Sylvie Patron

NARRATIVE FICTION BEFORE 1850: INSTANCES OF REFUTATION FOR POETIC THEORIES OF NARRATION? 1

1. In "Fiction, Pretense, and Narration", an article which has often been cited since its publication on the occasion of the French edition of Käte Hamburger"s Die Logik der Dichtung (The Logic of Literature), [START_REF][END_REF] Jean-Marie Schaeffer focuses on Hamburger"s narrative theory, which he considers to be incompatible with narratology, or at least with certain aspects of narratology. In the last part of his article, he offers the following argument: [...] contrary to what Hamburger thinks, the ideal model that emerges from her analyses is not that of fictional narrative in the third person, but a much more specific model  that of heterodiegetic narrative with internal focalization, that is, with a narrator who recounts a story from which he is absent while adopting the perspective of his characters or character. Indeed, the definition of a fictional narrative as a Here and Now presentation of characters and their inner life only seems to be realizable via strict internal focalization, any element of external focalization resulting fatally in a split of the narrative universe (a split that Hamburger's theory does not recognize). In pure form, this model of narrative (which, as a global model, scarcely dates back to before the second half of the nineteenth century) seems quite distant in fact from serious narrative: the idea that description of the interior life of a third person is prohibited in serious narrative seems unassailable at first glance. (160) I will single out one point only from this passage (notably leaving to one side the issue of whether the description of the inner life of a third person is prohibited in serious narrative or not). [START_REF] Cf | 160-161) for a negative answer using the example of Freud"s case histories and, for an opposite view using the same examples[END_REF] Hamburger put forward a theory which claimed to be a general theory of fictional narrative. According to Schaeffer, Hamburger"s theory would not account for fictional narratives before 1850; in other words, the latter would refute it, or would at least refute some of its essential propositions. In this article I would like to examine the legitimacy of his assertion, in the case not only of Hamburger"s theory, but also of theories by S.-Y. Kuroda and Ann Banfield, which I propose, following the latter two theorists, to call "poetic theories of narration". 4 As is clear from the passage quoted, like many other narratologists, Schaeffer is less interested in what Hamburger "thinks" or expresses explicitly, than in what "emerges" or is supposed to emerge from her analyses, which she neither thought nor expressed. However, it should be noted that not all narratologists have the same view of the model emerging from Hamburger"s analyses. Dorrit Cohn writes, for example, that: "[...] starting out from textual observations, [Hamburger] demonstrates that certain language patterns are unique to fiction, and dependent on the presence of fictional minds within the text" (1978: 7) and also that: [...] where Hamburger"s theory differs from all of these typologies [Norman Friedman"s, Franz Stanzel"s, Gérard Genette"s, etc.] is that for her, psychic omniscience is not a narrative type or mode or device or technique, but the pivotal structural norm that rules the realm of third-person fiction and that is logically ruled out in all other discursive realms. (1999: 25) For Cohn, the model which emerges from Hamburger"s analyses is not that of "heterodiegetic narrative with internal focalization", but that of the kind of narrative which is traditionally termed "omniscient", because it contains information unavailable to individuals with ordinary cognitive abilities (on the other hand, Cohn is right to say that, for Hamburger, the concept of psychic omniscience is not simply an issue of taxonomy).

It should also be noted that Schaeffer (1998: 162) is a relativist when it comes to Hamburger"s theory ("this model of narrative [...] scarcely dates back to before the second half of the nineteenth century" and, later, "Once we look further than the historical field that the author of The Logic of Literature privileges [...] the phenomena of fictionalization are far from being so omnipresent as her theory leaves it understood to be"). 5 By contrast, he is hardly a relativist in his use of narratological categories: the opposition between "homodiegetic" and "heterodiegetic" narrative (ib. 163, 166), which supports the narratorial theory of narrative, in other words, the theory of a narrator in all narrative; 6 "focalization" (ib. 167), defined as "the question of knowing from what viewpoint events are narrated" (despite the fact that narratology also use the term "focalization" to refer to "non-focalization" or the absence of point of view); the opposition between "internal focalization" and "external focalization". It is as though it were taken for granted that in any narrative with point of view, "a narrator [...] recounts a story from which he is absent while adopting the perspective of his characters or character" and that "any element of external focalization result[s] fatally in a split of the narrative universe". Yet Schaeffer may be considered, particularly with the benefit of hindsight, to be too ready to take for granted an issue which in fact begs further investigation.

The two points I have just cited make it possible to explain, in part, why Schaeffer pays only scant attention to the law which, according to Hamburger (83), governs third-person fictional narrative, to wit the disappearance of the author as the point of origin of the referential values and anchors of deixis ("I-Origo" in Hamburger"s terminology), correlative with the possibility of representing "the I-originarity (or subjectivity) of a third-person qua third-person". Schaeffer (1998: 150) quotes Hamburger"s famous statement regarding the use of verbs expressing internal processes in the third person ("She thus concludes that "Epic fiction is the sole epistemological instance where the I-originarity (or subjectivity) of a third-person qua third person can be portrayed""); he raises the issue on several occasions with reference to temporal and spatial deixis and to the epic preterit. However, he does not give the proposition any real theoretical status and, in particular, he does not discuss it in the final part of his article, which is devoted to a comparison between Hamburger"s narrative theory and narratology. To my mind, by contrast, the proposition is at the heart of Hamburger"s theory as well as Kuroda"s and Banfield"s. It is fundamental to the comparison which should be made between poetic theories of narration and narratology.

2. I will now briefly sketch a version of narratological theory in order to determine the extent to which Hamburger"s, Kuroda"s and Banfield"s theories are compatible or incompatible with it. 7 Narratology can be summarized by the following propositions:

(i) In any narrative, the story (the series of events told in the narrative) can be distinguished from the narrative itself (the spoken or written discourse which recounts the events). This distinction traditionally relies on the existence of discordances between the chronological order of the recounted events and the order of their disposition in the narrative.

(ii) The narrative is always recounted by somebody addressing somebody else (even in the case of written narrative: here, "recounted" signifies "produced in an oral or written verbal form"): this is narrating in Genette"s terminology.

(iii) In the case of fictional narrative, the story and the narrating (and thus the narrator and the narratee) are fictional. More precisely, a fictional act of narrating duplicates the author"s real act, which narratology does not analyse, although without it there would simply be no narrative. [START_REF] Schaeffer | confuses the author and the narrator from this point of view[END_REF] The fictional narrator recounts a series of events, with which he or she is familiar before recounting them, to the fictional narratee. It is the narrator who makes use of the categories of time (order, duration, frequency), mood and voice in Genettian narratology. It is the narrator who is responsible for the selection and presentation (sometimes termed "focalization", cf. Bal 1997: 8 et passim) of narrative information in other narratologies.

(iv) Genette distinguishes between homodiegetic fictional narratives (whose narrator is present as a character in the story he or she recounts) and heterodiegetic fictional narratives (where the narrator is absent from the story recounted). However, this distinction is only a secondary distinction within the category of fictional narrative, defined as the discourse of a fictional narrator. Other narratologists make an equivalent distinction between fictional narratives narrated by an internal narrator or an external narrator (cf. Chatman 155, 158, 170 and Stanzel 4-5 et passim) and by an external narrator or a narrator-character (cf. Bal 1997: 22 et passim, Chatman 234 and Stanzel 90).

(v) However, Genette makes no distinction between heterodiegetic fictional narratives in which the narrator refers to himself or herself using a first-person pronoun and those in which the narrator never refers to himself or herself and is not only absent as a character in the story recounted, but totally invisible in the narrative. 9 (vi) These propositions are not linked to any given narrative tradition. In any given tradition, individual author"s work or any particular narrative, the importance given to particular elements of time, mood or voice may vary; by contrast, the story, the narrative and the narrating, the fictional nature of the story and its narrating in fictional narrative, as well as the distribution of fictional narratives into the categories of homodiegesis or heterodiegesis are all invariable.

(vii) The fact that the fictional narrator makes use of the categories of time, mood and voice, or the fact that the narrator is homodiegetic in certain fictional narratives and heterodiegetic in others, or again, that certain heterodiegetic narrators never refer to themselves, are not responsible for any special "semantic effects" which might differ from the contents and relations determined in (i) and (ii) and play a role in the interpretation of a particular fictional narrative. On this point, Genette"s narratology differs from other narratologies which are more concerned with the issue of interpretation. [START_REF] Cf | notably, Chatman (234, on the "unreliable narrator[END_REF] (viii) Narratology claims a similarity to John Searle"s theory of fictional discourse (cf. Ryan 1981: 518-19, Genette 1993: 70-71 and Schaeffer 1994: 53 among others). It sees as equivalent Searle"s proposition that "the author of a work of fiction pretends to perform a series of illocutionary acts of assertion" and the proposition which can be deduced from (iii), that "the fictional narrator performs illocutionary acts of assertion". In doing so, it extends the description reserved by Searle for the first-person fictional narrative, or a version of it, to all fictional narratives. I have tried elsewhere to show that this extension is abusive and indeed contradictory from Searle"s perspective (cf. Patron 2009: 100-106, 123-25, 126-27, 130-33).

This presentation could give rise to long commentaries on the origins of such propositions, their formulation, which can vary from one author to another (certain propositions may even remain implicit) and on their retention by the school of narratology which is now called "postclassical". 11 I will simply offer two comments.

The theory (i)-(viii) is a minimal, non-linguistic version of narratological theory. More content might be afforded to the theory through interpreting the propositions (ii), (iii), (iv) and (v) from a linguistic point of view. Insofar as it is a narratorial theory of narrative, narratology implies that any sentence of a narrative is considered as a message communicated by a narrator, whether real or fictional, to its receiver or narratee, whether real or fictional (the issue of knowing precisely what "fictional" means will be left to one side for the time being). For fictional narratives, one could speak in terms of double communication or communication on two levels, the first one real, the second fictional, by forming the hypothesis of a relation between the two levels similar to quoting direct speech without introduction or comment clause; however, narratology does not analyse the first level, so that the fictional level takes the place of the two levels by itself. The distinction between firstperson fictional narratives (homodiegetic narratives, in Genette"s terminology) and third-person fictional narratives (heterodiegetic narratives) has no pertinence within this communicational framework: any sentence in a fictional narrative is considered as a message which is communicated either by a character who refers to himself or herself as I, or by a speaker who can never refer to himself or herself, but is nevertheless still a first person who is not actual but can be actualized as such.

As I have just pointed out, the theory from (i) to (viii) contains a blind spot. Narratology affirms that, in fictional narrative, the narrator and the narratee are fictional, yet it has no concept of fictionality. It follows quite simply from its affirmations that it applies a double standard, one holding for the narrator of first-person fictional narratives, as well as the other characters, places, etc. in the fiction which are often termed "fictional entities" and are considered to have the same properties as entities in the real world, and another holding for the narrator of thirdperson fictional narratives (I am thinking in particular of those in which the narrator never refers to himself and is totally invisible and absent from his narrative). If the first narrator is undoubtedly fictional, the second might more legitimately be termed "postulated" (in the theoretical sense) or simply "theoretical". 12 3. I shall now turn to poetic theories of narration. It is much more difficult to present an acceptable version of these theories in a succinct manner than it is for narratology. It is also difficult to rectify the misreadings and misrepresentations to which they have given rise and potentially provide more content to elements which remain undetermined, without being able to discuss them at length (cf. Patron 2009: ch. 7-9). I shall limit myself to emphasizing three aspects: their calling into question of the idea of communication; their refutation of the narratorial hypothesis in certain precise cases and their reinterpretation of an old theory in new theories.

