then E Z 2 0 (x) converges to f (x) R K 2 (u)du and sup i∈Z d \{0} E|Z 0 (s)Z i (t)| = o(M -d n ).
Combining [START_REF] Machkouri | A central limit theorem for stationary random fields[END_REF] and Lemma 3, we derive

f n (x) -Ef n (x) 2 2 = O (|Λ n |b n ) -1 . Hence, EI n,2 (A) ≤ κA |Λ n |b n . (13)
Combining ( 9), ( 10), ( 11) and ( 13), we obtain

EJ n ≤ κ 1 A 2 + A b n + 1 |Λ n |b n .
Optimizing in A, we derive [START_REF] Carbon | Kernel density estimation for random fields: the l 1 theory[END_REF]. The proof of Theorem 1 is complete.

Proof of Theorem 2

Without loss of generality, we consider only the case k = 2 and we refer to x 1 and x 2 as x and y (x = y). Let λ 1 and λ 2 be two constants such that λ 2 1 + λ 2 2 = 1 and note that

λ 1 (|Λ n |b n ) 1/2 (f n (x) -Ef n (x)) + λ 2 (|Λ n |b n ) 1/2 (f n (y) -Ef n (y)) = i∈Λn ∆ i |Λ n | 1/2 , λ 1 (|Λ n |b n ) 1/2 (f n (x) -Ef n (x)) + λ 2 (|Λ n |b n ) 1/2 (f n (y) -Ef n (y)) = i∈Λn ∆ i |Λ n | 1/2 ,
where

∆ i = λ 1 Z i (x) + λ 2 Z i (y) and ∆ i = λ 1 Z i (x) + λ 2 Z i (y) and for all z in R, Z i (z) = 1 √ b n (K i (z) -EK i (z)) and Z i (z) = 1 √ b n K i (z) -EK i (z)
where K i (z) and K i (z) are defined by (4). Applying Lemma 1 and Lemma 2, we know

that 1 |Λ n | 1/2 i∈Λn ∆ i -∆ i 2 ≤ κ(|λ 1 | + |λ 2 |) (m d n b n ) 3/2 |i|>mn |i| 5d 2 δ i = o(1). (14)
So, it suffices to prove the asymptotic normality of the sequence |Λ n | -1/2 i∈Λn ∆ i n≥1 . We are going to follow the Lindeberg's type proof of Theorem 1 in [START_REF] Dedecker | A central limit theorem for stationary random fields[END_REF]. We consider the notations

η = (λ 2 1 f (x) + λ 2 2 f (y))σ 2 and σ 2 = R K 2 (u)du. ( 15
)
Lemma 4 E(∆ 2 0 ) converges to η and sup

i∈Z d \{0} E|∆ 0 ∆ i | = o(M -d n ).
On the lattice Z d we define the lexicographic order as follows: if i = (i 1 , ..., i d ) and j = (j 1 , ..., j d ) are distinct elements of Z d , the notation i < lex j means that either i 1 < j 1 or for some k in {2, 3, ..., d}, i k < j k and i l = j l for 1 ≤ l < k. We let ϕ denote the unique function from {1, ..., |Λ n |} to Λ n such that ϕ(k) < lex ϕ(l) for

1 ≤ k < l ≤ |Λ n |. For all real random field (ζ i ) i∈Z d and all integer k in {1, ..., |Λ n |}, we denote S ϕ(k) (ζ) = k i=1 ζ ϕ(i) and S c ϕ(k) (ζ) = |Λn| i=k ζ ϕ(i)
with the convention S ϕ(0

) (ζ) = S c ϕ(|Λn|+1) (ζ) = 0.
From now on, we consider a field (ξ i ) i∈Z d of i.i.d. standard normal random variables independent of (X i ) i∈Z d . We introduce the fields Y and γ defined for all i in Z d by

