Laure Gardelle

Laurence Vincent-Durroux

Hélène Vinckel-Roisin

Speakers, addressees and the referential process: a pragmatic approach [running head: Speakers, addressees and the referential process]

Keywords:

In pragmatics, reference is defined by Abbott (2010: 2) as "the three-place relation that holds when speaker x uses expression y to identify entity z." This is the everyday definition of the verb refer, as in "When you said 'that jerk from the Dean's office,' who were you referring to?" (ibid.). In this approach, the relation between an expression and a referent "is essentially controlled by the referential intentions of the speaker" (Gunkel et al. 2017: 34). 2 Saying that that jerk from the Dean's office "refers to" someone, for instance, is a short-cut for the idea that the phrase enables the speaker to

refer to that personsimilarly, saying that this knife cuts well does not evoke a self-cutting knife, but implies users (Abbott 2010: 2). Such a short-cut, while common and convenient, can be misleading because it easily leads to downplaying, or even discarding, the active role of speakers in establishing or processing reference. 3 This is reflected in definitions of reference such as the following: "[Reference] concerns the ability of linguistic expressions to refer to real-world entities" (Finegan 2015: 209), or "Reference is a two-place relation. One relata refers, the other (the referent) is referred to" (Biggs & Geirsson 2021: 1). This twoplace description of reference as a relationship between a referential expression and an extralinguistic entity also echoes the semantic approach to reference laid out in Frege's seminal paper "Über Sinn und Bedeutung" (1892). Let's take Abbott's example of that jerk from the Dean's office.

"Sinn" (translated in the 1948 English version as "sense"; a more common translation nowadays is "literal meaning") is conveyed compositionally by the descriptive elements. "Bedeutung" ("reference" in the 1948 translation) has to do with what or who the NP designates -let's say, Jeff Peterson. That jerk from the Dean's office and Jeff Peterson have the same reference, although they do not share the same sense (for more detail on the semantic approach, see for instance [START_REF] Abbott | Reference[END_REF].

The aim of the present volume is to show that the difference between the three-place and the two-place description is in fact crucial. Reference is first and foremost the result of choices in a given context, based on a number of linguistic instructions, conventions, cognitive status, but also speaker and addressee mediation, contextual representations and a Theory of Mind [START_REF] Premack | Does the chimpanzee have a theory of mind?[END_REF]. [START_REF] Strawson | On referring[END_REF]'s classic "On referring" pointed out that expressions do not refer inherently, but are produced by speakers in a given context and interpreted by addressees. In this, he was taking one step further Saussure's idea (1916) that language is based on a system of signs defined by the relation between the signifier and the signified ("signification"), which has to be distinguished from the relation between a sign and the object in the world. In this volume, we would like to take one more step. Our aim is to fill a relative gap in existing research not just by acknowledging the role of speakers and addressees, but by giving them pride of place, through a focus on the relationship between conventions and pragmatic strategies in context. The studies in this volume are therefore based on authentic occurrences in their contexts of production (and reception), except for one chapter due to its theoretical framework. A number of chapters also include multimodal environments. This focus on speakers and addressees, in turn, shows how important it is to understand reference by linguistic means as being linguistic in nature. A referential expression is not so much access to a real-world entity as a set of instructions chosen by a speaker at a particular point to access a discourse object deemed accessible within a discourse model, and which has to be interpreted by an addressee.

Section 1 first provides evidence that speakers and addressees matter in the referential process. Section 2 will then focus on a number of conventions that aid them in this task, as established by recent research on reference.

Finally, section 3 will consider how the contributions to this volume further help towards a better understanding of the complex relationship between speakers, addressees, language and the extralinguistic world.

Evidence that speakers and addressees matter

Referents are mental objects more than real-world entities

In semantic approaches to reference, an expression is referential if it points to a specific real-world entity (or set of entities)or an entity in an imaginary world, such as a unicorn or a fictional character. As a consequence, a number of linguists such as von Heusinger (2002) or Trask (2007: 245) regard generics (cats), quantifier phrases (every problem) and phrases used attributively (e.g. the fastest runner in Brazil used with no knowledge of who that is, though with the conviction that a real-world entity fits that description) as non-referential. To them, these NPs are definite, but do not have reference. This, however, we take to be a restrictive view of reference, because it fails to capture the linguistic nature of referential expressions. There are more advantages, we suggest, to taking a unifying view of referential expressions as referring to mental objects, created through discoursehence the notion of "discourse referents."

The concept of "discourse referent," or "discourse entity" was first introduced by [START_REF] Karttunen | Discourse referents[END_REF] in his study of indefinite NPs (according to [START_REF] Corblin | Les Chaînes de Référence dans le Discours[END_REF]. As noted by Kleiber (1997: 121), the concept does not mean that entities only exist in discourse; rather, they are created through discourse. A discourse referent often corresponds to a real-world entity, but it does not have to. It may be part of an imaginary world, but more importantly, indefinite NPs may establish "short-term" referents, which have no specific reference in the extralinguistic world (they are "nonspecific"), but are temporarily "interpreted specifically" in discourse, within a limited domain [START_REF] Karttunen | Discourse referents[END_REF]. One example given by Karttunen (1976: 375) is epistemic scenarios, as in (2): "[a]t least with modals (and the future will), it is possible to continue discussing a thing that actually does not yet exist, provided that the discourse continues in the same mode." Other cases include short-term referents constructed within the scope of a quantifier, as in (3), or suppositions, as in (4).

(2) You must write a letter to your parents. It has to be sent by airmail.

The letter must get there by tomorrow.

(3) Harvey courts a girl at every convention. She always comes to the banquet with him. The girl is usually also very pretty.

(4) Suppose Mary has a car. She takes me to work in it. I drive the car, too.

To [START_REF] Karttunen | Discourse referents[END_REF], what an expression needs in order to be referential is to provide access to a specific discourse entity. When no specific discourse construct is meant, then an expression is not referential. For instance, a car is not referential in Bill doesn't have a car because it could not be followed by *It is black. Or a Norwegian interpreted non-specifically in Janet wants to marry a Norwegian ('she does not have anyone special in mind, but it is a matter of principle to her'), is non referential because it would not license the continuation She introduced him to her mother yesterday. This is also the conclusion, for instance, of [START_REF] Huddleston | The Cambridge Grammar of the English Language[END_REF]), Trask (2007: 245) or Finegan (2015: 209). But here again, [START_REF] Karttunen | Discourse referents[END_REF] points out that a short-term referent might be established by the same expression in the same sentence if the speaker then elaborates on the hypothetical scenario, for instance with the continuation Suppose that it is true, then she will certainly introduce him to her mother. In other words, within this scenario, a Norwegian and him are interpreted specifically, despite the absence of a corresponding specific entity in the extralinguistic world when the sentence is produced. As shown from these examples, an NP "establishes a discourse referent" when "there may be a coreferential pronoun or definite noun phrase later in the discourse" (Karttunen 1976: 385).

The notion of discourse referents as mental constructs can be (and has been) further dissociated from extralinguistic entities than in Karttunen (1976)'s proposal: a first mention may be considered referential even when there is no anaphoric pronoun in the continuation. Hedberg et al. (2019), in the Givenness Hierarchy framework, regard generics and expressions used attributively as referential because even with non-specific expressions, the speaker expects the addressee to construct or access a representation. For Aguilar-Guevara, Le Bruyn & Zwarts (2014: 1), non-specific readings display "weak referentiality" because they license an anaphoric pronoun. As summed up by Langacker (2008: 270), "[f]rom a linguistic standpoint these nominals are indeed referential, as they single out a grounded instance of a type as their referent. Their special property is that they profile a virtual instance rather than an actual one." Not all linguists will agree with this conclusion. Bach (2008: 16, 29), for instance, considers that "so-called discourse reference is not genuine reference." In non-specific Janet wants to marry a Norwegian, he concludes that as the first mention (e.g. a Norwegian) is not referential (the speaker does not have a specific person in mind), neither are the anaphoric pronouns.

To him, even reference to a fictional character or a unicorn is "pseudoreference" rather than genuine reference. In the wake of Langacker (2008: 15), we claim instead that "reference to 'virtual' entities (e.g. any cat) [is] pervasive," and exemplifies "the diverse array of mental constructions that help us deal withand in large measure constitutethe world we live in and talk about." A conception of reference such as Bach's makes "truth" and truth conditionality all important, whereas the extralinguistic world is only a human-filtered view of "reality". Similarly, not all referential expressions are meant to have clearly identifiable referents, and yet a specific referent exists. One example is the French impersonal pronoun on or German man (cf. English one) (see for example [START_REF] Fløttum | La présence de l'auteur dans les articles scientifiques: étude des pronoms je, nous et on[END_REF][START_REF] Truan | Generisch, unpersönlich, indefinit? Die Pronomina man, on, one und generisches you im politischen Diskurs[END_REF][START_REF] Delaborde | Analyse en Corpus de Chaînes de Coréférence[END_REF]): its fuzzy reference entails that, in (5), there is no evidence that the three occurrences are co-referential):

(5) après un quart d'heure d'attente [...], on vint enfin lui apprendre que mademoiselle Pauline D... n'était point mariée, et qu'elle habitait toujours la ville. Aussitôt l'étrangère ordonna qu'on mît sa voiture sous la remise et qu'on lui préparât une chambre.

('after a quarter-of-an-hour wait […], ['on'] came at last to tell her that miss Pauline D… was not married, and that she was still living in town.

At once the stranger ordered that ['on'] put her carriage under the shed and that ['on'] get a room ready for her.' (our translation) (Sand 1881example cited by Delaborde 2020: 105) We suggest that denying reference in fiction (fictional characters) the status of "genuine reference" is similarly too restrictive a view of cognition: the referential expressions and their processing involve the same mechanisms.

The notion of "discourse referent" usefully highlights the speaker (and addressee) mediation at work in any form of communication. A referential expression is invariably interpreted against a "Mental model" [START_REF] Johnson-Laird | Mental Models. Towards a Cognitive Science of Language, Inference and Consciousness[END_REF], or a "text-world model" (Textweltmodell, [START_REF] Schwarz | Indirekte Anaphern in Texten: Studien zur Domänengebundenen Referenz und Kohärenz im Deutschen[END_REF][START_REF] Schwarz | Establishing coherence in text. Conceptual continuity and textworld model[END_REF][START_REF] Schwarz-Friesel | Indirect anaphora in text: a cognitive account[END_REF]) or, for Webber (2016[1979]: 2), a dynamic "discourse model", which "contains a collection of entities, recording their properties and the relations they participate in."4 When using a definite pronoun, for instance, the speaker has the referent in his/her own discourse model, "assumes that on the basis of the discourse thus far, a similar entity will be in the addressee's (partially formed) model," (Webber 2016(Webber [1979]]:

2), and presumes that the addressee is able to access it. A key notion, therefore, is that of speaker "intention": a speaker intends to point the addressee to the discourse entity (ibid., see also Gunkel et al. 2017: 34).

Importance of speaker intentions

As noted by Hanks (2019: 16), "[i]n ordinary conversation we never just refer to something. We always do so as part of a broader speech act"assertions, promises, orders, and so on. Addressees therefore expect speakers to have intentions.

Intentions imply control over the use of reference (O'Rourke 2019: 10).

As a consequence, if a 20 th -century speaker states The king of France is wise, an addressee will probably not immediately discard it as uninterpretable ('there is no king in France'). Because speaker control is expected, the addressee will rather try to understand how there may be a referent to the king of France in the speaker's mindconsidering this as a description for a work of fiction, for instance, or evidence that the speaker believes that there is a king in France, or gives the statement a generic (Strawson 1950: 329-330) or a metaphoric value.

Because speech itself is "initiated, monitored, and concluded intentionally," and because "its contours reflect conscious, intentional influence if not outright control," (O'Rourke 2019: 22), in some contexts the form of a referring expression is a crucial way for a speaker to express stance, possibly with domination issues involved. Salazar [START_REF] Orvig | La co-construction: une facette dialogale du dialogisme? Talk given at the Colloque international Dialogisme: Langue[END_REF] show how in a conversation between a French patient and her psychiatrist, the latter's use of ces rêves-là 'those dreams' where the patient uses cauchemars 'nightmares,' shows distancing of the professional from the patient's anguish. Speaker positioning is also evidenced among children.

