Mohammadhadi Farzanehkaloorazi

Mehdi Tale

Stéphane Caro
email: stephane.caro@irccyn.ec-nantes.fr

Collision-free workspace of parallel mechanisms based on an interval analysis approach

Keywords: Parallel mechanism, Collision-free workspace, Mechanical interference, Interval analysis

This paper proposes an interval-based approach in order to obtain the obstacle-free workspace of parallel mechanisms containing one prismatic actuated joint per limb, which connects the base to the end-effector. This approach is represented through two cases studies, namely a 3-RPR planar parallel mechanism and the so-called 6-DOF Gough-Stewart platform. Three main features of the obstacle-free workspace are taken into account: mechanical stroke of actuators, collision between limbs and obstacles and limb interference. In this paper, a circle(planar case)/spherical(spatial case) shaped obstacle is considered and its mechanical interference with limbs and edges of the end-effector is analyzed. It should be noted that considering a circle/spherical shape would not degrade the generality of the problem, since any kind of obstacle could be replaced by its circumscribed circle/sphere. Two illustrative examples are given to highlight the contributions of the paper.

Introduction

Parallel mechanisms (PMs) are known to be more precise and able to carry heavier loads than the serial manipulators. [START_REF] Merlet | Parallel Robots[END_REF][START_REF] Gosselin | Determination of the workspace of planar parallel manipulators with joint limits[END_REF] Besides several advantages, they are restricted by their limited workspace. Therefore, in practice, the presence of an obstacle inside their workspace should be taken into consideration in order to alleviate this limitation. Obtaining the obstacle-free workspace of PMs leads to have a conservative workspace for which all actions inside the aforementioned workspace are free of collision. Furthermore, obtaining the obstacle-free workspace is a definite asset in path planning and obstacle avoidance while controlling PMs.

The problems of path planning and obstacle avoidance have been frequently investigated in the literature. [START_REF] Bohigas | Planning singularity-free paths on closed-chain manipulators[END_REF] In ref. [4], Zi et al. considered the possible collision of a cooperative cable-driven parallel robot for multiple mobile cranes and used the sensor technology in order to avoid obstacles. Laliberte et al. in ref. [5] calculated the motion of the manipulator using the velocity inversion of a redundant manipulator, which optimizes the distance to obstacles. The algorithm includes joint limit constraints, collision detection and heuristics for the solution of typical difficult cases, thereby leading to a high success rate. Brooks et al. proposed an efficient algorithm to generate collision-free paths for a manipulator with five or six revolute joints. Yang et al. investigated dynamic collision-free trajectory generation in a non-stationary environment, using biologically inspired neural network approaches. Brocks and Khatib [START_REF] Brock | Elastic strips: A framework for motion generation in human environments[END_REF] represented elastic strip framework which enables the execution of a previously planned motion in a dynamic environment for robots with many degree of freedom (DOF). Khatib et al. [START_REF] Khatib | Robots in human environments: Basic autonomous capabilities[END_REF] presented developments of models, strategies and algorithms dealing with a number of autonomous capabilities that are essential for robot operations in human environments. These capabilities include integrated mobility and manipulation, cooperative skills between multiple robots, interaction ability with humans and efficient techniques for real-time modification of collision-free paths. In ref. [8], Komainda and Hiller presented a concept for motion control of redundant manipulators in a changing environment. In ref. [9], Jiménez et al. described a general approach to cover all distance computation algorithms, hierarchical object representations, orientation-based pruning criteria and space partitioning schemes. In ref. [10], Wenger and Chedmail illustrated the collision-free workspace of serial manipulators. Caro et al., in ref. [11], introduced a new method, based on numerical constraint programming, in order to compute a certified enclosure of the generalized aspects. In ref. [12], the collision-free workspace of a planar PM is investigated. This paper aims at extending the latter study by proposing a general concept to be applicable to 6-DOF PMs, such as Gough-Stewart platform.

Most of the approaches presented in the literature are case dependent and cannot be generalized and extended to other cases. In ref. [13], the workspace of 3-RRR PMs has been investigated. Even for this particular PMs, the approach could be applied upon considering some assumptions, where in the latter paper, the mechanisms should be symmetric. It should be noted that in the foregoing paper, the workspace is only investigated without considering mechanical interferences and obstacles within the workspace.

The mathematical framework used in this paper is based on interval analysis. [START_REF] Moore | Methods and Applications of Interval Analysis[END_REF][START_REF] Jaulin | Set-membership localization with probabilistic errors[END_REF][START_REF] Kaloorazi | Determining the maximal singularity-free circle or sphere of parallel mechanisms using interval analysis[END_REF] Interval analysis is a reliable method to evaluate functions and is used frequently in the field of robotics. [START_REF] Merlet | Interval analysis and robotics[END_REF] An interval variable of [x] = [x, x] is a set of all real numbers from the lower bound, x, to the upper bound, x. More details about the application of interval analysis in kinematic investigation of robotic mechanical system are discussed in Section 2 and in ref. [14].

In the context of application of interval analysis in investigating the kinematic properties of PMs, several papers have been published. Most of them dealt with solving the Forward Kinematic Problem (FKP) [START_REF] Merlet | Solving the forward kinematics of a gough-type parallel manipulator with interval analysis[END_REF] and obtaining the workspace. [START_REF] Kaloorazi | Determining the maximal singularity-free circle or sphere of parallel mechanisms using interval analysis[END_REF][START_REF] Kaloorazi | Interval-Analysis-Based Determination of the Singularity-Free Workspace of Gough-Stewart Parallel Robots[END_REF][START_REF] Farzaneh Kaloorazi | On the Maximal Singularity-free Workspace of Parallel Mechanisms via Interval Analysis[END_REF] Merlet, in ref. [21], investigated the workspace of PMs. Several papers have been published in the context of obtaining the singularity-free workspace using geometrical approach, [START_REF] Kaloorazi | Determination of the maximal singularity-free workspace of 3-dof parallel mechanisms with a constructive geometric approach[END_REF] and using interval analysis [START_REF] Kaloorazi | Determining the maximal singularity-free circle or sphere of parallel mechanisms using interval analysis[END_REF] in which the novelty is about demonstrating the state of the art of applying interval analysis in solving the foregoing problems. In ref. [23], authors used interval analysis in order to investigate the orientation workspace of a parallel kinematic machine. Merlet, in ref. [24], proposed an algorithm that enables one to determine almost all the geometries of a simplified Gough platform whose workspace should include an arbitrary set of poses. There have been few study conducted on analyzing the workspace of PMs by considering mechanical interference and also in the presence of obstacle. This is a definite asset both in design and control of PMs, which is the main concern of this paper.

Few studies have been conducted on obtaining the collision-free workspace of PMs and this is mainly due to the fact that this mechanisms have complex kinematic expressions comparing to their serial counterpart. Interval analysis has been used more in solving the FKP and in obtaining theoretical workspace, i.e., workspace in which mechanical interference and obstacle are not taken into account. Some papers worked on obtaining the collision-free workspace of cable-driven PMs which is out of the scope of the present paper. In the latter papers, the approach is mostly based on numerical techniques in which all the configurations and poses of the end-effector (EE) are tested in order to obtain configurations for which collision may occur. [START_REF] Otis | Determination and management of cable interferences between two 6-dof foot platforms in a cable-driven locomotion interface[END_REF] For instance, interval analysis has been used in investigating the workspace of cable-driven PMs. [START_REF] Gouttefarde | Interval-analysis-based determination of the wrench-feasible workspace of parallel cable-driven robots[END_REF] The main contribution of this paper deals with the use of interval analysis techniques in order to obtain the collision-free workspace of parallel manipulators. It should be noted that obtaining the collision-free workspace of robotic mechanical systems, PMs among others, is a very delicate task which is elusive to classical methods. Therefore, more elaborated methods should be used such as interval analysis techniques. To the best of author's knowledge, few studies have been conducted in this regard.

The remainder of this paper is organized as follows. First, the mathematical framework of interval analysis is broadly reviewed. Then, the concept of the algorithm developed to obtain the obstacle-free workspace and the corresponding pseudo-code are explained in detail. Finally, the results obtained for a 3-RPR and a 6-UPS PM are given and the corresponding obstacle-free workspaces are illustrated.

Interval Analysis and Mathematical Framework

Several people had the idea to bound rounding errors with intervals: e.g. Dwyer (1951), [START_REF] Dwyer | Computation with approximate numbers[END_REF] Sunuga (1958), [START_REF] Sungana | Theory of Interval Algebra and Application to Numerical Analysis[END_REF] Warmus (1956) [START_REF] Warmus | Calculus of approximations[END_REF] and Wilkinson (1980). [START_REF] Wilkinson | Turings work at the national physical laboratory and the construction of pilot ace, deuce, and ace[END_REF] However, interval mathematics has been widespread in the research community with Moore's book "Interval Analysis" in 1966. [START_REF] Moore | Interval Analysis. Series in Automatic Computation[END_REF] Moore's book transformed this simple idea into a viable tool for error analysis. Instead of merely treating rounding errors, Moore extended the use of interval analysis to bound the effect of errors from all sources, including approximation and errors in data. [START_REF] Hansen | Global Optimization Using Interval Analysis: Revised and Expanded[END_REF] In the literature, interval analysis is regarded as a powerful numerical method to solve a wide range of problems such as, among others, circumventing round-off errors, [START_REF] Merlet | Solving the forward kinematics of a gough-type parallel manipulator with interval analysis[END_REF] solving system of equations, optimization problem [START_REF] Hansen | Global Optimization Using Interval Analysis: Revised and Expanded[END_REF] and proper workspace presentation, etc.. [START_REF] Merlet | Interval analysis and robotics[END_REF][START_REF] Hansen | Global Optimization Using Interval Analysis: Revised and Expanded[END_REF][START_REF] Hao | Multi-criteria optimal design of parallel manipulators based on interval analysis[END_REF][START_REF] Chablat | An interval analysis based study for the design and the comparison of three-degrees-of-freedom parallel kinematic machines[END_REF] Furthermore, interval analysis provides an interactive visualization in the progress of calculation which is a definite asset in 2D and 3D representations of manipulator workspaces. Recently, upon revealing some remarkable features of interval analysis, such as finding the solution of a problem within some finite domains and taking into account the numerical computer round-off errors, it has stimulated the interests of many researchers in the robotic community to deal with complicated problems such as FKP and inverse kinematic problem (IKP), [START_REF] Merlet | Solving the forward kinematics of a gough-type parallel manipulator with interval analysis[END_REF][START_REF] Rao | Inverse kinematic solution of robot manipulators using interval analysis[END_REF] calibration [START_REF] Fusiello | Globally convergent autocalibration using interval analysis[END_REF][START_REF] Wabinski | Time Interval Analysis of Interferometer Signals for Measuring Amplitude and Phase of Vibrations[END_REF] and the determination of the singularity-free workspace of parallel manipulators. [START_REF] Kaloorazi | Determining the maximal singularity-free circle or sphere of parallel mechanisms using interval analysis[END_REF][START_REF] Kaloorazi | Interval-Analysis-Based Determination of the Singularity-Free Workspace of Gough-Stewart Parallel Robots[END_REF] In the evolutionary techniques, the chance of being trapped in a local optimum is highly dependent of the initial population and initial search space. However, in the case of interval analysis, the only parameter to obtain the global optimum is to choose the proper search space. In order to compute the maximum singularity-free workspace of PMs and other kinematic properties, [START_REF] Merlet | Interval analysis and robotics[END_REF][START_REF] Hao | Multi-criteria optimal design of parallel manipulators based on interval analysis[END_REF] interval analysis entails following advantages: (a) Contrary to other tools, which would result in a lengthy computation process and may converge to a local optimum, interval analysis is not a black box, since it requires combination of heuristics and numerical concepts to be effective; (b) it allows us to find all the solutions with inequalities within a given search space; [START_REF] Merlet | Solving the forward kinematics of a gough-type parallel manipulator with interval analysis[END_REF][START_REF] Oetomo | Certified Workspace Analysis of 3RRR Planar Parallel Flexure Mechanism[END_REF] (c) for two and three-dimensional problems, it leads to see the evolution of the solutions and to monitor the procedure in order to have better insight to the problem; (d) it allows us to consider uncertainties in the model of the robot.

