Liliane Campos

Keywords: Bulgakov, McBurney, Complicite, aesthetics, fragmentation, narrative, metalepsis, postmodernism, technology

In this study Liliane Campos links Complicite's Master and Margarita (2011) to the company's previous productions, from The Street of Crocodiles (1992) to Shun-Kin (2008). She develops a close analysis of The Master and Margarita as it was staged at the Avignon Festival in July 2012, arguing that the company's aesthetic is characterized by a tension between narrative fragmentation and visual connections. While their shows overflow with postmodernist multiplicity and division, the urge to connect these 'shards of stories' is a driving force in McBurney's artistic direction. This dynamic is explored here both on a semantic level, as a consequence of Complicite's physical language, and on a narrative level, in their use of framing and frame-breaking devices. The article highlights the company's recurrent themes and the defining traits of their performance style.

to please the authorities. The lovers escape the madness and unhappiness of 1930s Moscow on horseback, following the Devil and his retinue through the airs to a place where they will at last find peace. In the version of the Master and Margarita presented at the Avignon Festival in July 2012, Simon McBurney answered the first of these two challenges with a relatively simple stage 'trick': as the wall of her building was projected onto the stage around Margarita, she teetered on her window sill and then plunged downwards without ever leaving the boards, and her acceleration was provided by the moving image. The second flight, however, was a breath-taking moment which formed the climax of the final scenes: lying on their sides with their knees bent, the couple flew away with the help of another projected image, this time of a gigantic horse. Yet this was no flesh and blood animal, but the shape of a galloping horse composed of dozens of chairs. At the end of a technologically-rich production, in which projections of Jerusalem or Moscow covered the gigantic walls of Avignon's Palais des Papes, the fiery horse of Bulgakov's imagination was created by combining computer animation with the theatre's simplest prop, the chair.

For any spectator familiar with Complicite's work, this striking image brings back memories of a whole chain of previous shows, above all Mnemonic (1999), in which a chair was used, Bunraku-style, as a puppet representing the body of a Neolithic man. McBurney confesses to being 'amoureux des chaises' ('in love with chairs') and their capacity to represent the space of the human body. [START_REF] Mcburney | in a post-show discussion following a projection of The Three Lives of Lucie Cabrol[END_REF] It is not surprising, then, that the horse of Complicite's imagination should be made of chairs: the chair is to the company what the flying horse was to Bulgakov, the embodiment of art's freedom and its ability to escape from naturalistic constraints. It is typical of Complicite's creative style, both technologically inventive and yet continuing to speak in the ingenious language of bodies and props that first distinguished the company in the early 1980s and led to their work being labelled 'physical theatre'.

The material Complicite has adapted for the stage over the past twenty years is extremely diverse. Many of their pieces have been derived from twentieth-century fiction: The Street of Crocodiles (1992) was inspired by the life and writings of Bruno Schulz, The Three Lives of Lucie Cabrol (1994) by John Berger's novel Pig Earth, The Elephant Vanishes (2003) was a stage adaptation of three short stories by Haruki Murakami, and Shun-Kin (2008) wove together a tale by Junichiro Tanizaki and his essay "In Praise of Shadows". Poetry, philosophical essays and scientific texts are also fed into the company's devising process, and experts in related areas invited to their workshops. Sometimes the devising work does not focus on a fictional text but on a historical enquiry or a scientific idea, such as the neurology of memory in Mnemonic or the mathematics of infinity in A Disappearing Number (2007). Yet the thematic and stylistic continuities between all these pieces are striking, and stand out once again in The Master and Margarita.

This study links The Master and Margarita to Complicite's past body of work, arguing that the tension between fragmentation and connection that structures this show is characteristic of their aesthetic. While the company's use of narrative overflows with postmodernist multiplicity and division, the urge to connect these 'shards of stories' is a driving force in McBurney's artistic direction. [START_REF] Mcburney | Intricate Collisions[END_REF] These connections are explored here both on a semantic level, as a consequence of Complicite's physical language, and on a narrative level, in the company's use of framing and frame-breaking devices.

Diffracted Narratives

Although Hans-Thies Lehmann included Complicite in the list of 'postdramatic' artists outlined in the introductory chapter of his Postdramatic Theatre, most of their work is difficult to categorize as such, since it remains structured around clear storylines and characters. [START_REF] Lehmann | Postdramatic[END_REF] They do, however, favour fragmented narrative pluralities which resonate with postmodern concepts such as the loss of a unified idea of history and the weakening of structuring meta-narratives. [START_REF]The Transparent Society[END_REF] Amongst the themes that The Master and Margarita shares with the company's previous shows, the most remarkable is that of narrative itself, since the action is triggered by the narrative account of the death of Jesus, or rather the 'vagrant philosopher Yeshua', by Satan and the Master. This narrative, which focuses on Pontius Pilate and his fascination for Yeshua, provides a counterpoint for the Master and Margarita's tale of separation, persecution and final reunion. Despite their final escape, which is only possible after their death, both the lovers' tale and the embedded story of Pontius Pilate are stories of irretrievable loss, in which the past is bitterly regretted and the clock cannot be turned back.

These characters' estrangement from their former lives echoes the themes of displacement and disorientation developed by most of Complicite's shows, which explore the loss of traditional ways of life (The Three Lives of Lucie Cabrol), the alienation of urban life (The Elephant Vanishes), and individual stories of migration and lost identities which intertwine with collective historical trauma, such as World War I and II in The Street of Crocodiles, Mnemonic and A Disappearing Number. These plays invariably contain characters that no longer belong to the world they inhabit. Unsurprisingly, one of the characters that stand out in their adaptation of Bulgakov's novel is the poet Bezdomny ('the homeless'), who meets the Master in the psychiatric clinic and takes up the task of 'continuing the story' after he leaves. Whereas in the novel Bezdomny simply becomes a professor of history who vaguely remembers his wild adventure once a month, McBurney turns him into the final storyteller of the show, typing out and narrating its ending.

In order to explore the 'sense of dis-placement' and 'loss of continuity' that pervade Complicite's work, McBurney favours fragmented dramatic forms. [START_REF] Mcburney | Prologue[END_REF] His Master and Margarita begins with a chaotic series of textual and visual fragments taken from the whole show: standing on one side of the stage the Master describes his separation from Margarita, who stands on the other side in a separate rectangle of light. Agitation grows around them as the whole cast criss-crosses the stage and acts out snippets of their interaction to come. The effect recalls similar introductory scenes in Shun-Kin and A Disappearing Number, and creates a sense of confusion which announces the mayhem that Satan's visit to Moscow will cause.

Bulgakov's readers may find the structure of the novel somewhat disconcerting in its constant shifts from one set of characters to another, and this fragmentation is greatly amplified by Complicite's adaptation. Whereas Bulgakov inserts the Master's account of Pilate's encounter with Jesus into the Moscow storyline through isolated chapters, which occasionally take us away from with the main narrative, McBurney cuts them up and splices them with the Moscow chapters, rapidly shifting from one story to the other and relocating the action through simple visual indications -a bright white light and beds for the psychiatric clinic, or a square of carpet and a chair for Margarita's flat. The result is a whirlwind of rapid scenes which emphasize the ironic parallels between the different narratives, as for example when the literary critics who will cause the Master's downfall condemn his work, and then remain on stage in a stern circle while Pilate argues with the High Priest of Judaea about Yeshua's sentence and announces it to the crowd.

Bulgakov's two plots are thus further divided and intertwined, and this adaptation foregrounds the novel's narrative voice by distributing it between different characters. By starting the evening with the Master's narration of his downfall, instead of the novel's anonymous narration and dialogue between two writers, the company emphasizes his voice, whereas Bulgakov delays his entrance till the thirteenth chaptersomewhat ironically entitled 'Enter the hero'. Complicite's Master and Margarita, like Shun-Kin before it, relies heavily on narration and highlights the storyteller's role. In Shun-Kin, the cruel love story of Sasuke and his blind mistress Shun-Kin, set in the nineteenth century, was read out by a twenty-first-century narrator recording it for the radio. The nineteenth-century action was framed by scenes which isolated the narrator in the middle of a vast darkened stage, emphasizing the themes of light and shadows that ran through the show (partly inspired by Tanizaki's In Praise of Shadows). When, at the end of The Master and Margarita, Satan asks 'what would the world look like without shadow?', the line sounds as if it was borrowed from Shun-Kin.

But the company's manipulation of the narrative voice is more complex in this case: throughout the evening they replace Bulgakov's ironic, anonymous narrator, who only hints that he is himself a Muscovite, by the characters themselves. Whereas only a few episodes of the novel are narrated by characters (mostly Satan and the Master), Complicite distribute the whole narrative between the Master, Satan (or rather Professor Woland, his incarnation in Moscow), his assistant Koroviev, Bezdomny and Margarita.

As the story shifts from one narrator to another, the ambiguity of its status increases. Bulgakov already introduced ambivalence by having Woland and the Master give the same account of Yeshua's trial. Complicite split this ambivalent enunciation into yet more speakers, making sure that the different phases of the plot are mediated by different voices. As a result, the telling itself stands out as a situated act that cannot be neutral, and which frames the action. As Bulgakov suggests, the question of who writes, or rather who is permitted to write the official (hi)story is brought to the fore by Staline's iron rule. His Master is cast out for his writings about Jesus, which will be continued by his 'disciple' Bezdomny, and his Satan describes himself as 'a historian.' The question of who controls the story is paramount, and emphasized in McBurney's production by the circulating narrative. In the face of totalitarian monologism, Complicite's splintering of the narrative voice and the narrative structure acts both as a counterpoint to the single tale of the regime and a reminder of the inescapability of narrative frames. When asked about his choice of narrators, McBurney points out that Bulgakov invites us to ask ourselves what story we are living in, and to question the narratives we take for granted, such as that of capitalism. 6 This remark suggests an interesting paradox: using a satire of Communist Russia to question our own capitalistic context -and indeed this context may resonate with the materialist greed that characterizes Bulgakov's Muscovites.

Visual Connections

These multiple narrators reorganize the novel's ambivalent enunciation into a postmodernist paradigm of proliferating narratives, whose juxtaposition highlights the parallels between these stories of sacrifice and absolution. Despite their thematic and structural assertion of fragmentation, Complicite's dramatic forms always cohere through repeated motifs and metaphorical connections. Most of their shows begin with a prologue which suggests a structuring image. In The Street of Crocodiles, which was devised from the essays and short stories of Bruno Schulz, the actor playing Schulz opened a book and a feather fell out of it, triggering the memories that formed the body of the show and a temporary flight from the reality of his situation as a Polish Jew in 1942. In his introductory note to Mnemonic, McBurney remarks that the devising process at the root of this creation was a search for 'another form to tell our stories', and in several subsequent pieces this 'form' was drawn from science: a scientific idea was outlined in the prologue and became a structuring image for the plot -the biochemistry of memory in Mnemonic, the increasing disorder and irreversibility of the second law of thermodynamics in The Elephant Vanishes, and the symmetries and regularities of mathematical series in A Disappearing Number. [START_REF] Campos | Searching for resonance: scientific patterns in Complicite's Mnemonic and A Disappearing Number[END_REF] Each of these images suggests an overarching dynamic of connective fragmentation. In his introductory lecture to Mnemonic, for example, McBurney described the biochemical processes of memory, in which the act of remembering does not activate a fixed store of memories but depends on the constant sprouting and recreation of synaptic connections in the human brain. In self-consciously postmodern terms, McBurney thus appropriated this physical process as a metaphor for the form of Mnemonic, noting that 'we no longer live in a world of the single tale. So the shards of stories we have put together, some longer some shorter, collide here in the theatre, reflecting, repeating, and revolving like the act of memory itself.' [START_REF] Mcburney | Intricate Collisions[END_REF] One of the structuring images put forth by The Master and Margarita's 'prologue', before the frenzied criss-crossing and the first scene of the novel, is a nightmare the Master describes: as he sinks into depression and irrational fear, a 'cold, supple octopus' strangles his heart. This opening nightmare, which expresses the fears of the censored author and points to Bulgakov's own situation, suggests a possible reading of the ensuing performance along the textual structure of dreams, in which we can expect substitutions and condensations of meaning. Accordingly Moscow and Jerusalem are interchangeable, the actor playing the Master also plays Woland in most of his scenes, and the martyrdom of Yeshua offers a metaphorical parallel to the persecution of the Russian author and the suffering of his lover.