Salazar Orvig (2019: 302)'s study of a conversation between a French child aged 1;11 and her father finds that the child uses dislocations when she disagrees with him (il peut pas monter le bonhomme 'he can't get on(,) the man'), as opposed to just a pronoun when she shares his perspective (il est un peu gros 'he is a bit fat'). This example illustrates a more general tendency in children to use dislocation to "assert their own positioning as speaker" (Salazar Orvig et al. 2010: 397-398), with the possible addition of "yes" or a negation. More generally, children tend to use third-person pronouns in subject position for plain continuity, and nouns (or demonstratives) in cases of contrast (Salazar Orvig 2019: 302) with possible (re)assertion.

Importance of addressees

Reference is about providing access to a linguistically established entity.

Consequently, addressees are crucial for reference, as too seldom pointed out in studies on reference, in at least two ways. 5First, as speaker reference is an act of communication, it is "essentially audience-oriented", in that "a speaker uses an expression to refer his audience to an individual" (Bach 2008: 17). Reference is thus not just a three-place relation, as suggested by Abbott (2010)'s definition given at the beginning of this chapter, but a four-place relation, "between a speaker, an expression, an audience, and a referent: you use an expression to refer someone to something" (Bach 2008: 15). This importance of addressees is evidenced by foundational processes in language acquisition, such as establishing common communicative ground and joint attention (Tomasello 2003: 65). Studies in the field of language acquisition have shown that manual pointing is of paramount importance in acquisition, as it creates the relation between toddler, addressee and objects, a "referential triangle" [START_REF] Tomasello | The Cultural Origins of Human Cognition[END_REF] which often leads the adult to provide the name of the object pointed at by the child. Tomasello (2003:65) reports further findings on "a very high correlation between the amount of time infants spent in joint engagements with their mothers (…) and the size of the infants's vocabulary".

A cooperative speaker has to take into account the interlocutors' Common Ground, that is, the "body of information which interlocutors have reason to believe that each other have access to in the context of utterance" (Roberts 2019: 274). The Common Ground contains not only explicitly used term, was dispreferred here because it implies a more passive attitude than listener or asserted propositions, but also all kinds of (supposedly) shared information, including shared encyclopaedic knowledge, conforming to the Theory of Mind. Reference is largely based on inferences, and reference processing requires constant "dynamic update and downdate" (Roberts 2019: 277). For instance, Paul bought a Toyota because they're such robust cars (adapted from Kleiber 1991) requires an addressee to understand they as referring to Toyotas despite the lack of a plural antecedent. Another example is indirect anaphora, as in I have been angling for hours, but they just won't bite (Ich angele jetzt schon seit Stunden, aber sie wollen einfach nicht anbeißen, Schwarz 2000: 123).

Speakers are typically cooperative, following Grice's cooperation principle: they are expected to consider the Common Ground, and adjust to addressees as necessary. Cooperation, however, is just one convention, as shown by the cauchemars ('nightmares') example above. A speaker may choose to ignore elements of the Common Ground, which may require addressee adjustment (example of the king of France above) or lead to failure to achieve reference (the addressee may then have to ask for clarification or rephrasing).

Based on these facts, [START_REF] Consten | Anaphorisch oder Deiktisch? Zu einem Integrativen Modell Domänen-gebundener Referenz[END_REF] proposes to define reference as a "hearer-centred process" (hörerseitiger Prozess, literally 'a process on the hearer's side'), "triggered by language reception" (6). 1995: 466). (Consten 2004: 56) ('I understand reference as a process taking place on the hearer's side, which is triggered by language reception andtaking situational factors into accountaims at identifying a reference object. A corresponding mental image of the referent is activated and integrated into the current discourse representation. I expressly include the latter process in the definition of reference, because the ability to be integrated into the current discourse is the most important criterion for selection from various possible referential objects, for example in the case of ambiguous reference and for the instantiation of referents (see Schwarz 1995: 466).') Addressees are also essential to reference for a second reason: they are actors in the processing of referential expressionsin spoken communication, they are "listeners" rather than just "hearers." When processing messages, they do not always process referential expressions in depth. This is known as "good-enough" comprehension [START_REF] Ferreira | Goodenough representations in language comprehension[END_REF], or "shallow processing" (Kaiser & Fedele 2019: 332), and is the result of limited attentional resources (Kaiser & Fedele 2019: 331). An example of shallow processing is provided by [START_REF] Stewart | Shallow processing of ambiguous pronouns: evidence for delay[END_REF], who measured whole-sentence reading times for fabricated sentences with ambiguous or unambiguous pronouns, such as Paul lent {Rick / Kate} the CD before {he / she} left for the holidays. When they asked comprehension questions, they found that when the questions did not require pronoun resolution (e.g. Did Paul lend Rick the CD?), reading times were not affected by whether the pronoun was ambiguous or not. This is evidence of shallow processing: ambiguous pronouns are left underspecified at this stage, and will be resolved later if necessary, for instance if there is another question or if disambiguating information is provided (see Kaiser & Fedele 2019 for similar findings from other studies). When the comprehension questions did require pronoun resolution, on the other hand, informants had to engage in deep processing, so that ambiguous pronouns led to longer reading times.

Good-enough processing may lead addressees to be blind to errors in referential expressions. One example is the Moses Illusion: people, even when they are familiar with the Bible, tend to answer "two" to the question how many of each type of animal did Moses take on the ark?, not noticing that "Moses" is used instead of "Noah" [START_REF] Erickson | From words to meaning: a semantic illusion[END_REF]. This example is evidence that comprehension relies very much on prediction: the information structure of a wh-question presupposes that everything apart from the wh-phrase (here, "Moses took x animals on the ark") is considered true. Due to expectations of speaker control and cooperation, there is no reason for the addressee to doubt the truth of that presupposed information.

Another example is provided by eye-tracking experiments: a highly predictable word is more likely to be skipped than other words (Ferreira & Lowder 2016: 224). [START_REF] Clark | Whatever next? Predictive brains, situated agents, and the future of cognitive science[END_REF] therefore describes the brain as a "prediction engine": rather than passively process the information as it is received, the addressee constantly predicts what might be said next and adjusts his/her representations when necessary (Ferreira & Lowder 2016: 224). This process, also known as "forward modeling" [START_REF] Pickering | Toward a mechanistic psychology of dialogue[END_REF], is facilitated by the prototypical information structure in discourse. Typically, information is ordered from given to new. In addition, in Centering Theory, [START_REF] Grosz | Centering: a framework for modeling the local coherence of discourse[END_REF] have established the importance of "forward-looking" and "backward-looking" "centers" (that is, entities, referents) in building local coherence over several segments of discourse: they help speakers predict the most likely continuations. For example, (7) illustrates a type of transition called "center retaining" ("retention of the center from one utterance to the next," but not to subsequent utterances, Grosz, Joshi & Weinstein 1995: 210). (7a) exhibits reference to two entities, Susan and Betsy, called two "forward-looking centers." The highest-ranking one is Susan (as it is in subject position, in accordance with the Centering Theory's grammar-based ranking principles), who consequently becomes the preferred backward-looking center in (7b). In (7b), Susan is still the highest-ranking forward-looking center (pronoun in subject position). This makes her the backward-looking centre of (7c), hence reference via a personal pronoun there (her). In (7c), however, Susan is no longer the highest-ranking forward-looking center for what will come next: it is now Betsy's turn, as she is referred to by a proper name in subject position. This predicts that if there was a segment (7d), it would probably exhibit a "smooth shift," with Betsy now referred to with a personal pronoun [START_REF] Cornish | L'accessibilité cognitive des référents, le centrage d'attention, et la structuration du discours: une vue d'ensemble[END_REF]Vinckel-Roisin 2012). By generating a set of likely candidates for coming referential expressions, prediction processes facilitate the integration of new information in discourse processing (Ferreira & Lowder 2016: 241).

Prediction failures, which are also part of the process, allow addressees to update their models of a given domain (Ferreira & Lowder 2016: 240-241).

In addition to predicting, addressees restore, as shown by several experiments [START_REF] Handel | Listening. An Introduction to the Perception of Auditory Events[END_REF]). Contrary to prediction, which is a form of anticipation, the restoration process works backwards. For example, missing phonemes may be automatically restored, as in "It was found that the -eel was on the orange," for which participants were convinced that they had actually heard peel. The whole context facilitates meaning retrieval.

The constraints of cognitive limitations on reference processing are further evidenced by studies of Alzheimer's Disease patients, both for reference interpretation and production. Due to working memory impairments, AD patients "are better able to maintain information about referents when full NPs are used", unlike healthy subjects [START_REF] Almor | Why Do Alzheimer Patients Have Difficulty with Pronouns? Working Memory, Semantics, and Reference in[END_REF]. They find pronouns more difficult to process because they lack descriptive content, thus requiring more reliance on the semantic information stored in the working memory. Conversely, when producing narrative discourse, they use fewer of "the referential expressions expected at the introduction and shift stages" than healthy aged adults, probably due to lexical retrieval difficulties or, here again, working memory impairments [START_REF] Sandoz | Referential adjustment during discourse production in Alzheimer's disease[END_REF]. Now that we have established that speakers and addressees are central to the selection and processing of referential expressions, we take a closer look at a number of conventions that aid them in this task.

Conventions underlying the referential process

The linguistic material of a referential expression is an obvious influence on what it may or may not select in context: it provides instructions that help to identify the referent. More specifically, lexical elements contribute descriptions (e.g. black cat in that black cat), while determiners or pronouns give processing instructions (e.g. that expresses ostension and, in contrast with this, a form of distance). As a consequence, when in a given context, a number of expression types might be acceptable, they have different implications. For instance, in (8), the demonstrative is a way to "appeal to some shared experience, attempting to bring the past to life" (Ariel 1990: 199), whereas a definite article would just uniquely identify the referent. This, we suggest, is due to the extra "pointing" that the demonstrative provides, here pointing to shared memories.

(8) (Ariel 1990: 199) That holiday we spent in Cyprus was really something, wasn't it?

But beyond this linguistic material, as summed up by Aguilar-Guevara, Le Bruyn & Zwarts (2014: 1), "[m]any different factors can play a role in reference apart from the noun and the determiner, like the linguistic and non-linguistic context, the salience of the referent, world knowledge, and the syntactic position and information status of the noun phrase." It would be impossible to consider them all here. Within the linguistic material alone, beyond the referential expressions themselves, there are many influences.

One of them is coherence-based relations between constituents, as pointed out in section 1 with Centering Theory (see also alternative frameworks, such as the functional model of centering of [START_REF] Strube | Functional centering[END_REF][START_REF] Tomasello | The Cultural Origins of Human Cognition[END_REF].

Discourse structure may establish topics or centres of attention, through information structure (including the choice of syntactic subject, or suprasegmental features such as intonation), but also in relation to the semantics of the verb. For instance, with because, some verbs favour subject continuation, others object continuation. The prisoner confessed to the guard because he… projects the prisoner as the likely referent of he, whereas the mother punished her daughter because she… projects the daughter as the likely referent of she (Garvey & Garamazza 1974, Kaiser & Fedele 2019: 327). Free indirect discourse (FID) might also create its own bias for pronoun resolution, depending on the "perspectival structure" it establishes [START_REF] Kaiser | Perspective-shifting and free indirect discourse: experimental investigations[END_REF]. For example, Mary looked woefully at Elizabeth.

Poor girl; she was sick leads the reader to interpret she as referring to Elizabeth, which is not the case without the FID cue poor girl: in Mary looked woefully at Elizabeth. She was sick, the reference of she is ambiguous.

In this section, we will take a closer look at two types of conventions that have been the focus of a lot of recent research: the cognitive status of the referent, and genre.