Interval analysis is a branch of mathematics that basically works with closed intervals instead of accurate numbers. An interval [x] is a set of real numbers between two bounds and can be represented as follows:

[x] = [x, x] = {x ∈ R | x ≤ x ≤ x} , (x ≤ x), (1)
where x and x are lower bound and upper bound, respectively. All mathematical operations such as addition and multiplication can be performed on intervals. For instance, [START_REF] Moore | Methods and Applications of Interval Analysis[END_REF] [x]

+ [y] = [x, x] + [y, y] = [x + y, x + y], (2)
Moreover, a function of real numbers such as f (x) can be evaluated as an interval from a given interval, [x], which results in an interval [f ] = f ([x]). For example, for a monotonic function like

f (x) = x 3 , [f ] = f ([x]) = [f (x), f (x)] = [x 3 , x 3 ]. ( 4
)
The whole concept of interval analysis is based on bisecting a box (or a hyper-box in a higher dimensional space), called branch & prune approach, [START_REF] Van Hentenryck | Solving polynomial systems using a branch and prune approach[END_REF] upon considering some well-defined algebra on intervals and natural evaluations, in such a way that the latter box will converge toward the desired solution.

Obstacle Avoidance Formulation via Geometrical Concept

To the sake of understanding the proposed method, the latter is first fully described for 3-RPR PM. It should be noted that, without loss of generality, the obstacle is considered as a circle (planar case) or a sphere (spatial case) since any kind of obstacle could be replaced by its circumscribed circle or sphere. In order to avoid an obstacle, in general, the links of limbs and the edges of the EE are considered to be straight line segments. Moreover, all links can be considered to be perfect cylinder (rectangle in planar case). In the case of more complicated shape of a link, one should simplify it to a cylinder which contains all parts of the link. Even though it may cause to eliminate some non-collision part of the workspace, but it depends on the trade-off preference of the user to keep a balance between the computational time and precision of the result. This is a common approach to compute the constant-orientation workspace of parallel robots. In the case of 6-DOF PMs, since here is no human visualization for depicting six-dimensional space, thus one should fix the rotational DOF to have a graphical representation of the workspace.

A 1 A 2 A 3 B 1 B 2 B 3 φ l x l y ρ 1 ρ 2 ρ 3 x y x y P (Obstacle) O O (a)
A 1 P 1 P 2 [B 1 ] Obstacle Obstacle (a) [B 1 ] P 1 P 2 Obstacle Obstacle [B 2 ] (b)
As it can be observed in Fig. 1, in the case of planar PMs, the obstacle is a circle; x P and y P being the Cartesian coordinates of its center point and r P its radius. The problem of obtaining the obstacle-free workspace via interval analysis can be divided into two cases: the collisions between the obstacle and (a) a line passing through one point and one box (b) through two boxes. These two cases are depicted in Fig. 2. The first case is applicable for links that are connected to the fixed frame via a revolute joint; for instance, links A 1 B 1 , A 2 B 2 and A 3 B 3 in the 3-RPR planar PM shown in Fig. 1(a). The second case is more general and is applicable for those links which are the medial or distal links of the limb; there is not such a link in the 3-RPR planar PM. Moreover, the edges of the EE should be categorized in the second case, i.e., in the 3-RPR planar PM, B 1 B 2 , B 2 B 3 and B 3 B 1 .

Interval distance to obstacle

A simple solution to obtain the obstacle-free workspace is to write the equation from the distance of a point to a line in the 2D space, which can be written as follows: where d stands for the distance from point P to line L, x P and y P are the x and y coordinates of the obstacle P , respectively, m stands for the slope of the line and c is a constant. In the case of implementing an interval line using a point and a box, the first aforementioned case, one has

d = x P +my P -mc m 2 +1 -x P 2 + m x P +my P -mc m + c -y P 2 L : y = mx + c, (5)
m = y A -[y B ] x A -[x B ] , c= y A -mx A , (6)
in which x A and y A stand for the coordinates of point A, [x B ] and [y B ] stand for the components of box [B]. These lines are referred to as the collision lines, which are assigned to the links of mechanisms and in some configurations may interfere with the obstacle.

Collision of limbs with obstacle

Resorting to interval functions, [START_REF] Merlet | Parallel Robots[END_REF] one can apply interval variables to Eq. ( 5) and obtain an interval of distances, [d]. For example, in the case of the 3-RPR PM shown in Fig. 1(a), having geometric parameters represented in Table I, for the first limb, the collision line passes through A 1 , which is a fixed point, and a box in the search space of the interval algorithm. The task of the algorithm is to determine the distance from obstacle P to the collision line. If [d] > r P , i.e., the distance of all possible lines passing through the fixed point and the box under investigation will be higher than the obstacle radius, then the corresponding box will be fully inside the obstacle-free workspace. On the other hand, if [d] < r P , the box will be fully outside of the aforementioned workspace. Eventually, if 0 ∈ [d]r P the box goes for further bisection. The result of the above procedure is illustrated in Fig. 3(a). It is noteworthy that the green circles in the figures correspond to the lower limits of the actuators.

Collision of edges of EE with obstacle

The procedure of obtaining the obstacle-free workspace is not complete yet. Indeed, the RPR limb should be regarded as a segment instead of a line. Therefore, if

A -[B] 2 < A -P 2 , then [B]
does not interfere with the obstacle and should be a member of obstacle-free workspace, Fig. 3(b). In addition to the limbs, the interference between the edges of the EE and the obstacle should also be taken into account. In the case of 3-DOF PMs, a simple EE can be regarded as a triangle. Therefore, its edges belong to those lines passing through two intervals, i.e., the second case. In this case, obtaining the distance from the obstacle to the line, using Eq. ( 5), leads to a very time consuming and inefficient process. Figure 4 represents the obstacle-free workspace in the case that only the collision of the edge B 1 B 2 with the obstacle is taken into consideration. The result shown in Fig. 4 is not a desirable result. The white area, which shows unassigned boxes, comparing to Fig. 3(b), is a vast area. Furthermore, there are too many red boxes and green boxes in the middle of the workspace. From the previous results, we expected that the number of boxes would be lower. This shows that computational load of these boxes is high. The latter problem is known as blow up. Blow up is a common phenomenon in interval analysis, which appears when some intervals are multiplied together and when degree of complexity of the interval function becomes high. [START_REF] Hansen | Global Optimization Using Interval Analysis: Revised and Expanded[END_REF] Hence, a geometrical methodology is proposed in order to eliminate those parts of the workspace for which edges of the EE collide with the obstacle. Since the workspace is represented for a constant orientation, the slope of the interval line is constant. In other words, by determining the dimensions (width and height) and position of [B 1 ], the position and dimensions of [B 2 ] are known. The result of considering a constant slope for the interval line passing through two boxes is represented in Fig. 5. In the latter figure, the edge B 1 B 2 is regarded as a line and the workspace is obtained for a constant orientation of φ = π/4. However, the result is not complete because B 1 B 2 should be regarded as a segment line.

In order to find the obstacle-free workspace, only considering B 1 B 2 , a geometrical approach, based on the fact that the workspace is obtained for a constant orientation, is combined with interval analysis. For the sake of better understanding, first a simple example is discussed here. The example can be observed in Fig. 6. Assuming that the problem consists in obtaining a shape within the workspace which should be subtracted from the full workspace. In the constant orientation case, the slope of edge B 1 B 2 of the EE is constant, in this example φ = π/4. The workspace consists of the set of locations that point B 1 can reach in the fixed frame, because the moving frame is attached to the EE in B 1 . As it can be observed from Fig. 6, the shape is a rounded rectangle. A rounded rectangle is a shape which is generated by sweeping the center of a circle along a segment line. It is a rectangle having a rounded cap instead of right angle. An example of rounded rectangle is shown in Fig. 6. The next step is to define the shape using geometrical reasoning. By considering three geometrical constraints for the center of obstacle circle, the shape can be obtained: The center is subject to . This is a modified result of Fig. 4. A better solution is represented in Fig. 7(a).

P B 1 B 2 φ = π 4 P [x C ] [y C ]
x y Fig. 6. Technique to find the shape that should be subtracted from the full workspace. This rounded rectangle is the shape within the workspace that shows the locations of B 1 where edge B 1 B 2 collides with the obstacle circle.

along y-component. For the case study of this paper, one has

[x C ] = x P -[0, B 1 B 2 cos φ] ( 7 ) [y C ] = y P -[0, B 1 B 2 sin φ]. (8)
Figure 7(a) illustrates the results of the foregoing methodology, considering only the collision of edge B 1 B 2 with the obstacle, for φ = 0. In this figure, a rounded-end rectangle inside the workspace is eliminated. Upon considering all the three edges of the EE, Fig. 7(b) represents the obstacle-free workspace of the 3-RPR planar PM for φ = π/4.