3.1. For the representatives of poetic theories of narration (precisely, Kuroda and Banfield), there is nothing obvious in the idea that the relation between author and reader in fictional narrative is one of communication in any essential or interesting sense of the term "communication". The adjectives "essential" and "interesting" are Noam Chomsky"s (56, 57), quoted by Kuroda (1980: 69). 13 In order to speak of communication in an interesting manner, it is necessary to be able to rely on a linguistic and, potentially, a pragmatic analysis of that which pertains to communication, as opposed to that which can be understood not to pertain to it. Neither is there anything obvious in the idea that there is a fictional communicational relationship in all fictional narratives which implies a fictional narrator and narratee (Kuroda and Banfield take the same position as Hamburger here).

Based on certain characteristics of the Japanese language (usage of the adjectival form of adjective/verb pair for expressing feeling, usage of the word zibun in certain structures), Kuroda"s linguistic analyses establish:

(i) homogeneity between sentences in ordinary communication and sentences in first-person fictional narrative, 14 as well as sentences in certain third-person fictional narratives which could be seen as narrated by a narrator who never refers to himself or herself (examples might be "neutral" or "behaviourist" narratives such as certain short stories by Hemingway); 15 (ii) heterogeneity, by contrast, between sentences in ordinary communication and certain sentences in certain third-person fictional narratives (third-person fictional narratives with the representation of the "point of view" of one or several characters: Joyce"s Portrait of the Artist as a Young Man or Woolf"s To the Lighthouse, for example). 16 Sentences which use the adjectival form of the adjective/verb pair for expressing feeling (kanasii instead of kanasigaru ["sad"]) or the word zibun ("self") for representing "point of view" or the "lived experience" (Erlebnis) of one or more characters in the third person cannot be described adequately within the framework of the communicational theory of linguistic performance which is based on the concepts of speaker and receiver. On the one hand, Kuroda states that:

[t]he role of the writer of a non-first-person story is to assemble (in fact, create) such information and set it in order. This is in no way identifiable with the role of the "speaker" in the paradigm of linguistic performance [...]. (1973: 382-3) On the other hand, he uses precise linguistic tests (insertion of yo at the end of the segment, rewriting of the sentence in the first person), to show that such sentences cannot be seen to be narrated by a fictional narrator adopting the viewpoint of one or more characters or taking on the task of presenting their Erlebnis.

Kuroda then refers to Benveniste on histoire and discours, Hamburger on fiktionales Erzählen and Banfield on free indirect discourse (represented speech and thought) to fuel his critique of communicational theories of narrative. He formulates Genette"s revision of Benveniste"s opposition between histoire and discours very clearly (Kuroda 1976: 116, 125, 138). 17 He also puts forward a new theory of linguistic performance to replace communicational theory and provide access to a unified conception of narrative and discourse in general. His theory is based on the identification and characterization of the objective function, as he puts it, of the sentence, as distinct from its communicative function. The distinction points to the fact that any sentence which has been created as a real entity in the world calls up a meaning in the mind of the person hearing or reading it -and does so whether the sentence is materialized in a situation of communication or not. Kuroda writes, for example, about the sentence containing zibun:

Reading this sentence we obtain an image or knowledge of an event, but we do not assume the existence of any consciousness which has judged the occurrence of this event and communicated it to someone. Simply the sentence creates in us the image or knowledge of the event. This and this much is the function of the sentence vis-à-vis the reader. (1976: 134) Based on the comparison between the effects produced by speech acts like orders or information (as distinct from assertions) on an outside listener and on the speaker"s addressee in the strict sense, Kuroda"s pragmatic analyses lead to the identification and characterization of three linguistic functions: the objective function (which is the most essential, in the sense that it is an essential component of the second and third functions); the objectifying function which consists, for a speaker, in objectifying the content of his mind (this is an component in the third function) and the communicative function which consists, for a speaker, in communicating the objectified content to the addressee (this it the least essential function in the sense that it presupposes the others but is not presupposed by them). A speech act is only an act of communication if the communicative function is activated. Kuroda indicates in passing that in Japanese, the communicative function can be recognized by specific formal traits (stylistic traits, terms and honorific constructions). He comes to the conclusion that the crucial characteristic of "the communicational theory of narration" (the quotation marks are in his text) is more closely related in the end to the objectifying function than the communicative one:

The core of the theory is the assumed existence of a narrating agent. A narrative is assumed to be a product of the objectifying act of the narrator. Opinions concerning the role of the audience or the reader can vary. (1979: 10-11) To this he opposes his own poetic theory of narration:

The supposition of a narrator who objectifies the content of his consciousness in the form of a narrative is quite unnatural, especially in certain types of modern fiction where the inner experiences of multiple protagonists are simultaneously depicted. The function of language in such fiction cannot be accounted for in terms of the communicative or objectifying function. I have thus identified the function that makes such use of language possible as the objective function of language, the function such that an attentive consciousness cannot fail to respond to objectively materialized sentences and make out of them whatever sense can be constructed from them, without any necessity of recognizing or hypothesizing the existence behind those sentences of a consciousness objectifying what it perceives or judges. (ib. 11)

Banfield"s linguistic analyses, to which I shall return in the next section, make use of Kuroda"s and further them on a certain number of points: bringing the axiomatic nature of communication in the communicational theory of narration into question; affirming the necessity of defining communication in rigorous linguistic terms; defining linguistic subjectivity (this definition is based on a comparison between direct discourse, indirect discourse and free indirect discourse, or represented speech and thought, which enables the formal marks of the expression of subjectivity and those of communication to be isolated); introducing, alongside the idea of the speaker, the new idea of the subject of consciousness or SELF to designate the origin of deixis and subjectivity; characterization of sentences in third-person free indirect discourse as sentences which are not governed by the framework structured by the relation of communication between an I and a you.

Reflection on the role of writing can also be found in Banfield"s work, in the sense of written linguistic realization (as distinct from the transcription of oral language) in the production and reception of fictional narratives. In Banfield"s theory, writing is the extra-linguistic factor which enables certain potentialities of language to be actualized in performance.

3.2. This section is devoted to the refutation of the narratorial hypothesis in several specific cases. Banfield offers both the most powerful counter-examples and the most refined theory enabling explanation both of counter-examples to the initial hypothesis and of the facts it explained. I shall simply recall the following points:

(i) the principles she formulated for free indirect discourse (represented speech and thought): the principle of the uniqueness of the subject of consciousness, according to which "[f]or every node E, there is at most one referent, called the "subject of consciousness" or SELF, to whom all expressive elements are attributed" 18 (this principle is a reformulation of the "one expression/one speaker" principle for direct and indirect discourse); the principle of the priority of the speaker, according to which "[i]f there is an I, I is coreferential with the SELF" (Banfield 1982: 93).

(ii) the test enabling these principles to be justified: if an I is added to a sentence of third-person free indirect discourse, it becomes clear that it is no longer possible to attribute expressive elements and constructions to a subject other than the referent of the I (in other words, it is no longer possible to consider them as sentences of third-person free indirect discourse):

Where were her paints, she wondered? Her paints, yes. (Virginia Woolf, To the Lighthouse 168.)

Where were my paints, *she wondered? My paints, yes. 19 As Banfield puts it:

Since no first person may appear in represented speech and thought except one interpretable as the E"s SELF and since that first person must also appear in any parenthetical attached to the represented E, this means that represented Es cannot be simultaneously attributed to a covert or "effaced" narrator. (Banfield 1982: 97) 20 Note that in Banfield"s view, if sentences of free indirect discourse do not necessarily have a narrator, they do have an author who is responsible for their existence as real entities in the world as well as their disposition and role within the text.

The same test could be used and the same reasoning put forward in relation to sentences which, according to Banfield, represent subjectless subjectivity:

The sun had now sunk lower in the sky. (Virginia Woolf, The Waves 129) 21 It is clear that these sentences cannot be attributed to a covert or effaced narrator (or one revealed indirectly by the use of the adverb "now") without losing their essential characteristics: the combined occurrence of "now" and a past tense; the absence of any subject of consciousness to whom the "now" might be attributed.

3.3. This section concerns the question of the reinterpretation of a old theory within new theories. I shall only deal with Hamburger"s theory, but my comments can be extrapolated to Kuroda"s and Banfield"s.

The conceptualization of the difference between third-person fictional narrative, in which the author himself recounts the story and first-person fictional narrative (in the form of an embedded narrative 22 ), in which a character fictionally recounts the story, dates back to Aristotle. The concept of a narrator distinct from the author made its appearance much later, in the early 19th century, in the first descriptions of first-person fictional narrative without embedding (fictional memoirs or first-person novels in the proper sense). 23 The issues it encapsulates are the following: an I which does not belong to the author but to a fictional character; a pact of truthfulness within the fiction (the narrator speaks of the past in a truthful manner, which does not mean that he cannot lie or deform the truth, but simply that he does not relate a fictional narrative); restriction of narrative information to knowledge open to the I and to his memory; potentially a restriction of thought and style proper to the ability of the I; some degree of contrast in terms of their age, social status, general knowledge and rational ability between the narrated I in the story and the I narrating the story.

In Hamburger"s theory, the following reappear:

-the difference between third-person fictional narrative, considered as a general or standard case and first-person fictional narrative, considered as a particular case; 24 -the traditional view of the narrator reserved for first-person fictional narrative: Hamburger claims that "only in cases where the narrative poet actually does "create" a narrator, namely the first-person narrator of the first-person narrative, can one speak of the latter as a (fictive) narrator" (140); 25 she thus rejects the Stanzelian view of the (fictitious or fictional) narrator, extending to third-person fictional narratives with intrusions such as "I", "we", "our hero" and, from there, to all third-person fictional narratives; 26 -the issues implied by the traditional concept of the narrator, in particular the issue of the narrator belonging to the fictional world and the restriction of information to knowledge (of the inner life of the other characters) and memory available to the narrator.

By contrast, Hamburger is the first theoretician to conceptualize:

-the fact that the author of a third-person fictional narrative does not narrate in the same way as the author of a factual narrative or the narrator of a first-person fictional narrative within the fiction: he or she appears to narrate events which exist prior to the act of narration (which Hamburger expresses by provisionally retaining the term of "narrator") (cf. 62 et sq.), 27 but in reality, he or she "creates" such events, or makes them exist fictionally (which Hamburger conceptualizes in terms of "functional correspondence" between the narration and that which is narrated (cf. 136), while the narrator of a first-person fictional narrative "does not "engender" that which he narrates, but narrates about it in the same manner as in every reality statement [...]" (317) (which Hamburger conceptualizes in terms of "feigned reality statements") (cf. 311, 313 et sq.); -the fact that the author of a third-person fictional narrative disappears as the point of origin for referential values and deictic markers (real "I-Origo"), which allows the apparition of "fictive I-Origines" whose "I-originarity (or subjectivity)" may be developed to a greater or lesser degree (cf. 73 and 83), 28 while the narrator of a first-person fictional narrative "is always present" as I-Origo of the narrative and "never disappears, which [...] would result in the emergence of fictive I-origines in its place" (315); 29 -the fact that the third-person fictional narrative bears linguistic and textual signs of this appearance/disappearance: verbs describing inner processes in the third person, monologue and free indirect discourse, past tenses combined with present and future deictic adverbs, situational verbs combined with indications of date and long passages of dialogue (the list should not be considered exhaustive); in principle, such signs should not be used in first-person fictional narratives; in practice, they are often found combined with signs of first-person narration (few authors renounce long passages of dialogue, for example) (cf. 332, 337);

-the fact that third-person fictional narrative is a narrative with no narrator, since the hypothesis of an omniscient narrator as a corollary to that of a fictional narrator in third-person fictional narratives is so vague and devoid of independent justification that it can only be viewed as a pseudo-hypothesis (cf. 141). 30 I have two final comments to make.