Y i = ∆ i |Λ n | 1/2 and γ i = √ ηξ i |Λ n | 1/2
where η is defined by [START_REF] Hallin | Density estimation for spatial processes: the l 1 theory[END_REF]. Note that Y is an M n -dependent random field where M n = 2m n + 1 and m n is defined by (3). Let h be any function from R to R. For

0 < k ≤ l ≤ |Λ n |, we introduce h k,l (Y ) = h(S ϕ(k) (Y ) + S c ϕ(l) (γ)).
With the above convention we have that h k,|Λn|+1 (Y ) = h(S ϕ(k) (Y )) and also h 0,l (Y ) = h(S c ϕ(l) (γ)). In the sequel, we will often write h k,l instead of h k,l (Y ). We denote by B 4 1 (R) the unit ball of C 4 b (R): h belongs to B 4 1 (R) if and only if it belongs to C 4 (R) and satisfies max 0≤i≤4 h (i) ∞ ≤ 1. It suffices to prove that for all h in B 4 1 (R),

E h S ϕ(|Λn|) (Y ) -----→ n→∞ E (h ( √ ηξ 0 )) .
We use Lindeberg's decomposition:

E h S ϕ(|Λn|) (Y ) -h ( √ ηξ 0 ) = |Λn| k=1 E (h k,k+1 -h k-1,k ) . Now, we have h k,k+1 -h k-1,k = h k,k+1 -h k-1,k+1 + h k-1,k+1 -h k-1
,k and by Taylor's formula we obtain

h k,k+1 -h k-1,k+1 = Y ϕ(k) h ′ k-1,k+1 + 1 2 Y 2 ϕ(k) h ′′ k-1,k+1 + R k h k-1,k+1 -h k-1,k = -γ ϕ(k) h ′ k-1,k+1 - 1 2 γ 2 ϕ(k) h ′′ k-1,k+1 + r k where |R k | ≤ Y 2 ϕ(k) (1 ∧ |Y ϕ(k) |) and |r k | ≤ γ 2 ϕ(k) (1 ∧ |γ ϕ(k) |). Since (Y, ξ i ) i =ϕ(k) is inde- pendent of ξ ϕ(k) , it follows that E γ ϕ(k) h ′ k-1,k+1 = 0 and E γ 2 ϕ(k) h ′′ k-1,k+1 = E η |Λ n | h ′′ k-1,k+1
Hence, we obtain

E h(S ϕ(|Λn|) (Y )) -h ( √ ηξ 0 ) = |Λn| k=1 E(Y ϕ(k) h ′ k-1,k+1 ) + |Λn| k=1 E Y 2 ϕ(k) - η |Λ n | h ′′ k-1,k+1 2 + |Λn| k=1 E (R k + r k ) . Let 1 ≤ k ≤ |Λ n | be fixed. Since E|∆ 0 | = O √ b n and ∆ 2 0 b n n≥1 is uniformly integrable, we derive |Λn| k=1 E|R k | ≤ E ∆ 2 0 1 ∧ |∆ 0 | |Λ n | 1/2 = o(1)
and

|Λn| k=1 E|r k | ≤ η 3/2 E|ξ 0 | 3 |Λ n | 1/2 = O |Λ n | -1/2 .
Consequently, we obtain

|Λn| k=1 E (|R k | + |r k |) = o(1)
.

Now, it is sufficient to show lim n→∞ |Λn| k=1 E(Y ϕ(k) h ′ k-1,k+1 ) + E Y 2 ϕ(k) - η |Λ n | h ′′ k-1,k+1 2 = 0. ( 16
)
First, we focus on

|Λn| k=1 E Y ϕ(k) h ′ k-1,k+1 . Let the sets {V k i ; i ∈ Z d , k ∈ N\{0}} be defined as follows: V 1 i = {j ∈ Z d ; j < lex i} and for k ≥ 2, V k i = V 1 i ∩ {j ∈ Z d ; |i -j| ≥ k}. For all n in N\{0} and all k in {1, ..., |Λ n |}, we define E (n) k = ϕ({1, .., k}) ∩ V Mn ϕ(k) and S Mn ϕ(k) (Y ) = i∈E (n) k Y i .
For all function h from R to R, we define h Mn k-1,l = h S Mn ϕ(k) (Y ) + S c ϕ(l) (γ) . Our aim is to show that