Algorithm 1, in the appendix, represents the pseudo-code of the introduced method. It is based on a branch and prune algorithm. [START_REF] Van Hentenryck | Solving polynomial systems using a branch and prune approach[END_REF] Note that f , which indicates the number of actuated joint of the mechanism, is 3 in the case of planar PM. In the upcoming section, we will extend the algorithm to a higher DOF mechanism, i.e. 6-UPS PM and in that case f = 6. There is another variable which is indicated by g, referring to the dimension of the box. In the case of planar PM, it is 2 and in the case of spatial PM it is 3. In lines 5-7, the position of the three distal joints is determined as intervals in the fixed frame. In line 9, for all limbs, the distance from the distal to the proximal joints, i.e., the length of the actuator, is evaluated in order to be checked in line 15. In line 10, Line(A, B) is a function which creates a collision line that passes through points A and B. Since the second argument is an interval [B i ], hence the collision line is an interval line. Distance(L, P ) computes the Euclidean distance from point P to line L. Therefore, [d i ], i = 1, 2, 3, are intervals of possible distances from Algorithm 1 The pseudo-code of the algorithm to obtain the obstacle-free workspace of a PM with prismatic actuation. Lines followed by % are comments. and stand for logical OR and logical AND, respectively.

1: Input: Design parameters of the mechanism; x P , y P and r P for properties of the obstacle; [B] as search space; ρ min and ρ max as mechanical strokes; as the desired accuracy; f , the degree of freedom of the mechanism (i.e. 3 for 3-RPR planar PM and 6 6-UPS spatial PM); g dimension of boxes. 2: Output: L in as the constant-orientation obstacle-free workspace of the mechanism, L out as boxes, which are outside the aforementioned workspace. for i from 1 to f do 6: for i from 1 to f do 9:

[B i ] = L(1) + (b i -b i-
[ρ i ] = A i -[B i ] ;
% Length of prismatic actuator ρ i 10:

[d i ] = Distance(Line(A i , [B i ]),
L in ←-[B 1 ] 17: else if (ρ max < [ρ 1 ] . . . ρ max < [ρ f ]) . . . ([ρ 1 ] < ρ min . . . [ρ f ] < ρ min ) . . . (d 1 < r P A 1 -[B 1 ] 2 > A 1 -P 2 ) . . . . . . (d f < r P A f -[B f ] 2 > A f -P 2 ) . .
L out ←-[B 1 ] 19: else if Size([B 1 ] > then 20: L(end -1, end) ←-Bisect([B 1 ]) % Bisect [B 1 ]

ShiftLeft(Empty(L(1)))

% Erase data of L( 1) and shift one cell to the left 23: end while P to the corresponding line of the ith actuator. Lines 12-14 are interval lines passing through two intervals. Line 15 is the general if-clause for which if all [ρ i ] are in the acceptable range and if all [d i ] distances from P to the collision lines are higher than r P , then the box under study will be a member of the obstacle-free workspace, L in . There is an extra condition to ascertain that if the distance from the box [b i ] to the fixed point A i is lower than the distance from P to A i , hence the box should be inside the obstacle-free workspace. On the other hand, if only one of the aforementioned criteria is violated, then the box will be moved to the outer boxes list, L out . In line 19, if the box under study is partially inside the obstacle-free workspace and at least one of its dimensions is still larger than the desired precision, then it will be bisected by the largest edge (line 20) and two new boxes will be added at the end of list L. In line 22, ShiftLeft(Empty(L(1))), the first column of list L will be erased and all other elements of L will be moved one cell back to fill the gap. In other words, in each loop of the while-clause one box from list L will be investigated. In line 22, it is known that the aforementioned box is either located inside, outside or on the boundaries of the desired workspace;

B 3 B 1 , φ = π/4
therefore, there is no need to keep the box and erasing it will help the computer to free up memory. The algorithm continues until the prescribed precision is reached.

Results for 3-RPR PM

So far, an interval-based method to obtain the obstacle-free workspace of planar PMs, for a 3-RPR planar PM as a case study, has been introduced and the results of obstacle-free workspace, by considering only the collisions of one limb with its environment, Fig. 3(b), and only considering the collision of the edges of the EE, Fig. 7(b), are depicted. The next step is to put together all obstacle-free workspaces of all limbs. Figure 8(a) represents the obstacle-free workspace of the mechanism for a prescribed orientation, φ = π/4, by considering the collisions of all limbs with the obstacle. It should be noted that in Fig. 8(a), the obstacle is only inside the collision space of the first limb and as it can be observed, for the second and third limbs, the obstacle is located outside the corresponding red boxes. The latter is due to the fact that the represented workspace is depicted in the fixed coordinate frame and since a constant orientation of the EE is considered, thus other limbs should be translated into the fixed coordinate frame via the transformation of the moving frame. For instance, in the case of the 3-RPR planar PM for φ = π/4, if we move the obstacle along the vector of -{l x cos(π/4), l y sin(π/4)} T , which is the negative of position of joint B 2 represented in the moving frame, the obstacle will be located in the red boxes caused by considering the collisions of the second limb and the obstacle. The final step to obtain the obstacle-free workspace is to merge the obtained results and intersect them, i.e., considering the limb collisions, Fig. 8(a), and all edges of the EE, Fig. 7(b), with the obstacle. The final result is illustrated in Fig. 8 and as it could be expected, the obstacle divides the workspace into three separate parts, which are not connected to each other. The obtained obstacle-free workspace can be used in obstacle avoidance problems. As it is obtained via interval analysis, for all paths, whose points are located inside the obstacle-free workspace, it is guaranteed that the paths are free of obstacle collision.

The proposed interval-based method to obtain the obstacle-free workspace for planar PMs can be readily extended to more complicated and spatial PMs. Indeed, one can solve the IKP and use the obtained equations to determine the interval position of all joints and define the aforementioned segment lines to obtain the distances to the obstacle. It takes approximately 3 min to compute the obstacle-free workspace of the 3-RPR planar PM shown in Fig. 1(a), precision of 10 -6 percent of initial search space, with a 2 GHz processor, using INTLAB 6 toolbox. In the case of higher DOF mechanisms having more links, collision computation of each limb will be added into the computational time. For the sake of simplicity, the workspace of the mechanisms studied in this paper is represented for a constant orientation. This simplification does not affect the generality of the method, because it can be done for different orientations and represent a higher dimension collision-free workspace (3D workspace in case of planar mechanism for which z-axis is the orientation of the EE about z-axis).

Results for a 6-UPS PM

In this section, the aforementioned method is applied to obtain the obstacle-free workspace of a 6-UPS spatial PM. The proposed method is free from dimensional restriction and it can be applied to PM with more than two DOF, by only paying the computational time cost. A 6-UPS spatial PM is shown in Fig. 9(a) and its geometric parameters are represented in Table II. As it can be observed from Fig. 9, a sphere obstacle, P , is located inside the workspace of the mechanism. The workspace of the mechanism is obtained by using interval analysis, regardless of the obstacle, Fig. 9(b). [START_REF] Merlet | Interval analysis and robotics[END_REF][START_REF] Oetomo | Certified Workspace Analysis of 3RRR Planar Parallel Flexure Mechanism[END_REF] The workspace is depicted considering mechanical stroke of prismatic joints and spherical and universal joint limits.

The main advantage of using this method is that it could be readily extended to 6-DOF; in fact, it is a feature of the proposed method. Algorithm 1 can be directly used to determine the obstacle-free workspace of the 6-UPS spatial PM. By choosing the number of DOF f and the dimension of the box g, the algorithm computes the collision-free workspace of the corresponding PM, i.e., 3-RPR or 6-UPS. In line 6, the positions of all vertexes of the EE are obtained. Other lines of the algorithm is the same as before, by having in mind that the command Line creates a spatial line in the current case. Such 3D interval lines are illustrated in Fig. 10. In the case of a spatial PM, corresponding interval lines are derived from 3D expressions, and the distance from line to the obstacle is expressed as follows:

d = (x 2 -x 1 ) × (x 1 -x 0 ) x 2 -x 1 , (9)
in which, x 1 and x 2 are two points on the line, in our case, positions of the U joint and S joint. Moreover, x 0 stands for the center point of the obstacle.

Considering the limited workspace of the 6-UPS spatial PM, only two limbs of the mechanism clash with the obstacle, which are A 12 B 16 and A 56 B 16 . By pursuing the same procedure, as introduced in the previous section for 3-RPR planar PM, the obtained workspace could be computed in which only the collisions of one limb with the obstacle is considered, Fig. 11. The workspace is shown in two views, namely, top view, Fig. 11(a), and sectioned isometric view, Fig. 11(b). In this case, a conical part of the workspace is removed.

The final workspace of the mechanism, while considering the collisions of one limb with the obstacle, is represented in Fig. 12. Since the workspace is obtained for a constant orientation and the workspace of the 6-UPS spatial PM is very limited, the moving platform will not collide with the obstacle. The workspace is in a constant orientation because there is no graphical representation for a workspace in a dimension higher than three. In Fig. 12, two views of the obstacle-free workspace is represented, which are top view, Fig. 12(a), and sectioned isometric view, Fig. 12(b). In the case that the obstacle is bigger than the one considered here, other limbs may collide with it and there will be more subtracted conical area in the workspace.

Computational time and required memory of the process is highly dependent of the desired precision. It can be improved by choosing a proper initial search box to avoid unnecessary computation of the area outside of the workspace. Moreover, a symmetric and simple design of the robot may lead to a lower computing time. For this case study, using a 2 GHz CPU and 4 GB of ram and INTLAB toolbox in Matlab, it took almost three minutes to obtain the final result.

Conclusion

In this paper, an interval-based methodology was introduced in order to obtain the obstacle-free workspace of two Parallel Manipulators (PMs), namely a 3-RPR PM and 6-DOF Gough-Stewart platform. The proposed approach for the obstacle-free workspace and the corresponding results were represented. First, the collisions of proximal segment lines which pass through one point and one interval with the obstacle were investigated. Then a more general case in which the segment line passes through two intervals was used to determine the collisions between the obstacle and medial and distal limbs and also with edges of the EE. Indeed, the collision-free workspace of a 6-UPS spatial parallel manipulator was traced by using the proposed approach. To the best of authors' knowledge, this paper can be regarded as the first study on the collision-free workspace of parallel manipulator using interval analysis and opens an avenue to extend it to more complex parallel manipulators. However, the proposed approach in this paper could be well extended to parallel manipulator with one prismatic joint connecting the base to the EE directly. In fact, due to the limitation of interval analysis, although it is very reliable, it is a computationally intensive approach and in the case of high degree and complex equations, it may lead to very high computing time. Ongoing works deal with the collision detection of limbs between themselves and EE which is an important issue for the determination of the collision-free workspace of cable-driven parallel robots.

  [x] [y] = [min(S) , max(S)] , S = {xy, xy, xy, xy}.

Fig. 1 .