Hamburger"s theory does not deny those aspects of fictional narrative which were examined by narratology under the category of time (order, duration, frequency). 31 However, her theory, like those of Kuroda and Banfield, attributes the use of such categories, or techniques of composition, not to the narrator, but to the author -as was current before Genette and the advent of narratology. Their theories distinguish elements pertaining to fictional content (characters, places, etc. and the narrator if there is one) from the means put to work in constructing the fiction (language, style, techniques of composition). 32 Searle"s and Hamburger"s theories are not fundamentally incompatible. 33 Notably, they share elements stemming from the traditional view: the difference between third-person fictional narrative and first-person fictional narrative, the view of the narrator, etc. They also share a study of how reference works in fictional narrative (in Speech Acts, Searle (1969: 78) even evokes sentences which cannot be written or said: "Sherlock Holmes is coming to my house for dinner tonight", for the reference to "my house" puts us back in real world talk). It is true that Searle claims categorically that "[t]here is no textual property, syntactical or semantic, that will identify a text as a work of fiction " (1979: 65). But the necessary and sufficient properties for identifying a text as a work of fiction (Searle) should be distinguished from signs pointing to the fictional character of the existence of that which is narrated (Hamburger). 34 Besides, if Searle had paid closer attention and taken his reading further, he would have found numerous signs of fictionality in Hamburger"s sense in his chosen example of a fictional work (Iris Murdoch"s The Red and the Green), some of which could have been considered sufficient to identify the text as a work of fiction (cf. Wildekamp et al. 1980: notably 559-561, 565). As for Hamburger"s theory, she does not deny that many sentences or passages in fictional narratives are formally identical to those found in either oral or written factual narratives. 35 4. I mentioned at the outset that Hamburger put forward a theory which claimed to be a general theory of fictional narrative. However, her generalizing aim is not achieved in the same way in all the domains she examines. In the case of signs of fictionality, for example, globally speaking the theory attains a certain degree of generality -Hamburger does not claim that all the signs discovered and explained are found in all third-person fictional narratives, whatever their genre, period or the author under consideration. 36 By contrast, Hamburger does put forward a theory of the history of third-person fictional narratives (essentially incarnated in the novel as a genre), which is that of the ever greater autonomy in the presentation of characters in relation to that which a real subject could say or write of people belonging to a shared reality (this is what she understands by forming characters "as fictive persons or subjects") (59). She writes, for example:

We already have pointed out on several occasions that in the course of the nineteenth century the techniques of fictionalization had become more and more refined, the presentation of the psychic life came increasingly more to employ the devices of directly subjectifying the characters. That is the fictive I-originarity of the figures became more and more explicitly developed, culminating in the bold methods of Joyce. (168)

In the same way, Kuroda"s and Banfield"s theories aim to be theories of fictional narrative and not of a given corpus of fictional narratives, even if this may appear less obvious than in Hamburger's case. Remember the sentence from Kuroda (1979:11) quoted above:

The supposition of a narrator who objectifies the content of his consciousness in the form of a narrative is quite unnatural, especially in certain types of modern fiction where the inner experiences of multiple protagonists are simultaneously depicted. (my italics) I do not have time to present the "deictic shift theory" developed by a group of researchers in cognitive science at the State University of New York at Buffalo, which forms an attempt at generalization, as well as an empirical validation, of poetic theories of narration (cf. Duchan et al., eds.: notably ch. 1, 2 and 6).

Is it the case that fictional narratives prior to 1850 are not accounted for by poetic theories of narration -in other words, that they refute them, or at least refute some of their essential propositions? For this to be demonstrated, it would first need to be shown that a particular fictional narrative prior to 1850, chosen for a particular emblematic reason, formed an authentic counter-example to the theories or to some of their essential propositions. Next, multiple analyses of as wide and varied a range as possible of pre-1850 fictional narratives would need to be carried out. Finally, as part of the comparison between narratology and poetic theories of narration, it should be ascertained that the theory which esteems the counter-examples to be recognized as such is indeed narratology (one of the essential propositions of which, to wit "fictional narrative is the discourse of a fictional narrator", would need to be considered unrefuted). To the extent that this programme of work has not yet been carried out, I shall conclude by saying that if poetic theories of narration may perhaps be refutable by fictional narratives prior to the latter half of the 19 th century, we cannot yet consider them refuted.

Translated by Susan Nicholls

double standard mentioned. 13 Cf. also Banfield"s "Linguistic Competence" (1983: 229). 14 Kuroda (1973: 383) does not use "first-person story" in its traditional sense (a narrative in which a character tells his or her own story, or one in which he or she has taken part as a witness, and refers to himself or herself using a first-person pronoun) but in the sense of a narrative in which a narrator, who "may or may not be a character in the story", refers to himself or herself using a first-person pronoun. By contrast, Kuroda later (1979:10) offers a description of first-person fictional narrative (as "fictional autobiography") which coincides with the traditional description. 15 "Heterodiegetic narratives with external focalization" in Genette"s terminology. 16 "Heterodiegetic narratives with internal focalization" in Genette"s terminology. It is clear that Kuroda"s linguistic analyses lead to a very different taxonomy from Genette"s. Kuroda and Genette have a common reference, Norman Friedman. Cf. Kuroda (1973: 383) and Genette (1980: 187-188). 17 Cf. Genette (1982: 137-143). Cf. also Patron (2011, forthcoming). 18 The node E designates the initial (non-recursive) symbol of the basic rules in Banfield"s grammar. 19 This example is cited in Banfield (1982: 72, 94). 20 The first person interpretable as the SELF of the E refers in the case of free indirect discourse to the first person. 21 This example is cited in Banfield (1987: 273). 22 "Intradiegetic-homodiegetic narrative" in Genette"s terminology. 23 "Extradiegetic-homodiegetic narrative" in Genette"s terminology. 24 Hamburger uses "first-person narrative" in its traditional sense: "an autobiographical form which reports events and experiences referred to by the first-person narrator" (311). She explicitly leaves to one side the case of first-person narrative frames which introduce narratives which in fact function as third-person fictional narratives, either because the author did not manage to maintain the formal constraint of first-person narrative or because the author deliberately chose to play with it. 25 The sentence is quoted by Banfield in American translation (1978: 297 and 1982: 185). 26 Hamburger (1993: 142-175) accounts for third-person fictional narratives with authorial intrusions in the context of a study of objectivity and subjectivity in narrative (cf. also ib. 337-38). 27 The elimination of the term "narrator" to designate the author of third-person fictional narrative occurs on pages 139-40. 28 Cf. also 137, where Hamburger presents "the absence of the real I-Origo" and "the functional character of fictional narration" as two different formulations of a same fact. 29 The same idea occurs in Banfield"s theory of free indirect discourse. 30 Cf. also Kuroda (1973: 389, 390 and 1976: 133) and Banfield (1982: 211, 219). 31 Cf. Hamburger (1993: 226) on flashback technique in cinema and the novel. Cf. also Banfield (1985: 389, 392, 205 and 210), on iterative frequency in Proust. 32 Cf. Banfield (1982: 248, 253) and also Galbraith (1995: 49-50), who develops some of Banfield"s intuitions. I have tried to show the benefits to be gained from the distinction in analyzing and interpreting one particular fictional narrative. Cf. Patron (2010: 267-270). 33 Cf. also Schaeffer (2009: 110), who revises some of the propositions expressed in his earlier articles and books. 34 The distinction is expressed very clearly by Schaeffer (1994: 33-34, 41). 35 Cf. Hamburger (1993: 60-61, 68-70, 89-90, 134-135, etc.). Cf. also Banfield (1982: 257 et sq.) even though Banfield"s remarks only concern true, written stories, which use the simple past tense in French. 36 Cf. Hamburger (1993: 81-82, 149, 150, 177-179, 185). Page 150 contradicts Searle"s assertion that Hamburger"s theory does not account for what he terms "external focalization" (which Hamburger herself calls the presentation of characters "from without", in quotation marks in the text).


S Delattre

E Roquain

On the false discovery proportion convergence under Gaussian equi-correlation

Keywords: False discovery rate, Donsker theorem, equi-correlation, functional Delta method, p-value. 2000 MSC: 62G10, 62J15, 60F05

We study the convergence of the false discovery proportion (FDP) of the Benjamini-Hochberg procedure in the Gaussian equi-correlated model, when the correlation ρ m converges to zero as the hypothesis number m grows to infinity. By contrast with the standard convergence rate m 1/2 holding under independence, this study shows that the FDP converges to the false discovery rate (FDR) at rate {min(m, 1/ρ m )} 1/2 in this equi-correlated model.

Introduction

When testing simultaneously a large number m of null hypotheses, a popular global type I error, that can be traced back to [START_REF] Seeger | A note on a method for the analysis of significances en masse[END_REF], is the false discovery proportion (FDP), defined as the ratio of the number of erroneous rejections to the number of rejections. The average of this random variable, called the false discovery rate (FDR, introduced by [START_REF] Benjamini | Controlling the false discovery rate: a practical and powerful approach to multiple testing[END_REF]), has been studied by a considerable number of works, see for instance [START_REF] Sarkar | On methods controlling the false discovery rate[END_REF] and [START_REF] Blanchard | Adaptive FDR control under independence and dependence[END_REF] for a review. However, studying the FDR is not sufficient to catch the full behavior of the FDP, for instance a FDR control does not prevent the FDP from having large variations. Therefore, some other studies aim to directly control the upper-quantile of the FDP distribution, see e.g. [START_REF] Genovese | Exceedance control of the false discovery proportion[END_REF]; [START_REF] Lehmann | Generalizations of the familywise error rate[END_REF], or to directly compute the distribution of the FDP, either non-asymptotically [START_REF] Chi | Positive false discovery proportions: intrinsic bounds and adaptive control[END_REF]; [START_REF] Roquain | Exact calculations for false discovery proportion with application to least favorable configurations[END_REF], or asymptotically [START_REF] Genovese | A stochastic process approach to false discovery control[END_REF]. Recently, [START_REF] Neuvial | Asymptotic properties of false discovery rate controlling procedures under independence[END_REF][START_REF] Neuvial | Corrigendum to "Asymptotic properties of false discovery rate controlling procedures under independence[END_REF] computed the asymptotic distribution of the FDP actually achieved by the Benjamini-Hochberg (BH) procedure (and some other adaptive procedures) under independence of the p-values. It is proved that the FDP converges to the FDR at the parametric rate √ m. Furthermore, [START_REF] Farcomeni | Some results on the control of the false discovery rate under dependence[END_REF] showed that this convergence is unchanged under a specific shortrange dependency between the p-values.