This dream-logic reaches its climax in the final sequence, which contains a series of striking condensations and substitutions. At the end of Satan's ball, the image of Margarita's exhausted body is projected and magnified on the wall, superimposed over Yeshua's crucifixion. When Bezdomny takes over the story of Pontius Pilate from the Master, he narrates and enacts it in turns, becoming Yeshua's follower, Mathew the Levite. And as the performance draws to its end, when the Master is allowed to conclude Pilate's story and grant him his reunion with Yeshua, the actor playing Woland (at this stage) steps out of his long dark cloak, takes off his dark glasses, and walks towards Pilate as the bruised Yeshua of the beginning. Like many of the other transformations of the evening, this final substitution makes Bulgakov's verbal suggestions visually explicit: in this case the interdependence of good and evil, light and shadows.

The search for paradigmatic continuities within a fragmented syntagm is characteristic of Complicite's aesthetic and correlated to their particular brand of physical theatre. Up until the late 1990s, the visual ingenuity of their work lay mostly in a simple yet effective use of the semantic versatility of bodies and stage props. This produced memorable images, such as the chair / man in Mnemonic, the fluttering pages / birds in The Street of Crocodiles and Shun-Kin, or Simon McBurney and his acolytes energetically climbing a table / mountain with their chairs / backpacks on their backs in The Three Lives of Lucie Cabrol. Such visual language enhances the 'generative capacity' of the theatrical sign, shifting rapidly through different denotations. [START_REF] Elam | The Semiotics of Theatre and Drama[END_REF] But it also allows the actors to activate visual metaphors: when the company manipulated the chair like a puppet in Mnemonic, they enacted the manipulation of the past by the present, and the appropriation and reification of the Iceman as a 'National monument'. And when, in one of the first scenes of The Three Lives of Lucie Cabrol, the actors lay across the stage to form the furrows of earth that Lucie's father was ploughing, the image conveyed the organic ties binding this isolated peasant community to the earth.

This semiotic versatility is also to be found in The Master and Margarita: on a bare stage, a desk, two beds, a dozen chairs, an opaque screen and a glass booth are used to recreate the many different locations of the novel. The company use the same simple set of sticks to form Margarita's window sill or the Christ's cross, and chairs held over their heads to figure the olive-grove where Judas will be murdered. The minimalistic use of props contrasts with the wealth of images framing the stage: projected directly onto the walls of the Palais des Papes, a satellite map of Moscow allows us to situate each successive episode of the novel, the hills of Jerusalem frame the encounter between Pontius Pilate and Yeshua, a gigantic portrait of Stalin hangs behind the management committee meeting of the writers' club Massolit, and when Woland shows Margarita his terrifyingly accurate globe, images of the Irak war appear on the wall. On-stage cameras produce magnified images of the actors, and clever tricks are plays with projections onto the windows of the main wall behind the stage, in which Margarita and Satan occasionally appear, watching the action from above.

Amidst this profusion of ingenious effects, moments of pure physical creativity stand out as traces of the company's former economy of means. One of the most striking images is produced when the Master succumbs to paranoia and depression and burns his manuscript: the other actors first become black shadows closing in on him, and then enact the flames devouring the pages. This visual metaphor embodies his fears, but also suggests the political origins of his torment, inflicted upon him by the repressive context of Soviet Moscow.

The energy of these isolated moments leads us to wonder to what extent technological sophistication risks crowding out the company's original physical inventiveness. Yet the show's most memorable projections are not those of other spaces or times but amplifications and distortions of the space before our eyes (through the combined efforts of Es Devlin's design, Paul Anderson's lighting, Finn Ross's videos and Luke Halls's 3D animations). The bare, austere walls of the Palace, first of all, which serve as the walls of Moscow's buildings during Margarita's flight. They are later projected onto themselves, creating a dizzying trompe-l'oeil effect of depth for Satan's ball, and finally acquire huge, growing cracks throughout the last scenes, ending with their collapse when the Master speaks to Pontius Pilate and frees him from his eternal suffering. Or the sweating face of Pilate, crippled by a migraine during his encounter with Jesus, and Margarita's legs as her lover watches for her arrival from his basement flat. And of course the equine chairs which carry them to freedom. The most striking effects are those which recreate the fantastical visions of Bulgakov's novel from the materiality of the theatrical space: its walls, props, actors, and briefly, when Woland addresses the audience of Moscow's Variety theatre, Complicite's own audience.

The company's fluid scenic language thus continues to transmogrify the basic ingredients of theatre -albeit with a little more technological help than formerly -and to emphasize parallels and equivalences, both between its different narratives and between these narratives and their theatrical frame.

Stepping through the Frame

The other recurring source of connections in the fragmented syntagm of the show is the use of metalepsis: 'frame-breaking' moments which transgress the boundaries between different levels of the diegesis (for instance the story and a story within the story) or between the diegesis and its extra-diegetic frame. [START_REF] Gérard Genette | Metalepsis as a Transgeneric and Transmedial Phenomenon. A Case Study of the Possibilities of 'Exporting' Narratological Concepts[END_REF] These moments are characteristic of Complicite's prologues, which typically set up a frame for the main narrative and immediately transgress its boundaries. In the version of Shun-Kin presented in Paris, the main actor Yoshi Oida began by describing, in French, the Japan of his childhood and his ritual visits to his father's tomb, before the rest of the company dressed him in nineteenth-century clothing and turned him into the Japanese-speaking character Sasuke. In the first few minutes of A Disappearing Number, a mathematical lecture given by the character Ruth was interrupted by another character, Aninda, who addressed the audience and joked about the difficulties of mathematics and its reality effect in the fictional setting of a play. Similarly Mnemonic began with a lecture addressed to the audience, in the middle of which the speaker, Simon McBurney, became Virgil, a character listening to McBurney's lecture.

However The Master and Margarita uses metalepsis throughout the show, as a way of weaving together its different threads. These occurrences can be divided into four categories, the first of which results from the coexistence of different narrative levels on stage: the narrators or readers of the imbedded stories thus share the stage with their characters, as for example when Pontius Pilate is seated on the desk Margarita is seated at, reading about him. These transgressions in corpore draw on the semantic versatility conferred to the props.

Whereas these moments can be seen as frame-enhancing rather than truly framebreaking devices, a second type of metalepsis occurs when the narrators interact with their characters: when he takes over the story from the Master, Bezdomny both types out the account of the night of the execution and gets up from his desk to take Yeshua off his cross. This type of transgression is present in one of the final chapters of the novel, when the Master is allowed to see Pontius Pilate and to grant him absolution, but Complicite make it systematic whenever Bezdomny is narrating the story. In contrast to these intra-diegetic transgressions, a third type of metalepsis crosses the threshold between Bulgakov's narrative and his existence, when Bezdomny tells the audience about the censorship the author endured and his private readings of his manuscript. And a final type of boundary-crossing occurs when the production refers to the contemporary world, as for instance when Woland addresses the audience directly, referring sarcastically to iPhones and other wonders of modern technology, or indeed when Complicite's take on Bulgakov's diabolical cat enters the scene, as a Bunraku-style puppet spouting obscene jokes.

A monumental visual equivalent to these transgressions appears after Satan's ball, when Margarita's first plea for compassion causes a huge crack to appear in the (projected) wall behind the stage. The crack gradually widens in the final scenes, until Pontius Pilate and Yeshua walk away through the gap and the image of the whole wall finally collapses, leaving the company surrounded by constellations of dust. This spectacular effect literalizes the idea of frame-breaking, and recalls a similar moment in Mnemonic. Throughout Mnemonic, the body of the Neolithic man discovered in the Alps was represented by not one but two signifiers: on the one hand the puppet-chair, on the other the body of the character Virgil, whose fascination and growing empathy for this man was expressed by his taking his place on stage. In the final scene, the assembled cast watched the body of Virgil/ the Iceman through the frame of his refrigeration unit in the museum in Bolzano. One by one, they slipped under the frame and took his place, replacing each other in an accelerating, fluid movement just as, according to the stage directions, 'generation succeeds generation in a never-ending cycle'. 11 The concluding moments of both shows thus drew our attention to their frames by negating them.

In The Master and Margarita, one of the effects of the resulting de-hierarchizing of narrative levels is to question the distinctions between reality and fiction. As Borges once pointed out, the idea of Hamlet watching Hamlet disturbs us because if characters may be spectators then we, readers, may in fact be the characters of someone else's fiction. [START_REF] Borges | Magies partielles du Quichotte[END_REF] Stalin's Moscow is as much a narrative construction as Bulgakov's novel, and McBurney highlights the postmodernist potential of this 'pan-fictionality' by extending it to our contemporary world. However, the suggested parallels are not only paradoxical equivalences between the real and the fictional, but also metaphorical bridges between the narratives and their frames, highlighting the similarities and allegorical potential of different sufferings and persecutions, including that of Bulgakov himself.

Such systematic bridging is not without its dangers, since its metaphorical flexibility can flatten its subject matter through too many equivalences and correspondences. This may be a somewhat self-defeating potential of Complicite's semiotic and narrative fluidity, which sometimes suggests so much interchangeability that it veils the specificity of our contemporary position towards the past. [START_REF]Janelle Reinelt makes a similar point about Mnemonic's 'return to universalism' in 'Performing Europe: Identity Formation for a "New" Europe[END_REF] This is one extreme of their aesthetics, at which equivalences and connections erase the fractures and displacements they explore. But this extreme is only approached by the ending of Mnemonic and by The Master and Margarita. Most of their work has moved along a shifting scale, constantly redefining the balance between the framing and framebreaking elements of their devising -a balance which determines whether metalepsis emphasizes temporal dis-jointedness, or a sense of continuity in which distinctions between frame and content no longer hold.

In his review of the version presented at the Barbican (four months before it came to Avignon), Michael Billington described Complicite's Master and Margarita as 'a production which suggests Bulgakov's novel is a metaphor for a universe made up of unreliable narrators'. [START_REF] Billington | The Master and Margarita[END_REF] Perhaps it would be more accurate to call it a universe in which the distinction between reliable and unreliable narration no longer holds, since there is nothing beyond narrative, no lasting distinction between stories and their frames. Complicite's work continues to be unashamedly dramatic and absorbed by the act of story-telling. Increasingly, however, it focuses on the weaving together of narrative threads through verbal and visual parallelisms, supported by an inventive use of film, animation and projection, rather than the physical metaphors of their early days. Liliane Campos, Université Sorbonne Nouvelle -Paris 3


domain: stat math

text:
Nadia Morsli
email: nadia.morsli@imag.fr

Keywords: Georgii-Nguyen-Zessin formula, kernel estimator non-parametric estimation, pairwise interaction point process, Papangelou conditional intensity, strong uniform consistency, Orlicz spaces