 1 Fig. 1. (a) Schematic representation of a 3-RPR planar PM and a circle-shaped obstacle located at point P , ρ min = 5 and ρ max = 50. and (b) the constant-orientation workspace of a 3-RPR planar PM for φ = π/4 via interval analysis, collisions are ignored. A 1 = {-10, -5}, A 2 = {50, -5} and A 3 = {15, 40}; x P = 20, y P = 0 and r P = 3; l x = l y = 10. Green boxes are inside and red boxes are outside the workspace.

Fig. 2 .

 2 Fig. 2. Two types of interval lines. In both cases, there is no collision if the obstacle is located at either point P 1 or point P 2 . (a) Interval line passing through one point, A 1 , and one box, [B 1 ]; link A 1 B 1 in Fig. 1(a). (b) Interval line passing through two boxes, [B 1 ] and [B 2 ]; edge B 1 B 2 of the EE in Fig. 1(a).

Fig. 3 .

 3 Fig. 3. The obstacle-free workspace of a 3-RPR planar PM shown in Fig. 1(a), for φ = 0, only considering the first limb A 1 B 1 . Green boxes are inside and red boxes are outside the obstacle-free workspace. (a) A 1 B 1 as a line satisfying [d] > r P . (b) A 1 B 1 as a segment line satisfying [d] > r P and A -[B] 2 < A -P 2 .

Fig. 4 .

 4 Fig. 4. The obstacle-free workspace, considering only edge B 1 B 2 of EE, for φ = π/4. The result is undesirable due to interval blow up.

Fig. 5 .

 5 Fig. 5. The obstacle-free workspace, considering only edge B 1 B 2 of EE, for φ = π/4. This is a modified result of Fig. 4. A better solution is represented in Fig. 7(a).

3 :

 3 L(1) = [B] g % Position of the EE in g dimensional space; 4: while IsEmpty(L) = 1 do 5:

Fig. 7 .

 7 Fig. 7. The obstacle-free workspace of the mechanism shown in Fig. 1(a) by considering interval segment lines passing through two intervals. Green boxes are inside and red boxes are outside the obstacle-free workspace. (a) The case for which only B 1 B 2 collides with the obstacle, φ = 0. (b) Regarding all edges of the EE, i.e. B 1 B 2 , B 2 B 3 and B 3 B 1 , φ = π/4

Fig. 8 .

 8 Fig. 8. The obstacle-free workspace of the 3-RPR planar PM shown in Fig. 1, for φ = π/4, (a) considering limb collision only (b) final result considering all collisions. Green boxes are inside and red boxes are outside the obstacle-free workspace, respectively. (a) Only limbs collisions are taken into account, φ = π/4. (b) Intersection of all obstacle-free workspaces including all limbs and all edges of the EE.

Fig. 9 .

 9 Fig. 9. (a) 6-UPS spatial PM and (b) the corresponding workspace considering actuation stroke, via interval analysis. P = [100, 0, 500], r P = 10. (a) A 6-UPS spatial PM and a sphere obstacle P . 19 (b) The sliced constant-orientation workspace of a 6-UPS spatial PM, via interval analysis.

Fig. 10 .

 10 Fig. 10. Two types of interval lines in case of 3D workspace. (a) Interval line passing through one point, A 1 , and one 3D box, [B 1 ]; link A 12 B 16 in Fig. 9(a). (b) Interval line passing through two 3D boxes, [B 1 ] and [B 2 ]; edge B 16 B 23 of the EE in Fig. 9(a).

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Obstacle-free workspace of a 6-UPS Spatial PM, considering the collisions of one limb with the obstacle. (a) Top view (b) Sectioned isometric view.

Table I .

 I Geometric parameters of a 3-RPR PM.

	i	x Ai	y Ai	ρ min	ρ max	x B i	y B i	unit
	1	-10	-5	5	50	0	0	cm
	2	50	-5	5	50	10	0	cm
	3	15	40	5	50	0	10	cm

  1 ) O xy ; % Position of box [B i ] in the fixed frame O xy

	7:	end for
	8:

  P ); % Distance from P to the line passing through A i and [B i ]

	11:	end for
	12:	for i from 1 to f do
	13:	[t i ] = Distance(Line([B i ], [B i+1 ]), P );
	14:	end for
	15:

if ρ min < [ρ 1,...,f ] < ρ max (r P < d 1,...,f A 1,...,f -[B 1,...,f ] 2 < A 1,...,f -P 2 )

r P < t 1,...,f then % • stands for supreme of and interval 16:

  by the largest edge and add two new boxes at the end of L

	21:	end if
	22:

Table II .

 II Geometric parameters of the 6-UPS Spatial PM under study (all lengths are given in mm).

	i	1	2	3	4	5	6
	x ai	92.58	132.58	40.00	-40.00	-132.58	-92.58
	y ai	99.64	30.36	-130.00	-130.00	30.36	99.64
	z ai	23.10	23.10	23.10	23.10	23.10	23.10
	x bi	30.00	78.22	48.22	-48.22	-78.22	-30.00
	y bi	73.00	-10.52	-62.48	-62.48	-10.52	73.00
	z bi	-37.10	-37.10	-37.10	-37.10	-37.10	-37.10
	ρ imin	454.5	454.5	454.5	454.5	454.5	454.5
	ρ imax	504.5	504.5	504.5	504.5	504.5	504.5


Frédéric Eyi-Minko

Clément Dombry
email: clement.dombry@univ-fcomte.fr

Extremes of independent stochastic processes: a point process approach

Keywords: extreme value theory, partial maxima process, superextremal process, functional regular variations, weak convergence. AMS Subject classification. Primary: 60G70 Secondary: 60F17

For each n ≥ 1, let {X in , i 1} be independent copies of a nonnegative continuous stochastic process X n = (X n (s)) s∈S indexed by a compact metric space S. We are interested in the process of partial maxima

where the brackets [ • ] denote the integer part. Under a regular variation condition on the sequence of processes X n , we prove that the partial maxima process Mn weakly converges to a superextremal process M as n → ∞. We use a point process approach based on the convergence of empirical measures. Properties of the limit process are investigated: we characterize its finite-dimensional distributions, prove that it satisfies an homogeneous Markov property, and show in some cases that it is max-stable and self-similar. Convergence of further order statistics is also considered. We illustrate our results on the class of log-normal processes in connection with some recent results on the extremes of Gaussian processes established by Kabluchko.

Introduction

A classical problem in extreme value theory (EVT) is to determine the asymptotic behavior of the maximum of independent and identically distributed (i.i.d.) random variables (Z i ) i≥1 . What are the assumptions that ensure the weak convergence of the rescaled maximum max

1≤i≤n Z i -b n a n , a n > 0, b n ∈ R,
and what are the possible limit distributions? These questions were at the basis of the development of EVT and found their answers in the Theorem by Fisher and Tippett [START_REF] Fisher | Limiting forms of the frequency of the largest or smallest member of a sample[END_REF] characterizing all the max-stable distributions and in the description of their domain of attraction by Gnedenko [START_REF] Gnedenko | Sur la distribution limite du terme maximum d'une série aléatoire[END_REF] and de Haan [START_REF] De Haan | On regular variation and its application to the weak convergence of sample extremes[END_REF]. Another stimulating point of view developped by Lamperti [START_REF] Lamperti | On extreme order statistics[END_REF] is to introduce a time variable and consider the asymptotic behavior of the partial maxima

max 1≤i≤[nt] Z i -b n a n , t ≥ 0.
The corresponding limit process is known as an extremal process: it is a pure jump Markov process which is also max-stable, see Resnick [START_REF] Resnick | Weak convergence to extremal processes[END_REF].

Since then, EVT has known many developments. Among several other directions, the extension of the theory to multivariate and spatial settings is particularly important, as well as the statistical issues raised by the applications on real data sets. For excellent reviews of such developments, the reader is invited to refer to the monographies by Resnick [START_REF] Resnick | Extreme values, regular variation and point processes[END_REF], de Haan and Fereira [START_REF] De Haan | Extreme value theory[END_REF] or Beirlant et al [START_REF] Beirlant | Statistics of extremes[END_REF] and the references therein.

Our purpose here is to focus on the functional framework and investigate the asymptotic behavior of the partial maxima processes based on a doubly infinite array of independent random processes. Let S be a compact metric space and, for each n ≥ 1, let {X in , i 1} be independent copies of a sample continuous stochastic process (X n (s)) s∈S . Without loss of generality, we will always suppose that X n is non-negative (otherwise consider X n (s) = e Xn(s) ) and we denote by C + = C + (S) be the set of non-negative continuous functions on S. We are mainly interested in the process of pointwise maxima

M n (s) = max{X in (s), 1 i n}, s ∈ S, (1)
and the process of partial maxima

Mn (t, s) = max{X in (s), 1 i [nt]}, t ≥ 0, s ∈ S. (2)
We use the convention that the maximum over an empty set is equal to 0, so Mn (0, s) ≡ 0.

Clearly, we also have Mn (1, s) = M n (s). In this framework, the parameter s ∈ S is thought as a space parameter and t ∈ [0, +∞) as a time variable.

Our approach relies on the convergence of the following empirical measures

β n = n i=1 δ X in and βn = i≥1 δ (X in ,i/n) (3)
on C + and C + ×[0, +∞) respectively, where δ a denotes the Dirac measure at point a. The maxima processes M n and Mn can be written as functionals of the empirical measure β n and βn respectively. We will prove the continuity of the underlying functionals and then use the Continuous Mapping Theorem to deduce convergence of the maxima processes from convergence of empirical measures.

Connections between EVT and point processes are well known. In the case when the underlying state space is locally compact, the following result holds (see Proposition 3.21 in [START_REF] Resnick | Extreme values, regular variation and point processes[END_REF]): if nP[X n ∈ • ] vaguely converges to some measure µ, the empirical measures β n and βn converge to Poisson random measures with intensity µ and µ ⊗ respectively, being the Lebesgue measure on [0, +∞). However, in our framework the state space C + is not locally compact and we need a suitable generalization of the above result. To this aim, we follow the approach by Davis and Mikosch [START_REF] Davis | Extreme value theory for space-time processes with heavy-tailed distributions[END_REF] based on the notion of boundedly finite measures and -weak convergence detailed by Daley and Vere-Jones in [START_REF] Daley | An introduction to the theory of point processes[END_REF].

The paper is organized as follows. In section 2, we introduce the technical material needed on boundedly finite measures, -weak convergence and convergence of empirical measures. Our main result is the convergence of the partial maxima process Mn which is stated and proved in Section 3. Then, in Section 4, we investigate some properties of the limit process M , known as a superextremal process. A brief extension of our results to further order statistics is considered in Section 5. The last section is devoted to an application of our results to the class of log-normal processes, based on a recent work by Kabluchko [START_REF] Kabluchko | Extremes of independent Gaussian processes[END_REF] on the extremes of Gaussian processes.