In this paper, we are interested in studying the convergence of the FDP of the BH procedure in the model where the test statistics have exchangeable Gaussian errors, with equi-correlation ρ (allowing for instance long-range dependencies). This model has become quite standard in multiple testing (see e.g. [START_REF] Benjamini | Adaptive linear step-up procedures that control the false discovery rate[END_REF][START_REF] Finner | Dependency and false discovery rate: asymptotics[END_REF]), as it is a very simple instance of dependent p-value model. From an intuitive point of view, the test statistics can be seen as independent test statistics plus a disturbance variable whose importance depends on the value of ρ. When ρ ∈ (0, 1) is fixed with m (and in the "ideal" setting where the p-values under the alternative are all equal to zero), [START_REF] Finner | Dependency and false discovery rate: asymptotics[END_REF] proved that the FDP of the BH procedure converges to a non-deterministic random variable that still depends on the disturbance variable. When ρ m → 0, we show here that this disturbance variable has no effect on the limit of the FDP anymore, which equals π 0 α (where π 0 is the proportion of true nulls), but can still have an effect on the asymptotic variance of the FDP or even on the convergence rate. More precisely, when ρ m → 0 our main result states that {min(m, 1/ρ m )} 1/2 (FDP mπ 0 α) N(0, V) holds for a given V > 0; in comparison with the independent case, we may distinguish the two following cases, recovering all the possible convergence regimes of ρ m to zero:

• when lim m mρ m = θ ∈ [-1, +∞), the limit of the FDP and the convergence rate are the same as in the independent case. The asymptotic variance V is larger if θ > 0, smaller if θ < 0 and is the same whenever θ = 0 (i.e. mρ m → 0).

• when lim m mρ m = +∞ and lim m ρ m = 0 the convergence rate is

ρ -1/2 m instead of m 1/2 .
On the one hand, this shows that the FDP of the BH procedure is still well concentrated around π 0 α under weak equi-correlation such that ρ m = O(1/m). On the other hand, this puts forward that the concentration of the FDP of the BH procedure around the FDR may be arbitrarily slow when ρ m → 0, which is a striking result that has not been reported before to the best of our knowledge. Therefore, our recommendation is that the BH procedure can be used under Gaussian equi-correlation when ρ m = O(1/m) (including the case of a negative equi-correlation) but should be used carefully as soon as mρ m → ∞, as the actual convergence rate of the FDP to the FDR might be much slower.

The paper is organized as follows: Section 2 presents the model, the notation and the main result. The latter is proved in Section 3, including a generalization to any "regular" thresholding procedure, recovering the so-called π 0 -adaptive procedures studied in [START_REF] Neuvial | Asymptotic properties of false discovery rate controlling procedures under independence[END_REF]. Finally, some further points in connexion with our methodology are discussed in Section 4.

Setting and main result

We observe X i = τ i + Y i , 1 ≤ i ≤ m, where the parameter of interest is (τ i ) i ∈ {0, µ} m (for a given µ > 0) and the (unobservable) error vector (Y 1 , ..., Y m ) is an exchangeable Gaussian vector with EY 1 = 0 and Var Y 1 = 1. We let

ρ m = Cov(Y 1 , Y 2 ) ∈ [-(m -1) -1 , 1].
We consider the problem of the one-sided testing of the null "τ i = 0" against the alternative "τ i = µ", simultaneously for any 1 ≤ i ≤ m. To test each null, we define the p-value p i = Φ(X i ), where Φ(z) = P(Z ≥ z) is the standard Gaussian upper-tail function. The c.d.f. of each p-value is denoted by G 0 (t) = t under the null and by G 1 (t) = Φ(Φ -1 (t)µ) under the alternative. The number of true nulls is denoted by m 0 (m) = |{i | τ i = 0}| and is assumed to be of the form mπ 0 for a given proportion of true null π 0 ∈ (0, 1) independent of m. The "mixture" c.d.f. of the p-values is denoted by

G(t) = π 0 G 0 (t) + (1 -π 0 )G 1 (t). Next, we define the e.c.d.f.'s G 0,m (t) = (m 0 (m)) -1 m i=1 1{τ i = 0}1{p i ≤ t}, G 1,m (t) = (m -m 0 (m)) -1 m i=1 1{τ i > 0}1{p i ≤ t} and G m (t) = m -1 m i=1 1{p i ≤ t}.
Given a pre-specified level α ∈ (0, 1), the procedure of [START_REF] Benjamini | Controlling the false discovery rate: a practical and powerful approach to multiple testing[END_REF] can be defined as the procedure rejecting the nulls corresponding to p i ≤ T BH ( G m ) where the (datadriven) threshold

T BH ( G m ) is max{t ∈ [0, 1] | G m (t) ≥ t/α}.
Next, the false discovery proportion at a given threshold t ∈ [0, 1] is defined as the proportion of true nulls among the hypotheses having a p-value smaller than or equal to t:

FDP m (t) = |{1 ≤ i ≤ m | τ i = 0, p i ≤ t}| |{1 ≤ i ≤ m | p i ≤ t}| ∨ 1 = m 0 (m) m G 0,m (t) G m (t) ∨ m -1 ,
where | • | denotes the cardinality function.

We now state our main result.

Theorem 2.1. There is a unique point t ∈ (0, 1) such that G(t ) = t /α and we have

(i) if lim m mρ m = θ ∈ [-1, +∞), then √ m FDP m (T BH ( G m )) -π 0 α N       0, π 0 α 2 1 -t t + θ π 2 0 α 2 2π(t ) 2 e -(Φ -1 (t )) 2       ;
(1)

(ii) if lim m mρ m = +∞ and lim m ρ m = 0, then ρ -1/2 m FDP m (T BH ( G m )) -π 0 α N       0, π 2 0 α 2 2π(t ) 2 e -(Φ -1 (t )) 2       .
(2)

3. Proof of Theorem 2.1

A more general result

In what follows, we denote the space of functions from [0, 1] to R which are right-continuous and with left-hand limits (Skorokhod's space) by D(0, 1) and the space of continuous functions from [0, 1] to R by C(0, 1). The method for proving our result relies on the methodology let down by [START_REF] Neuvial | Asymptotic properties of false discovery rate controlling procedures under independence[END_REF] which consider the case of a general threshold function T : D(0, 1) → [0, 1] which is Hadamard differentiable at G, tangentially to C(0, 1) (see van der Vaart (1998) for a formal definition). The proof of Theorem 2.1 is presented here as a consequence of a more general theorem, true for any such threshold. The derivative of the threshold T at G, which is a continuous linear form on C(0, 1), is denoted by ṪG . According to the Riesz representation theorem, the continuous linear form ṪG can be written as ṪG (F) = 1 0 F(t) ṪG (dt), where we identified the linear form ṪG and the corresponding signed measure.

Theorem 3.1. Let T : D(0, 1) → [0, 1] be Hadamard differentiable at G, tangentially to C(0, 1), with derivative ṪG . Let q(t) = π 0 t/G(t) for t > 0, let t = T (G) and assume t > 0. We set

ζ 0 = q(t )(1-q(t )) t δ t + q(t )π 0 ṪG , ζ 1 = -q(t )(1-q(t )) G 1 (t ) δ t + q(t )(1 -π 0 ) ṪG and c(T ) = (2π) -1/2 1 0 e -1 2 (Φ -1 (t)) 2 ζ 0 (dt) + (2π) -1/2 1 0 e -1 2 (Φ -1 (t)-µ) 2 ζ 1 (dt) ; σ 2 (T ) = π -1 0 [0,1] 2 (s ∨ t -st)ζ 0 (ds)ζ 0 (dt) + (1 -π 0 ) -1 [0,1] 2 (G 1 (s ∨ t) -G 1 (s)G 1 (t))ζ 1 (ds)ζ 1 (dt) .
Then the following holds:

(i) if lim m mρ m = θ ∈ [-1, +∞), √ m FDP m (T ( G m )) -q(t ) N 0, σ 2 (T ) + θc(T ) 2 ; (3) (ii) if lim m mρ m = +∞ and lim m ρ m = 0, ρ -1/2 m FDP m (T ( G m )) -q(t ) N 0, c(T ) 2 . ( 4
)
Let us now check that Theorem 2.1 follows from Theorem 3.1. From [START_REF] Neuvial | Asymptotic properties of false discovery rate controlling procedures under independence[END_REF] Corollary 7.12, T BH :

F → max{t ∈ [0, 1] | F(t) ≥ t/α} is Hadamard differentiable at G, tangentially to C(0, 1), with derivative Ṫ BH G = (1/α -Ġ(t )) -1 δ t . Moreover, t = max{t ∈ [0, 1] | G(t)
≥ t/α} is positive, because lim t→0 + t/G(t) = 0. Also, since G(t ) = t /α and q(t ) = (1/α -Ġ(t ))π 0 α 2 /t , we may check that ζ 1 = 0 and ζ 0 = (π 0 α/t )δ t in the above theorem, which leads to Theorem 2.1.

Proof of Theorem 3.1

Let us now prove Theorem 3.1. First write FDP m (T (

G m )) = (m 0 (m)/m)π -1 0 Ψ( G 0,m , G 1,m )
, where for any F 0 , F 1 in D(0, 1) with F(T (F)) > 0 (letting

F = π 0 F 0 + (1 -π 0 )F 1 ), we put Ψ(F 0 , F 1 ) = π 0 F 0 (T (F))
F(T (F)) . From standard computations, Ψ is Hadamard differentiable at (G 0 , G 1 ), tangentially to C(0, 1) 2 and the derivative takes the form, for (H 0 , H 1 )

∈ C(0, 1) 2 , ΨG 0 ,G 1 (H 0 , H 1 ) = q(t )(1 -q(t )) H 0 (t ) t -H 1 (t )
G 1 (t ) + q(t ) ṪG (H), where

H = π 0 H 0 + (1 -π 0 )H 1 .
Applying the functional Delta method, this leads to the following useful result, which was essentially stated in [START_REF] Neuvial | Asymptotic properties of false discovery rate controlling procedures under independence[END_REF].

Proposition 3.2. Let T : D(0, 1) → [0, 1] be Hadamard differentiable at G, tangentially to C(0, 1), with derivative ṪG . Let q(t) = π 0 t/G(t) for t > 0, let t = T (G) and assume t > 0. If for a given sequence a m → ∞ with a m = o(m),

a m       G 0,m -G 0 G 1,m -G 1       W 0 W 1 , (5)
where the convergence in distribution is relative to the Skorokhod topology and where W 0 and W 1 are processes with continuous paths, then we have

a m FDP m (T ( G m )) -q(t ) X, (6)
where X = ζ 0 (W 0 ) + ζ 1 (W 1 ) and ζ 0 , ζ 1 are defined as in Theorem 3.1.

A convergence of the type (5) in the particular Gaussian equi-correlated model is stated in Lemma 3.3. Using Proposition 3.2, this proves that (6) holds both in the cases (i) and (ii) with a m = √ m and a m = ρ -1/2 m , which respectively leads to (3) and (4) (the variance computations are straightforward).