Non-parametric inference for stationary pairwise interaction point process

Nadia Morsli

Introduction

The theory of spatial models is a growing field over the last decade, with various applications in several domains such as ecology (Diggle [8]), forestry (Matérn [20]), spatial epidemiology (Lawson [START_REF] Lawson | Statistical Methods in Spatial Epidemiology[END_REF]) and astrophysics (Neyman and Scott [START_REF] Neyman | Statistical Approach to Problems of Cosmology[END_REF]). Gibbs point processes arose in statistical physics as models for interacting particle systems. Gibbs point processes in R d can be defined and characterized through the Papangelou conditional intensity (Møller and Waagepetersen [START_REF] Møller | Statistical Inference and Simulation for Spatial Point Processes[END_REF]). It is also an important tool in the context of simulations of Gibbs processes. Examples of Markov and non-Markov Gibbs point process models and their conditional intensities are presented in Baddeley et al [START_REF] Baddeley | Residual analysis for spatial point processes[END_REF], Møller and Waagepetersen ([21], [START_REF] Møller | Modern statistics for spatial point processes[END_REF]). In this paper, we are concerned with non-parametric statistics for stationary pairwise interaction point processes (a special case of a Gibbs process) which describes the interaction between pairs of points by a function (called a pairwise interaction function). They have been introduced in statistical literature by Ripley and Kelly [START_REF] Ripley | Markov point processes[END_REF], Daley and Vere-Jones [START_REF] Daley | An introduction to the Theory of Point Processes[END_REF] and Georgii [START_REF] Georgii | Gibbs measures and Phase Transitions[END_REF]. These provide a large variety of complex patterns starting from simple potential functions (or pairwise interaction functions) which are easily interpretable as attractive or repulsive forces acting among points, and they are of practical importance because of their ability to model a wide variety of spatial point patterns, especially those displaying some degree of spatial regularity. Our objective of this work is to study estimating nonparametric interaction function in the Papangelou conditional intensity. We suggest a new non-parametric estimate of the pairwise interaction function in the Papangelou conditional intensity for stationary pairwise interaction point process. Sufficient conditions to strong uniform consistency are obtained for the resulting estimator, by stationary field of dependent random variables. Note also that the main results of this work are obtained via assumptions of belonging to Orlicz spaces induced by exponential Young functions for stationary real random fields. Our results also carry through the most important particular case of Orlicz spaces random fields. Many attempts have been tried to estimate the potential function from point pattern data in a parametric framework: maximization of likelihood approximations (Ogata and Tanemura [START_REF] Ogata | Estimation of interaction potentials of spatial point patterns through the maximum likelihood procedure[END_REF], Ogata and Tanemura [START_REF] Ogata | Likelihood analysis of spatial point patterns[END_REF], Penttinen [START_REF] Penttien | Modelling interaction in spatial point partterns: parameter estimation by maximum likelihood method[END_REF]), pseudolikelihood maximization (Besag et al. [START_REF] Besag | Point process limits of lattice processes[END_REF], Jensen and Møller [START_REF] Jensen | Pseudolikelihood for exponential family models of spatial point processes[END_REF]) and also some ad hoc methods (Strauss [32], Ripley [START_REF] Ripley | Markov point processes[END_REF], Hanisch and Stoyan [START_REF] Hanisch | Remarks on statistical inference and prediction for hard-core clustering model[END_REF], Diggle and Gratton [START_REF] Diggle | Monte Carlo methods of inference for implicit statistical models[END_REF], Fiksel [START_REF] Fiksel | Estimation of interaction potentials of Gibbsian point processes[END_REF], Takacs [START_REF] Takacs | Estimator for the pair-potential of a Gibbsian point process[END_REF], Billiot and Goulard [START_REF] Billiot | An estimation Method of the Pair Potential Function for Gibbs Point Processes on Spheres[END_REF]).

The rest of the paper is organized as follows. Section 2 sets up the generic notation and the basic tools of point processes in R d . We present the model for stationary pairwise interaction point process and the assumptions which will be considered in the sequel in Section 3. In Section 4, we present our main results; on both the estimation method and strong uniform consistencies for the resulting estimator. The last section is devoted to the proofs. O ⊆ B d be the system of all bounded Borel sets. A point process X in R d is a locally finite random subset of R d , i.e. the number of points N(Λ) = n(X Λ ) of the restriction of X to Λ is a finite random variable whenever Λ is a bounded Borel set of R d (see Daley and Vere-Jones [START_REF] Daley | An introduction to the Theory of Point Processes[END_REF]). We define the space of locally finite point configurations in

Generic notation and Basic tools

R d as N l f = {x ⊆ R d ; n(x Λ ) < ∞, ∀Λ ∈ B d 0 }, where x Λ = x ∩ Λ. We equip N l f with σ -algebra N l f = σ {{x ∈ N l f : n(x Λ ) = m}, m ∈ N 0 , Λ ∈ B d 0 }, where N 0 = N ∪ {0} = {0, 1, 2, 3, . . .}.
The volume of a bounded Borel set Λ of R d is denoted by |Λ| and o = (0, . . . , 0). Let ||t|| be the Euclidean norm for a point t ∈ R d and ∑ = signifies summation over distinct pairs. Let x is a realization of a Gibbs point process X in Λ (window). In the context of finite point processes, a point process X is a Gibbs process if and only if X has a probability density f (x) with respect to the distribution of the unit rate Poisson point process on Λ, such that if x, y are two possible configurations with x ⊂ y, then f (y) > 0 implies f (x) > 0. The Papangelou conditional intensity of X can then be calculated as λ (u, x) = f (x ∪ {u})/ f (x) for any configuration x and any point u ∈ Λ with u / ∈ x. The Papangelou conditional intensity can be interpreted as follows: for any u ∈ R d and x ∈ N l f , λ (u, x)du corresponds to the conditional probability of observing a point in a ball of volume du around u given the rest of the point process is x. In general it is not possible to deal with densities of infinite point processes. However, the Papangelou conditional intensity of Gibbs point processes X in R d (Møller and Waagepetersen [START_REF] Møller | Statistical Inference and Simulation for Spatial Point Processes[END_REF]) is a function λ : R d ×N l f → R + and characterized by the Georgii-Nguyen-Zessin (GNZ) formula (see Papangelou [START_REF] Papangelou | The Armenian Connection: Reminiscences from a time of interactions[END_REF] and Zessin [START_REF] Zessin | Der Papangelou prozess[END_REF] for historical comments and Georgii [START_REF] Georgii | Canonical and grand canonical Gibbs states for continuum systems[END_REF] or Nguyen and Zessin [START_REF] Nguyen | Integral and differential characterizations Gibbs processes[END_REF] for a general presentation). The Georgii-Nguyen-Zessin (GNZ) formula states that for any non-negative measurable function

h on R d × N l f E ∑ u∈X h(u, X \ u) = E R d h(u, X)λ (u, X)du. (2.1)
Let X ⊗ X be the point process on R d × R d consisting of all pairs (u, v) of distinct points of X. It follows immediately from the GNZ formula (2.1) that X ⊗ X is a Gibbs point process with (two-point) Papangelou conditional intensity λ (u, v, x) = λ (u, x)λ (v, x ∪ {u}), for u, v ∈ R d , x ∈ N l f , meaning that the GNZ formula in the form

E = ∑ u,v∈X h(u, v, X \ {u, v}) = R d R d E h(u, v, X)λ (u, v, X)dudv (2.2)
is satisfied for any non-negative measurable function h

(u, v, x) on R d × R d × N l f .

Model

The pairwise interaction point process is characterized by its conditional intensity Papangelou defined by

λ (u, x) = g 0 (u) exp -∑ v∈x\u g 0 ({u, v})
and note that g 0 (u, v) = g 0 (v, u) (i.e. symmetric pairwise interaction).

If we consider a special case where g 0 (u) is a constant and g 0 ({u, v}) = g(vu) is translation invariant. In this case, the pairwise interaction point process is called stationary.

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

λ β (u, x) = β exp -∑ v∈x\u g(v -u) (3.1)
where β is the true value of the Poisson intensity parameter, g represents the nonnegative pairwise interaction potential defined on R d . The pairwise interaction between points may also be described in terms of the pairwise interaction function G = exp(-g).

In this semi-parametric model (3.1), the estimator of the Poisson intensity parameter β represented the first step in our procedure in the paper [START_REF] Coeurjolly | Poisson intensity parameter estimation for stationary Gibbs point processes of finite interaction range[END_REF], we have established its strong consistency and asymptotic normality, we also considered its finite-sample properties simulation study. Now, we develop a method of non-parametric estimation of the pairwise interaction function G, with the class of model defined by (3.1), under assumptions as general as possible.

The basic assumption throughout this paper is the Papangelou conditional intensity has a finite range R, i.e.

λ β (u, x) = λ β (u, x B(u,R) ), (3.2)
for any u ∈ R d , x ∈ N l f , where B(u, R) is the closed ball in R d with center u and radius R.

Main results

Suppose that a single realization x of a point process X is observed in a bounded window

Λ n ∈ B d O where (Λ n ) n≥1
is a sequence of cubes growing up to R d . We face a missing data problem, which in the spatial point process literature is referred to as a problem of edge effects, we can avoid this problem by reducing the window by introducing the 2R-interior of the cubes Λ n , i.e.

Λ n,R = {u ∈ Λ n : B(u, 2R) ⊂ Λ n }
and assume this has non-zero area. The choice of Λ n,R is induced by the fact that when u is on the edge of Λ n,R and v ∈ B(u, R), then we can observe the point v. We have assumed a known interaction range R. In practice, R is often estimated by maximizing a profile pseudolikelihood over a grid. We assume that the support of the interaction function

G = exp(-g) is T = {t ∈ R d ; g(t) > 0, for ||t|| < R}. Throughout in this paper, h is a non-negative measurable function defined for all w ∈ R d , x ∈ N l f , by h(w, x) = 11(x ∩ B(w, R) = / 0), (4.1)
and we also introduce the following function

F(o, w) = E[h(o, X)h(w, X)] = P(X ∩ B(o, R) = / 0, X ∩ B(w, R) = / 0). (4.2)
To estimate the function β F(o,t), we propose an empirical estimator Fn (t) defined for t ∈ T by Fn (t) = 1

|Λ n,R | ∑ u∈X Λ n,R h(u, X \ {u})h(t + u, X \ {u}). (4.3)
To estimate the function β 2 G(t) F(o,t), we propose a kernel-type estimator H n (t) defined for t ∈ T by The kernel function K : R d -→ R is non-negative and bounded with bounded support, and satisfies:

H n (t) = 1 b d n |Λ n,R | = ∑ u,v∈X v-u∈B(o,R) 11 Λ n,R (u)h(u, X \ {u, v})h(v, X \ {u, v})K v -u -t b n , (4.4
R d K(u)du = 1. Let u = (u 1 , . . . , u d ) , u i ∈ R, R d u i 1 1 . . . u i d d K(u 1 , . . . , u d )du 1 . . . du d = 0, for 0 < d ∑ j=1 i j < m.
To establish the strong uniform consistency of the kernel-type estimator (4.4) over some compact set T 0 ⊂ T we need a further smoothness condition on the kernel function.

Condition Z : The kernel function K satisfies a Lipschitz condition, i.e. there exists a constant η > 0 such that

K(u) -K(v) ≤ η||u -v|| for any u, v ∈ T.
We also assume additionally that:

Condition C (T ): F(o,t) and G(t) are continuous on any fixed compact set T 0 ⊂ T . Condition Z :

∃α 1 > 0, β F(o,t) ≥ α 1 and ∃α 2 > 0, β G(t) ≤ α 2 , ∀t ∈ T 0 .
The following theorem gives an asymptotic representation for the mean of the kerneltype estimator (4.4).

Theorem 4.1. Consider a stationary pairwise interaction point process X in R d with Papangelou conditional intensity (3.1) satisfying condition (3.2). Furthermore the kernel function K satisfies Condition K(d, 1). We have

lim n→∞ E H n (t) = β 2 G(t) F(o,t)
at any continuity point t ∈ T of G F.