Preliminaries on boundedly finite measures and point processes

In this section, we present general results on boundedly finite measures and point processes that will be useful in the sequel. The reader should refer to Appendix 2.6 in Daley and Vere-Jones [START_REF] Daley | An introduction to the theory of point processes[END_REF] or to section 2 of Davis and Mikosch [START_REF] Davis | Extreme value theory for space-time processes with heavy-tailed distributions[END_REF]. This has also close connections with the theory of regular variation, see Hult and Lindskog [START_REF] Hult | Extremal behavior of regularly varying stochastic processes[END_REF][START_REF] Hult | Regular variation for measures on metric spaces[END_REF].

Boundedly finite measures

Let (E, d) be a complete separable metric space and E the Borel σ-algebra of E. We denote by M b (E) the set of all finite measures on (E, E). A sequence of finite measures {µ n , n 1} is said to converge weakly to µ ∈ M b (E) if and only if f dµ n → f dµ for all bounded continuous function f on E. We write µ n w → µ to denote weak convergence. It is well known that this notion of convergence is metrized by the Prokhorov metric

p(µ 1 , µ 2 ) = inf{ε > 0; ∀A ∈ E, µ 1 (A) µ 2 (A ε ) + ε and µ 2 (A) µ 1 (A ε ) + ε} where A ε = {x ∈ E; ∃a ∈ A, d(a, x) < ε} is the ε-neighborhood of A. Endowed with this metric, M b (E) is a complete separable metric space.
A measure µ on (E, E) is called boundedly finite if it assigns finite measure to bounded sets, i.e. µ(B) < ∞ for all bounded B ∈ E. Let M b (E) be the set of all boundedly finite measures µ on (E, E). A sequence of boundedly finite measures (µ n ) n≥1 is said to converge -weakly to µ ∈ M b (E) if and only if f dµ n → f dµ for all bounded continuous function f on E with bounded support. There exists a metric p on M b (E) that makes M b (E) a complete and separable metric space and such that convergence in the metric sense coincides with -weak convergence. Such a metric can be constructed as follows. Fix an origin e 0 ∈ E and, for r > 0, let Br = {x ∈ E; d(x, e 0 ) ≤ r} be the closed ball of center e 0 and radius r. For any µ 1 , µ 2 ∈ M b (E), let µ

p (µ 1 , µ 2 ) = ∞ 0 e -r p r (µ (r) 1 , µ (r) 2 ) 1 + p r (µ (r) 1 , µ (r) 2 )
dr.

There are several equivalent characterizations of -weak convergence, see Daley and Vere Jones [START_REF] Daley | An introduction to the theory of point processes[END_REF] Proposition A2.6.II. Let (µ n ) n≥1 and µ be boundedly finite measures on E, the following statements are equivalent: i ) f dµ n → f dµ for all f bounded continuous real valued function on E with bounded support;

ii ) p (µ n , µ) → 0;

iii ) there exist a sequence r k +∞ such that, for all k 1, µ

(r k ) n w → µ (r k ) ; iv ) µ n (B) → µ(B) for all bounded B ∈ E such that µ(∂B) = 0.
We write µ n w → µ to denote -weak convergence.

A +∞ such that for all k ≥ 1, there exist N 0 and m 0 such that

∀n N, µ (r k ) n = m i=1 δ x n,k i and µ (r k ) = m i=1 δ x k i for some {x n,k i , x k i ; 1 ≤ i ≤ m, n ≥ N } in Br k such that x n,k i → x k i , where µ (r k ) n
and µ (r k ) are the restrictions of µ n and µ to Br k and the x n,k i 's and x k i 's are suitable relabellings of the points falling in Br k .

Convergence of the empirical measures

Let (Ω, F, P) be a probability space. A boundedly finite point process on E is a measurable mapping N : Ω → M (b,p) (E). A typical example of boundedly finite point process on E is a Poisson point process Π ν with intensity ν ∈ M b (E). We will also consider the following empirical point processes β n and βn defined by Equation ( 3), where, for each n ≥ 1, {X in , i ≥ 1} are independent copies of an E-valued random variable X n . The random variable β n is a finite point process on E, while βn is a boundedly finite point process on Ẽ = E × [0, +∞) endowed with the metric

d((x 1 , u 1 ), (x 2 , u 2 )) = d(x 1 , x 2 ) + |u 2 -u 1 |, x 1 , x 2 ∈ E, u 1 , u 2 ∈ [0, +∞).
The following Proposition will play a key role in the sequel.

Proposition 2.2. The following statements are equivalent, where the limits are taken as n → ∞: iii ) βn ⇒ Πν with Πν a Poisson point process on E ×[0, +∞) with intensity ν ⊗ , being the Lebesgue measure on [0, +∞), and

i ) nP[X n ∈ • ] w → ν; ii ) β n d → Π ν with Π ν a
d → standing for convergence in distribution in M (b,p) (E × [0, +∞)).
The proof of this Proposition follows by an adaptation of Proposition 3.21 in Resnick [START_REF] Resnick | Extreme values, regular variation and point processes[END_REF] which states a similar result in the case of a locally compact state space and in terms of vague convergence. As noticed by Davis and Mikosch [START_REF] Davis | Extreme value theory for space-time processes with heavy-tailed distributions[END_REF], the proof remains valid for a complete separable metric space E if we change vague convergence by -weak convergence. See also Theorem 4.3 in Davydov, Molchanov and Zuyev [START_REF] Davydov | Strictly stable distributions on convex cones[END_REF].

In the terminology of Hult and Lindskog [START_REF] Hult | Extremal behavior of regularly varying stochastic processes[END_REF][START_REF] Hult | Regular variation for measures on metric spaces[END_REF], the condition i) in Proposition 2.2 means that the sequence X n is regularly varying. A particularly important case is when E is endowed with a strucure of cone, i.e. a multiplication by positive scalars. If there exists a sequence (a n ) n≥1 of positive reals such that

nP[a -1 n X ∈ • ] w → ν
for some nonzero ν ∈ M b (E), then the random variable X is said to be regularly varying.

Under some technical assumptions on the structure of the convex cone E (e.g. continuity properties of the multiplication), the limit measure ν is proved to be homogeneous of order -α < 0, i.e.

ν(λA) = λ -α ν(A), λ > 0, A ∈ E.
Furthermore, the sequence a n is regularly varying with index 1/α. In this framework, if {X i , i ≥ 1} are i.i.d. copies of a regularly varying random variables X, then the triangular array

X in = a -1 n X i satisfies condition i) since nP[X 1n ∈ • ] = nP[a -1 n X ∈ • ] w → ν.

Extremes of independent stochastic processes

We go back to our original problem where the X in 's are non negative continuous processes on S. We endow the set C + = C + (S) of non negative continuous functions on S with the uniform norm x = sup s∈S |x(s)|, x ∈ C + . Recall that S is assumed to be a compact metric space and hence (C + , • ) is a complete separable metric space. It turns out that, when working with maxima, the uniform metric is not adapted, mainly because sets such as {x ∈ C + ; x ≥ ε} are not bounded for this metric. For this reason, following the seminal work of de Haan and Lin [START_REF] De | On convergence toward an extreme value distribution in C[0, 1[END_REF], we introduce C + 0 = (0, +∞] × S C + where S C + = {x ∈ C + : x ∞ = 1} is the unit sphere. We define the metric

d((r 1 , s 1 ), (r 2 , s 2 )) = |1/r 1 -1/r 2 | + s 1 -s 2 , (r 1 , s 1 ), (r 2 , s 2 ) ∈ C + 0 .
The metric space (C + 0 , d) is complete and separable as the product of the complete and separable metric spaces (0, +∞] (with distance |1/r 1 -1/r 2 |) and S C + . The polar decomposition defined on

C + 0 = C + \ {0} by T : C + 0 → (0, ∞) × S C + x → ( x , x/ x )
.

is an homeomorphism and we identify in the sequel C + 0 and (0, +∞) × S C + . In this metric, a subset B of C + 0 is bounded if and only if it is bounded away from zero, i.e. included in a set of the form {x ∈ C + ; x ≥ ε} for some ε > 0.

Spatial maximum process

For the sake of clarity, we present first our results for the spatial maximum process M n defined by Equation [START_REF] Beirlant | Statistics of extremes[END_REF]. The following theorem is a natural extension of Theorem 2.4 from de Haan and Lin [START_REF] De | On convergence toward an extreme value distribution in C[0, 1[END_REF] where X in = n -1 X i is obtained from an i.i.d. sequence of continuous (or càd-làg) processes (X i ) i≥1 and the limit maximum process M is simple max-stable. In our triangular array setting, we obtain in the limit a max-infinitely divisible process. We recall that a random element ξ of C(S) is called max-infinitely divisible if for every n ≥ 1, there exists independent and identically distributed elements ξ 1n , . . . , ξ nn in C(S) such that ξ Furthermore, M is a max-infinitely divisible process with exponent measure ν, i.e. it admits the representation M (s)

+ (S); ii) nP[X n ∈ • ] w → ν in M b (C + 0 ) with ν(C + 0 \ C + 0 ) = 0; iii) β n d → Π ν with Π ν a
d = sup i≥1 Y i (s), s ∈ S, (5)
where i≥1 δ Y i is a Poisson point process on C + 0 with intensity ν, and the exponent measure ν is uniquely determined by the relations

P[M (s 1 ) ≤ y 1 , . . . , M (s k ) ≤ y k ] = exp -ν{f ∈ C + 0 ; f (s i ) > y i for some i = 1, . . . , k} (6)
for all k ≥ 1, s 1 , . . . , s k ∈ S and y 1 , . . . , y k > 0.

Remark 3.2. The normalisation condition (4) might be stronger than necessary but is required in the proof. Since Remark 3.3. Working on the space C(S) of continuous functions is important in Theorem 3.1. The case of the space of càd-làg functions D[0, 1] endowed with the Skorokhod J 1 topology is studied by Gentric [START_REF] Gentric | Convergence of the normalized maximum of regularly varying random functions in the space D[END_REF]. He shows that regular variation does not always imply convergence of the partial maxima process. This is related to the fact that the operation (f, g) → max(f, g) is not continuous on D[0, 1]. We believe that the good framework to work with non-continuous process is the space of semi-continuous functions as is the seminal work by Giné et al. [START_REF] Giné | Max-infinitely divisible and max-stable sample continuous processes[END_REF]. For the sake of simplicity, we do not consider this general framework here.