3.3. Convergence of the e.c.d.f.'s in the Gaussian equi-correlated model

Lemma 3.3. Let (Z 0 , Z 1 , Z) be a random variable such that Z 0 (d) = π -1/2 0 B, Z 1 (d) = (1 -π 0 ) -1/2 B • G 1 , B being a standard Brownian bridge on [0, 1], Z 0 is independent from Z 1 , Z ∼ N(0, 1), Cov(Z, Z 0 (t)) = (2π) -1/2 exp (-{Φ -1 (t)} 2 /2) and Cov(Z, Z 1 (t)) = (2π) -1/2 exp (-{Φ -1 (t) -µ} 2 /2).
Let also U ∼ N(0, 1) be independent of the vector (Z 0 , Z 1 , Z). Then we have the following convergences in law for the Skorokhod topology:

(i) if lim m mρ m = θ ∈ [-1, +∞), √ m       G 0,m -G 0 G 1,m -G 1       Z 0 + (Z - √ 1 + θU) Φ • Φ -1 Z 1 + (Z - √ 1 + θU) Φ • (Φ -1 -µ) ; (7) (ii) if lim m mρ m = +∞ and lim m ρ m = 0, ρ -1/2 m       G 0,m -G 0 G 1,m -G 1       U Φ • Φ -1 U Φ • (Φ -1 -µ) . (8)
To prove Lemma 3.3, first remark that the distribution of the X i 's may be realized as

X i = 1 -ρ m (ξ i -ξ) + (1 + (m -1)ρ m )/m U + µ1{τ i > 0}
, where (ξ 1 , ..., ξ m , U) are all i.i.d. N(0, 1) variables and ξ denotes the empirical mean of the ξ

i 's. Let G 0,m (t) = (m 0 (m)) -1 i:τ i =0 1{Φ(ξ i ) ≤ t}, G 1,m (t) = (m -m 0 (m)) -1 i:τ i >0 1{Φ(ξ i + µ) ≤ t} and f m (t, U, ρ m ) = (1 -ρ m ) -1/2 Φ -1 (t) -(1 + (m -1)ρ m )/m U . The process ( G 0,m -G 0 , G 1,m -G 1 ) is then equal to V m + W m where V m (t) =        ( G 0,m -G 0 )(Φ( f m (t, U, ρ m ) + ξ)) ( G 1,m -G 1 )(Φ( f m (t, U, ρ m ) + ξ -µ(1 -ρ m ) -1/2 + µ))        W m (t) = Φ( f m (t, U, ρ m ) + ξ) -t Φ( f m (t, U, ρ m ) + ξ -µ(1 -ρ m ) -1/2 ) -Φ(Φ -1 (t) -µ) .
Next, applying Donsker's theorem, we derive

√ m( G 0,m -G 0 , G 1,m -G 1 , ξ) (Z 0 , Z 1 , Z), where (Z 0 , Z 1 , Z) is defined as in Lemma 3.3. Since ρ m → 0, the inverse functions of t → Φ( f m (t, U, ρ m ) + ξ) and t → Φ( f m (t, U, ρ m ) + ξ -µ(1 -ρ m ) -1/2 + µ) converge uniformly on [0, 1]
to the identity a.s. Therefore, applying the Skorokhod's representation theorem, we get

√ m(V m , ξ) (Z 0 , Z 1 , Z). (9)
Let us now consider the case (i), in which lim m mρ m = θ ∈ [-1, +∞). In that case, a standard reasoning involving Taylor expansions of Φ and y → Φ(yΦ -1 (t)) leads to

√ mW m (t) = Φ(Φ -1 (t))( √ m ξ - √ 1 + θ U) Φ(Φ -1 (t) -µ)( √ m ξ - √ 1 + θ U) + R 0,m (t) R 1,m (t) ,
with remainder terms satisfying ||R 0,m || ∞ ∨ ||R 1,m || ∞ → 0 in probability. Since U is independent of all the other variables, we derive from (9

) that √ m(V m , W m ) Z 0 , Z 1 , (Z - √ 1 + θU)) Φ • Φ -1 , (Z - √ 1 + θU)) Φ • (Φ -1 -µ) .
This implies (7). Consider now the case (ii), in which lim m mρ m = +∞ and lim m ρ m = 0. In that situation, we deduce from (9) that ρ -1/2 m V m converges in probability to 0. Furthermore, using that ρ -1/2 m ξ tends to zero in probability, we obtain that

ρ -1/2 m W m (t) = Φ(Φ -1 (t))(-U) Φ(Φ -1 (t) -µ)(-U) + T 0,m (t) T 1,m (t) ,
with remainder terms satisfying ||T 0,m || ∞ ∨ ||T 1,m || ∞ → 0 in probability. This implies (8).

4. Discussion: FDP convergence in the case ρ m = ρ ∈ (0, 1)

When ρ m = ρ ∈ (0, 1), we cannot expect that the FDP concentrates around the FDR as in Theorem 2.1 (see e.g. [START_REF] Finner | Dependency and false discovery rate: asymptotics[END_REF] Theorem 2.1). As a consequence, even if the FDP has a mean below π 0 α (because the false discovery rate of the BH procedure is below π 0 α for each m for PRDS statistics, see Theorem 1.2 in [START_REF] Benjamini | The control of the false discovery rate in multiple testing under dependency[END_REF]), the FDP can exceed π 0 α + ε (ε > 0) with a probability that does not vanish when m grows to infinity.

We claim here that in the ideal situation where the parameters of the model π 0 , µ, ρ are perfectly known, it is possible to modify the p-values so that the FDP convergence to the FDR keeps the parametric convergence rate √ m. For this, we replace each test statistic X i by X i = m/((m -1)(1ρ))(X i -X + (1π 0 )µ), so that ( X 1 , ..., X m ) is a Gaussian vector with variances equal to 1, equi-correlation ρ m = -(m -1) -1 and means E X i = m/((m -1)(1ρ))τ i . We build the corresponding p-values by letting p i = Φ( X i ), which are uniform under the null and have the c.d.f. G 1,m (t) = Φ(Φ -1 (t)µ m ) for µ m = (m/(m -1)) 1/2 µ(1ρ) -1/2 under the alternative. Although the latter depends (slightly) on m, we easily check that our methodology applies using G 1 (t) = Φ(Φ -1 (t)µ) for µ = µ(1ρ) -1/2 and that the following convergence holds:

√ m FDP m -π 0 α N       0 , π 0 α 2 1 -t ρ t ρ - π 2 0 α 2 2π(t ρ ) 2 e -(Φ -1 (t ρ )) 2       ,
where FDP m denotes the FDP of the BH threshold T BH used with the p-values p i 's and where t ρ ∈ (0, 1) is the unique point t ∈ (0, 1) satisfying π 0 t+(1-π 0 )Φ(Φ -1 (t)-µ(1-ρ) -1/2 ) = t/α (which depends on ρ). Of course, while this p-value modification greatly improves the concentration of the FDP, this approach is oracle because π 0 , µ, ρ are generally unknown. A correct estimation of the model parameters within such a procedure stays an open issue.


Claudio Macci
email: macci@mat.uniroma2.it

Large deviations for estimators of unknown probabilities, with applications in risk theory

Keywords: Large deviations, Varadhan's Lemma, level crossing probability, compound Poisson process, Brownian motion

) and we cannot say that the Bayesian approach is more conservative as in that paper.

Introduction

In this paper we prove asymptotic results for estimators of unknown probabilities and, motivated by potential applications in risk theory, we present several examples where these probabilities concern the occurrence of level crossing events.

The unknown probabilities are (ρ n (q|θ)), where q = (q 1 , . . . , q d ) ∈ (0, ∞) d for some d ≥ 1 and the unknown parameter θ belongs to some parameter space Θ. One may wish to asses the unknown probabilities based on experience. Several estimation methods separate the point estimation of the unknown parameter θ and the estimation of the unknown probabilities but, as pointed out in [START_REF] Ganesh | Bayesian network management[END_REF] for the gambler's ruin probability, these methods can lead to a very misleading inference. In this paper we propose two alternative approaches which provide a wide full statistical analysis for the unknown probabilities, capturing all the uncertainties on the unknown parameter θ: the Bayesian approach proposed in Ganesh et al. (1998) (see also Ganesh and[START_REF] Ganesh | An inverse of Sanov's Theorem[END_REF][START_REF] Asmussen | Ruin Probabilities[END_REF] based on estimates of the predictive probabilities Θ ρ n (q|θ)π n (dθ), where π n is the posterior distribution given the n-sample; the classical approach based on estimates of the expected values E θ [ρ n (q| Θ n )], where Θ n is an empirical estimator based on the n-sample. The latter is different from the frequentist approach presented in Ganesh et al. (1998) and here we cannot say that the Bayesian approach is more conservative as in that paper. Typically ( Θ n ) will be the sequence of the maximum likelihood estimators (MLEs from now on).

In this paper we refer to theory of large deviations which gives an asymptotic computation of small probabilities on exponential scale; see e.g. [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF] as a reference on this topic. We provide some extensions of the results in [START_REF] Macci | Mixture of conjugate prior distributions and large deviations for level crossing probabilities[END_REF]: the large deviation principles for posterior distributions are proved without considering finite mixtures of conjugate prior distributions; we consider a further statistical model based on underlying Inverse Gaussian distributed samples concerning independent replications of Brownian motions; we consider level crossing probabilities concerning some bivariate risk models, i.e. the model in Avram et al. (2008) and a version of it based on the Brownian motion.

We also remark that we present a unique result (we mean Proposition 3.2 for the classical approach and Proposition 3.3 for the Bayesian approach) which works well for all the level crossing probabilities presented in this paper. This unique result is formulated in terms of sequences of probabilities ((ρ n (q|θ)) : θ ∈ Θ) which satisfy the exponential decay condition in Definition 3.1 below; this condition allows to obtain large deviation estimates with standard applications of the Varadhan's Lemma.

We conclude with the outline of the paper. Section 2 is devoted to recall some preliminaries. In section 3 we present large deviation results for estimators of unknown probabilities with exponential decay; moreover we discuss the comparison between classical and Bayesian approaches. The applications in risk theory are presented in section 4 (standard risk models) and in section 5 (bivariate risk models).

Preliminaries

We start with some preliminaries on large deviations and we refer to [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]. A rate function is a lower semi-continuous function

I : Ω → [0, ∞],
where Ω is a Hausdorff topological space with Borel σ-algebra B Ω . A sequence of probability measures (ν n ) on (Ω, B Ω ) satisfies the large deviation principle (LDP from now on) with rate function

I if lim sup n→∞ 1 n log ν n (F ) ≤ -inf ω∈F I(ω) for all closed sets F ⊂ Ω and lim inf n→∞ 1 n log ν n (G) ≥ -inf ω∈G I(ω) for all open sets G ⊂ Ω.
A rate function I is good if all the level sets {{ω ∈ Ω :

I(ω) ≤ η} : η ≥ 0} are compact. A sequence of Ω valued random variables (Z n ) satisfies the LDP if (ν n ) does with ν n = P (Z n ∈ •).
We refer to the definition for random variables when we consider the classical approach, where (Z n ) is a sequence of estimators. We refer to the definition for probability measures when we consider the Bayesian approach, where (ν n ) is the sequence of posterior distributions; in this case we have a family of random probability measures which satisfies the LDP almost surely. In some sense we should say that we have a conditional LDP, indeed this LDP holds assuming that a sequence of sufficient statistics converges to some limit value (which determines the true value of the parameter).

Throughout this paper we consider the following well known large deviation results in the literature, and we refer again to [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF]: the principle of the largest term (Lemma 1.2.15), the contraction principle (Theorem 4.2.1), the Varadhan's Lemma (Theorem 4.3.1) and the Sanov's Theorem (Theorem 6.2.10).

All the rate functions in this paper can be expressed in terms of the relative entropy between two probability measures (also called Kullback-Leibler divergence). Given two probability measures ν 1 and ν 2 on the same measurable space (Ω, B Ω ), the relative entropy of ν 1 with respect to ν 2 is

H(ν 1 |ν 2 ) = Ω log dν 1 dν 2 (ω) ν 1 (dω) if ν 1 ν 2 ∞ otherwise,
where ν 1 ν 2 means that ν 1 is absolutely continuous with respect to ν 2 , and dν 1 dν 2 is the corresponding density.