Influence of cognitive status on the form of referential expressions

Cognitive research on reference in the past twenty years has shown a relationship between expression types and cognitive statuses, a convention which facilitates reference processing. The exact relationship is not unanimously established to date, except for personal pronouns. In particular, it still requires further research into the link between cognitive requirements and linguistic material. But existing studies do exhibit some definite trends.

We will consider two approaches here: Ariel's (1990) Accessibility Marking Scale, and Gundel et al.'s (1993) Givenness Hierarchy.

In Ariel (1990)'s model, the Accessibility Marking Scale (9), referring expressions are ranked according to the degree of Accessibility of the referent that they signal: "[t]he choice of a referring expression is dependent on the Accessibility status the mental representation of the referent is assumed to have for the addressee at the current stage of the discourse" (Ariel 1990: 69). As Ariel (1990: 198-99) herself stresses, a speaker is not constrained by the degree of Accessibility signalled by the expression type: in some cases, "despite an assessed degree of Accessibility x, a speaker chooses a marker associated with either a lower or a higher degree of Accessibility than x." However, such discrepancies are meant to "encourage an addressee to derive specific additional contextual implications." This is confirmed for instance by (10), in which topic continuation might have triggered he rather than the NP the animal. Going for a phrase that marks a lower degree of Accessibility suggests a change of perspective, from a person-like treatment of the cat, making decisions and interacting with guests, to distancing as the speaker looks for an explanation (It was not that...).

(10) [START_REF] Morris | Catlore[END_REF]) Thomas Huxley, the great biologist, whose household was dominated by a long series of cats over a period of forty years, described how one of them, a young tabby tom-cat, developed the alarming game of jumping on the shoulders of his dinner-guests and refusing to dismount until they fed him some titbit. It was not that the animal was hungry. It was the shock impact of the game that provided the reward. [START_REF] Gundel | Cognitive status and the form of referring expressions in discourse[END_REF], rather than make predictions about degrees of Accessibility, sought to establish minimal cognitive requirements for each expression type, and to characterize cognitive statuses. They proposed the Givenness Hierarchy (11). Again, there is no one-to-one mapping. A given cognitive status implies all those to its right (for instance, a referent that is activated is also familiar, uniquely identifiable and so on), so that often, in theory, an expression type might be replaced by one of those to its right.

The Hierarchy only establishes minimal requirements, which are also preferences. For instance, in English, a personal pronoun requires a referent that is "in focus", while that + N requires a referent that is not just uniquely identifiable, like the + N, but at least familiar.

(11) [START_REF] Gundel | Cognitive status and the form of referring expressions in discourse[END_REF] A given expression type might be used for cognitive statuses to the left of its preferred status. For instance, even though that + N requires a referent that is at least familiar, it may be used for one that is activated or in focusbut it may not be used for a status to its right, for instance for an entity that is uniquely identifiable, but not familiar. A personal pronoun has to be used for a referent "in focus". [START_REF] Gundel | Cognitive status and the form of referring expressions in discourse[END_REF] propose that deviations from the preferred expression type for a given cognitive status is the result of specific conversational implicatures, in relation to Grice's maxim of quantity. 6 There are probably other factors as well, one of which is genre -see 2.2 below. But the association between expression types and preferred or minimal cognitive statuses does provide a refined understanding of speaker choices in context.

For instance, because a personal pronoun implies a referent "in focus", a seemingly antecedentless pronoun may reflect the fact that the referent is a major continuing concern of the focaliser's, as in ( 12) (Emmott 1995: 246).

(12) (Rankin 2000: 173-4) It was late evening, and Rebus was driving.

Not the M90 into Fife: tonight, he was on the M8, heading west, heading for Glasgow. He'd spent half an hour at the hospital, followed by an hour and a half with Rhona and Jackie Platt, their guest for dinner at the Sheraton. He'd worn a fresh suit and shirt.

He hadn't smoked. He'd drunk a bottle of Highland Spring.

They were planning yet more tests on Sammy. The neurologist had taken them into his office and talked them through the procedures.

Here, they does not refer to Rebus and his wife Rhona, or Rhona and their guest, despite the previous mention. Rebus's "overriding concern" in the novel is Sammy's medical situation, so that it remains in focus in internal focalisation passages, as an arch-topic. The move away from Rhona and Q2: Do not make your contribution more informative than required.

Jackie Platt at the end of the first paragraph (He), together with the paragraph break and the shift from the past perfect to the past progressive (were planning), prepares readers for a possible return to that major concern of Rebus's. Reference processing of They might involve initial repair on readers' part, but this is easily achieved once they read tests on Sammy (Emmott 1995).

Another example of the influence of perspective on cognitive status, and consequently, on the choice of referential expressions, is "override" reflexive pronouns in English, that is, the use of reflexives where the default use would be a personal pronoun (e.g. Ann claimed that junior lecturers like herself were being exploited. / Paul was determined to be promoted ahead of Sue. That profile of himself in the company newsletter would certainly help.). These third-person override reflexives are often found in free indirect style, as in these examples, when the antecedent refers to the focaliser [START_REF] Huddleston | The Cambridge Grammar of the English Language[END_REF]: 1495).

Influence of genre on referential expressions and referential chains

As noted by [START_REF] Thurmair | Referenzketten im Text: Pronominalisierungen, Nicht-Pronominalisierungen und Renominalisierungen[END_REF], the influence of genre ('Textsortenspezifik') has long been underestimated in linguistics. Recent research has established that genre definitely has an influence on the properties of referential chains (e.g. [START_REF] Thurmair | Referenzketten im Text: Pronominalisierungen, Nicht-Pronominalisierungen und Renominalisierungen[END_REF][START_REF] Schnedecker | Les chaînes de référence: présentation[END_REF][START_REF] Schnedecker | Les Chaînes de Référence en Français[END_REF]. It is more generally one of the five external constraints on the composition of referential chains, as defined by Schnedecker (2021: 149-199), along with the period in which the discourse was produced, the language or family of languages in which it is produced, the ontological nature of the entities, and the medium of communication. There are additional internal constraints related to discourse structure (ibid.).

Some data show an influence of genre on expression types, regardless of issues of cognitive status. For instance, Condamines (2005: 45) finds that in French, hyperonymic anaphoric expressions are significantly more common in technical texts than in novels. Press articles commonly exhibit what Ranger (2002: 87) calls "dispensable lexical anaphoric expressions", that is, NPs whose lexical content is not directly relevant at the point when it is used. For instance, in (13), the information contained in the underlined NP (age, titles of famous films) is irrelevant to understanding why Polanski pleaded guilty (Baumer et al. 2021: 38). This generic convention therefore ignores principles of relevance, necessity of the linguistic material for identification, or cognitive status. A reader who is not very familiar with

Polanski might even find it difficult to process reference here, because the information might be new to him/her. The director had travelled to Switzerland to accept a lifetime achievement award at the Zurich film festival, the organisers of which expressed "great consternation and shock" at his detention.

The 76-year-old director of Rosemary's Baby and Chinatown pleaded guilty to the assault at the time but jumped bail and fled the US the following year to avoid a lengthy jail sentence.

Similarly, in advertising discourse in France, it has become acceptable to use a low Accessibility marker for a highly accessible referent. Schnedecker (1995) calls these referential expressions "rebel redenominations" (redénominations rebelles). One instance is an advert for the skin care product Extra-Vieille. ('Mistakes on the football pitch are the players', but the original mistake lies with their coach. With his choices, Löw posed as a master above the reality of the championship that his team had built so far.

Löw likes to do unexpected things, to act in an unconventional way, to ignore traditional, often trivial, certainties. His ideas have often proved productive: […])

Conventions might evolve over time. For instance, in Anglo-Norman judicial documents, proper names were repeated even for referents in focus because it was a convention of the genre (Capin 2014: 73).

Beyond an influence on preferences for individual referential expressions, genre has an effect on the overall properties of reference chains, such as distance between links of the chains or proportions of pronouns. 7 Tutin (2002), for instance, concludes from an exploratory corpus of French texts that referential density, defined as the proportion of referential expressions against the overall number of words, is higher in novels than in human sciences research papers; or that long-distance pronouns (pronouns whose closest non-pronominal antecedent is at a long 7 Distance calculations are themselves a major issue in studies of reference chains. One complex question is where a reference chain begins and ends. For instance, it would not be realistic to regard all references to the protagonist of a novel as a single reference chain (Schnedecker & Landragin 2014: 6). But where exactly to end one chain is no easy matter. Another, related issue is that of the most relevant criteria for measurement. If what matters is that the referent should be activated in the discourse model, for instance, Combettes & Tomassone (1988: 14) ask whether time of activation should be measured in terms of reading time, or of number of sentences between two links in the chain, or even of speaker (as opposed to addressee) activation of the referent.

distance) are much more common in the former than in the latter. Similar comparative studies have been carried out for other genres, such as dosage instructions or user manuals [START_REF] Thurmair | Referenzketten im Text: Pronominalisierungen, Nicht-Pronominalisierungen und Renominalisierungen[END_REF], recipes and introductions to news items [START_REF] Schnedecker | Impact des genres sur la composition des chaînes de référence: le cas des faits divers[END_REF], encyclopaedic articles [START_REF] Schnedecker | Chaînes de référence et variations selon le genre[END_REF], or short stories and journalistic portraits (Baumer 2015). A study of a homogeneised corpus, viz. soft news articles with a title in the form "topiccomment" (e.g. Museum's £3m Cezanne stolen), suggests that "referential chains are still heterogenous", but that there are indeed genre-related properties (Baumer et al. 2021). Many issues deserve further investigation, in particular the interaction between these data and cognitive status, linguistic material; or which genres are relevant for categorisation, beyond very specific ones such as recipes. Part IV focuses on the pragmatics of reference in contexts of interaction.

Manfred Consten first brings new insights as to how crucially important addressees are in referentialization strategies. From a corpus of oral conversations in German, he shows that reference is not just about activating referents and making them part of a mental discourse representation. Rather, it is a "collaborative, interactive procedure".

Participants may agree about the concept to which they refer, but they may also cooperatively look for the right concept, or find that even though they use the same referential expression, they are not targeting the same concept.

As a consequence, reference should not be considered as a "chain of single acts", but as an "ongoing communicative activity". from L2 learners, but moves away from nominal reference to consider mention), type of activity (e.g., playing, reading), and discourse type (e.g., action description, event description). They find that in both children aged 2 and adults, it would be wrong to claim that just one of these parameters has an influence. They also find that the strength of these parameters differs partly in the two sets of speakers. They suggest that children might not be considering the cognitive status of the entity, but because they repeat or connect to the adult's uses, they experience the use of pronouns in adequate positions, which enables them to form generalizations.

  Ich fasse Referenz als hörerseitigen Prozess auf, der durch Sprachrezeption ausgelöst wird undunter Einbeziehung situativer Faktorenauf die Identifikation eines Referenzobjektes abzielt. Ein entsprechendes Referenzkonzept wird aktiviert und in die aktuelle Diskursrepräsentation integriert. Letzteren Prozess beziehe ich ausdrücklich in die Referenzdefinition ein, denn die Integrierbarkeit in den aktuellen Diskurs ist das wichtigste Kriterium für die Auswahl aus verschiedenen möglichen Referenzobjekten, etwa im Falle ambiger Referenz sowie für die Instanziierung von Referenten (vgl. Schwarz

  gave Betsy a pet hamster. b. She reminded her that such hamsters were quite shy. c. Betsy told her that she really liked the gift.

  Markers (gaps, including pro, PRO and wh traces, reflexives, and Agreement) High Accessibility

(

  13)(Addley & Connelly 2009) But, it was revealed today, the past has finally caught up with Roman Polanski. The film director was arrested in Switzerland on Saturday on a decades-old warrant relating to the rape of a 13-year-old girl in 1977.