If Condition K(d, m) is satisfied and G F has bounded and continuous partial derivatives of order m in an open ball B o (t, δ ) (for some δ > 0) for t ∈ T , then

E H n (t) = β 2 G(t) F(o,t) + O(b m n ) as n → ∞. (4.6)
The estimator (4.3) turns out to be an unbiased estimator of β F(o,t) and if X is ergodic, strongly consistent (the uniform strong consistency) as n tends infinity, since a classical ergodic theorem for spatial point processes obtained in [START_REF] Nguyen | Ergodic theorems for Spatial Process[END_REF]. This implies the following:

Proposition 1. Consider a stationary pairwise interaction point process X in R d with Papangelou conditional intensity (3.1) satisfying condition (3.2) and we assume that Condition Z is fulfilled. For any fixed compact set T 0 ⊂ T , we have sup

t∈T o | G n (t) -β G(t)|-→0 P-a.s. iff sup t∈T o | H n (t) -β 2 G(t) F(o,t)|-→0 P-a.s..
To get a strong uniform consistency of the estimator H n (t) to the function β 2 G(t) F(o,t), we assume that the domain Λ n is divided into a fixed number of subdomains as follows Λ n = ∪ i∈I n Λ i , following Preston [START_REF] Preston | Random fields[END_REF], Klein [START_REF] Klein | Dobrushin uniqueness techniques and the decay of correlation in continuum statistical mechanics[END_REF], Jensen and Künsch [START_REF] Jensen | On asymptotic normality of pseudo likelihood estimates of pairwise interaction processes[END_REF] we will describe a point process in R d as lattice field by means of this decomposition Λ i = {ξ ∈ R d ; q(i j - 1 2 ) ≤ ξ j ≤ q(i j + 1 2 ), j = 1, . . . , d} for a fixed number q > 0, i = (i 1 , . . . , i d ), and setting X i = X Λ i , i ∈ Z d , this becomes a Gibbs lattice field. We will consider estimation of β 2 G(t) F(o,t) from H n (t), where Λ n = ∪ i∈I n Λ i , and where the process is observed in Λ n,R = ∪ i∈ I n Λ i , where I n = {i ∈ I n ; |i -j| ≤ 1, for all j ∈ I n }, and the norm is | j| = max{| j 1 |, . . . , | j d |} and assume that I n increases towards Z d and write

H n (t) = 1 b d n |Λ n,R | ∑ i∈ I n = ∑ u∈X i ,v∈X v-u∈B(o,R) h(u, X \ {u, v})h(v, X \ {u, v})K v -u -t b n . (4.7)
To shorten the notation we introduce the random variables

Z i = = ∑ u,v∈X v-u∈B(o,R) 11 Λ i (u)h(u, X\{u, v})h(v, X\{u, v})K v -u -t b n and Zi = Z i -E Z i for all i ∈ I n .
Recall that a Young function ψ is a real convex nondecreasing function defined on R + which satisfies lim t→∞ ψ(t) = +∞ and ψ(0) = 0. We define the Orlicz space L ψ as the space of real random variables Z defined on the probability space (Ω, F , P) such that E[ψ(|Z|/c)] < +∞ for some c > 0. The Orlicz space L ψ equipped with the so-called Luxemburg norm . ψ defined for any real random variable Z by

Z ψ = inf{ c > 0 ; E[ψ(|Z|/c)] ≤ 1 }
is a Banach space. For more about Young functions and Orlicz spaces one can refer to Krasnosel'skii and Rutickii [START_REF] Krasnosel | Convex Functions and Orlicz Spaces[END_REF]. Let θ > 0. We denote by ψ θ the exponential Young function defined for any x ∈ R + by

ψ θ (x) = exp((x + ξ θ ) θ ) -exp(ξ θ θ ) where ξ θ = ((1 -θ )/θ ) 1/θ 11{0 < θ < 1}. ( 4
.8) 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 p in {2, 3, . . . , d}, i p < j p and i q = j q for 1 ≤ q < p. Let the sets {V k i ; i ∈ Z d , k ∈ N * } 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} where |i -j| = max 1≤l≤d |i l -j l |.
For any subset I n of Z d define F I n = σ ( Zi ; i ∈ I n ) and set

E |k| ( Zi ) = E( Zi |F V |k| i ), k ∈ V 1 i . (4.9)
Denote θ (q) = 2q/(2q) for 0 < q < 2.

Theorem 4.2. We assume that Conditions K(d, m), C (T ) and Z are fulfilled. Furthermore, we also assume that G F has bounded and continuous partial derivatives of order m on a compact set T 0 ⊂ T . If there exists 0 < q < 2 such that the centered random variable Z0 ∈ L ψ θ (q) . Then

sup t∈T o H n (t) -β 2 G(t) F(o,t) = O a.s. (log n) 1/q (b n √ n) d + O(b m n ) as n → ∞.
Our results also carry through the most important particular case of Orlicz spaces random fields, i.e. For ψ p (t) = |t| p , for any t ∈ R + , the Luxembourg norm is nothing but the L p -norm. Theorem 4.3. We assume that Conditions K(d, m), C (T ) and Z are fulfilled. Furthermore, we also assume that G F has bounded and continuous partial derivatives of order m on a compact set T 0 ⊂ T . If there exists p > 2 such that the centered random variable Z0 ∈ L p . Assume that b n = n -q 2 (log n) q 1 for some q 1 , q 2 > 0. Let a, b ≥ 0 be fixed and if a(p + d)d 2 /2q 2 d > 1 and b(p

+ d) + q 1 d > 1. Then sup t∈T o H n (t) -β 2 G(t) F(o,t) = O a.s. n a (log n) b (b n √ n) d + O(b m n ) as n → ∞.

Proofs

Keep in mind h is given by (4.1) and the function F(o,t) defined by (4.2).

Proof of Theorem 4.1

H n (t) as a double sum and by means of the GNZ formula (2.2) with

h(u, v, X) = 11 Λ n,R (u)11(v -u ∈ B(o, R))h(u, X)h(v, X)K v -u -t b n , we get E H n (t) = 1 b d n |Λ n,R | E = ∑ u,v∈X v-u∈B(o,R) 11 Λ n,R (u)h(u, X \ {u, v})h(v, X \ {u, v})K v -u -t b n = 1 b d n |Λ n,R | E R 2d 11 Λ n,R (u)11(v -u ∈ B(o, R))h(u, X)h(v, X)K v -u -t b n λ β (u, v, X)dudv.
We remember the second order Papangelou conditional intensity by:

λ β (u, v, x) = λ β (u, x)λ β (v, x ∪ {u}) for any u, v ∈ R d and x ∈ N l f .
Using the finite range property (3.2) for each function λ β (u, x) and λ β (v, x ∪ {u}), we have

λ β (u, X) = λ β (u, X ∩ B(u, R)) = β when X ∩ B(u, R) = / 0 and λ β (v, X ∪ {u}) = λ β (v, (X ∪ {u}) ∩ B(v, R)) = β G(v -u) when X ∩ B(v, R) = / 0 and v -u ∈ B(o, R).
In this way, by stationarity of X, we obtain

E H n (t) = β 2 b d n |Λ n,R | E R 2d 11 Λ n,R (u)11(v -u ∈ B(o, R))h(u, X)h(v, X)K v -u -t b n G(v -u)dudv = β 2 b d n |Λ n,R | R 2d 11 Λ n,R (u)11(v -u ∈ B(o, R)) F(o, v -u)K v -u -t b n G(v -u)dudv = β 2 R d 11(b n z + t ∈ B(o, R)) F(o, b n z + t)K(z)G(b n z + t)dz.
The continuity of FG in t and the boundedness conditions on the kernel function yield the desired result with the first statement of Theorem 4.1 by dominated convergence theorem.

For the results of the second part of Theorem 4.1, we consider B 0 (t, δ ) an open ball in R d , the function G(t) F(o,t) has bounded and continuous partial derivatives of order m in B o (t, δ ) (for some δ > 0 ) for t ∈ T , then for any point z in R d , there exists θ ∈ (0, 1), such that by Taylor-Lagrange formula, we get

G(t + b n z) = G(t) + m-1 ∑ k=1 ∑ |α|=k (b n z) α α! ∂ k G(t) ∂ x α + b m n R m (z,t), where R m (z,t) = ∑ |α|=m z α α! ∂ m G ∂ x α (t + θ b n z). F(o,t + b n z) = F(o,t) + m-1 ∑ k=1 ∑ |α|=k (b n z) α α! ∂ k F(o,t) ∂ x α + b m n R m (z,t), where R m (z,t) = ∑ |α|=m z α α! ∂ m F ∂ x α (o,t + θ b n z
). So we multiply two such functions, their product equals the product of their m th Taylor polynomials plus terms involving powers of t higher than m. In other words, to compute the m th Taylor polynomial of a product of two functions, find the product of their Taylor polynomials, ignoring powers of t higher than m. So we denote this product by T (t)(b n z) α , then we have

E H n (t) = β 2 G(t) F(o,t) R d K(z)dz + β 2 T (t)b α n R d z α K(z)dz + O(b m n ) as n → ∞.
Together with Condition K(d, m), we get the asserted rate of convergence.

Proof of Theorem 4.2 and Theorem 4.3

To establish rates of the uniform P-a.s. convergence for the estimator (4.7), we apply a triangle inequality decomposition allows for

sup t∈T 0 H n (t) -β 2 G(t) F(o,t) ≤ sup t∈T 0 H n (t) -E H n (t) + sup t∈T 0 E H n (t) -β 2 G(t) F(o,t) .
(5.1) Let (r n ) n≥1 be sequence of positive numbers going to zero. Following Carbon and al. [START_REF] Carbon | Kernel density estimation for random fields[END_REF], the compact set T 0 can be covered by v n cubes T k having sides of length l n = r n b d+1 n and center at c k . Clearly there exists c > 0, such that v n ≤ c/l d n . Define

A 1 = max 1≤k≤v n sup t∈T k H n (t) -H n (c k ) A 2 = max 1≤k≤v n sup t∈T k E H n (t) -E H n (c k ) A 3 = max 1≤k≤v n H n (c k ) -E H n (c k ) then sup t∈T o | H n (t) -E H n (t)| ≤ A 1 + A 2 + A 3 .
(5.2)

We study the following lemmas which significantly improve the desired result.

Lemma 1. For j=1,2, we have

A j = O a.s. (r n ) as n → ∞.
Proof. For any t ∈ T k , by Condition Z , we derive that there exists constant η > 0 such that as n → ∞

H n (t) -H n (c k ) ≤ 1 b d+1 n η t -c k 1 |Λ n,R | = ∑ u,v∈X v-u∈B(o,R) 11 Λ n,R (u)h(u, X \ {u, v})h(v, X \ {u, v}) ≤ 1 b d+1 n ηr n b d+1 n 1 |Λ n,R | = ∑ u,v∈X v-u∈B(o,R) 11 Λ n,R (u)h(u, X \ {u, v})h(v, X \ {u, v}) ≤ ηr n 1 |Λ n,R | = ∑ u,v∈X v-u∈B(o,R) 11 Λ n,R (u)h(u, X \ {u, v})h(v, X \ {u, v}) .
From the last inequality and the Nguyen and Zessin [START_REF] Nguyen | Ergodic theorems for Spatial Process[END_REF] ergodic theorem, it easily follows

A 1 = O a.s. (r n ) as n → ∞.
Then one gets with the same arguments as before and by L 1 -version of the ergodic theorem of Nguyen and Zessin [START_REF] Nguyen | Ergodic theorems for Spatial Process[END_REF]:

A 2 = O a.s. (r n ) as n → ∞.
The proof of Lemma 1 is complete.

We now concentrating on the stochastic part.

Lemma 2. Assume that for Z0 ∈ L ψ θ (q) , for some 0 < q < 2 and r n = (log n) 1/q /(b n √ n) d holds. Then we have,

A 3 = O a.s. (r n ) as n → ∞.
Proof. To establish the proof of Lemma 2 we need the following result. Consider the following assumption:

∃q ∈]0, 2[ ∃c > 0 E[exp(c|X 0 | θ (q) )] < +∞ (5.3) θ (q) = 2q/(2 -q).
Theorem 5.1. (El Machkouri, [START_REF] Machkouri | Kahane-Khintchine inequalities and functional central limit theorem for stationary real random fields[END_REF]). Let (X i ) i∈Z d be a zero mean stationary real random field which satisfies the assumption (5.3) for some 0 < q < 2. There exists a positive universal constant M 1 (q) > 0 depending only on q such that for any family (a i ) i∈Z d of real numbers and any finite subset

Γ in Z d , ∑ i∈Γ a i X i ψ q ≤ M 1 (q) ∑ i∈Γ |a i |b i,q (X) 1/2 where b i,q (X) = |a i | X 0 2 ψ θ (q) + ∑ k∈V 1 0 |a k+i | X k E |k| (X 0 ) 2 ψ θ (q)
. Now, in the sequel, the letter C denotes any generic positive constant. We consider the exponential Young function defined by (4.8). Let ε > 0 and t ∈ T 0 be fixed

P | H n (t) -E H n (t)| > εr n = P   | ∑ i∈ I n Zi | > εr n (nb n ) d   ≤ exp - ε r n (nb n ) d || ∑ i∈ I n Zi || ψ q + ξ q q E exp | ∑ i∈ I n Zi | || ∑ i∈ I n Zi || ψ q + ξ q q .
Therefore, we assume that there exists a real 0 < q < 2, such that Z0 ∈ L ψ θ (q) . Applying Theorem 5.1 to the sequence Zi , i ∈ I n , we have

P | H n (t) -E H n (t)| > εr n ≤ P   | ∑ i∈ I n Zi | > εr n (nb n ) d   ≤ (1 + e ξ q q ) exp - ε r n (nb n ) d M(∑ i∈ I n b i,q ( Z)) 1/2 + ξ q q where b i,q ( Z) = Z0 2 ψ θ (q) + ∑ k∈V 1 0 Zk E |k| ( Z0 ) 2 ψ θ (q)
, M is a positive constant depending only on q.