We recall the following formulas. If N [µ, σ 2 ] is the Normal distribution with mean µ and variance σ 2 , we have

H(N [µ 1 , σ 2 ]|N [µ 2 , σ 2 ]) = (µ 1 -µ 2 ) 2 2σ 2 ; if G[α, β] is the Gamma distribution with continuous density g α,β (y) = β α Γ(α) y α-1 e -βy 1 (0,∞) (y) for α, β > 0, we have H(G[α, β 1 ]|G[α, β 2 ]) = A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT α β 2 β 1 -1 -log β 2 β 1 ; if P [λ]
is the Poisson distribution with discrete density g λ (y) = λ y y! e -λy 1 N∪{0} for λ > 0, we have

H(P [λ 1 ]|P [λ 2 ]) = λ 1 log λ 1 λ 2 -λ 1 + λ 2 .
Note that, if we consider the law of the constant 0 as P [0] and the equality 0 log 0 = 0, the latter formula also holds with λ 1 = 0.

In view of the applications in risk theory it is useful to recall some known results: firstly the LDPs for MLEs (Lemmas 2.1-2.2-2.3), secondly the LDPs for posterior distributions (Lemmas 2.4-2.5-2.6). We cite some old references for the results for MLEs: see e.g. [START_REF] Bahadur | On deviations of the sample mean[END_REF] and [START_REF] Bahadur | Large deviations of the sample mean in general vector space[END_REF]. A more recent reference for the results for both MLEs and posterior distributions is Macci (2010, section 4) which deals with exponential families: the mean parameter γ coincides with the parameter θ for Normal and Poisson samples, whereas for Gamma samples we have to consider a change of variable (via the contraction principle) to transfer the LDP to the family of posterior distributions on θ = α βγ . We always use the standard notation

Z n = 1 n n k=1 Z k .
Lemma 2.1 (Normal samples) Let (Z n ) be a sequence of i.i.d. N [θ, 1/r] distributed random variables, where θ ∈ R is unknown. Then the sequence of MLEs ( Θ n ) coincides with (Z n ) and satisfies the LDP with good rate function I θ defined by

I θ ( θ) = H(N [ θ, 1/r]|N [θ, 1/r]).
Lemma 2.2 (Gamma samples) Let (Z n ) be a sequence of i.i.d. G[α, βθ] distributed random variables, where θ > 0 is unknown. Then the sequence of MLEs ( Θ n ) coincides with (α/(βZ n )) and satisfies the LDP with good rate function I θ defined by

I θ ( θ) = H(G[α, θ]|G[α, θ]) if θ > 0, and I θ ( θ) = ∞ otherwise. Lemma 2.3 (Poisson samples) Let (Z n ) be a sequence of i.i.d. P [θ] distributed random vari- ables,
where θ > 0 is unknown. Then the sequence of MLEs ( Θ n ) coincides with (Z n ) and satisfies the LDP with good rate function I θ defined by

I θ ( θ) = H(P [ θ]|P [θ]) if θ ≥ 0, and I θ ( θ) = ∞ otherwise.
In view of Lemmas 2.4-2.5-2.6 presented below we need some further preliminaries. We denote the posterior distribution given (Z 1 , . . . , Z n ) = (z 1 , . . . , z n ) by π n , namely we set π n = π 0 (•|Z 1 = z 1 , . . . , Z n = z n ) where π 0 is the prior distribution. Moreover we denote the support of the prior distribution π 0 by S(π 0 ); obviously S(π 0 ) is a closed subset of the parameter space Θ.

In each lemma we shall assume the following condition (C) on θ n , where θ n is the sampled value of Θ n as in Lemmas 2.1-2.2-2.3 above:

(C): θ n → θ as n → ∞ for some θ ∈ S(π 0 ).
Then θ can be actually interpreted as the true value of the parameter. Lemma 2.4 (Normal samples) Consider the same situation of Lemma 2.1 and assume that (C) holds. Then (π n ) satisfies the LDP with good rate function We start with the definition of sequence of probabilities ((ρ n (q|θ)) : θ ∈ Θ) with an exponential decay (as n → ∞). Definition 3.1 The sequences of probabilities ((ρ n (q|θ)) : θ ∈ Θ) have exponential decay if, for some d ≥ 1, there exist continuous functions w 1 , . . . ,

I(•| θ) defined by I(θ| θ) = H(N [ θ, 1/r]|N [θ, 1/r]) if θ ∈ S(π 0 ), and I(θ| θ) = ∞ otherwise.
w d : Θ → [0, ∞) and numbers c 1 , . . . , c d ∈ (0, 1] and M ∈ [0, ∞) such that c i e -(nq i +M )w i (θ) ≤ ρ n (q|θ) ≤ d j=1 e -nq j w j (θ) (1)
for all i ∈ {1, . . . , d}, θ ∈ Θ, q = (q 1 , . . . , q d ) ∈ (0, ∞) d and n ≥ 1.

Now we prove the large deviation estimates for any sequence of probabilities with an exponential decay. We start with the classical approach. Proposition 3.2 (Classical approach) Let ((ρ n (q|θ)) : θ ∈ Θ) be sequences of probabilities with exponential decay. Moreover let w θ i (q i ) be defined by

w θ i (q i ) := inf θ∈Θ {q i w i ( θ) + I θ ( θ)}.
Let ( Θ n ) be a sequence of random variables which satisfies the LDP with good rate function I θ , which uniquely vanishes at θ (the true value of the parameter). Then we have

lim n→∞ 1 n log E θ [ρ n (q| Θ n )] = -min{w θ i (q i ) : i ∈ {1, . . . , d}}.
Proof. We have to check the upper bound lim sup

n→∞ 1 n log E θ [ρ n (q| Θ n )] ≤ -min{w θ i (q i ) : i ∈ {1, . . . , d}}
and the lower bound

lim inf n→∞ 1 n log E θ [ρ n (q| Θ n )] ≥ -min{w θ i (q i ) : i ∈ {1, . . . , d}}.
Proof of the upper bound. For each fixed i ∈ {1, . . . , d} we have lim sup

n→∞ 1 n log E θ [e -nq i w i ( Θn) ] = sup θ∈Θ {-q i w i ( θ) -I θ ( θ)} = -w θ i (q i ) (2)
by the Varadhan's Lemma; thus we obtain lim sup

n→∞ 1 n log E θ [ρ n (q| Θ n )] ≤ lim sup n→∞ 1 n log d j=1 E θ [e -nq j w j ( Θn) ] = = max lim sup n→∞ 1 n log E θ [e -nq i w i ( Θ n ) ] : i ∈ {1, . . . , d} = -min{w θ i (q i ) : i ∈ {1, . . . , d}}
by the second inequality in (1), the principle of the largest term and (2). Proof of the lower bound. Let i ∈ {1, . . . , d} and ε > 0 be arbitrarily fixed. Then we have lim inf

n→∞ 1 n log E θ [ρ n (q| Θ n )] ≥ lim inf n→∞ 1 n log E θ [c i e -(nq i +M )w i ( Θ n ) ] = A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT = lim inf n→∞ 1 n log E θ [e -n(q i +M/n)w i ( Θn) ] ≥ lim inf n→∞ 1 n log E θ [e -n(q i +ε)w i ( Θn) ] = -w θ i (q i + ε)
by the first inequality in (1) and the Varadhan's Lemma. We notice that -w θ i (•) is continuous on [0, ∞); indeed it is easy to check that it is convex on the open set (0, ∞) (and therefore continuous on (0, ∞); see e.g. Theorem 3.2 in Rudin, 1986), and lim q i ↓0 -w θ i (q i ) = -w θ i (0) = 0 since 0 ≥ -w θ i (q i ) ≥ -q i w i (θ) (by taking into account the equality I θ (θ) = 0). Then we have lim inf

n→∞ 1 n log E θ [ρ n (q| Θ n )] ≥ -w θ i (q i )
letting ε go to zero, and we obtain the lower bound taking max{-w θ i (q i ) : i ∈ {1, . . . , d}} at the right hand side.

We can also prove a Bayesian version of Proposition 3.2. We have the same proof (we have to replace the classical approach items with the analogous items of the Bayesian approach) and therefore we omit it. We shall denote again the sampled value of Θ n by θ n . Proposition 3.3 (Bayesian approach) Let ((ρ n (q|θ)) : θ ∈ Θ) be sequences of probabilities with exponential decay. Consider a prior distribution π 0 on θ and let π n be the posterior distribution given n sampled values. Assume that there exists a sequence of sufficient statistics ( Θ n ) such that, if θ n → θ as n → ∞ for some θ ∈ Θ, the sequence (π n ) satisfies the LDP with good rate function I(•| θ), which uniquely vanishes at θ (the true value of the parameter). Moreover let w i (q i | θ) be defined by

w i (q i | θ) := inf θ∈Θ {q i w i (θ) + I(θ| θ)}. (3)
Then we have

lim n→∞ 1 n log Θ ρ n (q|θ)π n (dθ) = -min{w i (q i | θ) : i ∈ {1, . . . , d}}.
Comparison between classical and Bayesian approaches. In the next sections the sequences of probabilities with exponential decay are suitable level crossing probabilities. Thus, motivated by potential applications in risk theory, now we investigate whether the Bayesian approach is more or less conservative than the classical approach. More precisely, by taking into account Propositions 3.2 and 3.3, we can say the Bayesian approach is more (less) conservative than the classical approach if min{w i (q i | θ) : i ∈ {1, . . . , d}} is smaller (larger) than min{w θ i (q i ) : i ∈ {1, . . . , d}} for each θ ∈ Θ. In all the examples in this paper we can check the equality

I θ ( θ) = I(θ| θ) for all θ, θ ∈ Θ (4) if S(π 0 ) = Θ;
this is what typically happens, even if there exists a counterexample (see Proposition 6.1 in [START_REF] Macci | Censored exponential data: large deviations for MLEs and posterior distributions[END_REF]. Then, if we restrict the attention on the case S(π 0 ) = Θ, (4) yields

w θ i (q i ) = inf θ∈Θ {q i w i (θ) + I θ (θ)} = inf θ∈Θ {q i w i (θ) + I( θ|θ)},
and we cannot say if it is larger or smaller than w i (q i | θ) in (3). Thus, in general, we cannot compare min{w i (q i | θ) : i ∈ {1, . . . , d}} and min{w θ i (q i ) : i ∈ {1, . . . , d}}, but they coincide if I(θ| θ) = I( θ|θ) for all θ, θ ∈ Θ. For instance this happens for the statistical models based on the Brownian motion with unknown drift (and known variance parameter), where the rate functions are expressed in terms of the relative entropy between two Normal distributions with the same variance (indeed we have

H(N [µ 1 , σ 2 ]|N [µ 2 , σ 2 ]) = H(N [µ 2 , σ 2 ]|N [µ 1 , σ 2 ])).

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

A different comparison between the classical approach and the Bayesian approach has been also considered in [START_REF] Ganesh | Bayesian network management[END_REF]. The difference concerns the classical approach which is based on the limit lim

n→∞ 1 n log ρ n (q| θ) = -min{q i w i ( θ) : i ∈ {1, . . . , d}};
thus, in that case, the Bayesian approach is more conservative than the classical one since the inequality min{w i (q i | θ) : i ∈ {1, . . . , d}} ≤ min{q i w i ( θ) : i ∈ {1, . . . , d}} holds. In our opinion the classical approach presented in this paper captures all the uncertainty about the unknown parameter (as happens for the Bayesian approach), and therefore seems to be more appropriate than the one in Ganesh et al. (1998).