(

  14) (example cited by Schnedecker 1995: 317) Extra-Vieille est un accord parfait de notes hespéridées, délicatement soutenues par un coeur orange et agreste. Extra-Vieille existe en eau de Cologne, en bain moussant, en gel douche, en lait pour le corps, en savon et en déodorant. ('Extra-Vieille is a perfect mix of citrus notes, delicately enhanced by a rustic orange core. Extra-Vieille comes in cologne, foaming bath, shower gel, body lotion, soap and deodorant.') A last example, from present-day German, is the use of proper names in press comments. A journalist who takes an argumentative stance will very commonly repeat proper names for a highly accessible referent, even when reference is unambiguous (see Vinckel-Roisin 2018). In extract (15) below, which comments on the national German football team's defeat to Italy, the journalist's criticism of coach Joachim Löw is made explicit in the fifth paragraph. The first two sentences convey the journalist's claim; the third, in which the proper name Löw is repeated, marks a shift to the arguments that substantiate the claim. Repeating Löw therefore hints at a change in stages of stance-taking, in a "claim → arguments" sequence (Vinckel-Roisin 2018: 177-179). (15) (example cited by Vinckel-Roisin 2018: 178) Die Fehler im Spiel haben die Spieler begangen, aber der Ur-Fehler stammt von ihrem rainer w hat sich mit seinen Beschl ssen ber die Turnier-Wirklichkeit erhoben, die sein Team bis dahin aufgebaut hatte. w liebt es, das Unerwartete zu tun, unkonventionell zu handeln, sich ber die klassischen, oft ja auch platten Weisheiten hinwegzusetzen. Seine Ideen haben sich oft als produktiv erwiesen: [...] (sueddeutsche.de, 29.06.2012)

8

  We would like to thank the 33 anonymous reviewers for their very helpful comments on the drafts of the chapters.This volume brings together a cross-section of current research on referential conventions and pragmatic strategies, in a number of different fields (formal and theoretical linguistics, semantics, discourse analysis, psycholinguistics, interactional linguistics, natural language processing), different verbal and non-verbal languages (English, German, different varieties of French, Indonesian, Belgian sign language) and different contexts (language acquisition, second language learning, and various genres such as news articles, narratives, game playing or satire). It is meant as a series of thought-provoking studies which place speakers and addressees at the core of the referential act. We hope that the volume can modestly contribute to a refined understanding of the nature of conventions when encoding or decoding referential expressions, and of the relationship between internalised conventions and pragmatic choices.Part I introduces new theoretical insights into referential conventions.Manfred Krika & Fereshteh Modarresi consider the relationship betweensyntactic integration within a noun phrase (degrees of object incorporation, as illustrated for instance in German, for the activity of fishing, by viele Fische fangen, Fische fangen, am Fische fangen sein, beim Fischfang sein and fischen), degree of definiteness (strong definites, weak definites, bare nominals) and likelihood of anaphoric uptake. They conclude from experiments with constructed examples that even though highly incorporated objects (am Fischefangen sein) and weak definites (ins Kino gehen 'go to the pictures'), for instance, mention entities (fish and the pictures in these examples), syntax prevents those entities from reaching discourse referent status, and consequently, from being antecedents for subsequent mentions. This finding has theoretical implications beyond German: the authors also include references to Persian and pseudoincorporated objects. Elise Mathurin, working on English it in weather sentences, calls for a theoretical distinction between "referential" and "replaceable by a full NP". In the wake of Bolinger's notion of "ambience" and Langacker's Cognitive Grammar framework, she argues for a deictic dimension to it, and coins the concept of "referential intention": the addressee is expected to consider an abstract setting that is both obvious in nature and vague enough to be hard to delimit exactly. Laure Gardelle considers generic bare plurals in characterizing sentences, such as birds in birds fly, which she argues are referential. She focuses on how plurals may license exceptions in context, an issue which is not satisfactorily accounted for by existing quantification or probability models. Using the "generics-asdefault" approach of psychologists instead, she adds the notion of "negligibility" to account for exception licensing, with the crucial notion that pragmatic features such as speaker knowledge and beliefs play a role in the contextual evaluation of what is negligible or not. She further shows that negligibility is also at work with plurals in specific contexts, because plurals convey generalizations through a process which she terms "homogeneization".The next two chapters consider reference from the perspective of reference chains, an area which has only been partially investigated (see above).DominiqueDias, focusing on genre, seeks to understand what exactly, within a genre, has an influence on reference chains. From a corpus of 80 newspaper literary reviews in French and German, he concludes that a strictly global approach to genre is not satisfactory. Taking a more local, incontext perspective, he finds an influence of the type of referent (because referents of different types are involved in different processes), of editorial conventions in a given newspaper, and of writing perspectives. Silvia Federzoni, Lydia-Mai Ho-Dac and Cécile Fabre, working on data in French, also find an influence of conventions or conventionalised perspectives on the choice of referential expressions within chains. They foreground two dimensions: genre, and animacy, which regardless of text type provides further evidence of the anthropocentric bias in the way humans consider the world around them. The last chapter in Part I focuses on the role of gestures in reference, more specifically in constructed action, a referential strategy which implies using the body and the voice to depict referents and their actions. Sébastien Vandenitte seeks to determine whether signing and speaking linguistic communities use different articulators (body parts and voice) to denote referents, through the first-ever quantitative analysis of comparable data from LSFB (French Belgian Sign Language) signers and Belgian French speakers. The two language communities appear to be using the same articulators to depict referents, although LSFB signers use constructed action to a greater extent. Head and gaze are the most frequently used articulators, facial expression being prominent in LSFB. The data also shows differences in the use of facial expressions, hands and/or arms. Parts II to IV investigate the relationship between conventions and pragmatic exploitation. Part II focuses on argumentative contexts. Michelle Lecolle considers the argumentative potential of plural references to humans in domains for which there are social or political stakes. Plural reference may be achieved through either pluralised count nouns, like les Français 'the French/French people', or collective nouns (and corresponding NPs), such as l'opinion publique 'public opinion'. She finds that regardless of their insertion in discourse, the grammar of these nouns and NPs allows for a number of argumentative strategies, such as vagueness or ambiguities, generalizations, and existential presupposition. The grammar of these nouns also triggers partly different argumentative potentials: only collective nouns exhibit a double layer of conceptualisation (one whole / many parts), which in context might activate further strategies, such as foregrounding of cohesion. Elodie Vargas and Jérémy Machy show how referential expressions can be revelators of cultural representations, through a corpusbased study of references to a controversial technological invention: electric vehicles, in the French and the German press. They find very differentattitudes in the two countries, with corresponding differences, for instance, in the use of metaphors. The authors also show that a given referential expression can only be interpreted within the context in which it is used, so that even a seemingly "neutral" noun phrase may in fact be highly subjective, even argumentative, as it may carry strong ideological, economic, political and cultural orientations. It is a very different type of argumentative context that Novi Djenar explores in her contribution. She shows that in Indonesian argumentative practices, referring to the self and the addressee by means of pronouns, kin terms or other nouns, accomplishes more than simple referring. Based on a corpus of political interviews, she finds that making the first-person pronoun explicit in subject position, in cooccurrence with a speech or cognition verb, identifies the speaker as a claimant and agent of assertions. In addition, for second person reference, when a minimal, conventional form does not accomplish what it was meant to do, a more-than-minimal form can serve as a rhetorical device to elicit a response.Part III moves away from argumentative contexts to focus on creative uses of referential expressions. Stephen Skalicky & Victoria Chen investigate the use of this in a large corpus of headlines from three North American satirical newspapers. They are specifically interested in instances in which pronominal this is used without a clear referent; they designate this use as the "empty satirical this" (e.g.,None of This Would Have Happened Had You Flossed). The lack of a clear referent may lead to the co-construction of satirical meaning, the final resolution of which depends upon the reader's willingness and ability to provide a fitting referent. The empty satirical this was found in all three newspapers, representing a creative and unique use of this which is atypical in non-satirical headlines. Samia Ounoughi considers the coinage of oronyms (proper nouns that denote portions of mountain relief) by members of the Alpine Club of London in the 19 th century. She concludes that oronyms actually create places, as they carve out portions of relief from what is really a continuum. In this, they are a specific type of proper noun. Coining new oronyms requires cooperation and coconstruction among speakers, both to achieve toponymation and to reach "referential efficiency", that is, carve out a zone that may not be exactly similar, but that is deemed sufficiently similar in context for communication needs. Finally, Thomas Bertin's study of artère ('artery') in French literature shows that creativity may be recursive over time. This bodily metaphor, initially creative (it is first found in Victor Hugo's Hunchback of Notre-Dame), became dead over time; but it is now used in creative ways in literature, either to add explanations or justifications through modifiers (which may occasionally be at odds with the prototypical representation of artère), or to extend the notion beyond roads.

  The next chapter turns to the emergence of joint construction of reference in L2 French. Heike Baldauf-Quilliatre, Elizaveta Chernyshova, Isabel Colón de Carvajal, Carole Étienne, Lydia Heiden & Laurène Smykowski carry out a multimodal conversation analysis on data collected in situ during a card game session. In the situation under study, the participants need to find a common solution to compensate for a missing card, in order to continue their activity, without necessarily having the proper L2 resources at their disposal. Their creativity is therefore mobilized, and different verbal and non-verbal resources are temporally fine-tuned (among others, joint visual attention on the object gesturally put in focus). Once the referent-function association is established and grounded, it is "activated" later on thanks to an iconic gesture. The chapter by Caroline David, Laurence Vincent-Durroux, Kerry Mullan, Christine Béal & Cécile Poussard also deals with spoken data

Here, "speaker" is understood as "anyone producing language"be it spoken, written, signed, or other.

[START_REF] Schwarz-Friesel | Indirect anaphora in text: a cognitive account[END_REF]

5, footnote 3) "[The text-world model] of the text is the mental representation of the referential states-of-affairs. It can be described in terms of conceptual constellations including all participating referents, their relations among each other and the events and situations in which they are performing."

The term addressee is slightly too restrictive, as someone might overhear a conversation and still process the messageincluding referential expressions. It was preferred over listener, which, too, is commonly used in pragmatics, because addressee is perhaps less connoted for spoken verbal language (so that it easily accommodates sign language and written language), and because it is for the addressee alone (whether real, virtual, imagined or universal) that a speaker may adjust to a Common Ground. Hearer, another commonly

[START_REF] Grice | Logic and conversation[END_REF]) Maxims of quantity: Q1: Make your contribution as informative as required (for the current purposes of the exchange).

temporal reference. From recordings of French L2 learners of English and English native speakers retelling the events they watched in a short film extract, the authors show that such a common, unobtrusive word as and carries cultural conventions in the structuring of discourse. This could not have been expected from the strictly grammatical potential of French and English, as et / and might be used with the same values (including temporal values, which are the focus of the present study). Yet French and English participants use and in partly different ways, with a strong influence of French et for the former. While the influence of L1 is not novel, this study is innovative in that it brings to light a lack of awareness of conventional structuring among advanced learners, and also hypothesizes a possible cognitive load created by the task, which involves complex communication.

The social proximity between the speakers and the interviewer (all of them students) might also have a pragmatic influence.

In the final chapter, Marine Le Mené-Guigoures¸ Anne Salazar Orvig, Christine Da Silva-Genest & Haydée Marcos turn to L1, more specifically to the choice of personal and demonstrative pronouns in adultchild dialogues. They find that already for toddlers, personal pronouns are preferred in subject / topic position, which is not the case for demonstrative pronouns. In order to establish whether this is due to adjustment to the cognitive status of the referent, they consider several parameters: syntactic


Keywords: Systèmes de particules en interaction, formules de Feynman-Kac, U -statistiques, fluctuations, théorèmes limites

Convergence de U -statistiques pour des systèmes de particules en interaction

 dlPG99]. Pour des modèles de systèmes de particules en interaction de type Feynman-Kac ou autres, nous sommes confrontés à un nouveau type de problème. Pour être plus précis, dans de tels algorithmes les échantillons de N particules sont formés de variables aléatoires corrélées; bien que tout bloc de taille finie soit formé de variables asymptotiquement indépendantes lorsque la taille des populations tend vers l'infini. Dans cette étude, nous analysons finement la convergence de ces modèles et nous démontrons des théorèmes limites pour des U -statistiques associées aux mesures d'occupation de systèmes de particules en interaction.