P[ M n ≤ y] = P[ X n ≤ y] n , condition (4)
For the proof of Theorem 3.1, note that the equivalence ii)⇔ iii) is a straightforward application of Proposition 2.2. For the implication i)⇒ ii), we follows the lines of the proof of Theorem 2.4 in de Haan and Lin [START_REF] De | On convergence toward an extreme value distribution in C[0, 1[END_REF]. Finally, for the implication ii)⇒ i) we will need the following Lemma. With a slight abuse of notation, we define Proof of i)⇒ ii): this is a direct adaptation of the proof of Theorem 2.4 in de Haan and Lin [START_REF] De | On convergence toward an extreme value distribution in C[0, 1[END_REF] and we give only the main lines. We use the notation ν n = nP[X n ∈ •] and we want to prove ν n converges in M b (C + 0 ). The starting point is the convergence M n d → M which entails for all k ≥ 1, s 1 , . . . , s k ∈ S and y 1 , . . . , y k > 0 continuity points

M (b,p) (C + 0 ) = {µ ∈ M (b,p) (C + 0 ) : µ(C + 0 \ C + 0 ) = 0}. Note M (b,p) (C + 0 ) is an open subset of M (b,p) (C + 0 ).
P[M n (s 1 ) ≤ y 1 , . . . , M n (s k ) ≤ y k ] → P[M (s 1 ) ≤ y 1 , . . . , M (s k ) ≤ y k ]. (7)
Setting A s 1 ,y 1 ,...,s k ,y k = {f ∈ C + 0 ; f (s i ) ≤ y i , i = 1, . . . , k}, we have

P[M n (s 1 ) ≤ y 1 , . . . , M n (s k ) ≤ y k ] = P[X n (s 1 ) ≤ y 1 , . . . , X n (s k ) ≤ y k ] n = 1 - 1 n ν n (A c s 1 ,y 1 ,...,s k ,y k ) n
and we deduce from (7) that

lim n→∞ ν n (A c s 1 ,y 1 ,...,s k ,y k ) = lim n→+∞ -log P[M n (s 1 ) ≤ y 1 , . . . , M n (s k ) ≤ y k ] = -log P[M (s 1 ) ≤ y 1 , . . . , M (s k ) ≤ y k ] := ν(A c s 1
,y 1 ,...,s k ,y k ). This last quantity is finite thanks to the normalisation condition (4). This shows that if the limit ν exists, it is necessarily given by (6). To prove the existence of the limit, one needs to show that the sequence ν n is tight in M b (C + 0 ). This implies that the relations (4) define a measure ν ∈ M b (C + 0 ) and that ν n w → ν as n → ∞.

Tightness of the sequence ν n is the purpose of Lemma 3.2 in de Haan and Lin [START_REF] De | On convergence toward an extreme value distribution in C[0, 1[END_REF]. The proof there assumes that M n d → M in Skohorod space with the limit M simple max-stable. Here we consider convergence in C + (S) (which simplifies the proof thanks to simpler tightness criterion) but we do not assume max-stability of the limit. A close inspection of the proof reveals that max-stability is not needed but only that -log P( M ≤ y) < ∞ for all y > 0 and that is the reason why we require the normalization condition (4).

Proof of Lemma 3.4:

• First, we show that the mapping θ is well defined. This is not obvious since the pointwise supremum of a countable family of functions is not necessarily finite nor continuous.

Let µ ∈ M (b,p) (C + 0 ). For ε > 0, define µ (ε) as the restriction of µ to the set Bε = [ε, ∞]×S C and θ ε = θ(µ (ε) ). Since Bε is bounded for the metric d, the point measure µ (ε) has only a finite number of atoms and therefore θ(µ (ε) ) is the maximum of a finite number of non negative continuous functions and is hence a non negative continuous function.

Furthermore, for all s ∈ S, |θ(µ (ε) )(s) -θ(µ)(s)| ε. So θ(µ) is the uniform limit as ε → 0 of the continuous functions θ ε (µ), and hence θ(µ) ∈ C + and satisfies

θ(µ (ε) ) -θ(µ) ε. (8)
• Second, we show that the mapping θ is continuous.

Let {µ n = i∈In δ x n i , n 1} be a sequence of measures in M (b,p) (C + 0 ) converging to µ = i∈I δ x i . For all ε > 0, there exists ε < ε, N 0 and m 0 such that

∀n N, µ (ε ) n = m i=1 δ x nε i and µ (ε ) = m i=1 δ x ε i Lemma 3.6. The mapping θ : M (b,p) (C + 0 ×[0, +∞)) → D([0, +∞), C + ) defined by θ(0) ≡ 0 and θ i∈I δ (x i ,t i ) = (t, s) → sup{x i (s)1 [t i ,+∞) (t); i ∈ I}
is well defined and continuous on C.

Proof of Lemma 3.6:

The proof is similar to the proof of Lemma 3.4 and we give only the main lines.

• First we show that θ is well defined. Recall that

C + 0 × [0, +∞) is endowed with the metric d((r 1 , s 1 , t 1 ), (r 2 , s 2 , t 2 )) = d((r 1 , s 1 ), (r 2 , s 2 )) + |t 2 -t 1 |. Let µ ∈ M (b,p) (C + 0 × [0, +∞)). For ε > 0 and M > 0, let µ (ε,M ) be its restriction to Bε,M = [ε, +∞] × S C + × [0, M ].
Since Bε,M is bounded, µ (ε,M ) has only a finite number of atoms and we easily check that θ(µ (ε,M ) ) belongs to D([0, +∞), C + ). Furthermore, for t < M and s ∈ S,

| θ(µ (ε,M ) )(t, s) -θ(µ)(t, s)| ≤ ε so that θ(µ (ε,M ) ) converges uniformly on [0, M ] × T to θ(µ) as ε → 0.
The constant M being arbitrary, this implies that θ(µ) ∈ D([0, +∞), C + ) and that the application θ is well defined. It also holds that

sup (t,s)∈[0,M ]×S | θ(µ (ε,M ) )(t, s) -θ(µ)(t, s)| ≤ ε. ( 10
)
• Next, we show that θ is continuous on C.

Let {µ n = i∈In δ (x n i ,t n i ) ; n 1} be a sequence of M (b,p) (C + 0 × [0, +∞)) converging to µ = i∈I δ (x i ,t i ) ∈ C. For all ε > 0 and M > 0, there exist ε < ε, M > M , N ≥ 1 and some m ≥ 1, such that ∀n N, µ (ε ,M ) n = m i=1 δ (x n i ,t n i ) and µ (ε ,M ) = m i=1 δ (x i ,t i ) with x n i → x i and t n i → t i as n → ∞.
The condition µ ∈ C ensures that the t i 's are pairwise distinct and we can suppose without loss of generality that t 1 < • • • < t m . We then also have, for large enough n, t n

1 < • • • < t n m . Define δ M the metric associated with the J 1 -topology on D([0, M ], C + ) by δ M (x, y) = inf λ sup (t,s)∈[0,M ]×S |x(λt, s) -x(t, s)|
where the infinimum is taken over the set of non-decreasing homeomorphisms λ of [0, M ]. Since for large n the t i 's and the t n i 's are in the same relative order, there exists a non-decreasing homeomorphism

λ n M of [0, M ] such that λ n M (t n i ) = t i for all 1 ≤ i ≤ m. We then have δ M ( θ(µ (ε ,M ) n ), θ(µ (ε ,M ) )) ≤ δ M ( θ(µ n )), θ(µ (ε ,M ) n )) + δ M ( θ(µ (ε ,M ) n ), θ(µ (ε ,M ) )) + δ M ( θ(µ (ε ,M ) ), θ(µ)) ≤ sup (t,s)∈[0,M ]×S | max 1≤i≤m (x n i (•)1 [λ n M t n i ,+∞) (t)) -max 1≤i≤m (x i (•)1 [t i ,+∞) (t))| = max 1≤i≤m x n i -x i → 0 as n → ∞.
Hence, for sufficiently large n, δ M ( θ(µ

(ε ,M ) n
), θ(µ (ε ,M ) )) ≤ ε and thanks to Equation [START_REF] Dwass | Extremal processes[END_REF], this entails

δ M ( θ(µ n )), θ(µ)) ≤ δ M ( θ(µ n )), θ(µ (ε ,M ) n )) + δ M ( θ(µ (ε ,M ) n ), θ(µ (ε ,M ) )) + δ M ( θ(µ (ε ,M ) ), θ(µ)) ≤ 3ε.
Since M is arbitrary large and ε arbitrary small, this proves the convergence θ(µ n ) to θ(µ) in D([0, +∞), C + ).

Proof of Theorem 3.5 :

The proof is very similar to the proof of Theorem 3.1. Note that Mn = θ( βn ) where βn is defined by Equation (3). According to Proposition 2.2, the empirical measure βn weakly converges in M (b,p) (C + 0 × [0, +∞)) to a Poisson point process Πν with intensity ν ⊗ . The assumption ν(C + 0 \ C + 0 ) and the fact that the Lebesgue measure has no atoms ensure that Πµ lies almost surely in C. From Lemma 3.6, the mapping θ is continuous on C, so that the Continuous Mapping Theorem implies θ( βn ) ⇒ θ( Πν ) which is equivalent to Mn ⇒ M .

Properties of the limit process

In this section, we give some properties of the limiting process M defined by Equation ( 9) in Theorem 3.5. These results extend the classical theory on extremal processes (see Dwass [START_REF] Dwass | Extremal processes[END_REF][START_REF] Dwass | Extremal processes[END_REF][START_REF] Dwass | Extremal processes[END_REF], Tiago de Oliveira [START_REF] Tiago De Oliveira | Extremal processes: Definition and properties[END_REF], Resnick and Rubinovitch [START_REF] Resnick | The structure of extremal processes[END_REF], Resnick [START_REF] Resnick | Weak convergence to extremal processes[END_REF] ...) from a univariate or multivariate setting to a functional framework. Note also the work by Resnick and Roy [START_REF] Resnick | Superextremal processes, max-stability and dynamic continuous choice[END_REF] in a functionnal setting with applications to the dynamic continuous choice problem.

First, we characterize the finite dimensional distributions of M . We use vectorial notations: for l ≥ 1, s = (s 1 , . . . , s l ) ∈ S l , y = (y 1 , . . . , y l ) ∈ [0, +∞) l and t > 0, we write M (t, s) ≤ y if and only if M (t, s i ) ≤ y i for all i ∈ {1, . . . , l}.