4 Applications in risk theory (standard models)

In this section we consider two different sequences ((ρ n (q|θ)) : θ ∈ Θ) which concern level crossing probabilities for compound Poisson processes and Brownian motions. More precisely we deal with the statistical models in Macci and Petrella (2006) (where only Bayesian approach is discussed) and, for each one, we study the two following sequences of level crossing probabilities: firstly the level goes to infinity (we set Q = nq for some fixed q > 0 and n → ∞), secondly we consider a suitable scaling called slow Markov walk limit (we refer to the presentation in Asmussen and Nielsen, 1995; see also [START_REF] Bucklew | Large Deviations Techniques in Decision, Simulation and Estimation[END_REF]. For each statistical model we check the hypotheses of Propositions 3.2 and 3.3: firstly we check the exponential decay condition; secondly we provide a LDP as required in their statements.

Preliminaries on compound Poisson process. A compound Poisson process (S(t)) is defined by S(t) = N (t)

k=1 B k , where: (N (t)) is a Poisson process with intensity λ, i.e. N (t) = n≥1 1 T 1 +•••+T n ≤t , and (T k ) are i.i.d. exponentially distributed with mean 1/λ; (B k ) are i.i.d. and positive random variables with common law , and independent of (T k ). This model is often used in insurance where Q is the initial capital, p the premium rate, (N (t)) is the claim number process, (B k ) are the claims and (S(t)) is the aggregate claims process. The infinite horizon level crossing probability Ψ (p) (λ, ) (Q) = P ({∃t ≥ 0 : S(t) ≥ Q + pt}) represents the ruin probability. We have upper and lower bounds for Ψ (p) (λ, ) (Q) if is concentrated on a bounded set. Here we consider the set [0, M ] where M is the maximum claim size and we have the following refinement of the Lundberg's inequality, where w 1 (λ, ) is defined below in eq. ( 5):

e (Q+M )w 1 (λ, ) ≤ Ψ (p) (λ, ) (Q) ≤ e Qw 1 (λ, ) .
It is also known that, if = G[1, β] (i.e. is the exponential distribution with mean 1/β), we have Ψ Preliminaries on Brownian motion. We recall that (S(t)) is a Brownian motion with drift µ and variance parameter 1/r if, for each fixed h ≥ 1 and 0 < t 1 < . . . < t h , the random variables S(t 1 ), S(t 2 ) -S(t 1 ), . . . , S(t h ) -S(t h-1 ) are independent and normally distributed, with means µt 1 , µ(t 2 -t 1 ), . . . , µ(t h -t h-1 ) and variances t 1 /r, (t 2 -t 1 )/r, . . . , (t h -t h-1 )/r. If we consider the infinite horizon level crossing probability Ψ (µ,r) (Q) = inf{t ≥ 0 : S(t) ≥ Q}, we have Ψ (µ,r) (Q) = e -max{-2µr,0}Q (see e.g. Corollary 1.6 in [START_REF] Asmussen | Ruin Probabilities[END_REF]; note that there is a minor error in the expression presented in [START_REF] Macci | Mixture of conjugate prior distributions and large deviations for level crossing probabilities[END_REF].

(p) (λ,G[1,β]) (Q) = 1 -max{βp-λ,0}
The slow Markov walk limit scaling consists of the sequence of parameters (µ n , r n ) := (µ, nr), i.e. the variance parameter simply goes to zero.

Compound Poisson process with bounded jumps

Here we have θ = (λ, ) and Θ = (0, ∞) × M 1 [0, M ], where M 1 [0, M ] is the space of probability measures on [0, M ] for some M ∈ (0, ∞) equipped with the metric of weak convergence. Moreover we think to have ρ n (q|λ, ) = Ψ (p) (λ, ) (nq) or ρ n (q|λ, ) = Ψ (p) (nλ, [n]) (q). Note that we could consider two simplified statistical models in which only λ or is unknown, and the other parameter is known. In these cases the results can be obtained with minor changes of the results presented below and therefore we omit the details.

It is easy to check (for details see equation ( 4.3) and its consequence in Macci and Petrella, 2006) that, for both the choices of ρ n (q|λ, ), ( 1) holds with d = 1, c 1 = 1, and

w 1 (λ, ) = sup{γ ≥ 0 : λ(G (γ) -1) -pγ ≤ 0}, (5)
where G (γ) = M 0 e γx (dx) is the moment generating function of concerning ; in particular see Lemma 4.2 in [START_REF] Macci | Mixture of conjugate prior distributions and large deviations for level crossing probabilities[END_REF] for the continuity of w 1 (λ, ).

We consider the n-sample (Z 1 , . . . , Z n ) = ((T 1 , B 1 ), . . . , (T n , B n )). The LDP in the statement of Proposition 3.2. Let ( Θ n ) be the estimators defined by

Θ n = (T -1 n , n ), where: T n = 1 n n k=1 T k is the empirical mean of T 1 , . . . , T n ; n = 1 n n k=1 δ B k is the empirical law of B 1 , . . . , B n
, where δ x is the concentrated unit measure concentrated at x. Then the sequence ( Θ n ) satisfies the LDP with good rate function I (λ, ) defined by

I (λ, ) ( λ, ) = H(G[1, λ]|G[1, λ]) + H( | ) if ( λ, ) ∈ Θ ∞ otherwise by the LDP of (T - 1
n ) provided by Lemma 2.2 (with α = β = 1 and (T 1 , . . . , T n ) in place of (Z 1 , . . . , Z n )), by the LDP of ( n ) provided by the Sanov's Theorem and by Corollary 2.9 in Lynch and Sethuraman (1987) which provides the LDP for sequences of multivariate random variables with independent components. The LDP in the statement of Proposition 3.3. We consider the prior distribution π 0 = ν 0 ⊗ D η 0 on (λ, ) where ν 0 is the prior distribution on λ and D η 0 is the Dirichlet Process prior on with hyperparameter the finite measure η 0 on [0, M ] (see e.g. [START_REF] Ferguson | A Bayesian analysis of some nonparametric problems[END_REF]. Then the posterior distribution on (λ, ) given the n-sample is

π n = ν n ⊗ D ηn := ν 0 (•|T 1 . . . , T n ) ⊗ D η 0 (•|B 1 , . . . , B n ), where D η 0 (•|B 1 . . . , B n ) = D η 0 + n i=1 δ B i by Theorem 1 in Ferguson (1973). Then, if θ n → ( λ, )
as n → ∞ for some ( λ, ) ∈ S(ν 0 ) × S(D η 0 ), the sequence of posterior distributions (π n ) on (λ, ) satisfies the LDP with good rate function I(•| λ, ) defined by

I(λ, | λ, ) = H(G[1, λ]|G[1, λ]) + H( | ) if (λ, ) ∈ S(ν 0 ) × S(D µ ) ∞ otherwise
by the LDP of (ν n ) provided by Lemma 2.5, by the LDP of (D ηn ) provided by Theorem 1 in Ganesh and O'Connell (2000) and by Corollary 2.9 in Lynch and Sethuraman (1987) which provides the LDP for sequences of product measures.

Another statistical model where λ is unknown and is known. Here θ = λ and Θ = (0, ∞). We consider a different n-sample based on the claim numbers in consecutive unit time intervals, i.e. (Z 1 , . . . , Z n ) = (N (1), . . . , N (n) -N (n -1)).

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

The LDP in the statement of Proposition 3.

Compound Poisson process with exponential jumps

One could consider = G [1, β] and therefore the inference problem with θ = (λ, β) and Θ = (0, ∞) × (0, ∞). In such a case we should think to have

ρ n (q|λ, β) = Ψ (p) (λ,G[1,β]) (nq) or ρ n (q|λ, β) = Ψ (p) (nλ,G[1,nβ]) (q).
Unfortunately in this case we cannot prove the limits in Propositions 3.2-3.3 because we cannot check the lower bound in (1) (an explanation can be found in subsection 4.3 in [START_REF] Macci | Mixture of conjugate prior distributions and large deviations for level crossing probabilities[END_REF].

We do not have this problem assuming to have β = kλ for some known k > 0 (as in subsection 4.3 in [START_REF] Macci | Mixture of conjugate prior distributions and large deviations for level crossing probabilities[END_REF]. Then here we have θ = λ and Θ = (0, ∞). Moreover we think to have

ρ n (q|λ) = Ψ (p) (λ,G[1,kλ]) (nq) or ρ n (q|λ) = Ψ (p) (nλ,G[1,knλ]) (q).
It is easy to check (for details see equation (4.5) and the first part of subsection 4.3 in Macci and Petrella, 2006) that, for both the choices of ρ n (q|λ), (1) holds with d = 1, c 1 = min{(kp) -1 , 1}, M = 0 and w 1 (λ) = kλ max{1 -(kp) -1 , 0}.

We consider the n-sample (Z 1 , . . . , Z n ) = (kB 1 + T 1 , . . . , kB n + T n ). The LDP in the statement of Proposition 3.2. Let ( Θ n ) be the estimators defined by Θ n = 2Z

Brownian motion

In this subsection we consider two cases: µ is unknown and r is known; µ is known and r is unknown. We present a statistical model for the first case, and two statistical models for the second one.

A statistical model where µ is unknown and r is known. Here we have θ = µ and Θ = R (thus r is known). Moreover we think to have one of the two following cases:

ρ n (q|µ) = Ψ (µ,r) (nq) or ρ n (q|µ) = Ψ (µ,nr) (q).
It is easy to check that, for both the choices of ρ n (q|µ), (1) holds with d = 1, c 1 = 1, M = 0 and w 1 (µ) = max{-2µr, 0}.

We consider the n-sample (Z 1 , . . . , Z n ) = (S(1), . . . , S(n) -S(n -1)). The LDP in the statement of Proposition 3.

A C C E P T E D M A N U S C R I P T ACCEPTED MANUSCRIPT

A statistical model where µ is known and r is unknown. Here we have θ = r and Θ = (0, ∞); thus µ is known and, in order to avoid the trivial situation in which the level crossing occurs with probability 1, we should consider µ < 0. Moreover we think to have one of the two following cases: ρ n (q|r) = Ψ (µ,r) (nq) or ρ n (q|r) = Ψ (µ,nr) (q).

It is easy to check that, for both the choices of ρ n (q|r), (1) holds with d = 1, c 1 = 1, M = 0 and w 1 (r) = max{-2µr, 0}.

We consider the n-sample (Z 1 , . . . , Z n ) = ((S(1) -µ) 2 , . . . , (S(n) -S(n -1) -µ) 2 ). The LDP in the statement of Proposition 3.2. Let ( Θ n ) be the estimators defined by Θ n = Z Another statistical model where µ is known and r is unknown. Here we have θ = r and Θ = (0, ∞) as before, and µ < 0 is known. We consider a different n-sample based on the i.i.d. first passage times V 1 , . . . , V n to a barrier a < 0 concerning n independent replications of the Brownian motion; since µ < 0, the negativeness of a ensures that each passage time occurs within a finite time with probability 1. Then the random variables V 1 , . . . , V n are IG[a/µ, a 2 r] distributed, where IG[a/µ, a 2 r] is a suitable Inverse Gaussian distribution (see equation (1.2) and the beginning of section 3.2 in [START_REF] Seshadri | The Inverse Gaussian Distribution: Statistical Theory and Applications[END_REF]; in other words these random variables have continuous density

f (v) = |a| √ r √ 2πv 3/2 exp - rµ 2 2 (v -a/µ) 2 v 1 (0,∞) (v).
Furthermore it is known that the i.i.d. random variables (V i -a/µ) 2

V i : i ∈ {1, . . . , n} are G[1/2, µ 2 r/2] distributed (indeed each random variable Q = a 2 r (V i -a/µ) 2 (a/µ) 2 V i is χ 2 1 = G[1/2, 1/2] distributed by Proposition 1.3 in [START_REF] Seshadri | The Inverse Gaussian Distribution: Statistical Theory and Applications[END_REF].