Note that, the random field Zi = Z i -E Z i depends on X Λ i , for any i ∈ I n , i.e. |i -j| ≤ 1, for j ∈ I n only. Then we deduce from the construction of F V |k| i defined by (4.9), that the random field Z i is independent of F V |k| i , and we conclude that the conditional means of Zi given

F V |k| i is zero, i.e. E |k| ( Z0 ) = 0 . Then we derive that b i,q ( Z) = Z0 2 ψ θ (q)
. It results for Z0 ∈ L ψ θ (q) that there exists constant C > 0 and so if r n = (log n)

1/q /(b n √ n) d , such that sup t∈T 0 P | H n (t) -E H n (t)| > εr n ≤ (1 + e ξ q q ) exp - ε r n ( √ nb n ) d C + ξ q q ≤ (1 + e ξ q q ) exp - ε q log n C q . ( 5.4)
On the other hand, we have

P (|A 3 | > εr n ) ≤ v n sup t∈T 0 P | H n (t) -E H n (t)| > εr n .
Using (5.4), choosing ε sufficiently large, therefore, it follows with Borel-Cantelli's lemma P(lim

sup n→∞ |A 3 | > εr n ) = 0.
Now, we will accomplish the proof of Theorem 4.3.

Lemma 3. Assume Z0 ∈ L p for some p > 2 and b n = n -q 2 (log n) q 1 for some q 1 , q 2 > 0. Let a, b ≥ 0 be fixed and denote

r n = n a (log n) b /(b n √ n) d . If a(p + d) -d 2 /2 -q 2 d > 1 and b(p + d) + q 1 d > 1. Then we have A 3 = O a.s (r n ) as n → ∞.
Proof. Let p > 2 be fixed, such that Z0 ∈ L p and for any ε > 0,

P(| H n (t) -E H n (t)| > εr n ) = P   | ∑ i∈ I n Zi | > εr n (nb n ) d   ≤ ε -p E | ∑ i∈ I n Zi | p r p n (nb n ) pd ≤ ε -p r p n (nb n ) pd   2p ∑ i∈ I n c i ( Z)   p/2
.

The last inequality follows from Dedecker [START_REF] Dedecker | Exponential inequalities and functional central limit theorems for random fields[END_REF], where c i ( Z) = Zi We conclude the proof of Theorem 4.2 and Theorem 4.3 by combining the inequality (5.1) and the inequality (5.2).

Proof of Proposition 1

We consider the decomposition (5.5)

G
Additionally, assume that P is ergodic, the ergodic theorem (Nguyen and Zessin [START_REF] Nguyen | Ergodic theorems for Spatial Process[END_REF]) immediately gives, as n → ∞, Fn (t)β F(o,t) a.s.

-→ 0.

(5.6)

Note that there exists at least one stationary Gibbs measure. If this measure is unique, it is ergodic. Otherwise, it can be represented as a mixture of ergodic measures (see [START_REF] Georgii | Gibbs measures and Phase Transitions[END_REF], Theorem 14.10). Now, using the monotony of functions F and Fn , we can approach the functions β F and Fn by their values in a finite number of points. Bringing this remark and the result (5. Together with Theorem 4.2, the expression (5.7) and the expression (5.8), we complete the proof.

Let B d

  be the Borel σ -algebra (generated by open sets) in R d (the d-dimensional space) and B d

  ) where = over the summation sign means that the sum runs over all pairwise different points u; v in X and K : R d → R denotes a smoothing kernel function associated with a sequence (b n ) n≥1 of bandwidths satisfying the below Condition K(d, m) and Condition Z . Plugging in the above estimators (4.3) and (4.4) and with the convention c/0 = 0 for all real c, we suggest a new edge-corrected non-parametric estimator G n (t) for β G(t) by G n (t) = H n (t) Fn (t) , for t ∈ T. (4.5) To establish strong uniform consistency results for the non-parametric quantity G n (t) defined by (4.5), we have to impose certain natural restrictions on the kernel function K and the sequence (b n ) n≥1 : Condition K(d, m): The sequence of bandwidths b n > 0 for n ≥ 1, is chosen such that lim n→∞ b n = 0 and lim n→∞ b d n |Λ n,R | = ∞.

2 p +∑ k∈V 1 i 2 . 2 p

 2122 Zk E |k-i| ( Zi ) pThe conditional means E |k-i| ( Zi ) is zero, then we have c i ( Z) = Zi and with the stationarity of X, we derive that there exists C > 0 such thatP (|A 3 | > εr n ) ≤ v n sup t∈T 0 P(| H n (t) -E H n (t)| > εr n ) As v n ≤ c/l d n and l n = r n b 1+d n , therefore for r n = n a (log n) b /(b n √ n) d , it follows as n → ∞, P (|A 3 | > εr n ) ≤ Cε -p n a(p+d)-d 2 /2 (log n) b(p+d) b d n ≤ Cε -p n a(p+d)-d 2 /2-q 2 d (log n) b(p+d)+q 1 d . For a(p+d)-d 2 /2-q 2 d > 1 and b(p+d)+q 1 d > 1, we have ∑ n≥1 P (|A 3 | > εr n ) < ∞.The proof of Lemma 3 is complete.By strengthening the uniform continuity assumption (Condition C (T )) of G(t) F(o,t), one hand, by Theorem 4.1, we have supt∈T o | E H n (t)β 2 G(t) F(o,t)| = O(b m n ).

  [START_REF] Daley | An introduction to the Theory of Point Processes[END_REF], we have as n → ∞, By (5.6) and by Condition Z , we obtain at any point t ∈ T 0 Fn (t) > α 1 , Pa.s., furthermore, by relation (5.5), we also see thatG n (t)β G(t) ≤ α 1 -1 H n (t)β 2 G(t) F(o,t) + α 2 α -11Fn (t)β F(o,t) . (5.8)

	sup	Fn (t) -β F(o,t)	a.s. -→ 0.	(5.7)
	t∈T o

Acknowledgments

The research was supported by laboratory Jean Kuntzmann, Grenoble University, France.


Nadia Morsli
email: nadia.morsli@imag.fr

Keywords: Kernel-type estimator, Pairwise interaction point process, Rates of strong uniform consistency, Papangelou conditional intensity, Spatial statistics

Consistency and strong consistency of non-parametric estimate of the potential function of stationary and isotropic pairwise interaction point processes

Nadia Morsli

To cite this version:

Nadia Morsli. Consistency and strong consistency of non-parametric estimate of the potential function of stationary and isotropic pairwise interaction point processes. 2016.

Introduction

Gibbs point processes are a natural class of models for point patterns exhibiting interactions between the points. Fields of applications for point processes are image processing, analysis of the structure of tissues in medical sciences, forestry (see [START_REF] Matérn | Spatial Variation: Stochastic Models and their Applications to Some Problems in Forest Surveys and Other Sampling Investigations. 2nd[END_REF]), ecology (see [START_REF] Diggle | Statistical analysis of spatial point patterns[END_REF]), spatial epidemiology (see [START_REF] Lawson | Statistical Methods in Spatial Epidemiology[END_REF]) and astrophysics (see [START_REF] Neyman | Statistical Approach to Problems of Cosmology[END_REF]). Non-parametric estimation has been largely ignored by researchers.

One exception is the suggestion to use the non-parametric estimation of the pair correlation function and its approximate relation to the pair potential through the Percus-Yevic equation (see [START_REF] Diggle | A nonparametric estimator for pairwise-interaction point processes[END_REF]). The approximation is a result of a cluster expansion method, and it is accurate only for sparse data. Pairwise interaction point processes densities are intractable as the normalizing constant is unknown and/or extremely complicated to approximate. However, we can resort to estimates of parameters using the conditional intensity. In this present paper, we assume that the pairwise interaction point process is stationary and isotropic , so that its Papangelou conditional intensity involves two terms: a Poisson intensity parameter and a pair potential function (or pairwise interaction function). This paper follows a previous one, that was interested in the estimator of the Poisson parameter in the Papangelou conditional intensity. In this present paper, we propose a new non-parametric estimation of the potential function in the Papangelou conditional intensity. We establish consistency and strong consistency for the resulting estimator.

Our paper is organized as follows. Section 2 introduces basic definitions and notations. In Section 3, we briefly present some models satisfying the assumptions needed to prove our asymptotic results. In Section 4, we present our main results. Consistency of the non-parametric estimator for the potential function in the Papangelou conditional intensity is proved in Section 4.1, it is based on the knowledge of the Papangelou conditional intensity and the iterated Georgii-Nguyen-Zessin formula. Using Orlicz spaces we can obtain a strong consistency of the non-parametric estimator in Section 4.2. The proofs are given in Section 5.

Basic notations

Throughout the paper we adopt the following notations. Let B d be the Borel σalgebra (generated by open sets) in R d (the d-dimensional space) and B d O ⊆ B d denote the class of bounded Borel sets. We define a spatial point process X on R d as a locally finite random subset of R d , i.e. the number of points

N(W ) = n(X W ) of the restriction of X to W is a finite random variable whenever W ⊂ R d is a bounded region. N l f = {x ⊆ R d ; n(x W ) = n(x ∩W ) < ∞, ∀ W ∈ B d 0 }
is the space of locally finite configurations of points in R d and will be denoted by x. We equip N l f with σ -algebra

N l f = σ {{x ∈ N l f : n(x W ) = m}, m ∈ N 0 ,W ∈ B d 0 }. That is N l f is the smallest sigma algebra generated by {x ∈ N l f : n(x W ) = m}. The volume of a bounded Borel set W of R d is denoted by |W | and o denotes the origin of R d , i.e. o = (0, . . . , 0) ∈ R d . For any finite subset Γ of Z d , we denote |Γ| the number of elements in Γ. || • || denotes Euclidean distance on R d . σ d = 2π d/2 Γ(d/2)
is the measure of the unit ball in R d . Further, ∑ = ξ 1 ,...,ξ n means that the summation goes over the n-tuples of mutually distinct points ξ 1 , . . . , ξ n . Let S d-1 be the unit ball in R d .

For any Gibbs point processes in a bounded window, the Papangelou conditional intensity at a location u given the configuration x is related to the probability density f by λ (u, x) = f (x ∪ {u}/ f (x) (for u / ∈ x), the ratio of the probability densities for the configuration x with and without the point u added. The Papangelou conditional intensity can be interpreted as follows: for any u ∈ R d and x ∈ N l f , λ (u, x)du corresponds to the conditional probability of observing a point in a ball of volume du around u given the rest of the point process is x.

Gibbs point processes in R d can be defined and characterized through the Papangelou conditional intensity (see [START_REF] Møller | Statistical Inference and Simulation for Spatial Point Processes[END_REF]) which is a function λ : R d × N l f → R + . The Georgii-Nguyen-Zessin (GNZ) formula (see [START_REF] Papangelou | The Armenian Connection: Reminiscences from a time of interactions[END_REF], [START_REF] Zessin | Der Papangelou prozess[END_REF], [START_REF] Georgii | Canonical and grand canonical Gibbs states for continuum systems[END_REF], [START_REF] Nguyen | Integral and differential characterizations Gibbs processes[END_REF]) states that for any non-negative measurable function

h on R d × N l f E ∑ u∈X h(u, X \ u) = E R d h(u, X)λ (u, X)du. (1)
Using induction we obtain the iterated GNZ-formula: for non-negative func-

tions h : (R d ) n × N l f -→ R E = ∑ u 1 ,...,u n ∈X h(u 1 , . . . , u n , X \ {u 1 , . . . , u n }) = . . . E h(u 1 , . . . , u n , X)λ (u 1 , . . . , u n , X)du 1 . . . du n (2)
where λ (u 1 , . . . , u n , x) is Papangelou conditional intensity and is defined (not uniquely) by λ (u 1 , . . . ,

u n , x) = λ (u 1 , x)λ (u 2 , x∪{u 1 }) . . . λ (u n , x∪{u 1 , . . . , u n-1 }).