Proposition 4.1. For k 1, 0 = t 0 < t 1 < • • • < t k , s ∈ S l and y 1 , . . . , y k ∈ [0, +∞) l , it holds P M (t 1 , s) y 1 , ..., M (t k , s) y k = k i=1 exp[-(t j -t j-1 )ν(A j )] with A j = {f ∈ C + 0 ; ∃i ∈ 1, . . . , l, f (s i ) ≥ min k≥j y k,i }.
Proof of Proposition 4.1 : Let j ∈ {1, . . . , k}. Note that M (t j , s) y j if and only if Πν doesn't intersect the set

B j = {(f, t) ∈ C + 0 × [0, +∞); t ≤ t j and ∃i ∈ {1, • • • l} f (s i ) > y j,i }
So, we have

P M (t 1 , s) y 1 , ..., M (t k , s) y k = P Πν ∩ (∪ k j=1 B j ) = ∅] = exp[-(ν ⊗ )(∪ k j=1 B j )].
The B j 's are not pairwise disjoint. To compute the measure

(ν ⊗ )(∪ k j=1 B j ), we observe that ∪ k j=1 B j = ∪ k j=1 ([t j-1 , t j ) × A j ) ∪ ({t k } × A k )
where the sets in the right hand side are pairwise disjoint. From this, we deduce

(ν ⊗ )(∪ k j=1 B j ) = k j=1 (t j -t j-1 )ν(A j ).
This proves the Proposition.

Next we prove that the process t → M (t, •) is a C + (S)-valued homogeneous Markov process and consider its sample path properties. Let F t be the σ-algebra generated by { M (t , s); t ∈ [0, t], s ∈ S}. The symbol ∨ stands for pointwise maximum.

In the particular case when the measure ν is homogeneous of order -α < 0, the process M enjoys further interesting properties. Proposition 4.3. Suppose ν is homogeneous with index -α < 0. Then:

• M is max-stable with index α > 0, i.e. if M 1 , . . . , M n are independent copies of M , the maximum ∨ n i=1 M i has the same law as n 1/α M ;

• M is self-similar with index 1/α, i.e. for all c > 0, the rescaled process ( M (ct, •)) t≥0 has the same distribution as (c 1/α M (ct, •)) t≥0

Proof of Proposition 4.3 :

We check that the the two processes have the same finite dimensional distributions. To this aim, we use Proposition 4.1 and the notations of the proposition.

• Let M 1 , . . . , M n be independent copies of M . By Proposition 4.1 and the homogeneity of ν,

P ∨ n i=1
M i (t j , s) y j , ∀j ∈ {1, . . . , k} = P M (t j , s) y j , ∀j ∈ {1, . . . , k}

n = k i=1 exp[-(t j -t j-1 )ν(A j )] n = k i=1
exp[-(t j -t j-1 )ν(n -1/α A j )] = P n 1/α M (t j , s) y j , ∀j ∈ {1, . . . , k} .

This proves the max-stability.

• Similarly, the self-similarity is proven as follows:

P M (ct j , s) y j , ∀j ∈ {1, . . . , k} = k i=1 exp[-c(t j -t j-1 )ν(A j )] = k i=1 exp[-(t j -t j-1 )ν(c -1/α A j )]
= P c 1/α M (t j , s) y j , ∀j ∈ {1, . . . , k} .

An application to maxima of Gaussian processes

In geostatistics, Gaussian processes are often used as spatial models. In the finitedimensional setting, the asymptotic behavior of the maxima of Gaussian random vectors was first investigated by Huesler and Reiss [START_REF] Hüsler | Maxima of normal random vectors: between independence and complete dependence[END_REF]. Recently, Kabluchko, Schlather and de Haan [START_REF] Kabluchko | Stationary max-stable fields associated to negative definite functions[END_REF] and Kabluchko [START_REF] Kabluchko | Extremes of independent Gaussian processes[END_REF] consider the functional setting and prove convergence of the maxima of i.i.d. centered Gaussian processes under suitable conditions on their covariance structure. The family of Brown-Resnick processes has quickly become a popular model for spatial extremes due to its simple characterization by a negative definite functions.

Our purpose here is to revisit their results and apply the present framework: we put the emphasis on regular variations and point processes. In the sequel, we follow the approach by Kabluchko, see Theorems 2 and 6 in [START_REF] Kabluchko | Extremes of independent Gaussian processes[END_REF].

We suppose that (S, d) is a compact metric space satisfying the entropy condition

1 0 (log N (ε)) 1/2 dε < ∞,
where N (ε) is the smallest number of balls of radius ε needed to cover S.

Let Z n = {Z n (s); s ∈ S}, n ≥ 1, be a sequence of continuous zero-mean unit-variance Gaussian processes with covariance function

r n (s 1 , s 2 ) = E[Z n (s 1 )Z n (s 2 )], s 1 , s 2 ∈ S.
Define the scaling sequence

b n = (2 log n) 1/2 -(2 log n) -1/2 ((1/2) log log n + log(2 √ π)), n ≥ 1 (11)
and the rescaled process

Y n (s) = b n (Z n (s) -b n ), s ∈ S.
We consider the log-normal process

X n (s) = exp Y n (s), s ∈ S.
The following theorem provides a non trivial example of Gaussian random fields satisfying condition ii) in Theorem 3.1. Thanks to the equivalence i)⇔ ii), it essentially follows from the convergence of maxima due to Kabluchko [START_REF] Kabluchko | Extremes of independent Gaussian processes[END_REF] . Since convergence of point process is conceptually at a higher level than convergence of maxima, we wish to provide here a direct proof of the -convergence ν n w → ν without considering convergence of maxima M n d → M . To this aim, we use a criterion for -convergence due to Hult and Lindskog [START_REF] Hult | Regular variation for measures on metric spaces[END_REF].

Cov[Y w n (s 1 ), Y w n (s 2 )] = b 2 n (r n (s 1 , s 2 ) -r n (s 1 , s 0 )r n (s 2 , s 0 )) . (15)
Under assumption i), it holds

lim n→∞ µ w n (s) = w - 1 2 Γ(s, s 0 ) and lim n→∞ Cov[Y w n (s 1 ), Y w n (s 2 )] = 1 2 (Γ(s 1 , s 0 ) + Γ(s 2 , s 0 ) -Γ(s 1 , s 2 )).
This implies that lim n→∞ Var(Y w n (s 2 ) -Y w n (s 1 )) = Γ(s 1 , s 2 ) and proves the Lemma.

Proof of Theorem 6.1 :

Our proof relies on the following criterion for -convergence in C + 0 by Hult and Lindskog [START_REF] Hult | Regular variation for measures on metric spaces[END_REF]. According to [START_REF] Hult | Regular variation for measures on metric spaces[END_REF]Theorem 4.4], it is enough to prove that: is bounded by some constant M > 0.

i) for every r > 0, sup n≥1 nP[sup s∈S X n (s) > r] < ∞; ii) for every ε > 0, lim δ→0 sup n≥1 nP[ω Xn (δ) ε] = 0, where ω f (δ) = sup{|f (s 1 )-f (s 2 )|; s 1 , s 2 ∈ T, d(s 1 , s 2 ) ≤ δ} denotes the modulus of continuity of f ∈ C(S); iii) nP[X n ∈ • ] → ν in
• Proof of i): Let r > 0 and r = ln r. We have

nP[sup s∈S X n (s) > r] = nP[sup s∈S Y n (s) > r] = n √ 2πb n e b 2 n /2 R e -w-w 2 /2b 2 n P[sup s∈S Y w n (s) > r]dw ≤ M R e -w P[sup s∈S Y w n (s) > r]dw, so that sup n≥1 nP[sup s∈S X n (s) > r] ≤ M R e -w sup n≥1 P[sup s∈S Y w n (s) > r]dw. (16)
By tightness of the family {Y w n -µ w n ; w ∈ R, n ≥ 1} (see Lemma 6.2), there exists c 1 > 0 such that for all n ≥ 1 and w ∈ R,

P[sup s∈S (Y w n (s) -µ w n (s)) > c 1 ] < 1 2 .
Furthermore, Equations ( 14)-( 15) and assumption ii) of Theorem 6.1 together imply that there are some c 2 , c 3 > 0 and n 0 ≥ 1 such that for all n ≥ n 0 and all w < 0,

In view of Equation ( 16), this proves i).

• Proof of ii): For δ > 0, we have We will apply Lebesgue dominated convergence Theorem to prove that this has limit zero as δ → 0. Lemma 6.2 implies that for every fixed w, the sequence {e Y w n ; n 1} is tight so that lim We are left to prove a suitable domination condition: for all n ≥ 1, we have with r = min 1≤j≤k y j . Equation ( 19) yields the required domination condition.

  restriction of µ 1 and µ 2 to Br . Note that µ 1 and µ 2 are finite measures on Br and denote by p r the Prokhorov metric on M b ( Br ). Define

  Poisson point process on E with intensity ν and d → standing for convergence in distribution inM (b,p) (E).

d = ∨ n i=1 ξ in where d =

 d stands for equality in distribution(seeGiné et al. [16] orKabluchko and Stoev [23]). Theorem 3.1. Assume that for each n ≥ 1, {X in , i ≥ 1} are i.i.d. copies of a nonnegative continuous stochastic process X n . Then the following statements are equivalent, where the limits are taken as n → ∞: i) (M n (s)) s∈S d → (M (s)) s∈S where the limiting process M is assumed to satisfy the condition essinf M := inf{y > 0; P( M > y) < 1} ≡ 0 (4) and d → stands for convergence in distribution in C

  Poisson point process on C + 0 with intensity ν and d → standing for convergence in distribution in M (b,p) (C + 0 ).

  is fulfilled as soon as sup n≥1 nP[ X n > y] < ∞ for all y > 0.Giné et al consider the weaker condition essinfM (s) ≡ 0 and prove that every continuous max-infinitely divisible process M satisfying this weaker condition admits a Poisson point process representation[START_REF] Davis | Extreme value theory for space-time processes with heavy-tailed distributions[END_REF]. This weak condition is not restrictive because if b(s) = essinfM (s) is not identically equal to 0, one can consider M (s) -b(s). It remains here an open question whether the convergence M n d → M with essinfM (s) ≡ 0 implies essinf M = 0 or not.

Lemma 3 . 4 .

 34 The mapping θ : M (b,p) (C + 0 ) → C + defined by θ(0) ≡ 0 andθ i∈I δ x i = s → sup{x i (s); i ∈ I}is well-defined and continuous. Proof of Theorem 3.1: Proof of ii)⇒ i): Under the assumption nP[X n ∈ • ] w → ν, we know from Proposition 2.2 that the empirical measure β n defined by Equation (3) converges in distribution in M (b,p) (C + 0 ) to a Poisson point process Π ν with intensity ν. The assumption ν(C + 0 \C + 0 ) ensures that Π ν lies almost surely in M (b,p) (C + 0 ). Noting that M n = θ(β n ) with θ continuous (cf Lemme 3.4), the continuous mapping Theorem (see e.g. Theorem 5.1 in Billingsley [2]) entails θ(β n ) d → θ(Π ν ). This proves the convergence M n r → M with M = θ(Π ν ), i.e. i) holds together with the representation (5). The normalisation condition (4) is easily checked since P[ M > y] = 1 -exp -ν{f ∈ C + 0 ; f > y} < 1 for all y > 0.