We consider the n-sample (Z 1 , . . . ,

Z n ) = (V 1 -a/µ) 2 V 1
, . . . , (Vn-a/µ) 2

V n .

The LDP in the statement of Proposition 3. 5 Applications in risk theory (bivariate models)

In this section we consider sequences of probabilities ((ρ n (q|θ)) : θ ∈ Θ) having interest in risk theory and, as we shall see, the exponential decay condition holds with d = 2. More precisely we deal with the level crossing probabilities of a (possibly piecewise) linear barrier b(t; Q

Lemma 2 . 5 (

 25 Gamma samples) Consider the same situation of Lemma 2.2 and assume that (C) holds. Then (π n ) satisfies the LDP with good rate function I(•| θ) defined by I(θ| θ) = H(G[α, θ]|G[α, θ]) if θ ∈ S(π 0 ), and I(θ| θ) = ∞ otherwise.

Lemma 2 . 6 (

 26 Poisson samples) Consider the same situation of Lemma 2.3 and assume that (C) holds. Then (π n ) satisfies the LDP with good rate function I(•| θ) defined by I(θ| θ) = H(P [ θ]|P [θ]) if θ ∈ S(π 0 ), and I(θ| θ) = ∞ otherwise.

Q

  (see e.g. equation (5.3.8) in Rolski et al., 1999). The slow Markov walk limit scaling consists of the sequence of parameters (λ n , n ) := (nλ, [n]), where [n] is the law of the random variables {B k /n : k ≥ 1}. In particular, if = G[1, β], we have [n] = G[1, nβ].

2 .

 2 Let ( Θ n ) be the estimators defined by Θ n = Z n = N (n) n . Then, by Lemma 2.3, the sequence ( Θ n ) satisfies the LDP with good rate function I λ defined by I λ ( λ) = H(P [ λ]|P [λ]) if λ ≥ 0, and I λ ( λ) = ∞ otherwise. The LDP in the statement of Proposition 3.3. If θ n → λ as n → ∞ for some λ ∈ S(π 0 ), by Lemma 2.6 the sequence of posterior distributions (π n ) on λ satisfies the LDP with good rate function I(•| λ) defined by I(λ| λ) = H(P [ λ]|P [λ]) if λ ∈ S(π 0 ), and I(λ| λ) = ∞ otherwise.

  n + T n ) -1 , where T n = 1 n n k=1 T k is the empirical mean of T 1 , . . . , T n and B n = 1 n n k=1 B k is the empirical mean of B 1 , . . . , B n . Then, by Lemma 2.2 (with α = 2 and β = 1), the sequence ( Θ n ) satisfies the LDP with good rate function I λ defined by I λ ( λ) = H(G[2, λ]|G[2, λ]) if λ > 0, and I λ ( λ) = ∞ otherwise. The LDP in the statement of Proposition 3.3. If θ n → λ as n → ∞ for some λ ∈ S(π 0 ), by Lemma 2.5 the sequence of posterior distributions (π n ) on λ satisfies the LDP with good rate function I(•| λ) defined by I(λ| λ) = H(G[2, λ]|G[2, λ]) if λ ∈ S(π 0 ), and I(λ| λ) = ∞ otherwise.

2 .

 2 Let ( Θ n ) be the estimators defined by Θ n = Z n = S(n) n . Then, by Lemma 2.1, the sequence ( Θ n ) satisfies the LDP with good rate function I µ defined by I µ ( µ) = H(N [ µ, 1/r]|N [µ, 1/r]). The LDP in the statement of Proposition 3.3. If θ n → µ as n → ∞ for some µ ∈ S(π 0 ), by Lemma 2.4 the sequence of posterior distributions (π n ) on µ satisfies the LDP with good rate function I(•| µ) defined by I(µ| µ) = H(N [ µ, 1/r]|N [µ, 1/r]) if µ ∈ S(π 0 ), and I(µ| µ) = ∞ otherwise.

  Lemma 2.2 (with α = β = 1/2), the sequence ( Θ n ) satisfies the LDP with good rate function I r defined by I r ( r) = H(G[1/2, r]|G[1/2, r]) if r > 0, and I r ( r) = ∞ otherwise. The LDP in the statement of Proposition 3.3. If θ n → r as n → ∞ for some r ∈ S(π 0 ), by Lemma 2.5 the sequence of posterior distributions (π n ) on r satisfies the LDP with good rate function I(•| r) defined by I(r| r) = H(G[1/2, r]|G[1/2, r]) if r ∈ S(π 0 ), and I(r| r) = ∞ otherwise.

2 .

 2 Let ( Θ n ) be the estimators defined by Θ n = 1/(µ 2 Z n ). Then, by Lemma 2.2 (with α = 1/2 and β = µ 2 /2), the sequence ( Θ n ) satisfies the LDP with good rate function I r defined byI r ( r) = H(G[1/2, r]|G[1/2, r]) if r > 0,and I r ( r) = ∞ otherwise. The LDP in the statement of Proposition 3.3. If θ n → r as n → ∞ for some r ∈ S(π 0 ), by Lemma 2.5 the sequence of posterior distributions (π n ) on r satisfies the LDP with good rate function I(•| r) defined by I(r| r) = H(G[1/2, r]|G[1/2, r]) if r ∈ S(π 0 ), I(r| r) = ∞ otherwise.

  1 , p 1 , Q 2 , p 2 ) = min{Q 1 + p 1 t, Q 2 + p 2 t} (6) for Q 1 , p 1 , Q 2 ,p 2 > 0. These probabilities appear in the bivariate risk model in Avram et al. (2008) and can be reduced to one-dimensional level crossing probabilities (some details are recalled below in subsection 5.3); in Avram et al. (2008) only the compound Poisson process is considered and here we also present a version of the model based on the Brownian motion. We have to consider

Acknowledgements

I thank Mauro Piccioni for inspiring the classical approach in this paper and Florin Avram for telling me about the model in his paper. I also thank the referee for some useful comments and for pointing out an inexactness in the previous version of the paper. The financial support of the Research Grant PRIN 2008 Probability and Finance is gratefully acknowledged.

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again some items presented in the previous section; we have the same preliminaries, the same sequences of estimators ( Θ n ) and the same sequences of posterior distributions (π n ). Here we only check the exponential decay condition with d = 2 for the sequences ((ρ n (q|θ)) : θ ∈ Θ) concerning each statistical model.

Compound Poisson process

Let (S(t)) be the compound Poisson process in the previous section and let Ψ (λ, ) (Q 1 , Q 2 ) be the level crossing probabilities defined by

then, if we consider the level crossing probabilities Ψ (p) (λ, ) (Q) in the previous section, we have

Case with bounded jumps. We consider ρ n (q 1 , q 2 |λ, ) = Ψ (λ, ) (nq 1 , nq 2 ) or ρ n (q 1 , q 2 |λ, ) = Ψ (nλ, [n]) (q 1 , q 2 ). Arguing as in subsection 4.1 and by [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF], one can check the exponential decay condition for both the choices of ρ n (q 1 , q 2 |λ, ) with d = 2, c i = 1 for i ∈ {1, 2}, and

Case with exponential jumps. We consider

). Arguing as in subsection 4.2 and by [START_REF] Dembo | Large Deviations Techniques and Applications[END_REF], one can check the exponential decay condition for both the choices of

Brownian motion

Let (S(t)) be the Brownian motion in the previous section and we use the notation

and we have

The statistical model where µ is unknown and r is known. We consider ρ n (q 1 , q 2 |µ) = Ψ (µ,r) (nq 1 , nq 2 ) or ρ n (q 1 , q 2 |µ) = Ψ (µ,nr) (q 1 , q 2 ). Arguing as in subsection 4.3 and by [START_REF] Ferguson | A Bayesian analysis of some nonparametric problems[END_REF], one can check the exponential decay condition for both the choices of ρ n (q 1 , q 2 |µ)

The two statistical models where µ is known and r is unknown. We think to have ρ n (q 1 , q 2 |r) = Ψ (µ,r) (nq 1 , nq 2 ) or ρ n (q 1 , q 2 |r) = Ψ (µ,nr) (q 1 , q 2 ). Arguing as in subsection 4.3 and by [START_REF] Ferguson | A Bayesian analysis of some nonparametric problems[END_REF], one can check the exponential decay condition for both the choices of ρ n (q 1 , q 2 |r) with d = 2, c i = 1 for i ∈ {1, 2}, M = 0 and w i (r) = max{-2(µ -p i )r, 0} for i ∈ {1, 2}.

In order to avoid the trivial case where the level crossing occurs with probability 1, we should have µ < p 1 , p 2 (which is indeed equivalent to w 1 (r), w 2 (r) > 0). Anyway we remark that the condition µ < 0 is needed if we consider the n-sample Z 1 , . . . , Z n in terms of the first passage times V 1 , . . . , V n to a barrier a < 0 as in subsection 4.3.

A C C E P T E D M A N U S C R I P T

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5.3 Some connections with the geometric considerations in Avram et al. (2008) In this subsection we illustrate the relationship between the geometric considerations in Avram et al. (2008) and some inequalities between the exponential decay rates in this paper.

Following the description of the model in Avram et al. ( 2008), we assume to have two companies that divide between them both claims and premia in some specified proportions δ 1 , δ 2 ∈ (0, 1) with

where u 1 , u 2 > 0 are the initial capitals and v 1 , v 2 > 0 are the premium rates. Thus, if we think to the choices of (S(t)) in this paper, the compound Poisson process is more appropriate than the Brownian motion. We note that we have R 1 (t) ≤ 0 or R 2 (t) ≤ 0 for some t ≥ 0 if and only if we have

) is defined by ( 6) and

We also note that, if p 1 > p 2 , the two-dimensional ruin problem is equivalent to the ruin problem for the second company if

reduces to the linear barrier Q 2 + p 2 t). In other words, if p 1 > p 2 and if we consider the general notation

with θ ∈ Θ (instead of the specific notation of each example), we have

This fact has the following geometric interpretation: if p 1 > p 2 , the two-dimensional ruin problem is equivalent to the ruin problem for the second company if u 2 ≤ δ 2 δ 1 u 1 , i.e. if the initial capitals of the two companies is situated below of the line u 2 = δ 2 δ 1 u 1 (in the positive quadrant). In conclusion we illustrate how these geometric considerations in Avram et al. ( 2008) can be related with some inequalities between w θ 1 (q 1 ) and w θ 2 (q 2 ), and between w 1 (q 1 | θ) and w 2 (q 2 | θ). For all the examples presented in this section assume p 1 ≥ p 2 ; then, for all θ ∈ Θ, we have w 1 (θ) ≥ w 2 (θ) whence we obtain q 2 w 2 (θ) ≤ q 1 w 1 (θ) if q 2 ≤ q 1 . In conclusion, if p 1 ≥ p 2 and q 2 ≤ q 1 , we have: min{w θ 1 (q 1 ), w θ 2 (q 2 )} = w θ 2 (q 2 ) (or w θ 2 (q 2 ) ≤ w θ 1 (q 1 )) for all θ ∈ Θ; min{w 1 (q 1 | θ), w 2 (q 2 | θ)} = w 2 (q 2 | θ) (or w 2 (q 2 | θ) ≤ w 1 (q 1 | θ)) for all θ ∈ Θ.