lim n→∞ |Λn| k=1 E Y ϕ(k) h ′ k-1,k+1 -Y ϕ(k) S ϕ(k-1) (Y ) -S Mn ϕ(k) (Y ) h ′′ k-1,k+1 = 0. (17)
First, we use the decomposition

Y ϕ(k) h ′ k-1,k+1 = Y ϕ(k) h ′ Mn k-1,k+1 + Y ϕ(k) h ′ k-1,k+1 -h ′ Mn k-1,k+1 .
Applying again Taylor's formula,

Y ϕ(k) (h ′ k-1,k+1 -h ′ Mn k-1,k+1 ) = Y ϕ(k) S ϕ(k-1) (Y ) -S Mn ϕ(k) (Y ) h ′′ k-1,k+1 + R ′ k ,
where

|R ′ k | ≤ 2 Y ϕ(k) S ϕ(k-1) (Y ) -S Mn ϕ(k) (Y ) 1 ∧ |S ϕ(k-1) (Y ) -S Mn ϕ(k) (Y )| . Since (Y i ) i∈Z d is M n -dependent, we have E Y ϕ(k) h ′ Mn k-1,k+1
k | = 0. In fact, considering the sets W n = {-M n + 1, ..., M n -1} d and W * n = W n \{0}, it follows that |Λn| k=1 E|R ′ k | ≤ 2E   |∆ 0 |   i∈W * n |∆ i |     1 ∧ 1 |Λ n | 1/2 i∈W * n |∆ i |     ≤ 2M d n sup i∈Z d \{0} E(|∆ 0 ∆ i |) = o(1)
(by Lemma 4).

In order to obtain ( 16) it remains to control

F 1 = E   |Λn| k=1 h ′′ k-1,k+1 Y 2 ϕ(k) 2 + Y ϕ(k) S ϕ(k-1) (Y ) -S Mn ϕ(k) (Y ) - η 2|Λ n |   .
Applying again Lemma 4, we have

F 1 ≤ E   1 |Λ n | |Λn| k=1 h ′′ k-1,k+1 ∆ 2 ϕ(k) -E(∆ 2 0 )   + η -E ∆ 2 0 + 2 j∈V 1 0 ∩Wn E|∆ 0 ∆ j | ≤ E   1 |Λ n | |Λn| k=1 h ′′ k-1,k+1 ∆ 2 ϕ(k) -E(∆ 2 0 )   + o(1).
So, it suffices to prove that

F 2 = E   1 |Λ n | |Λn| k=1 h ′′ k-1,k+1 ∆ 2 ϕ(k) -E(∆ 2 0 )
  goes to zero as n goes to infinity. In fact, we have

F 2 ≤ 1 |Λn| |Λn| k=1 J (1)
k (n) + J (2) k (n) where J
(1)

k (n) = E h ′′ Mn k-1,k+1 ∆ 2 ϕ(k) -E ∆ 2 0 = 0 since h ′′ Mn k-1,k+1 and ∆ ϕ(k) are independent. Moreover, J (2)
k (n) = E h ′′ k-1,k+1 -h ′′ Mn k-1,k+1 ∆ 2 ϕ(k) -E ∆ 2 0 ≤ E         2 ∧ |i|<Mn i =0 |∆ i | |Λ n | 1/2     ∆ 2 0     ≤ 1 |Λ n |b n E     |∆ 0 | b n × |i|<Mn i =0 |∆ 0 ∆ i |     = o(1) since (|∆ 0 | √ b n ) n≥1 is uniformly integrable and |i|<Mn i =0 E|∆ 0 ∆ i | = o(1) by Lemma 4.
The proof of Theorem 2 is complete.