Examples of Papangelou conditional intensity

Examples of Papangelou conditional intensities are presented in [START_REF] Baddeley | Residual analysis for spatial point processes[END_REF], [START_REF] Møller | Statistical Inference and Simulation for Spatial Point Processes[END_REF], [START_REF] Møller | Modern statistics for spatial point processes[END_REF]. The following presents some examples. Let u ∈ R d , x ∈ N l f and R > 0.

1. A special case of pairwise interaction is the Strauss process. It has Papangelou conditional intensity

λ (u, x) = β Φ n [0,R] (u,x\u)
where

β > 0, 0 ≤ Φ ≤ 1 and n [0,R] (u, x) = ∑ v∈x 11( v -u ≤ R)
is the number of pairs in x with distance not greater than R.

2. Piecewise Strauss point process.

λ (u, x) = β p ∏ j=1 Φ n [R j-1 ,R j ] (u,x\u) j where β > 0, 0 ≤ Φ j ≤ 1, n [R j-1 ,R j ] (u, x) = ∑ v∈x 11( v -u ∈ [R j-1 , R j ]) and R 0 = 0 < R 1 < . . . < R p = R < ∞.
3. Triplets point process.

λ (u, x) = β Φ s [0,R] (x∪u)-s [0,R] (x\u)
where β > 0, 0 ≤ Φ ≤ 1 and s [0,R] (x) is the number of unordered triplets that are closer than R.

4. Lennard-Jones model.

λ (u, x) = β ∏ v∈x\u Φ( v -u
with log Φ(r) = θ 6 r -6θ 12 r -12 11 (0,R] (r), for r = ||v -u||, where θ > 0 and β > 0 are parameters.

Main results

The Papangelou conditional intensity (see [START_REF] Møller | Statistical Inference and Simulation for Spatial Point Processes[END_REF]) for a pairwise interaction point process is defined by

λ (u, x) = γ 0 (u) exp -∑ v∈x\u γ 0 ({u, v}) .
If γ 0 (u) = β is a constant and γ 0 ({u, v}) = γ(||u -v||) is invariant under translations and rotations, then a pairwise interaction point process is said to be stationary and isotropic or homogeneous. For convenience, throughout in this paper, we consider a stationary and isotropic pairwise interaction point process. Then its Papangelou conditional intensity at a location u is given by

λ (u, x) = β exp -∑ v∈x\u γ(||v -u||) , ∀u ∈ R d , x ∈ N l f ( 3
)
where β is the so-called Poisson intensity parameter and γ is the so-called the pair potential and is assumed a non-negative function; a name that originates in statistical physics: it measures the potential energy caused by the interaction among pairs of points (u, v) as a function of their distance ||v -u||. The pairwise interaction between points may also be described in terms of the pair potential function γ into the interaction function Φ = exp(-γ) which has the following interpretation.

For Φ > 1, λ (u, x) is increasing in x (the attractive case). For Φ < 1, λ (u, x) is decreasing in x (the repulsive case). It can be computed for the case Φ = 1 which corresponds to the homogeneous Poisson point process with intensity β . Usually a finite range of interaction R, is assumed such that

γ(||v -u||) = 0 whenever ||v -u|| > R.
In other words, λ (u, x) depends on x only through x ∩ B(u, R), i.e.

λ (u, x) = λ (u, x ∩ B(u, R)), (4)
where B(u, R) is the closed ball in R d with centered at u and radius R > 0. In this present paper, we propose a new non-parametric estimation of the pair potential γ (or more precisely for the pairwise interaction function Φ = exp(-γ)) in the Papangelou conditional intensity for a stationary and isotropic pairwise interaction point process. We establish the consistency and the strong consistency for the resulting estimator. Suppose that a single realization x of a point process X is observed in a bounded window W n ∈ B d 0 where (W n ) n≥1 is a sequence of cubes growing up to R d . Throughout in this paper, h is a non-negative measurable function defined for all

u ∈ R d , x ∈ N l f by h(u, x) = 11 inf v∈x ||v -u|| > R = 11 (d(u, x) > R) , (5)
note that for r ∈ (0, R],

F(o, rv) = E[ h(o, X) h(rv, X)] = P(d(o, X) > R, d(rv, X) > R)
and

J(r) = S d-1 F(o, rv)dv.
To estimate the function β 2 J(r)Φ(r), we suggest an edge-corrected kernel-type estimator R n (r) defined for r ∈ (0, R] by

R n (r) = 1 b n |W n 2R |σ d = ∑ u,v∈X ||v-u||≤R 11 W n 2R (u) ||v -u|| d-1 h(u, X\{u, v}) h(v, X\{u, v})K 1 ||v -u|| -r b n . (6)
will denote Minkowski substraction, with the convention that

W n 2R = W n B(u, 2R) = {u ∈ W n : ||u -v|| ≤ 2R for all v ∈ W n }
denotes the 2R-interior of the cubes W n , with Lebesgue measure |W n 2R | > 0. K 1 is an univariate kernel function associated with a positive sequence (b n ) n≥1 of bandwidths satisfying the following:

Condition K(1, α) : The sequence of bandwidths b n > 0 for n ≥ 1, is chosen such that lim n→∞ b n = 0 and lim n→∞ b n |W n 2R | = ∞.
The kernel function K 1 : R -→ R is non-negative and bounded with bounded support, such that:

R K 1 (ρ)dρ = 1, R ρ j K 1 (ρ)dρ = 0, j = 0, 1, ..., α -1, for α ≥ 2.
To estimate the function β J(r), we suggest an empirical estimator J n (r) defined for r ∈ (0, R] by

J n (r) = 1 |W n 2R | ∑ u∈X 11 W n 2R (u) h(u, X \ {u})h (u, X \ {u}), (7)
where h (u, x) = S d-1 h(rv + u, x)dv. Using the spatial ergodic theorem of [START_REF] Nguyen | Ergodic theorems for Spatial Process[END_REF], the estimator (7) turns out to be unbiased and strongly consistent. The natural estimator of the Poisson intensity β is

β n = ∑ u∈X 11 Λ n,R (u) h(u, X \ {u}) Λ n,R h(u, X)du . (8)
In [START_REF] Coeurjolly | Poisson intensity parameter estimation for stationary Gibbs point processes of finite interaction range[END_REF], we introduced a semi-parametric estimator (8) of the parameter β and studied its strong consistency, asymptotic normality and simulation study. Finally, the desired estimator for the interaction function Φ(r) = exp(-γ(r)) is defined by the ratio

Φ n (r) = R n (r) β n J n (r) , for r ∈ (0, R]. (9)
The strong consistency of the estimators ( 7) and ( 8) implies the following:

Proposition 1.
Let γ be a pairwise interaction potential defined in (3) satisfying condition (4). Let K 1 kernel function satisfying Condition K(1, α) and the function J(r)exp(-γ(r)) has bounded and continuous partial derivatives of order α for all α ≥ 1 in (rδ , r + δ ) for some δ > 0. Then as n → ∞

Φ n (r)-→ exp(-γ(r)) in probability P (resp.P-a.s.) iff R n (r)-→β 2 J(r) exp(-γ(r)) in probability P (resp.P-a.s.).

The convergence in probability (consistency) of the kernel-type estimator R n (r) (defined in (6)) will be discussed in Section 4.1. Conditions ensuring uniform Pa.s. convergence (strong uniform consistency) of the kernel-type estimator R n (r) will be discussed in Section 4.2.

Consistency of the kernel-type estimator

In this section we discuss the consistency of R n (r). For this it follows the interesting to determine the asymptotic behavior of E R n (r) and Var R n (r).

Asymptotic representation of the mean and the variance of the kerneltype estimator

In this section we derive asymptotic representations for the mean and the variance of the kernel-type estimator R n (r). We will use the Landau notation

f (n) = O(h(n)) as n → ∞ for error terms f (n) satisfying lim sup n→∞ f (n) h(n) < ∞. Theorem 1.
Let γ be a pairwise interaction potential defined in (3) satisfying condition (4). Let K 1 kernel function satisfying Condition K(1, 1). Then we have

lim n→∞ E R n (r) = β 2 J(r) exp(-γ(r)),
in every point of continuity r ∈ (0, R] of J × exp(-γ).

If Condition K(1, α) is satisfied and the function exp(-γ(r))J(r) has bounded and continuous partial derivatives of order α in (rδ , r + δ ) for some δ > 0 and for α ≥ 1. Then we have

E R n (r) = β 2 J(r) exp(-γ(r)) + O(b α n ) as n → ∞.
Theorem 2. Let γ be a pairwise interaction potential defined in (3) satisfying condition (4).

Let K 1 kernel function satisfying Condition K(1, α) for all α ≥ 1 such that R K 2 1 (ρ)dρ < ∞ . Then, we have, lim n→∞ b n |W n 2R | Var( R n (r)) = 2β 2 σ d r d-1 J(r) exp(-γ(r)) R K 2 1 (ρ)dρ,
in every point of continuity r ∈ (0, R] of J × exp(-γ).

Rates of strong uniform consistency of the kernel-type estimator

Before realizing the strong consistency for R n (r) (defined in ( 6)) we introduce some necessary definitions and notations. A Young function ψ is a real convex nondecreasing function defined on R + which satisfies lim t→∞ ψ(t) = +∞ and ψ(0) = 0. We define the Orlicz space L ψ as the space of real random variables Z defined on the probability space (Ω, F , P) such that E[ψ(|Z|/c)] < +∞ for some c > 0. The Orlicz space L ψ equipped with the so-called Luxemburg norm . ψ defined for any real random variable Z by

Z ψ = inf{ c > 0 ; E[ψ(|Z|/c)] ≤ 1 }
is a Banach space. For more about Young functions and Orlicz spaces one can refer to [START_REF] Krasnosel'skii | Convex Functions and Orlicz Spaces[END_REF]. Let θ > 0. We denote by ψ θ the Young function defined for any x ∈ R + by

ψ θ (x) = exp((x+ξ θ ) θ )-exp(ξ θ θ ) where ξ θ = ((1-θ )/θ ) 1/θ 11{0 < θ < 1}.
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 p in {2, 3, . . . , d}, i p < j p and i q = j q for 1 ≤ q < p. Let the sets {V k i ; i ∈ Z d , k ∈ N * } 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} where |i -j| = max 1≤l≤d |i l -j l |.
By a real random field we mean any family (ε i ) i∈Z d of real-valued random variables and for any subset

Γ of Z d define F Γ = σ (ε i ; i ∈ Γ) and set E |k| (ε i ) = E(ε i |F V |k| i ), k ∈ V 1 i . Denote θ (q) = 2q/(2 -q) for 0 < q < 2.
Next we list a set of conditions which are needed to obtain (rates of) strong uniform consistency over some compact set [r 1 , r 2 ] in (0, R] of the estimator R n (r) to the function β 2 J(r)Φ(r). The following assumption is imposed:

Condition L p : The kernel function K satisfies a Lipschitz condition, i.e. there exists a constant η > 0 such that

K 1 (ρ) -K 1 (ρ ) ≤ η|ρ -ρ | for all ρ, ρ ∈ [r 1 , r 2 ].
Now, we split up the window W n 2R into cubes such as W n 2R = ∪ i∈Γ n Λ i , following [START_REF] Preston | Random fields[END_REF], [START_REF] Klein | Dobrushin uniqueness techniques and the decay of correlation in continuum statistical mechanics[END_REF], [START_REF] Jensen | On asymptotic normality of pseudo likelihood estimates of pairwise interaction processes[END_REF] we will describe a point process in R d as lattice process by means of this decomposition Λ i = {ξ ∈ R d ; q(i j - 1 2 ) ≤ ξ j ≤ q(i j + 1 2 ), j = 1, . . . , d} for a fixed number q > 0, i = (i 1 , . . . , i d ), where the process is observed in

W n 2R = ∪ i∈ Γ n Λ i , where Γ n = {i ∈ Γ n ; |i -j| ≤ 1, for all j ∈ Γ n },
R n (r) = 1 b n |W n 2R |σ d ∑ i∈Γ n R i (r),
where

R i (r) = = ∑ u,v∈X ||v-u||≤R 11 Λ i (u) ||v -u|| d-1 h(u, X \ {u, v}) h(v, X \ {u, v})K 1 ||v -u|| -r b n . Note for all k ∈ Γ n , Rk = R k (r) -E R k (r) and S n = ∑ k∈Γ n Rk (r).
Strong uniform consistency for R n (r) is obtained via assumptions of belonging to Orlicz spaces induced by exponential Young functions for stationary real random fields which allows us to derive the Kahane-Khintchine inequalities by [START_REF] El Machkouri | Kahane-Khintchine inequalities and functional central limit theorem for stationary real random fields[END_REF].