Introduction

The convergence of U -statistics has been intensively studied for estimators based on families of i.i.d. random variables and variants of them. In most cases, the independence assumption is crucial. When dealing with Feynman-Kac and other interacting particle systems of Monte-Carlo type, one faces a new type of problem. Namely, in a sample of N particles obtained through the corresponding algorithms, the distributions of the particles are correlatedalthough any finite number of them is asymptotically independent with respect to the total number N of particles. It happens so (and this is the main contribution of the present article to show) that this asymptotic independence is enough in practice to insure the convergence of U -statistics based on interacting particle systems. In the following, we prove therefore the convergence of U -statistics for different particle systems under mild assumption that are satisfied by Feynmann-Kac particle systems. The case of Bird and Nanbu systems also fits in this framework and will be treated elsewhere by the third Author [START_REF] Rubenthaler | Expansion of the propagation of chaos for Bird and Nanbu systems[END_REF]. To study the asymptotics of Feynman-Kac systems, whose properties are crucial to ensure the convergence of the statistics, we will use a functional representation, as introduced in [START_REF] Del Moral | Coalescent tree based functional representations for some Feynman-Kac particle models[END_REF] in the framework of discrete interacting particle systems.

The article is organized as follows. To fix the notation and the general framework of interacting particle systems, we first recall the (Feynman-Kac, continuous) interacting particle model. The model first appeared in quantum physics, in the work of Feynman and Kac in the 1940-50's, as a way to encode the motion of a quantum particle evolving in a potential (e.g. the interaction potential of a quantum field theory, viewed as a perturbation of the free Hamiltonian) in terms of path-integral formulas. It was realized progressively that interacting particle systems could be used in incredibly many different settings in probability and statistics. A detailed list of the (still expanding) application areas of these models is contained in [Del04], to which we refer for further informations. Recall simply, since we focus here on its statistical features, that the model is mainly used, in applied statistics, as a Bayesian nonlinear filtering model: the motion of the particles is driven by a diffusion process and the potential encodes the likewood of the states with respect to observations or to some reference path.

We study then the associated empirical joint distributions of a finite number k of particles and study the convergence of the distributions in terms of the total number N of particles of the system. This is closely related to our previous joint work [START_REF] Del Moral | Coalescent tree based functional representations for some Feynman-Kac particle models[END_REF] on discrete Feynman-Kac models -although the continuous hypothesis we use in the present article leads to some simplification of the tricky combinatorics that showed up in the discrete framework.

We turn then to U -statistics for interacting particle systems and prove that under mild asumptions on the behavior of the system (satisfied e.g. by Feynman-Kac and Boltzmann systems) several asymptotic normality properties holds.

Feynman-Kac particle systems

Let us consider a E-valued Markov process X t , where E = R d (or an arbitrary metric space) with a time-inhomogeneous infinitesimal generator L t , continuous trajectories, and a positive bounded potential function V t , 0 ≤ V t (x) ≤ V ∞ . We assume that the distribution of X 0 is γ 0 = η 0 . Notice that these hypothesis are meaningful for most applications, but could be accomodated to more general ones, see [Del04]. We are interested in the unnormalized (resp. normalized) distribution flows γ t and η t that are solutions, for sufficiently regular test functions f and under appropriate regularity conditions, of the nonlinear equations:

d dt γ t (f ) = γ t (L t (f )) -γ t (f V t ) and d dt η t (f ) = η t (L t (f )) + η t (f (η t (V t ) -V t )).
In terms of X t , we have:

γ t (f ) = E f (X t ) exp - t 0 V s (X s )ds , η t (f ) = γ t (f ) γ t (1) .

Definitions and notation

Let us fix first of all some notation. For q ∈ N * , we write [q] := {1, . . . , q}. For q, N ∈ N * , we set q, N := {s ∈ [N ] [q] , s injective}. For q even, we write I q for the set of partitions of [q] in pairs. We have

#I q = q! 2 q/2 q 2 !
.

The set of smooth bounded (resp. smooth bounded symmetric) functions on E q is written B b (E q ) (resp. B sym b (E q )). We also write B sym 0 (E q ) for the set of symmetric functions:

B sym 0 (E q ) := F ∈ B sym b (E q ) : E F (x 1 , . . . , x q )γ t (dx q ) = 0 .
Notice that the set of functions B sym 0 (E q ) depends on t, so that a better notation would be B sym 0,t (E q ). However, since in practice the abbreviated notation should not lead to confusion, we decided not emphasize this dependency for notational simplicity. We write simply B 0 (E) for the centered functions:

F ∈ B b (E) : E F (x)γ t (dx) = 0
The empirical (possibly random) measure associated to a (possibly random) vector x = (x 1 , . . . , x N ) ∈ E N is given by

m(x) = 1 N N i=1 δ xi .
We have, for all q ∈ N * , F : E q → R:

m(x) ⊗q (F ) = 1 N q s∈[N ] [q]
F (x s(1) , . . . , x s(q) ) .

We also consider the corresponding U -statistics:

m(x) ⊙q (F ) = 1 (N ) q s∈ q,N F (x s(1) , . . . , x s(q) ) = N q -1 1≤i1<...<iq≤N F sym (x i1 , . . . , x iq ) ,
where (N ) q = N ! (N -q)! = # q, N and (F ) sym (y 1 , . . . , y q ) :=1 q! σ∈Sq F (y σ(1) , . . . , y σ(q) ) , with S q the symmetric group of order q.

The notion of differential for sequences of signed measures 1 will also be useful. The total variation norm is written || || tv , so that, for any linear operator L on B b (E q ),

||L|| tv := sup f ∈B b (E q ): f ≤1 |L(f )| Let (Θ N ) N ≥1
, be a uniformly bounded sequence of measures on E q , in the sense that sup N ≥1 Θ N tv < ∞. The sequence Θ N is said to converge strongly to some measure Θ, as

N ↑ ∞ if and only if ∀f ∈ B b (E) lim N ↑∞ Θ N (f ) = Θ(f )
Definition 1.1. Let us assume that Θ N converges strongly to Θ. The discrete derivative of the sequence (Θ N ) N ≥1 is the sequence of measures (∂Θ N ) N ≥1 defined by

∂Θ N := N Θ N -Θ
We say that Θ N is differentiable, if ∂Θ N is uniformly bounded, and if it converges strongly to some measure ∂Θ, called simply the derivative of Θ N , as N ↑ ∞.

The discrete derivative ∂Θ N of a differentiable sequence can itself be differentiable. In this situation, the derivative of the discrete derivative is called the second derivative of Θ N and it is denoted by ∂2 Θ = ∂ (∂Θ), and so on.

A sequence Θ N that is differentiable up to order (k + 1), has the following representation

Θ N = 0≤l≤k 1 N l ∂ l Θ + 1 N k+1 ∂ k+1 Θ N with sup N ≥1 ∂ k+1 Θ N tv < ∞ ,
and the convention ∂ 0 Θ = Θ, for l = 0.

A genetic particle model

A particle system approaching the measures η t (and therefore also γ t ) is the following. At t = 0, the random vector Ξ t = (ξ 1 t , . . . , ξ N t ) is a family of i.i.d. random variables distributed according to γ 0 = η 0 . Each entry ξ i t of the vector diffuses according to the generator L t , independently of the other entries. Each entry ξ i t has an exponential clock of parameter V ∞ (independent of all the other variables defined up to now). When the clock of ξ i t rings, say at τ , it can

• jump to a randomly (and uniformly) chosen particle in the family, including itself, with a probability

Vτ (ξ i t ) V∞
• stay where it is with probability 1 -

Vτ (ξ i t )
V∞ . Up to a renormalization of the jump type generators (which is convenient for our present purposes), this is the model described in [Del04, Sect. 1.5.2]. Notice that the law of ξ i t depends on N , so that a more consistent notation for ξ i t would be ξ N,i t -whenever we want to emphasize the dependency on N , we will switch to this second notation. The corresponding empirical 2 measures and empirical U-statistics are given (and related) by:

η N t = m(Ξ t ) = m(ξ 1 t , . . . , ξ N t ), γ N t = γ N t (1) • η N t , with γ N t (1) = exp - t 0 η N s (V s )ds , (γ N t ) ⊙q (F ) = γ N t (1) q • (η N t
) ⊙q (F ). These are the analogs in continuous time to the random measures and U-statistics defined and studied in [START_REF] Del Moral | Coalescent tree based functional representations for some Feynman-Kac particle models[END_REF] in a discrete time setting. We recall that η N t and γ N t are known to converge as N → ∞ to η t and γ t [Del04].

In oder to study the convergence of the (empirical) U-statistics (γ N t ) ⊙q and (η N t ) ⊙q , we are going to rewrite them by means of functional expansions (actually, Laurent series in the parameter N ). These expansions will allow us, later, to control the convergence and (among others) to extend to particle systems the classical central limits theorems for U-statistics. Deriving these expansions is the main purpose of the first part of the article, together with first results of convergence.

For these purposes, it is useful to introduce an auxiliary q-particle system. The reasons for its introduction will become clear later and stem from a backward analysis of Feynman-Kac trajectories of families of q particles. Definition 1.2. The auxiliary system of q particles, ξ1 t , . . . , ξq t is defined as follows: the random variables ξ1 0 , . . . , ξq 0 are independant of law η 0 . Moreover, the particles ξ1 t , . . . , ξq t diffuse according to L t (1.1) (one more time, independently of one another) and undergo the following jumps. For any (i, j) ∈ [q] 2 , i = j, there is an exponential clock of parameter V ∞ /N and a corresponding Poisson point process

T (i,j) 1 , T (i,j) 2 , . . . of parameter V ∞ /N . The T (i,j) 1 , T (i,j) 2
, . . . are named the ringing times. At a ringing time t ∈ {T

(i,j) 1 , T (i,j) 2 , . . . }, ξi t ← ξj t with proba. Vt( ξi t ) V∞ ← ξj t with proba. 1 - Vt( ξi t )
V∞ .

(1.2)

The notation ξi t ← ξj t means that ξi t jumps to (or is substituted by) ξj t . When q = N , the particle systems ( ξ1 t , . . . , ξN t ) t≥0 has the same law as (ξ 1 t , . . . , ξ N t ) t≥0 .

We set, for an arbitrary

F ∈ B b (E q ), F e (( ξ1 s , . . . , ξq s ) 0≤s≤t ) = F ( ξ1 t , . . . , ξq t ) exp - t 0 [V s ( ξ1 s ) + • • • + V s ( ξq s )]ds . and E t,k (F ) := E F e (( ξ1 s , . . . , ξq s ) 0≤s≤t )|k rings on[0; t] .
Notice, for further use, that E t,k (F ) does not depend on N . This is because, as a consequence of the general properties of Poisson point processes, conditionally to the hypothesis that there are k rings on [0, t], the distribution of the ringing times is uniform on [0, t] and therefore independent of the parameter V ∞ /N ; since the other parameters of the process (L t and the jump probabilities

Vt( ξi t )
V∞ ) are independent of N , the property follows.

Expansion of the unnormalized measure

Let us write for an arbitrary

F ∈ B b (E q ), Q N t,q (F ) := E((γ N t ) ⊙q (F )).
We refer to Q N t,q as to the empirical unnormalized measure associated to the particle system ξ i t . Since the joint distribution of the sequence ξ 1 t , . . . , ξ N t is invariant by permutations, Q N t,q (F ) = Q N t,q (F sym ), and we can assume without restriction that F is a symmetric function. The first question we adress is the (exact) computation of the speed of convergence of the empirical unnormalized measure Q N t,q to γ ⊗q t .

Theorem 1.3. For F ∈ B sym b (E q ), we have the Laurent expansion

Q N t,q (F ) = ∞ k=0 (λt) k e -λt k! E t,k (F ) (1.3) = γ ⊗q t (F ) + ∞ k,i≥0 k+i≥1
(-1) i (λt) k+i k!i! E t,k (F ) (1.4) = γ ⊗q t (F ) + ∞ r=1 1 N r r k=0 (-1) r-k k!(r -k)! (q(q -1)V ∞ t) r E t,k (F ) (1.5)
where λ := q(q -1) V∞ N . In particular, Q N t,q is differentiable up to any order with

∂ r Q t,q (F ) = r k=0 (-1) r-k k!(r -k)! (q(q -1)V ∞ t) r E t,k (F ) .
Notice in particular that, although the E t,k (F )s depend on the choice of the upper bound V ∞ for the potential function V , the coefficients of the development do not (they are the derivatives of Q t,q , that do not depend on V ∞ ).