Proof of Theorem 3

Let n be a fixed positive integer and let x be fixed in R. We have

U n (x) = U n (x)+R n (x)
where

U n (x) = |Λ n |b n f n (x) -Ef n (x) f (x) R K 2 (t)dt and R n (x) = |Λ n |b n f n (x) -f n (x) f (x) R K 2 (t)dt . Denote D n (x) = sup t∈R |P(U n (x) ≤ t) -Φ(t)
| and let p ≥ 2 be fixed. Arguing as in Theorem 2.2 in [START_REF] Machkouri | Berry-Esseen's central limit theorem for non-causal linear processes in Hilbert spaces[END_REF], we have

D n (x) ≤ D n (x) + R n p p+1 p . (18)
Denoting

σ 2 = f (x) R K 2 (t)dt and σ 2 n = E U 2 n , we have D n (x) = sup t∈R |P(U n (x) ≤ t) -Φ(t)| ≤ sup t∈R |P(U n (x) ≤ t) -Φ (t/σ n ) | + sup t∈R |Φ (t/σ n ) -Φ (t) | = sup t∈R |P(U n (x) ≤ tσ n ) -Φ (t) | + sup t∈R |Φ (t/σ n ) -Φ (t) |.
Applying the Berry-Esseen's type theorem for m n -dependent random fields established by Chen and Shao ([5], Theorem 2.6), we obtain

sup t∈R |P(U n (x) ≤ tσ n ) -Φ (t) | ≤ κ R |K(t)| τ f (x -tb n )dt m (τ -1)d n σ τ (|Λ n |b n ) τ 2 -1 . (19)
Arguing as in Yang et al. ( [START_REF] Yang | Berry-Esséen bound of sample quantiles for φ-mixing random variables[END_REF], p. 456), we have

sup t∈R |Φ (t/σ n ) -Φ (t) | ≤ (2πe) -1 2 (σ n -1) 1 1 σn≥1 + (2πe) -1 2 1 σ n -1 1 1 0<σn<1 ≤ (2πe) -1 2 max |σ n -1|, |σ n -1| σ n } ≤ κ max |σ n -1|, |σ n -1| σ n } × (σ n + 1) ≤ κ|σ 2 n -1|.
So, we derive

D n (x) ≤ κ R |K(t)| τ f (x -tb n )dt m (τ -1)d n σ τ (|Λ n |b n ) τ 2 -1 + κ|σ 2 n -1|. (20)
Using [START_REF] Machkouri | A central limit theorem for stationary random fields[END_REF], we have also

|σ 2 n -1| ≤ 1 σ 2 E(Z 2 0 (x)) -σ 2 + j∈Z d \{0} |j|<Mn E Z 0 (x)Z j (x) . (21)
Noting that K 0 (x

) 1 = O(b n ) and K 0 (x) 2 = O( √ b n
) and using the following lemma, Lemma 5 For all p > 1, any positive integer n and any x in R,

K 0 (x) -K 0 (x) p ≤ √ 2p b n |j|>mn δ j,p ,
we obtain

E(Z 2 0 (x)) -E(Z 2 0 (x)) = 1 b n E(K 2 0 (x)) -E(K 2 0 (x)) ≤ 1 b n K 0 (x) 2 K 0 (x) -K 0 (x) 2 ≤ κ b 3/2 n |j|>mn δ j and E(Z 2 0 (x)) -σ 2 = 1 b n E(K 2 0 (x)) -(E(K 0 (x)) 2 -f (x) R K 2 (t)dt ≤ 1 b n E(K 2 0 (x)) -f (x) R K 2 (t)dt + 1 b n (E(K 0 (x)) 2 ≤ R K 2 (v)|f (x -vb n ) -f (x)|dv + O(b n ) ≤ κ b n R |v|K 2 (v)dv + O(b n ) = O(b n ).
Hence,