Proposition 2. We assume that Conditions K(1, α) and L p are fulfilled. Furthermore, we also assume that J(r) exp(-γ(r)) has bounded and continuous partial derivatives of order α in [r 1δ , r 2 + δ ] for some δ > 0.

If there exists 0 < q < 2 such that R0 ∈ L ψ θ (q) and ∑

k∈V 1 0 Rk E |k| ( R0 ) 2 ψ θ (q) < ∞. (10)
Then

sup r 1 ≤r≤r 2 R n (r)-β 2 J(r) exp(-γ(r)) = O a.s. (log n) 1/q b n n d/2 +O(b α n ) as n → ∞.
Our results also carry through the most important particular case of Orlicz spaces random fields, for p-integrable (2 < p < +∞) real random fields. Proposition 3. We assume that Conditions K(1, α) and L p are fulfilled. Furthermore, we also assume that J(r) exp(-γ(r)) has bounded and continuous partial derivatives of order α in [r 1δ , r 2 + δ ] for some δ > 0.

If there exists p > 2 such that R0 ∈ L p and

∑ k∈V 1 0 Rk E |k| ( R0 ) p 2 < ∞. (11)
Assume that b n = n -q 2 (log n) q 1 for some q 1 , q 2 > 0. Let a, b ≥ 0 be fixed and if a(p + 1)d 2 /2q 2 > 1 and b(p + 1)

+ q 1 > 1. Then sup r 1 ≤r≤r 2 R n (r)-β 2 J(r) exp(-γ(r)) = O a.s. n a (log n) b b n n d/2 +O(b α n ) as n → ∞.

Proofs

Proof of Theorem 1. Using the notation

L(u 1 , . . . , u s , X) = h(u 1 , X) . . . h(u s , X), ( 12
)
where h is given by [START_REF] Diggle | Statistical analysis of spatial point patterns[END_REF].

F(u 1 , . . . , u s ) = E[ h(u 1 , X) . . . h(u s , X)], J(||u 1 ||, . . . , ||u s ||) = 11(||u 1 || ≤ R, . . . , ||u s || ≤ R).
The calculation of the expectation and the variance of R n (r) is based on the iterated Georgii-Nguyen-Zessin (GNZ) formula (2), i.e. applying the preceding formula (2) for s = 2 and with

h(u, v, X) = 11 W n 2R (u) ||v -u|| d-1 J(||v -u||) L(u, v, X)K 1 ||v -u|| -r b n , we derive E R n (r) = 1 b n |W n 2R |σ d E R 2d 11 W n 2R (u) ||v -u|| d-1 J(||v -u||) L(u, v, X)K 1 ||v -u|| -r b n λ (u, v, X)dudv.
We remember the second order Papangelou conditional intensity by:

λ (u, v, x) = λ (u, x)λ (v, x ∪ {u}) for any u, v ∈ R d and x ∈ N l f .
Using the finite range property (4) for each function λ (u, x) and λ (v, x ∪ {u}), we have

λ (u, X) = λ (u, X ∩ B(u, R)) = β when d(u, X) > R and λ (v, X ∪ {u}) = λ (v, (X ∩ B(v, R)) ∪ {u}) = β Φ(||v -u||) when d(v, X) > R.
Consequently, by the stationarity of X and from the definition of L given by ( 5), we get

E R n (r) = β 2 b n |W n 2R |σ d E R 2d 11 W n 2R (u) ||v -u|| d-1 J(||v -u||) L(u, v, X)K 1 ||v -u|| -r b n Φ(||v -u||)dudv = β 2 b n σ d R d J(||s||) ||s|| d-1 E[ L(o, s, X)]K 1 ||s|| -r b n Φ(||s||)ds.
Recall a property of the integration theory (see [START_REF] Briane | Théorie de l'Intégration[END_REF] or [START_REF] Rudin | Real and Complex Analysis[END_REF]). Let S d-1 be the unit ball in R d , i.e. S d-1 = {u ∈ R d : ||u|| = 1}, then for any Borel function

f : R d -→ R + , R d f (||u||)du = ∞ 0 S d-1 f (rz)r d-1 σ d drdz.
By combining the above result, we get so:

E R n (r) = β 2 σ d ∞ -r/b n S d-1 J(b n ρ + r) F(o, (b n ρ + r)v)K 1 (ρ)Φ(b n ρ + r)σ d dρdv.
With bounded support on the kernel function and by dominated convergence theorem, we get as n → ∞, E R n (r) -→ β 2 J(r) exp(-γ(r)). Now, we are going to prove the second part of Theorem 1. We have a product of two functions F(o, (b n ρ + r)v)Φ(b n ρ + r) and we approximate each one of them with a Taylor formula up to a certain α. For any point ρ in R, there exists θ ∈ (0, 1), such that by Taylor-Lagrange formula, we get

Φ(b n ρ + r) = Φ(r) + α-1 ∑ k=1 Φ (k) (r) k! (b n ρ) k + Φ (α) (r + b n ρθ ) α! b α n and F(o, (b n ρ + r)v) = F(o, rv) + α-1 ∑ k=1 F (k) (o, rv) k! (b n ρ) k + F (α) (o, (r + b n ρθ )v) α! b α n .
So we multiply two such functions, their product equals the product of their α th Taylor polynomials plus terms involving powers of r higher than α. In other words, to compute the α th Taylor polynomial of a product of two functions, find the product of their Taylor polynomials, ignoring powers of r higher than α. So we denote this product by T n (rv, r), then we have as n → ∞

F(o, (b n ρ + r)v)Φ(b n ρ + r) = F(o, rv)Φ(r) + α-1 ∑ k=1 T n (rv, r)(b n ρ) k + O(b α n ). It follows that, E R n (r) = β 2 J(r)Φ(r) + β 2 S d-1 α-1 ∑ k=1 T n (rv, r)b k n dv R ρ k K 1 (ρ)dρ + O(b α n ) as n → ∞.
Together with Condition K(1, α) implies the second assertion of Theorem 1.

Proof of Theorem 2. The proof of Theorem 2 makes use of the following corollary.

Corollary 1. Consider any Gibbs point process X in R d with Papangelou conditional intensity λ . For any non-negative, measurable and symmetric function f :

R d × R d × N l f -→ R, we have Var = ∑ u,v∈X f (u, v, X\{u, v}) = 2 E R 2d f 2 (u, v, X)λ (u, v, X)dudv + 4 E R 3d f (u, v, X) f (v, w, X)λ (u, v, w, X)dudvdw + E R 4d f (u, v, X) f (w, y, X)λ (u, v, w, y, X)dudvdwdy - R 4d E[ f (u, v, X)λ (u, v, X)] E[ f (w, y, X)λ (w, y, X)]dudvdwdy.
Proof. Consider the decomposition (see [START_REF] Jolivet | Upper bounds of the speed of convergence of moment density estimators for stationary point processes[END_REF] and [START_REF] Heinrich | Asymptotic Gaussianity of some estimators for reduced factorial moment measures and product densities of stationary Poisson cluster processes[END_REF])

= ∑ u,v∈X f (u, v, X\{u, v}) 2 = 2 = ∑ u,v∈X f 2 (u, v, X\{u, v}) + 4 = ∑ u,v,w∈X f (u, v, X\{u, v, w}) f (v, w, X\{u, v, w}) + = ∑ u,v,w,y∈X f (u, v, X\{u, v, w, y}) f (w, y, X\{u, v, w, y}). (13)
Applying the preceding (GNZ) formula (2) combining with (13), we obtain

Var = ∑ u,v∈X f (u, v, X\{u, v}) = E = ∑ u,v∈X f (u, v, X\{u, v}) 2 -E = ∑ u,v∈X f (u, v, X\{u, v}) 2 = 2 E R 2d f 2 (u, v, X)λ (u, v, X)dudv + 4 E R 3d f (u, v, X) f (v, w, X)λ (u, v, w, X)dudvdw + E R 4d f (u, v, X) f (w, y, X)λ (u, v, w, y, X)dudvdwdy - R 4d E[ f (u, v, X)λ (u, v, X)] E[ f (w, y, X)λ (w, y, X)]dudvdwdy.
We obtain the desired result.

Applying Corollary 1 to this function

f (u, v, X) = 11 W n 2R (u) ||v -u|| d-1 J(||v -u||) L(u, v, X)K 1 ||v -u|| -r b n , it is easily seen that Var R n (r) = A 1 + A 2 + A 3 -A 4 ,
where

A 1 = 2 b 2 n |W n 2R | 2 σ 2 d E R 2d 11 W n 2R (u) ||v -u|| 2(d-1) J(||v -u||) L(u, v, X)K 2 1 ||v -u|| -r b n λ (u, v, X)dudv, A 2 = 4 b 2 n |W n 2R | 2 σ 2 d E R 3d 11 W n 2R (u)11 W n 2R (v) ||v -u|| d-1 ||v -w|| d-1 J(||v -u||, ||v -w||) L(u, v, w, X) K 1 ||v -u|| -r b n K 1 ||v -w|| -r b n λ (u, v, w, X)dudvdw, A 3 = 1 b 2 n |W n 2R | 2 σ 2 d E R 4d 11 W n 2R (u)11 W n 2R (w) ||v -u|| d-1 ||w -y|| d-1 J(||v -u||, ||w -y||) L(u, v, w, y, X) × K 1 ||v -u|| -r b n K 1 ||w -y|| -r b n λ (u, v, w, y, X)dudvdwdy,
and

A 4 = 1 b 2 n |W n 2R | 2 σ 2 d E R d 11 W n 2R (u) ||v -u|| d-1 J(||v -u||) L(u, v, X)K 1 ||v -u|| -r b n λ (u, v, X)dudv 2 .
The asymptotic behavior of the leading term A 1 is obtained by applying the second order Papangelou conditional intensity given by:

λ (u, v, x) = λ (u, x)λ (v, x ∪ {u}) for any u, v ∈ R d and x ∈ N l f .
Using the finite range property (4) for each function λ (u, x) and λ (v, x ∪ {u}) and by stationarity of X, it results

A 1 = 2β 2 b 2 n |W n 2R | 2 σ 2 d R 2d 11 W n 2R (u) ||v -u|| 2(d-1) J(||v -u||) E[ L(o, v -u, X)]K 2 1 ||v -u|| -r b n Φ(||v -u||)dudv = 2β 2 b n |W n 2R |σ 2 d ∞ -r/b n S d-1 J(b n ρ + r) (b n ρ + r) d-1 F(o, (b n ρ + r)w)K 2 1 (ρ)Φ(b n ρ + r)dρdσ (w).
Dominated convergence theorem and assumption of K 1 imply for r ∈ (0, R]

lim n→∞ b n |W n 2R |A 1 = 2β 2 σ d r d-1 J(r)Φ(r) R K 2 1 (ρ)dρ.
We will now show that all other integrals to Var( R n (r)) converge to zero. For the asymptotic behavior of the second term A 2 , we remember the third order Papangelou conditional intensity by λ (u, v, w, x) = λ (u, x)λ (v, x ∪ {u})λ (w, x ∪ {u, v}) for any u, v, w ∈ R d and x ∈ N l f . Since X is a point process to interact in pairs, the interaction terms due to triplets or higher order are equal to one, i.e. the potential γ(y) = 0 when n(y) ≥ 3, for / 0 = y ⊆ x. Using the finite range property (4) for each function λ (u, x), λ (v, x ∪ {u}) and λ (w, x ∪ {u, v}) and after an elementary calculation, we have

λ (u, v, w, / 0) = β 3 Φ(||v -u||)Φ(||w -v||) if ||u -w|| < R β 3 Φ(||v -u||)
otherwise.