Proof of Theorem 1.3.

We have first, for

f ∈ B b (E) d dt E(η N t (f )) = E(η N t (L t (f ))) + N i,j=1 E   V t (ξ i t ) f (ξ j t ) -f (ξ i t ) N 2   and d dt E(γ N t (f )) = E(γ N t (L t (f ))) -E(η N t (V t )γ N t (f )) + E   γ N t (1) N i,j=1 V t (ξ i t ) f (ξ j t ) -f (ξ i t ) N 2   For F ∈ B sym b (E q ),
let us introduce the useful notation:

F i : E q-1 -→ B b (E), F i (x 1 , ..., x q-1 )(y) := F (x 1 , ..., x i-1 , y, x i , ..., x q-1 )
and let us extend L t to functions in B sym b :

L t (F )(x 1 , ..., x n ) := q i=1 L t (F i (x 1 , ..., x i-1 , x i+1 , ..., x n ))(x i ).
We get:

d dt E((γ N t ) ⊙q (F )) (1.6) = E((γ N t ) ⊙q (L t (F ))) -E(qη N t (V t )(γ N t ) ⊙q (F )) +E    γ N t (1) q s∈<q,N > q i=1 N k=1 V t (ξ s(i) ) × F i (ξ s(1) t , . . . , ξ s(i-1) t , ξ s(i+1) t , . . . , ξ s(q) t )(ξ k t ) -F (ξ s(1) t , . . . , ξ s(i) t , . . . , ξ s(q) t ) N • (N ) q
The second term in the right hand side of (1.6) reads:

E   -q γ N t (1) q N • (N ) q s∈<q,N > N k=1 F (ξ s(1) t , . . . , ξ s(q) t )V t (ξ k t )   = E   -q γ N t (1) q N • (N ) q s∈<q,N > F (ξ s(1) t , . . . , ξ s(q) t )(V t (ξ s(1) t ) + • • • + V t (ξ s(q) t
))

  + E   -q γ N t (1) q N • (N ) q s∈<q+1,N > F (ξ s(1) t , . . . , ξ s(q) t )V t (ξ s(q+1) t )   .
Similarly, making use of the symmetry properties of F , the last term in the right hand side of (1.6) reads:

E   q γ N t (1) q N • (N ) q s∈<q,N > N k=1 V t (ξ s(q) t )F (ξ s(1) t , . . . , ξ s(q-1) t , ξ k t )   - E   γ N t (1) q (N ) q s∈<q,N > F (ξ s(1) t , . . . , ξ s(q-1) t , ξ s(q) t )(V t (ξ s(1) t ) + • • • + V t (ξ s(q) t
))

 

The first term in this last sum decomposes then into (making use once again of the symmetry properties of F ):

E   γ N t (1) q N • (N ) q s∈<q,N > (V t (ξ s(1) t ) + • • • + V t (ξ s(q) t ))F (ξ s(1) t , . . . , ξ s(q-1) t , ξ s(q) t )   + E   q γ N t (1) q N • (N ) q s∈<q,N > q-1 k=1 V t (ξ s(q) t )F (ξ s(1) t , . . . , ξ s(q-1) t , ξ s(k) t )   + E   q γ N t (1) q N • (N ) q s∈<q+1,N > F (ξ s(1) t , . . . , ξ s(q-1) t , ξ s(q) t )V t (ξ s(q+1) t )  
Reorganizing the summands in these expansions, we get finally that the two last terms of (1.6) sum up to :

-E   γ N t (1) q (N ) q s∈<q,N > F (ξ s(1) t , . . . , ξ s(q) t )(V t (ξ s(1) t ) + • • • + V t (ξ s(q) t
))

  + E γ N t (1) q N • (N ) q s∈<q,N > q i,r=1 V t (ξ s(i) t ) F (ξ s(1) t , . . . , ξ s(i-1) t , ξ s(r) t , ξ s(i+1) t , . . . , ξ s(q) t ) -F (ξ s(1) t , . . . , ξ s(q) t
) .

We set

F Vt (x 1 , . . . , x q ) := (V t (x 1 ) + • • • + V t (x q ))F (x 1 , . . . , x q ) F (i,r)
Vt (x 1 , . . . , x q ) = V t (x i )(F (x 1 , . . . , x i-1 , x r , x i+1 , . . . , x q ) -F (x 1 , . . . , x q )).

Then the equations above give

d dt Q N t,q (F ) = Q N t,q L t (F ) -Q N t,q (F Vt ) + 1≤i,r≤q 1 N Q N t,q (F (i,r) Vt ) .
And so

Q N t,q (F ) = E F ( ξ1 t , . . . , ξq t ) exp - t 0 V t ( ξ1 s ) + • • • + V t ( ξq s )ds . (1.7)
We then obtain Equation (1.3) of the theorem by noticing that

P(k rings on[0, t]) = (λt) k e -λt k! .
We obtain Equation (1.4) by developping the term e -λt and by noticing that

E t,0 (F ) = γ ⊗q t (F ) .

Wick theorem for interacting particle systems

We say that two particles of the auxiliary system ξi , ξj , i = j interact at t if and only if ξi t jumps to ξj or ξj t jumps to ξi at t. We say that a trajectory of the auxiliary system of particles ξ1 t , . . . , ξq t is exactly Wick-coupled on [0, t] if and only each particle of the system has exactly one interaction with another particle between 0 and t (notice that the existence of such trajectories requires q to be even). We write W t for the set of Wick-coupled trajectories on [0, t]; the set W t embeds into the set of trajectories with exactly q/2 rings on [0, t].

Theorem 1.4. If F ∈ B sym 0 (E q ) then, for r < q/2, ∂ r Q N t,q (F ) = 0. Moreover, for q even, ∂ q/2 Q t,q (F ) = (q(q -1)V ∞ t) q/2 (q/2)! E t,q/2 (F )

In particular, lim sup

N →+∞ N q/2 |Q N t,q (F )| ≤ (q(q -1)V ∞ t) q/2 (q/2)! F ∞ .
(1.8)

Proof of Thm 1.4. Recall that F is symmetric and belongs to B sym 0 (E q ) so that, for any i ≤ q, E F (x 1 , ..., x q )γ t (dx i ) = 0.

The proof follows then from Thm 1.3 together with the observation that, if there are strictly less than q/2 rings, then at least one of the particles does not interact with the others so that, if k < q 2 , E t,k (F ) = 0. Corollary 1.5. With the assumptions of Thm 1.3 and q even, we have:

∂ q/2 Q t,q (F ) = q! (q/2)! (V ∞ t) q/2 E((F e ( ξ1 s , . . . , ξq s ) 0≤s≤t )|W t ).
If, furthermore, F = (f 1 ⊗ • • • ⊗ f q ) sym (with f 1 , . . . , f q centered with respect to γ t ) then we have the Wick-type expansion

E(F e (( ξ1 s , . . . , ξq s ) 0≤s≤t )|W t ) = 2 q/2 (q/2)! q! Iq∈Iq {i,j}∈Iq E ′ t,1 (f i ⊗ f j )
where the sum is over the set I q of partitions I q of [q] into pairs, i, j being paired if {i, j} ∈ I q and where, for a function G of two variables

E ′ t,1 (G) := E(G(μ 1 t , μ2 t ) exp - t 0 [V s (μ 1 s ) + V s (μ 2 s )]ds |one ring on [0, t])
with μ1 t , μ2 t an auxiliary system of two particles (defined with the same rules as a general auxiliary system of q particles).

Proof. Indeed, from the enumeration of possible interactions between particles and the general properties of Poisson processes, we have:

P(W t |q/2 rings on [0, t]) = q! (q(q -1)) q/2 .
The first part of the Corollary follows.

The last part follows from the observation that the distributions of two particles (or of blocks of particles) are independent conditionnally to the assumption that they do not interact (either directly or through interactions with other particles), so that quantities such as E(F e (( ξ1 s , . . . , ξq s ) 0≤s≤t )|W t ) may be computed by disjoint integrations over blocks (pairs in this particular case) of interacting particles. Since Card(I q ) = q! 2 q/2 (q/2)! , the Corollary follows.

Expansion of the normalized measure

The purpose of the present section is to prove for the measures associated to the empirical U -statistics (η N t ) ⊙q properties similar to the ones obtained in the unnormalized case. We conclude the section with a Wick formula in this setting.

We set, for F ∈ B b (E q ): P N t,q (F ) = E((η N t ) ⊙q (F )) . Theorem 1.6. The sequence (P N t,q ) is differentiable up to any order. Let us mention that an explicit formula (that we omit) for the derivatives ∂ l P t,q follows immediately from Fla 1.13 in our proof.

Proof. Let us expand first (η N t ) ⊙q in terms of (γ N t ) ⊙q . Let, in the following, F be a bounded symmetric function of q variables. We first have:

(η N t ) ⊙q (F ) = (γ N t ) ⊙q (F )γ N t (1) -q = (γ N t ) ⊙q (F ) 1 γ t (1) q 1 1 -1 - γ N t (1) γt(1)
q .

We set Ft := F γt(1) q and get:

(η N t ) ⊙q (F ) = (γ N t ) ⊙q ( Ft ) 1 1 -1 - γ N t (1) γt(1) q = (γ N t ) ⊙q ( Ft ) 1 (1 -u N t ) q , with u N t = 1 - γ N t (1) γ t (1) .
Recall the decomposition (that holds for any m, see [DPR09, Lemma 4.11]):

1 (1 -u N t ) q = 0≤k≤m (q -1 + k) k (u N t ) k k! + (u N t ) m 1≤k≤q C k+m q+m u N t 1 -u N t k
from which it follows that:

(η N t ) ⊙q (F ) = (γ N t ) ⊙q ( Ft ) 0≤k≤m (q -1 + k) k (u N t ) k k! + (γ N t ) ⊙q ( Ft )(u N t ) m 1≤k≤q C k+m q+m u N t 1 -u N t k =: (1) + (2) .
For an arbitrary function f on E, we set: θ t (f ) := f -ηt(f ) γt(1) . Then, recalling the equality γ t (1) = exp -t 0 η s (V s )ds , we get

γ N t (1) γ t (1) = e - R t 0 [η N u (Vu)-ηu(Vu)]du = 1 - t 0 [η N u (V u ) -η u (V u )] γ N u (1) γ u (1) du = 1 - t 0 γ N u (θ u (V ))du ,
where the last identity follows from the rewriting

[η N u (V u ) -η u (V u )] γ N u (1) γ u (1) = η N u V u -η u (V u ) γ u (1) γ N u (1) = γ N u (θ u (V u )).
In particular, we get, for an arbitrary k > 0,

1 - γ N t (1) γ t (1) k = s1,...s k ∈[0,t] k i=1 γ N si (θ si (V si ))ds 1 . . . ds k (1.9) = k! 0≤s1≤•••≤s k ≤t k i=1 γ N si (θ si (V si ))ds 1 . . . ds k .
We therefore have

(1) = 0≤k≤m (q -1 + k) k 0≤s1≤•••≤s k ≤t k i=1 γ N si (θ si (V si )) (γ N t ) ⊙q ( Ft )ds 1 . . . ds k = 0≤k≤m (q -1 + k) k × 0≤s1≤•••≤s k ≤t γ k,q,N s,t (S k,q s1,...,s k ,t (F ))ds 1 . . . ds k (1.10) with s := {s 1 , . . . , s k }, γ k,q,N s,t = γ N s1 ⊗. . . γ N s k ⊗(γ N t ) ⊙q , S k,q s,t (F ) = θ s1 (V s1 )⊗• • •⊗θ s k (V s k )⊗ Ft . We introduce the operators (for G of k + q variables, j ≤ k) D j G(x 1 , . . . , x k+q ) = N -q -k + j N G(x 1 , . . . , x k+q ) + 1 N j+1≤i≤k+q G j (x 1 , . . . , x k+q )(x i )
We then have, for any empirical measure m(x) = 1 N 1≤i≤N δ xi and for any function G of k + q ≤ N variables and any measure µ on