E(Z 2 0 (x)) -σ 2 ≤ κ b 3/2 n |j|>mn δ j + O(b n ). (22)
Now, let i = 0 be fixed. We have

E|Z 0 (x)Z i (x)| ≤ 1 b n E|K 0 (x)K i (x)| + 3 b n (E|K 0 (x)|) 2 . ( 23
)
Moreover, keeping in mind that ||α| -|β|| ≤ |α -β| for all (α, β) in R 2 and applying the Cauchy-Schwarz inequality, we obtain

E|K 0 (x)K i (x)| -E|K 0 (x)K i (x)| ≤ 2 K 0 (x) 2 K 0 (x) -K 0 (x) 2
and applying Lemma 5, we derive

E|K 0 (x)K i (x)| -E|K 0 (x)K i (x)| ≤ κ √ b n |j|>mn δ j . (24)
Combining ( 23) and ( 24), we have

E|Z 0 (x)Z i (x)| ≤ κ b 3/2 n |j|>mn δ j + 1 b n E|K 0 (x)K i (x)| + 3 b n (E|K 0 (x)|) 2 . ( 25
)
Using Assumption (B3), we obtain

E K 0 (x)K i (x) = R 2 K x -u b n K x -v b n f 0,i (u, v)dudv ≤ κb 2 n R |K(w)|dw 2 .
Since E|K 0 (x)| = O(b n ), we derive from [START_REF] Wu | Asymptotic theory for stationary processes[END_REF] that

j∈Z d \{0} |j|<Mn E Z 0 (x)Z j (x) ≤ κM d n b 3/2 n |j|>mn δ j + O(M d n b n ). (26)
Finally, combining [START_REF] Rugh | Nonlinear system theory[END_REF], ( 21), ( 22) and ( 26), for all α > 1, we obtain

D n (x) ≤ κm d(τ -1) n σ τ (|Λ n |b n ) τ 2 -1 + κ m d(α-1) n b 3/2 n |j|>mn |j| dα δ j + O(m d n b n ). (27)
Since there exist α > 1 and p ≥ 2 such that i∈Z d |i| dα δ i,p < ∞, we derive from Lemma 2 that

R n (x) p ≤ κ √ p σm d(α-1) n b 3/2 n i∈Z d |i| dα δ i,p . (28)
Combining ( 18), ( 27) and (28), we obtain

D n (x) ≤ κ m d(τ -1) n b n + 1 (|Λ n |b n ) τ 2 -1 + 1 m d(α-1) n b 3/2 n p p+1
(29) for all 2 < τ ≤ 3, all p ≥ 2 and all α > 1 such that i∈Z d |i| dα δ i,p < ∞. Optimizing in m n we derive

D n (x) ≤ κ b θ 1 n b n + 1 (|Λ n |b n ) τ 2 -1 θ 2
where

θ 1 = 3p(1 -τ ) 2(τ -1)(p + 1) + 2p(α -1)
and θ 2 = p(α -1) (τ -1)(p + 1) + p(α -1) .

Finally, choosing b

n = |Λ n | 2 τ -1 , we obtain D n (x) ≤ κ|Λ n | -θ where θ = 1 2 - 1 τ 3p(1 -τ ) + 2p(α -1) (τ -1)(p + 1) + p(α -1)
.

The proof of Theorem 3 is complete.

Appendix

Proof of Lemma 1. We follow the proof by Bosq et al. ( [START_REF] Bosq | Asymptotic normality for density kernel estimators in discrete and continuous time[END_REF], pages 88-89). First, m n goes to infinity since v n = b

m d n ≥ 1 bn (r (v n )) 1/3 ≥ 1 bn (r (m n )) 1/3 since v n ≤ m n . Finally, we obtain 1 (m d n b n ) 3/2 |i|>mn |i| 5d 2 δ i ≤ r(m n ) -----→ n→∞ 0.
The proof of Lemma 1 is complete.