Which ensures that λ (u, v, w, / 0) is a function that depends only variables ||v -u||, ||w -v||, denoted by Φ 1 (||v -u||, ||w -v||).

According to the stationarity of X, it follows that

A 2 = 4 b 2 n |W n 2R | 2 σ 2 d E R 3d 11 W n 2R (u)11 W n 2R (v) ||v -u|| d-1 ||v -w|| d-1 J(||v -u||, ||v -w||) L(u, v, w, X) × Φ 1 (||v -u||, ||w -v||)K 1 ||v -u|| -r b n K 1 ||v -w|| -r b n dudvdw = 4 |W n 2R |σ 2 d ∞ -r/b n ∞ -r/b n S d-1 S d-1 |W n 2R ∩ (W n 2R -(b n ρ + r)z)| |W n 2R | × F(o, (b n ρ + r)z, (b n ρ + r)z )Φ 1 (b n ρ + r, b n ρ + r)K 1 (ρ)K 1 (ρ )dρdρ dσ d (z)dσ d (z ).
The asymptotic behavior of the leading term A 2 is obtained by applying the dominated convergence theorem. When multiplied by b n |W n 2R |, we get lim n→∞ b n |W n 2R |A 2 = 0.

Next we introduce the finite range property (4) and reasoning analogous with the foregoing on λ (u, v, w, y, / 0), which ensures that λ (u, v, w, y, / 0) is a function that depends only variables ||v -u||, ||y -w||, ||w -u||, ||w -v||), denoted by Φ 2 (||v -u||, ||y -w||, ||w -u||, ||w -v||). We find that By dominated convergence theorem, we get lim n→∞ b n |W n 2R |A 3 = 0. For asymptotic behavior of the leading term A 4 , it then suffices to repeat the arguments developed previously to conclude the following result.

A 3 = 1 b 2 n |W n 2R | 2 σ 2 d E R 4 11 W n 2R (u)11 W n 2R (w) ||v -u|| d-1 ||w -y|| d-1 J(||v -u||, ||w -y||) L(u, v, w, y, X) × K 1 ||v -u|| -r b n K 1 ||w -y|| -r b n λ (u, v, w, y, X)dudvdwdy = 1 |W n 2R |σ 2 d R d ∞ -r/b n ∞ -r/b n S d-1 S d-1 |W n 2R ∩ (W n 2R -w)| |W n 2R | K 1 (ρ)K 1 (ρ ) × Φ 2 (b n ρ +
A 4 = β 4 b 2 n |W n 2R | 2 σ 2 d R 4d 11 W n 2R (u)11 W n 2R (w) J(||v -u||, ||w -y||) ||v -u|| d-1 ||w -y|| d-1 E[ L(u, v, X)] E[ L(w, y, X)] × Φ(||v -u||)Φ(||y -w||)K 1 ||v -u|| -r b n K 1 ||w -y|| -r b n dudvdwdy = β 4 b 2 n |W n 2R |σ 2 d R 3d |W n 2R ∩ (W n 2R -w)| |W n 2R |||v -u|| d-1 ||w -y|| d-1 J(||v||, ||w -y||) E[ L(o, v, X)] E[ L(w, y, X)] × Φ(||v||)Φ(||y -w||)K 1 ||v|| -r b n K 1 ||w -y|| -r b n dvdwdy = β 4 |W n 2R |σ 2 d R d ∞ -r/b n ∞ -r/b n S d-1 S d-1 |W n 2R ∩ (W n 2R -w)| |W n 2R | J(b n ρ + r, b n ρ + r)K 1 (ρ)K 1 (ρ ) × F(o, (b n ρ + r)z ) F(o, (b n ρ + r)z)Φ(b n ρ + r)Φ(b n ρ + r)dwdρdρ dσ d (z)dσ d (z ).
Then by dominated convergence theorem, we get lim n→∞ b n |W n 2R |A 4 = 0. Therefore, we have finished the proof of Theorem 2.

Proof of Proposition 2 and Proposition 3. The compact set

[r 1 , r 2 ] is covered by the intervals C i = [s i-1 , s i ],
where s i = r 1 + i(r 2r 1 )/N, i = 1, . . . , N. Choosing N as the largest integer satisfying N ≤ c/l n and l n = r n b 2 n . We apply a triangle inequality decomposition allows for sup

s i-1 ≤r≤s i R n (r) -E R n (r)| ≤ sup s i-1 ≤r,ρ≤s i R n (r) -R n (ρ) + sup s i-1 ≤r,ρ≤s i E R n (r) -E R n (ρ) + sup s i-1 ≤ρ≤s i R n (ρ) -E R n (ρ) .
By Lipschitz condition (Condition L p), we derive that there exists constant η > 0 such that n sufficiently large sup

s i-1 ≤r,ρ≤s i R n (r) -R n (ρ) ≤ 1 b 2 n η|r -ρ| 1
σ d |W n 2R | = ∑ u,v∈X ||v-u||≤R 11 W n 2R (u) ||v -u|| d-1 h(u, X \ {u, v}) h(v, X \ {u, v}). ≤ ηr n 1 σ d |W n 2R | = ∑ u,v∈X ||v-u||≤R 11 W n 2R (u) ||v -u|| d-1 h(u, X \ {u, v}) h(v, X \ {u, v}).
Follows from the last inequalities and the Nguyen and Zessin ergodic theorem [START_REF] Nguyen | Ergodic theorems for Spatial Process[END_REF], we get sup

s i-1 ≤r,ρ≤s i R n (r) -R n (ρ) = O a.s. (r n ) as n → ∞.
As well, we obtain sup

s i-1 ≤r,ρ≤s i E R n (r) -E R n (ρ) = O a.s. (r n ) as n → ∞.
We now concentrating on the stochastic part.

Lemma 1. Assume that either (10) holds for some 0 < q < 2 such that R0 ∈ L ψ θ (q) and r n = (log n)

1/q /b n ( √ n) d . Then sup s i-1 ≤ρ≤s i R n (ρ) -E R n (ρ) = O a.s. (r n ) as n → ∞.
Proof. We consider the exponential Young function define for any t ∈ R + by ψ q (t) = exp((t + ξ q ) q )exp(ξ q q ) where ξ q = ((1q)/q) 1/q 11{0 < q < 1}. Let ε > 0 and r ∈ [r 1 , r 2 ] be fixed

P | R n (r) -E R n (r)| > εr n = P |S n | > εr n b n n d ≤ exp - ε r n b n n d ||S n || ψ θ (q) + ξ q q E exp |S n | ||S n || ψ θ (q) + ξ q q .
Therefore, we assume that there exists a real 0 < q < 2, such that R0 ∈ L ψ θ (q) and using Kahane-Khintchine inequalities (cf. [START_REF] El Machkouri | Kahane-Khintchine inequalities and functional central limit theorem for stationary real random fields[END_REF], Theorem 1), we have

P | R n (r) -E R n (r)| > εr n = P |S n | > εr n b n n d ≤ (1 + e ξ q q ) exp - ε r n b n n d M(∑ i∈Γ n b i,q ( R)) 1/2 + ξ q q denote b i,q ( R) = R0 2 ψ θ (q) + ∑ k∈V 1 0 Rk E |k| ( R0 ) 2 ψ θ (q)
. We derive that if condition (10) holds, then there exists constant C > 0 and so if r n = (log n) 1/q /b n ( √ n) d sup r 1 ≤r≤r 2 P(| R n (r) -E R n (r)| > εr n ) ≤ (1 + e ξ q q ) exp -ε q log n C q .

Choosing ε sufficiently large, therefore, it follows with Borel-Cantelli's lemma

P(lim sup n→∞ sup s i-1 ≤ρ≤s i R n (ρ) -E R n (ρ) > εr n ) = 0.
Now, we will accomplish the proof of Proposition 3.

Lemma 2. Assume (11) holds for some p > 2 such that R0 ∈ L p and b n = n -q 2 (log n) q 1 for some constants q 1 , q 2 > 0. .

The last inequality follows from a Marcinkiewicz-Zygmund type inequality by [START_REF] Dedecker | Exponential inequalities and functional central limit theorems for random fields[END_REF], where

c i ( R) = Ri 2 p + ∑ k∈V 1 i Rk E |k-i| ( Ri ) p 2 .
Under assumption [START_REF] Jolivet | Upper bounds of the speed of convergence of moment density estimators for stationary point processes[END_REF] and with the stationarity of X, we derive that there exists C > 0 such that P sup

s i-1 ≤ρ≤s i R n (ρ) -E R n (ρ) > εr n ≤ N sup

≤

κε -p n a(p+1)-d/2-q 2 (log n) b(p+1)+q 1 .

For a(p + 1)d/2q 2 > 1 and b(p + 1) + q 1 > 1, we get for any ε > 0

∑ n≥1 P sup s i-1 ≤ρ≤s i R n (ρ) -E R n (ρ) > εr n < ∞.
Considering these arguments the proofs of Proposition 2 and Proposition 3 are completed, it results from a direct application of the theorem of Borel-Cantelli and by Theorem 1: sup

r 1 ≤r≤r 2 | E R n (r) -R(r)| = O(b α n ) as n → ∞.

  and the norm is | j| = max{| j 1 |, . . . , | j d |} and assume that Γ n increases towards Z d and we split up R n (r) as follows:

  r, b n ρ + r, ||w||, ||(b n ρ + r)z -w||)dwdρdρ dσ d (z)dσ d (z ), where Φ 2 (b n ρ + r, b n ρ + r, ||w||, ||(b n ρ + r)z -w||) = J(b n ρ +r, b n ρ +r) F(o, (b n ρ +r)z, (b n ρ +r)z )Φ 2 (b n ρ +r, b n ρ +r, ||w||, ||(b n ρ +r)z-w||).

  Let a, b ≥ 0 be fixed and denoter n = n a (log n) b /b n ( √ n) d . If a(p + 1)d/2q 2 > 1 and b(p + 1) + q 1 > 1, then sup s i-1 ≤ρ≤s i R n (ρ) -E R n (ρ) = O a.s (r n ) as n → ∞.Proof. Let p > 2 be fixed, such that R0 ∈ L p and for any ε > 0,P(| R n (r) -E R n (r)| > εr n ) = P |S n | > εr n b n n d ≤ ε -p E |S n |

r 1 ≤r≤r 2 P

 2 (| R n (r) -E R n (r)| > εr n ) .As N ≤ c/l n and l n = r n b 2 n , then forr n = n a (log n) b /b n ( √ n) d, it results for n sufficiently large, P sups i-1 ≤ρ≤s i R n (ρ) -E R n (ρ) > εr n ≤ κε -p n a(p+1)-d/2 (log n) b(p+1) b n

Acknowledgment

The research was supported by laboratory Jean Kuntzmann, Grenoble University, France.

Proof of Proposition 1. We consider the mean square error of R n (r),

The estimator R n (r) is asymptotically unbiased by Theorem 1 and so we have

and by Theorem 2, we have

Hence, R n (r) is consistent in the quadratic mean and hence consistent estimate of β 2 J(r) exp(-γ(r)). J n (r) and β n are strongly consistent estimates of β J(r) and β , thus the consistency (convergence in probability) of the estimator Φ n (t) of exp(-γ(r)). Then one gets with the same arguments as before and by Proposition 2 or Proposition 3, we conclude the strong consistency of the estimator Φ n (r) of the function exp(-γ(r)).