E j-1 µ ⊗ (m(x) ⊗ m(x) ⊙(k+q-j) )(G) = µ ⊗ m(x) ⊙(k+q-j+1) (D j G) .
For G function of q + k variables and s ≤ t and j ≤ k + q, we define then the Markov operator

R j s,t G(x 1 , . . . , x k+q ) = (1.11) E(G(x 1 , . . . , x k+q-j , ξ1 t-s , . . . , ξj t-s )e - R t-s 0 (Vs+u( ξ1 u )+•••+Vs+u( ξj u ))du | ξ1 0 = x k+q-j+1 , . . . , ξj 0 = x k+q ) ,
where the ξi are defined as before, excepted for the initial condition that reads now ξi 0 = x k+i . Reasoning as before (and using the auxiliary system ξi s as in Sect. 1.3), we get for s k ≤ t and H a function of k + q variables, µ a measure on E j-1 ,

E(µ ⊗ (γ N s k ⊗ (γ N t ) ⊙q )(H)|F s k ) = µ ⊗ (γ N s k ⊗ (γ N s k ) ⊙q )R q s k ,t (H) = µ ⊗ (γ N s k ) ⊙(q+1) D q+1 R q s k ,t (H) ,
where F t stands for the natural filtration on the probability space underlying the particle system. Here, we took advantage of the Markovian properties of the system and of Eq 1.7. So, by recurrence :

E(γ k,q,N s,t (G)) = (η 0 ) ⊗(k+q) R k+q 0,s1 D 1 R k-1+q s1,s2 . . . D k R q s k ,t (G) .
We proceed now as for the unnormalized measure and introduce still another more general auxiliary particle system ( ξ1 t , . . . , ξk+q t ) t≥0 . The system depends on k (and q), but we do not emphasize this dependency to abbreviate the notation. This should not create ambiguities since the dependency on k should be obvious from the context in the following formulas.

Let S 1 , S 2 , . . . , S k be the order statistics of k uniform random variables in [0, t] (we take k i.i.d. uniform variables in [0, t] and sort them). The S i are independent of all the other variables. We set S i := t for k < i ≤ k + q. The system (which agrees with the auxiliary system ξi t when k = 0) is defined as follows:

• The law of ( ξ1 0 , . . . , ξk+q 0

) is η ⊗k+q 0

• The particles ξi s diffuse according to L s and undergo the following jumps:

• Each couple (i, j) ∈ [q +k] 2 has an exponential clock of parameter V ∞ /N . At a ringing time s:

-If s > S i or s > S j , then nothing happens.

-Else,

V∞

-At s = S i , we make the following replacement (with U sampled uniformly in {i + 1, . . . , k + q}) ξU s ← ξi s with proba.

k -i + q N .
We say that there has been a parasite at s, so that the system has at most k parasites.

With the notation Šk,q S,t ( ξ1 )du , the equation (1.10) gives

E((1)) = 0≤k≤m (k -1 + q) k k! E( Šk,q S,t ( ξ1 S1 , . . . , ξk S k , ξk+1 t , . . . , ξk+q t )) = 0≤k≤m (k -1 + q) k k! × E +∞ r=0 (Λ S V ∞ /N ) r e -ΛsV∞/N
r! E( Šk,q S,t ( ξ1 S1 , . . . )|r rings , S 1 , . . . , S k ) with

Λ S := q(q -1)(t -S k ) + (q + 1)q(S k -S k-1 ) + • • • + (q + k)(q + k -1)(S 1 -0) ,
for which we will use the crude bound Λ S ≤ (q + k) 2 t. We set Ēk,r,l = E(Λ l S E( Šk,q S,t ( ξ1 s1 , . . . )|r rings , S 1 , . . . , S k )) , and notice that the parasites are sampled independently of the S i and of the ringing times, so that: Ēk,r,l = k i=0 P(i parasites)E(Λ l S E( Šk,q S,t ( ξ1 S1 , . . . )|i parasites, r rings, S 1 , . . . , S k )) .

The probability P(i parasites) depends on t, q and k, but we omit the corresponding indices to simplify the notation. The following lemma is crucial:

Lemma 1.7. For i + r < k 2 , we have: Dk,r,l,i := E(Λ l S E( Šk,q S,t ( ξ1 s1 , . . . )|i parasites, r rings, S 1 , . . . , S k )) = 0 , so that:

Ēk,r,l = k i=(⌈ k 2 ⌉-r) + P(i parasites) Dk,r,l,i .
where ⌈x⌉ stands for inf{k ∈ N : k ≥ x}.

Indeed, for i+r < k 2 , at least one of the ξj , j ≤ k does not interact with the other particles, so that, by Fubini's lemma, θ Sj (V Sj )( ξj Sj )e - R S j 0 Vu( ξj u )du can be integrated independently. However, conditionnally to the assumption that ξj does not interact with the other particles, we have:

E(θ Sj (V Sj )( ξj Sj )e - R S j 0 Vu( ξj u )du ) = η Sj (θ Sj (V Sj )) = 0,
and the lemma follows.

We then have (by developping the exponential in E((1)))

E((1)) = 0≤k≤m (k + q -1) k k! +∞ r=0 V r ∞ N r r! +∞ i=0 (-1) i i! V ∞ N i Ēk,r,r+i (1.12) = 0≤k≤m (k + q -1) k k! +∞ r=0 V r ∞ N r r! +∞ i=0 (-1) i i! V ∞ N i (1.13) k j=(⌈ k 2 ⌉-r) +
P(j parasites) Dk,r,r+i,j .

INRIA

By the construction of the auxiliary system of particles, P(j parasites) = O( 1 N j ). For further use, the explicit formula is:

P(j parasites ) = 1≤k1≤•••≤kj ≤k   (q + k 1 -1) N . . . q + k j -1 N l∈[k]\{k1,...,kj } 1 - q + l -1 N   ,
so that there exists coefficients (a l ) (depending on k and q, as usual we omit the corresponding indices for notational simplicity) such that

P(j parasites ) = k l=j a l N l .
Because of the condition i + r < k 2 in Lemma 1.7, it follows that the coefficient of any 1 N p in the expansion of E((1)) is a finite sum of coefficients which are independent of N . This is essentially all what we need in order to prove the Thm. 1.6, provided we are able (Step 1 below) to give a suitable upper bound for the remainder terms in the expansion of E((1)) at a given order, and (Step 2) to give a suitable upper bound to E((2)). This is the (cumbersome) purpose of the following conclusion to the proof of the Thm. 1.6, that we include for completeness sake but that the reader may skip, if she/he wishes.

Step 1: Upper bound for the remainder terms in the expansion of E((1)) at the order p, assuming that m is even and m = 2p.

Notice that for an arbitrary bounded function F , andso | Ēk,r,r+i | ≤ (t(q + k) 2 ) r+i 2 k V k ∞ e t(k+q)V∞ F ∞ . In the Fla 1.12 for the expectation E((1)), the sum of the terms with a factor 1/N r+i , r + i ≥ p is, in absolute value, equal to (we use here: ∀x ≥ 0, i≥j

F ∞ ≤ F ∞ e tV∞q , S k,q S,t ∞ ≤ 2 k V k ∞ e t(k+q)V∞ F ∞ ,
x i i! ≤ x j j! e x ) 0≤k≤2p (k + q -1) k k! +∞ r=0 V r ∞ N r r! i≥(p-r)+ (-1) i i! V ∞ N i Ēk,r,r+i ≤ 0≤k≤2p (k + q -1) k k! p-1 r=0 V r ∞ N r r! 2 k V k ∞ e t(k+q)V∞ F ∞ × 1 (p -r)! V ∞ t(q + k) 2 N p-r exp V ∞ t(q + k) 2 N (t(k + q) 2 ) r + +∞ r=p V r ∞ N r r! 2 k V k ∞ e t(k+q)V∞ F ∞ exp V ∞ t(q + k) 2 N (t(q + k) 2 ) r ≤ 1 N p 0≤k≤2p (k + q -1) k k! × p-1 r=0 V p+k ∞ 2 k e t(k+q)V∞ F ∞ 1 (p -r)!r! (t(q + k) 2 ) p e V∞ t(q+k) 2 N +2 k (t(k + q) 2 ) p V k+p ∞ p! e V∞t(k+q) F ∞ e 2 V∞ t(q+k) 2 N = O( 1 N p ) .
Since, by Fla 1.13, E((1)) is the sum of this term with a polynomial in 1 N , the proof of the Step 1 is concluded.

Step 2: Upper bound to E((2)).

Since γ t (1) = e - R t 0 ηu(Vu)du and γ N t (1) = e - R t 0 η N u (Vu)du , we get:

|E((2))| ≤ F ∞ e tqV∞   1≤k≤q C k+2p q+2p (e tV∞ + 1) k   E((u N t ) 2p ) .
We have, with the same auxiliary system of the ξ's as before but with q = 0, k = 2p and with S 1 , . . . , S 2p the order statistics of 2p uniform variables on [0, t],

E((u N t ) 2p ) = E(θ S1 (V S1 ) ⊗ • • • ⊗ θ S2p (V S2p )( ξ1 S1 , . . . , ξ2p S2p )e - R S 1 0 ... R S 2p 0 Vu 1 ( ξ1 u 1
)...Vu p ( ξ2p u 2p

)du1...du2p ) .

We get (with, now, Λ S = (2p

)(2p -1)S 1 + (2p -1)(2p -2)(S 2 -S 1 ) + • • • + 1 × (S 2p -S 2p-1 )) E((u N t ) 2p ) = E( +∞ r=0 Λ S V ∞ N r 1 r! e -Λ S V∞/N E(θ S1 (V S1 ) ⊗ • • • ⊗ θ S2p (V S2p ) ( ξ1 S1 , . . . , ξ2p S2p )e - R S 1 0 ... R S 2p 0 Vu 1 ( ξ1 u 1
)...Vu p ( ξ2p u 2p

)du1...du2p |r rings, S 1 , . . . , S 2p ))

= E( +∞ r=0 Λ S V ∞ N r 1 r! e -Λ S V∞/N 2p-1 i=0 P(i parasites) × E(θ S1 (V S1 ) ⊗ • • • ⊗ θ S2p (V S2p )( ξ1 S1 , . . . , ξ2p S2p ) e - R S 1 0 ... R S 2p 0 Vu 1 ( ξ1 u 1
)...Vu p ( ξ2p u 2p

)du1...du2p |r rings, i parasites, S 1 , . . . , S 2p )) .

We have

P(i parasites) ≤ C i 2p-1 2p -1 N i and know that, if r + i < p E(θ S1 (V S1 ) ⊗ • • • ⊗ θ S2p (V S2p )( ξ1 S1 , . . . , ξ2p S2p ) e - R S 1 0 ... R S 2p 0 Vu 1 ( ξ1 u 1
)...Vu p ( ξ2p u 2p

)du1...du2p |r rings, i parasites, S 1 , . . . , S 2p )) = 0 .

So that

E((u N t ) 2p ) ≤ 1 N p +∞ r=0 (t(2p) 2 V ∞ ) r r! 2p-1 i=(p-r)+ C i 2p-1 (2p -1) i e 2ptV∞ 2 2p V 2p ∞ and |E((2))| ≤ 1 N p × F ∞ e tqV∞   1≤k≤q C k+2p q+2p (e tV∞ + 1) k   × e t(2p) 2 V∞ 2p-1 i=0 C i 2p-1 (2p -1) i e 2ptV∞ 2 2p V 2p ∞ = O( 1 N p ),
and the proof of the Thm is complete.