Proof of Lemma 2. Let p > 1 be fixed. We follow the proof of Proposition 1 in [START_REF] Machkouri | A central limit theorem for stationary random fields[END_REF]. For all i in Z d and all x in R, we denote R i = K i (x) -K i (x). Since there exists a measurable function H such that R i = H(ε i-s ; s ∈ Z d ), we are able to define the physical dependence measure coefficients (δ

(n) i,p ) i∈Z d associated to the random field (R i ) i∈Z d . We recall that δ (n) i,p = R i -R * i p where R * i = H(ε * i-s ; s ∈ Z d ) and ε * j = ε j 1 1 {j =0} + ε ′ 0 1 1 {j=0} for all j in Z d .
In other words, we obtain R * i from R i by just replacing ε 0 by its copy ε ′ 0 (see [START_REF] Wu | Nonlinear system theory: another look at dependence[END_REF]). Let τ : Z → Z d be a bijection. For all l ∈ Z, for all i ∈ Z d , we denote

P l R i := E(R i |F l ) -E(R i |F l-1 ) where F l = σ ε τ (s) ; s ≤ l and R i = l∈Z P l R i .

Consequently,

i∈Λn a i R i p = l∈Z i∈Λn a i P l R i p and applying the Burkholder inequality (cf. [START_REF] Hall | Martingale limit theory and its application[END_REF], page 23) for the martingale difference sequence i∈Λn a i P l R i l∈Z , we obtain

i∈Λn a i R i p ≤   2p l∈Z i∈Λn a i P l R i 2 p   1 2 ≤   2p l∈Z i∈Λn |a i | P l R i p 2   1 2
.

Moreover, by the Cauchy-Schwarz inequality, we have

i∈Λn |a i | P l R i p 2 ≤ i∈Λn a 2 i P l R i p × i∈Λn P l R i p . (31)
Let l in Z and i in Z d be fixed.

P l R i p = E(R i |F l ) -E(R i |F l-1 ) p = E(R 0 |T i F l ) -E(R 0 |T i F l-1 ) p
where T i F l = σ ε τ (s)-i ; s ≤ l . Hence,

P l R i p = E H ((ε -s ) s∈Z d ) |T i F l -E H (ε -s ) s∈Z d \{i-τ (l)} ; ε ′ τ (l)-i |T i F l p ≤ H ((ε -s ) s∈Z d ) -H (ε -s ) s∈Z d \{i-τ (l)} ; ε ′ τ (l)-i p = H (ε i-τ (l)-s ) s∈Z d -H (ε i-τ (l)-s ) s∈Z d \{i-τ (l)} ; ε ′ 0 p = R i-τ (l) -R * i-τ (l) p = δ (n) i-τ (l),p . Consequently, i∈Z d P l R i p ≤ j∈Z d δ (n)
j,p and combining (30) and (31), we obtain

i∈Λn a i R i p ≤   2p j∈Z d δ (n) j,p i∈Λn a 2 i l∈Z P l R i p   1 2
.

Similarly, for all i in Z d , we have l∈Z P l R i p ≤ j∈Z d δ

(n) j,p and we derive i∈Λn

a i R i p ≤ 2p i∈Λn a 2 i 1 2 i∈Z d δ (n) i,p . (32)
Since K *

i = E K * i (x) F * n,i where F * n,i = σ ε * i-s ; |s| ≤ m n and K i (x) -K i (x) * = K * i (x) -K * i (x), we derive δ (n) i,p ≤ 2 K i (x) -K * i (x) p . Since K is Lipschitz, we obtain δ (n) i,p ≤ 2δ i,p b n (33)
where δ i,p = X i -X * i p . Morever, we have also

δ (n) i,p ≤ 2 K 0 (x) -K 0 (x) p . (34)
Combining (34) and Lemma 5, we derive

δ (n) i,p ≤ √ 8p b n |j|>mn δ j,p . (35)
Combining (33) and (35), we obtain

i∈Z d δ (n) i,p ≤ m d n √ 8p b n |j|>mn δ j,p + 2 b n |j|>mn δ j,p ≤ 2 √ 8pm d n b n |j|>mn δ j,p .
The proof of Lemma 2 is complete.