2πb n e b 2 n / 2

 22 the sense of finite-dimensional convergence.These three points are proven below. At several places, we will use that the scaling sequence b n defined by Equation[START_REF] Dwass | Extremal processes[END_REF] satisfies √ 2πb n e b 2 n /2 ∼ n as n → ∞. As a consequence, the sequence n √

Firste

  -w dw < ∞.

3 . 2 , w < 0 ,

 320 c 2 , and sup s∈S Var[Y w n (s)] c 2Applying Borell's inequality (see Theorem D.1 in[START_REF] Piterbarg | Asymptotic methods in the theory of Gaussian processes and fields[END_REF]), we obtainP[sup s∈S Y w n (s) > r] < 2ψ -r -w/2 -c 1 -where ψ is the tail of the standard Gaussian distribution. Consequently, ) > r]dw < ∞.

  e Y w n (δ) ε]dw.

δ→0e• 2 ne-w-w 2 2b 2 n 0 P

 220 -w sup n≥1 P[ω e Y w n (δ) ε] = 0.It remains to prove a suitable domination condition: since δ → P[ω e Y w n (δ) ε] is a nondecreasing function, it is enough to prove that for someδ 0 > 0 R e -w sup n≥1 P[ω e Y w n (δ 0 ) ε]dw < ∞.Sinceω e Y w n (δ 0 ) ≤ exp[sup t∈T Y w n (t)],we have e -w sup n≥1P[ω e Y w n (δ 0 ) ε] e -w sup n≥1 P[sup t∈T Y w n (t) ln ε].Equation (19) with r = ln ε provides the required domination condition. Proof of iii): We have to prove that for all set A ⊆ C + 0 of the formA = {f ∈ C + 0 ; ∃j ∈ [[1, k]], f (s j ) > x k }, k ≥ 1, s 1 , . . . , s k ∈ S, x 1 , . . . , x k ∈ (0, +∞), it holds lim n→∞ nP[X n ∈ A] = ν(A).Letting y j = ln x j , we havenP[X n ∈ A] = nP[∃j ∈ [[1, k]], X n (s j ) > x j ] = nP[∃j ∈ [[1, k]], Y n (s j ) > y j ] P[∃j ∈ [[1, k]], Y w n (s j ) > y j ]dw ∼ R P[∃j ∈ [[1, k]], Y w n (s j ) > y j ]dw.We will apply once again Lebesgue's dominated convergence Theorem. Lemma 6.3 entailsP[∃j ∈ [[1, k]], Y w n (s j ) > y j ] → P[∃j ∈ [[1, k]], w + W (s j ) -1 2 Γ(s j , s 0 ) > y j ], so that we expect nP[X n ∈ A] → R e -w P[∃j ∈ [[1, k]], w + W (s j ) -1 2 Γ(s j , s 0 ) > y j ]dw = ∞ [∃j ∈ [[1, k]], we W (s j )-1 2 Γ(s j ,s 0 ) > x j ]w -2 dw = ν(A).

e -w-w 2 2b 2 n

 2 P[∃j ∈ [[1, k]], Y w n (s j ) > y j ] ≤ e -w sup n≥1 P[sup s∈S Y w n (s) > r]

  boundedly finite point measure on E is a measure µ of the form Let {µ n = i δ x n i , n 1} and µ = i δ x i be elements of M (b,p) (E).

		µ =	δ x i
		i∈I
	with {x i , i ∈ I} a finite or countable family of points in E such that any bounded set
	B ⊆ E contains at most a finite number of the x i 's. The set of boundedly finite point
	measures on E is denoted by M (b,p) (E). It is a closed subset of M b (E) and hence is
	complete and separable when endowed with the induced metric (see Daley and Vere-Jones
	[4] Proposition 9.1.IV and Lemma 9.1.V).
		The following characterization of -weak convergence of boundedly finite point mea-
	sures will be useful.
	Lemma 2.1. Then
	µ n	w → µ if and only if there exist some sequence r k

with x nε i , x ε i ∈ Bε and x nε i → x ε i as n → ∞. Clearly, this implies that θ(µ (ε ) n ) converges in C + to θ(µ (ε ) ) as n → ∞. Hence there exists an integer n 0 such that θ(µ

) ≤ ε for all n ≥ n 0 . Then, thanks to Equation [START_REF] De Haan | Extreme value theory[END_REF], we obtain for

This proves that θ(µ n ) → θ(µ) as n → ∞ and that the mapping θ is continuous.

Spatio-temporal maximum process

We consider now convergence of the space-time process Mn defined by Equation [START_REF] Billingsley | Convergence of probability measures[END_REF]. For fixed t ≥ 0, the space process s → Mn (t, s) is sample continuous and non negative, i.e. a random elements of C + . Furthermore, the time process t → Mn (t, •) can be seen as a C + -valued càd-làg process on [0, +∞); it is indeed constant on intervals of the form

Hence, we will consider the process Mn as a random element of the Skohorod space D([0, +∞), C + ) endowed with the J 1 -topology (see for example Ethier and Kurtz [START_REF] Ethier | Markov processes[END_REF] for the definition and properties of Skohorod space).

Then, the process Mn weakly converges in D([0, +∞), C + ) as n → ∞ to the superextremal process M defined by

where

For the proof, we will need the following analog of Lemma 3.4 in the space-time framework. Let M b,p (C + 0 × [0, +∞)) be the subset of measures µ ∈ M (b,p) (C

In other words, a measure µ = i∈I δ (x i ,t i ) belongs to C if and only if r i < +∞ for all i ∈ I and the u i 's are pairwise distinct.

Proposition 4.2.

• (Markov property) For each t ≥ 0, the conditional distribution of ( M (t + h, •)) h≥0 given F t is equal to the distribution of ( M (t, •) ∨ M (h, •)) h≥0 where M is an independent copy of M .

• (regularity) The C + (S)-valued process t → M (t, •) is a pure jump process with finitely many jumps in each interval

The first point of the proposition states that the process t → M (t, •) has "independent and stationary increments" with respect to the maximum: for

where M 1 , . . . , M k are i.i.d. copies of M . This property is similar to the property of independence and stationarity of increments (with respect to the addition) of Lévy processes. For a fixed point s ∈ S, the process {M (t, s), t ≥ 0} is known as an extremal process (see Proposition 4.7 of Resnick [START_REF] Resnick | Extreme values, regular variation and point processes[END_REF]).

Proof of Proposition 4.2:

By the independence properties of the Poisson point process, Π[0,t] ν and Π(t,∞) ν are independent. Furthermore,

and the two terms in the r.h.s. are independent. It is easily seen that the first term is equal to M (t, •), and that the invariance of the Lebesgue measure implies that the second term has the same distribution as M (h, •).

• The definition [START_REF] De | On convergence toward an extreme value distribution in C[0, 1[END_REF] implies that conditionally on M (t 1 , •), the evolution of the process M (t, •) for t ∈ [t 1 , t 2 ] depends only on the points

Each of this point may give rise to a jump of M (t, •) in [t 1 , t 2 ]. The result follows since the number of such points follows a Poisson distribution and is hence finite.

Convergence of order statistics

The point process approach for extremes is powerful: the convergence of the empirical measure β n entails not only the convergence of the maxima M n but also of all the order statistics, and a similar result hold for the space-time version of these processes.

For 1 ≤ r ≤ n and s ∈ S, define M r n (s) as the r-th largest value among X 1n (s), . . . , X nn (s). Note M 1 n (s) = M n (s) is simply the maximum. For r > n, we use the convention M r n (s) = 0. The process M r n = (M r n (s)) t∈S is refered to as the r-th order statistic of the sample {X 1n , . . . , X 1n }. Similarly, for r ≥ 1, s ∈ S and t ≥ 0 define M r n (t, s) as the r-th largest value among X 1n (s), . . . , X [nt]n (s) with the convention M r n (t, s) = 0 if r > [nt]. An alternative definition in terms of the empirical measure βn is given by

≤ t} and the convention that the supremum over an empty set is equal to zero.

With these notations, we can strengthen Theorem 3.5 as follows.

with Πν a Poisson point process on C + 0 × [0, +∞) with intensity ν ⊗ .

The limit process M j corresponds to the r-th space-time order statistic associated to the point process Πν . is well defined and continuous on C. The proof is very similar to the proof of Lemma 3.6 and the details are omitted for the sake of brevity.

Theorem 6.1. Fix s 0 ∈ S and suppose that: i ) Uniformly in s 1 , s 2 ∈ S :

ii ) For all s 1 , s 2 ∈ S, there exists C > 0 such that

Then,

and {W (s), s ∈ S} a centered Gaussian process such that W (s 0 ) = 0 and with incremental variance Γ.

Following [START_REF] Kabluchko | Stationary max-stable fields associated to negative definite functions[END_REF][START_REF] Kabluchko | Extremes of independent Gaussian processes[END_REF], we define Y w n the process Y n conditioned by Y n (s 0 ) = w and we note µ w n (s) = E[Y w n (s)]. We will need the following two Lemmas. Lemma 6.2. Under conditions i) and ii) of Theorem 6.1, the family {Y w n ; n ≥ 1} is tight in C(S) for all fixed ω ∈ R, as well as the family {Y w n -µ w n ; w ∈ R, n ≥ 1}.

Lemma 6.2 follows from the proof of Theorem 6 in [START_REF] Kabluchko | Extremes of independent Gaussian processes[END_REF] (see also the proof of Theorem 17 in [START_REF] Kabluchko | Stationary max-stable fields associated to negative definite functions[END_REF]). Details are omitted here. Lemma 6.3. Under condition i) of Theorem 6.1, the following convergence in the sense of finite dimensional distributions holds: as n → ∞, {Y w n (t), s ∈ S} f.d.d.

-→ {w + W (s) -1 2 Γ(s, s 0 ), s ∈ S} with W defined in Theorem 6.1.

Proof of Lemma 6.3 : Standard computations for Gaussian processes entails that Y w n is a Gaussian process with mean and covariance µ w n (s) = wr n (s, s 0 ) + b 2 n (r n (s, s 0 ) -1), [START_REF] Fisher | Limiting forms of the frequency of the largest or smallest member of a sample[END_REF]