Corollary 1.8. If F ∈ B sym 0 (E q ), then • ∂ l P t,q (F ) = 0, ∀l < q/2
• for q even, ∂ q/2 P t,q (F ) simplifies to

∂ q/2 P t,q (F ) = V q/2 ∞ (q/2)! (q(q -1)t) q/2 E( F e ( ξ1 t , . . . , ξq t )| q 2 rings) , or ∂ q/2 P t,q (F ) = ∂ q/2 Q t,q (F ) γ t (1) q .
Proof of Corollary 1.8. Indeed, the same argument as in the proof of Lemma 1.7 shows that, for a centered function F ∈ B sym 0 , the terms D k,r,r+i,j vanish for j + r < k+q 2 (this is because at least one of the ξj , j ≤ k + q does not interact with the other particles if there are less than k+q 2 rings and parasites). It follows that, if q is even (resp. odd), q 2 (resp.

q+1

2 ) is the smallest power of 1 N appearing in the expansion of E((1)) for m > q and that the contribution of (2) is negligible with respect to 1/N q/2 , from which the first part of the Corollary follows.

The same reasoning shows that, for q even, only the case k = i = j = 0 and r = q 2 contributes to the coefficient of ( 1 N ) q 2 . As noticed before, for k = 0, the systems ( ξ) and ( ξ) have the same law. The second statement follows.

First results on asymptotic normality

The empirical measure η N t converges weakly to η t : for all bounded f , η N t (f ) a.s.

-→

N →+∞ η t (f ).
The Wick theorem (Cor. 1.8) allowed us to improve on this convergence result. Namely, we know from the previous section that

• For q odd and F ∈ B sym 0

(E q ): N q/2 E((η N t ) ⊙q (F )) -→ N →∞ 0,
• For q even, F ∈ B sym 0

(E q ): N q/2 E((η N t ) ⊙q (F )) -→ N →∞ ∆ q/2 (F ),
where ∆ q/2 is a shortcut for the signed measure:

∆ q/2 (F ) = (q(q -1)tV ∞ ) q/2 (q/2)! E( F e ( ξ1 t , . . . , ξq t )| q 2 rings) .
Furthermore, for F = (f 1 ⊗ ... ⊗ f q ) sym , with f i ∈ B 0 (E), this formula simplifies to:

∆ q/2 (F ) = (2V ∞ t) q/2 γ t (1) q Iq∈Iq {i,j}∈Iq E ′ t,1 (f i f j ) = Iq∈Iq {i,j}∈Iq W t (f i ⊗ f j ),
where W t := 2V∞t γt(1) 2 E ′ t,1 . A striking consequence of this Wick theorem for interacting particle systems and their U-statistics is that it leads immediately to an explicit result of asymptotic normality for the vector-valued random measures (η N t ) q . Notice that the same properties hold for the system studied in [START_REF] Rubenthaler | Expansion of the propagation of chaos for Bird and Nanbu systems[END_REF] and so we have the same asymptotic normality result for this system. More precisely: Theorem 2.1. ∀f 1 , . . . , f q ∈ B 0 (E),

N 1/2 (η N t (f 1 ), . . . , η N t (f q )) law -→ N →+∞ N (0, K) with K(i, j) = η t (f i f j ) + E(W t (f i ⊗ f j )).
Proof. For any u 1 , . . . , u q , we have:

E exp N η N t log 1 + iu 1 f 1 + • • • + iu q f q √ N = E   exp   k≥1 (-1) k+1 k N 1-k/2 η N t ((iu 1 f 1 + . . . iu q f q ) k )     ∼ N →+∞ E(exp( √ N (iu 1 η N t (f 1 ) + • • • + iu q η N t (f q )))) × exp - 1 2 η t ((u 1 f 1 + • • • + u q f q ) 2 )) ,
where the last equivalence follows from the differentiability of P N t,1 .

We also have

E exp N η N t log 1 + iu 1 f 1 + . . . iu q f q √ N = E   N j=1 1 + iu 1 f 1 (ξ j t ) + • • • + iu q f q (ξ j t ) √ N   = E   0≤k≤N 1 N k/2 1≤j1,...,j k ≤q i k u j1 . . . u j k 1≤i1<•••<i k ≤N f j1 (ξ i1 t ) . . . f j k (ξ i k t )   = E   0≤k≤N (N ) k N k/2 1≤j1,...,j k ≤q i k u j1 . . . u j k 1 k! (η N t ) ⊙k (f j1 ⊗ • • • ⊗ f j k )   -→ N →+∞ k≥0,k even (-1) k/2
1≤j1,...,j k ≤q u j1 . . . u j k k!

I k ∈I k {a,b}∈I k W t (f a ⊗ f b ) = k≥0,k even
(-1) k/2 2 k/2 (k/2)! 1≤j1,...,j k ≤q u j1 . . . u j k E(W t (f j1 ⊗ f j2 )) . . . E(W t (f

j k-1 ⊗ f j k )) = k≥0,k even (-1) k/2 2 k/2 (k/2)!   1≤j1,j2≤q u j1 u j2 E(W t (f j1 ⊗ f j2 ))   k/2 = exp   - 1 2 1≤j1,j2≤q u j1 u j2 E(W t (f j1 ⊗ f j2 ))  
The Theorem follows.

3 Convergence of empirical U -statistics.

Hoeffding's decomposition

We want here to study the convergence under fairly general assumptions of a sequence of empirical U-statistics (η N t ) ⊙q (F ) when N → +∞, where the η N t s are empirical measures on some space E and F is a bounded symmetrical function on E q .

Let us first write an analog of Hoeffding's decomposition (cf. [START_REF] Lee | U -statistics[END_REF], [dlPG99]). We fix some measure η t . For 1 ≤ k ≤ q and any F bounded symmetrical on E q , we define F (k) (x 1 , . . . , x q ) = E q-k F (x 1 , . . . , x k , x k+1 , . . . , x q )η ⊗(q-k) t (dx k+1 , . . . , dx q ) . Notice that F (q) = F and that, for an arbitrary k, F (k) is symmetrical on E k . We define recursively (the function F being fixed -for notational simplicity, we do not stress the dependency of θ, h (i) ... in the following formulas): θ = E q n F (x 1 , . . . , x q )η ⊗q t (dx 1 , . . . , dx q ) h (1) (x 1 ) = F (1) (x 1 )θ h (k) (x 1 , . . . , x k ) = F (k) (x 1 , . . . , x k ) -k-1 j=1 (k,j) h (j)θ , where (k,j) h (j) is an abbreviation for 1≤i1<•••<ij ≤k [h (j) (x i1 , . . . , x ij )]. The h (k) are bounded and symmetrical. They are constructed so as to be centered and to satisfy, for l < k, (h (k) ) l = 0, see e.g. [Lee90, Sect. 1.6].

s

  with proba. 1 -Vs( ξi )

RR n°6966

From now on, measure will have the more general meaning of signed measure INRIA

The adjective empirical refers, in the present article, to any measure, process or statistics obtained from a particle system approximation. RR n°6966

Asymptotic normality for empirical U-statistics.

As for classical U-statistics, the Hoeffding decomposition can be used to study the convergence of the empirical U-statistics (η N t ) ⊙q . Notice first that, ∀N , (η N t ) ⊙q (F ) = (η N t ) ⊙q (h (q) ) + (η N t ) ⊙q ( q-1 j=1 (q,j)

(h (j) )) + θ .

(3.1)

Notice that, ∀1 ≤ j ≤ q -1, (q,j) h (j) = 1 j! b∈ j,q h (j) (x b(1) , . . . , x b(j) ) .

So, with the notation of the first sections:

(η N t ) ⊙q ( (q,j) h (j) ) = 1 (N ) q a∈ q,N 1 j! b∈ j,q h (j) (ξ

Theorem 3.1. Suppose we have, for all j ≥ 2 and f ∈ B sym 0

for some constant C depending on t, N, j, f ∞ . Assume further that, with F as above,

The Theorem follows from Hoeffding's decomposition, together with the identity (η N t ) ⊙q ( (q,1)

Corollary 3.2. For the Feynman-Kac particle systems of the first sections, we have:

) 2 )). Indeed, notice first that the Eq 3.4 holds for Feynman-Kac particle systems e.g. as a Corollary of Th. 2.1. The Corollary follows then from the following Lemma.

Lemma 3.3. For the particle system described in Section 1, we have, ∀j, ∀f ∈ B sym 0

Notice first that, because of the very definition of (η N t ) ⊙2j (as an average over arbitrary configurations of distinct systems of 2j particles), although

, where Sym stands for the canonical symmetrization map from B sym 0

This function g k has the particular feature that there are α = 2j -2k indexes i such that

As for the proof of Corollary 1.8, we go back to the proof of Lemma 1.7. For the function g k defined above, the terms D k ′ ,r,r+i,j ′ vanish for j ′ + r < k ′ +α 2 . So jk is the smallest power of N appearing in the Laurent-type expansion of E((

This finishes the proof.

Wiener integrals expansions

The convergence of symmetric statistics to multiple Wiener integrals has been studied intensively, see e.g. the seminal [START_REF] Dynkin | Symmetric statistics, Poisson point processes, and multiple Wiener integrals[END_REF] or the account of the classical theory in [START_REF] Lee | U -statistics[END_REF]. We are interested here in proving that the main statements of the theory still hold for empirical U-statistics and focus on a particular example, namely the one of a symmetrical and bounded kernel constructed as a product of centered functions.

From the technical point of view, the key issue w.r. to the classical theory of limit distributions for U-statistics based on i.i.d. assumptions is related to the existence of interactions between the particles of the system. Taking these interactions into account amounts in practice to replace the central limit theorem by the analogous statement for interacting particle systems, namely Thm. 2.1.

Let us consider a kernel (a symmetrical bounded fonction on E q ) F of the form F = (f 1 ⊗ ... ⊗ f q ) sym with f i ∈ B 0 (E). Then, F (k) = h (k) = 0 for k = 1, ..., q -1, and we expect N q 2 (η N t ) ⊙q (F ) to converge in law. This is indeed the case. Let us start with some classical results (see e.g. [START_REF] Rota | Stochastic integrals: a combinatorial approach[END_REF] for a systematical approach by means of the poset of partitions). For an empirical signed measure m(x) =

where f V , V = {v 1 , ..., v k }, stands for f v1 ...f v k and P runs over the set of partitions of [q] into disjoint subsets.

Assume now that m

Let us consider now a partition P of [q] with j 1 sets with 1 element, j 2 sets with 2 elements, ..., j q sets with q elements. We notice that j 1 + 2j 2 + 2j 3 + ... + 2j p < j 1 + 2j 2 + ... + qj q = q excepted if j 3 = ... = j q = 0 (and then the two terms are equal). In particular,

) ⊙q (F ) converges in law, it converges to the same limit as

, where the sum is restricted now to partitions P of [q] with j 3 = ... = j q = 0. The next theorem follows.

Theorem 3.4. With the above asumptions, we have the convergence in law:

Here, J runs over the partitions of [q] -{i 1 , ..., i q-2k } in ordered pairs J 1 = {j 1 , j ′ 1 }, ..., J k = {j k , j ′ k } (where j i < j ′ i and j 1 < ... < j k ). Besides, ⌈x⌉ stands for the integer part of a real number x (the highest integer less or equal to x). The I(f i ), i = 1, ..., q are Wiener integrals of the functions f i . They form a Gaussian family with E[I(f i )] = 0 and E

Corollary 3.5. In the particular case F = f ⊗q with η t (f 2 ) = 1, we get the convergence in law:

where H k stands for the k-th Hermite polynomial. Indeed, in that case, since the number of partitions of [q] with q -2k singletons and k pairs is q! 2 k (q-2k)!k! , the Theorem simplifies to a convergence in law of N -q 2 (η N t ) ⊙q (F ) to

where one recognizes the expansion of the qth ("probabilistic") Hermite polynomial:

x q-2k .

[Del04] Pierre Del Moral, Feynman-Kac formulae, Probability and its Applications (New York), Springer-Verlag, New York, 2004, Genealogical and interacting particle systems with applications.

[dlPG99] Víctor H. de la Peña and Evarist Giné, Decoupling, Probability and its Applications (New York), Springer-Verlag, New York, 1999, From dependence to independence, Randomly stopped processes. U -statistics and processes. Martingales and beyond.