Proof of Lemma 3. Let s and t be fixed in R. Since E K 0 (s)K 0 (t) = E K 0 (s)K 0 (t) , we have

E K 0 (s)K 0 (t) -E (K 0 (s)K 0 (t)) ≤ K 0 (s) 2 K 0 (t) -K 0 (t) 2 .
Keeping in mind that K 0 (s) 2 = O( √ b n ) and using Lemma 5, we have

E K 0 (s)K 0 (t) -E (K 0 (s)K 0 (t)) ≤ κ √ b n |j|>mn δ j .
Moreover, keeping in mind Assumptions (A1), (A2) and (A4), we have

lim n 1 b n E (K 0 (s)K 0 (t)) = lim n R K (v) K v + t -s b n f (s-vb n )dv = u(s, t) f (s) R K 2 (u)du (37)
where u(s, t) = 1 if s = t and u(s, t) = 0 if s = t. We have also Let x be fixed in R. Choosing s = t = x and combining (36), (37), (38) and Lemma 1, we obtain E(Z 2 0 (x)) goes to f (x) R K 2 (u)du as n goes to infinity. In the other part, let i = 0 be fixed in Z d and let s and t be fixed in R. We have

E|Z 0 (s)Z i (t)| ≤ 1 b n E K 0 (s)K i (t) + 3 b n E K 0 (s) E K 0 (t) . (39)
Keeping in mind that ||α| -|β|| ≤ |α -β| for all (α, β) in R 2 and applying the Cauchy-Schwarz inequality, we obtain E|K 0 (s)K i (t)|-E|K 0 (s)K i (t)| ≤ K 0 (s) 2 K 0 (t)-K 0 (t) 2 + K 0 (t) 2 K 0 (s)-K 0 (s) 2 (42)

Moreover, using Assumption (B3), we have

E K 0 (s)K i (t) = R 2 K s -u b n K t -v b n f 0,i (u, v)dudv ≤ κb 2 n R |K(w)|dw
The proof of Lemma 3 is complete. For all integer n, let a n = n j=0 |Γ j | and let τ : N\{0} → Z d be the bijection defined by τ (1) = (0, ..., 0) and

Lemma 1

 1 ] denotes the integer part function. The following technical lemma is a spatial version of a result by Bosq et al. ([1], pages 88-89). If (A4) holds then

  and we observe the L 1 -convergence of ft to the true density function f of X 0,0 . In order to illustrate the asymptotic normality of the estimator (8), we put x = -1, t = 20 and b 20 = 0.7 and we calculate the expectation E ft (-1) of ft (-1) by taking again the arithmetic mean value of 100 replications of ft (-1). Finally, noting that R K 2 (x)dx = 4/5 and f (-1) = 1/100, we consider 1500 replications of √ 400 × 0.7 f20 (-1) -E f20 (-1)1/100 × 4/5

Figure 1 :

 1 Figure 1: Asymptotic normality of the kernel density estimator.

  and m n ≥ v n . For all positive integer m, we consider r(m) = |i|>m |i| 5d 2 δ i . Since (A4) holds, r(m) converges to zero as m goes to infinity. Moreover, m d n b n ≤ max √ b n , κ r(v n ) 1/3 + b n -----→ n→∞ 0 and

  Since b n |E(Z 0 (s)Z 0 (t)) -E(Z 0 (s)Z 0 (t)| = |E (K 0 (s)K 0 (t)) -E K 0 (s)K 0 (t) |, we have M d n |E(Z 0 (s)Z 0 (t)) -E(Z 0 (s)Z 0 (t)| ≤ κ (m d n b n )

  )f (svb n )dv R K(w)f (twb n )dw = 0. (38)

  E|K 0 (s)K i (t)| -E|K 0 (s)K i (t)| ≤ κ (m d n b n ) 3/2 |j|>mn |j| 5d 2 δ j .(41)Since Assumptions (A1) and (A4) hold andM d n b n = o(1), we have M d n b n E K 0 (s) E K 0 (t) = M d n b n R |K(u)|f (sub n )du R |K(v)|f (tvb n )dv = o(1).

2 .

 2 So, using again Assumption (A4) andM d n b n = o(1), we derive M d n b n E K 0 (s)K i (t) = o(1).(43)Combining (39), (41), (42), (43) and Lemma 1, we obtainM d n sup i∈Z d \{0} E|Z 0 (s)Z i (t)| = o(1).

Proof of Lemma 4 . 2 0 2 0

 422 Let x and y be two distinct real numbers. Noting that)) + 2λ 1 λ 2 E(Z 0 (x)Z 0 (y)) (y)) + 2λ 1 λ 2 E(Z 0 (x)Z 0 (y))and using (36) and Lemma 1, we obtain lim n→∞ and (45), we derive thatE(∆ ) converges to η = (λ 2 1 f (x) + λ 2 2 f (y)) R K 2 (u)du. Let i = 0 be fixed in Z d . Combining (44) and E|∆ 0 ∆ i | ≤ λ 2 1 E|Z 0 (x)Z i (x)|+λ 2 2 E|Z 0 (y)Z i (y)|+λ 1 λ 2 E|Z 0 (x)Z i (y)|+λ 1 λ 2 E|Z 0 (y)Z i (x)|, (46) we obtain M d n sup i∈Z d \{0} E|∆ 0 ∆ i | = o(1). The proof of Lemma 4 is complete.Proof of Lemma 5. Let p > 1 be fixed. We consider the sequence (Γ n ) n≥0 of finite subsets of Z d defined by Γ 0 = {(0, ..., 0)} and for all n in N\{0}, Γ n = {i ∈ Z d ; |i| = n}.

  is the true density function of X 0,0 and the bandwith b t is being set to |Λ t | -1/3 with |Λ t | denoting the number of elements in Λ t . Hence, we derive its expectation E

	value of 100 replications of of t in the following table	100 -100 | ft (x) -f (x)|dx by taking the arithmetic mean -100 | ft (x)-f (x)|dx. The results are given for several values 100
	t 10	|Λ t | = t 2 b t = |Λ t | -1/3 E 100 0.215	100 -100 | ft (x) -f (x)|dx 0.0171
	20	400		0.136	0.0163
	50	2500	0.074	0.0157
	100	10000	0.046	0.0153

1, 1[. In order to illustrate the result obtained in Theorem 1, we calculate (Monte Carlo method) 100 -100 | ft (x)f (x)|dx where f

• for all n in N\{0}, if l ∈ ]a n-1 , a n ] then τ (l) ∈ Γ n ,

• for all n in N\{0}, if (i, j) ∈ ]a n-1 , a n ] 2 and i < j then τ (i) < lex τ (j) Let (m n ) n≥1 be the sequence of positive integers defined by [START_REF] Carbon | Kernel density estimation for random fields[END_REF]. For all n in N\{0}, we recall that F n,0 = σ (ε -s ; |s| ≤ m n ) (see ( 4)) and we consider also the σ-algebra G n := σ ε τ (j) ; 1 ≤ j ≤ n . By the definition of the bijection τ , we have

) for all l in Z. Let p > 1 be fixed. Since (D l ) l∈Z is a martingale-difference sequence, applying Burkholder's inequality (cf. [START_REF] Hall | Martingale limit theory and its application[END_REF], page 23), we derive

, we obtain

The proof of Lemma 5 is complete.

Acknowledgments. The author is grateful to two anonymous referees for their careful reading and constructive comments. He is also indebted to Yizao Wang for pointing an error in the proof of a first version of Theorem 3.