Evangelia Adamou

Deixis and Temporal Subordinators in Pomak (Slavic, Greece)

Deixis expressed by three definite articles is a well-known characteristic of the Pomak varieties, given the absence of grammaticalized definite markers in most Slavic languages. In this paper, we present a practically unknown use of deixis in Slavic languages, namely the use of deictic suffixes in the formation of temporal subordinate conjunctions in Pomak. The deictic suffixes in the temporal subordinator indicate the relation of the process to the situation of utterance, and the choice of the deictic suffix depends on the type of event encoded in the clause. The free temporal subordinator indicates absence of anchoring of the event to the speech time. This study is based on first-hand data of a practically undescribed Pomak variety spoken in Greece.

Introduction

In a Pomak variety spoken in Greece, in the Xanthi area, three suffixes specify the location of an entity in relation to the speaker's sphere (-s-), the addressee's sphere (-t-), or away from the speaker and the addressee (-n-). All three suffixes are used in the formation of definite articles and demonstratives (see §3). Two of them, namely -t-and -n-, are also used with temporal values when the space and time are different from that of the situation of utterance:t-is no longer used for an entity in the addressee's sphere, but for an entity in a past time relative to the situation of utterance 1 .

Those uncommon uses allow us to understand the equally uncommon uses of those two deictic suffixes that form temporal subordinate conjunctions which anchor the event to the situation of utterance (for the theoretical framework and terminology of this analysis see [START_REF] Culioli | À propos d'opérations intervenant dans le traitement formel des langues naturelles[END_REF][START_REF] Culioli | À propos d'opérations intervenant dans le traitement formel des langues naturelles[END_REF], 1990and Robert 2006, also §4.2.1). Ag´ato/kug´ato 'when (past)' is used for a past moment in relation to the situation of utterance. Ag´ano/kug´ano 'when, whenever', is employed for future moments in relation to the situation of utterance, as well as for habitual events. Absence of these deictics indicates absence of anchoring to the situation of utterance; ag´a 'when (no anchoring)' is thus used in fictional narratives (see §4). Immediate anteriority is expressed by prefixing li 'immediately, all the time' to the subordinators presented above: li ga 'as soon as (no anchoring)', li kug´ato 'as soon as (past)' and li kug´ano 'as soon as (future)'. As far as the use of deixis in the temporal subordinators is concerned, Pomak holds a special place within the Slavic languages. Among the closest related languages, Macedonian does not make any use of the three deictic suffixes in order to form subordinating conjunctions. Bulgarian, on the other hand, uses the -t-suffix (also used for the definite article) to form a number of subordinators, but those conjunctions do not enter a system of oppositions related to deictics. The data available for the Rhodope dialects in Bulgaria show a similar use of the deictic suffixes in the subordinators as those described in this paper, but no study is available on their precise values and uses.

Table 1: Semantics and uses of deictic suffixes in Pomak1

Even though the grammaticalization of a definite article is considered to be an areal feature in the Balkans, the extensive use of deixis observed in Pomak temporals cannot be explained through language contact, despite the Pomak speakers' trilingualism with Greek and Turkish.

Pomak: sociolinguistic information

Pom´atsko 'Pomak' is the name used for the South Slavic variety spoken by Muslim inhabitants of the Rhodope Mountains in Greece (cf. map 1) who often migrated to other cities or countries during the second half of the 20 th century. This study 2 focuses on a Pomak variety spoken in a village of the Xanthi area I will refer to as Pomak1 3 .

The language in the village under study is still transmitted to children, contrary to other traditional Pomak speaking villages in Greece, where a shift to Turkish has been generalized. The majority of the speakers in this village are trilingual. The younger generation, men and women, have learnt Greek and Turkish at school, within the "Minoritariste" primary school educational system provided to the Muslim Minority since the Lausanne Treaty (1923). Elder women are sometimes monolingual or, most frequently, have basic communicational skills in Greek and Turkish (see [START_REF] Adamou | Slave. In Le patrimoine plurilingue de la Grèce[END_REF].

Pomak is often described as a conservative South Slavic group since it has for example preserved a case system, in contrast with the most closely related South Slavic languages that developed an analytical system for those functions. This feature is important since loss of the case system is one of the features distinguishing Bulgarian and Macedonian from Serbian for example.

Naturally, this approach can be questioned given that on other points the system can be described as an innovative one, as for example in the case of the overt expression of deixis and its use for the temporal conjunctions described in this paper. Moreover, Pomak also shares some Balkan Sprachbund properties, such as "will" future, subjunctive, dative/genitive merger (observed in the Pomak1 variety), postposed articles, and to some extent evidentiality (see [START_REF] Adamou | Slave. In Le patrimoine plurilingue de la Grèce[END_REF]; for evidentiality as a Balkan feature see Friedman 2004).

The Pomak spoken in Greece has a special interest for the study of South Slavic because it had little contact with modern Bulgarian, standard or dialectal, during the greatest part of the 20 th century (except for the few years of Occupation during the Second World War when education in Bulgarian was obligatory), and therefore did not undergo any changes due to standardisation practices, as did the varieties spoken in Bulgaria (Kanevska-Nikolova 2001).

Pomak varieties in Greece are still practically undescribed because, within a context of shift to Turkish, the Pomak language is submitted to political and ideological conflicts that make fieldwork research extremely difficult to conduct. Still, educated Pomak speakers, most often in collaboration with Greek authors, have participated in the publication of dictionaries, grammars, teaching methods for foreigners and other material such as folktales and songs (see [START_REF] Theoharidis | Pomakoi. Xanthi: Pakethra[END_REF], 1996a, 1996b, 1996c[START_REF] Rogo | Pomakika dhimotika tragoudhia tis Thrakis[END_REF], Kokkas 2004a, 2004b).

Deixis and noun modification

The Pomak variety under study has a three-term person-oriented deictic system, the deictic centre being not only the speaker but the addressee as well (see [START_REF] Anderson | Deixis. In Language, Typology and Syntactic Description[END_REF].

Innovating temporal uses also occur for the definite articles and are described in 3.1.2. non-standardized varieties spoken in Greece), while their grammaticalization in some dialects of North Russia is controversial (see [START_REF] Breu | Der Faktor Sprachkontakt in einer dynamischen Typologie des Slavischen[END_REF][START_REF] Kasatkina | Artikli, predartikli i psevdoartikli v russkom iazyke[END_REF]). In an areal perspective, postposed articles are a well-known Balkan Sprachbund property -shared by Romanian, Aromanian and Meglenoromanian, as well as Albanian-even though their development is probably equally influenced by each language's diachrony as well as by their mutual reinforcement through multilingualism [START_REF] Asenova | Balkansko ezikoznanie[END_REF].

As is generally admitted, definite articles are grammaticalized from demonstratives [START_REF] Lyons | Definitiness[END_REF]). In the case of South Slavic, written sources show that, by the 13 th century, the postposed Old Church Slavonic 4 demonstratives were grammaticalized into clitic demonstratives, and then into clitic articles: see [START_REF] Gălăbov | Za člena v bălgarskija ezik[END_REF], [START_REF] Svane | [END_REF], 1962[START_REF] Mirčev | Za člennite formi v srednobălgarskite pametnici[END_REF] or more recently Dimitrova-Vulchanova & Vulchanov (in press); for a discussion of the status in the modern languages see [START_REF] Elson | The definite article in Bulgarian and Macedonian[END_REF]; for a detailed diachronic model of definiteness in Bulgarian see [START_REF] Mladenova | Definiteness in Bulgarian[END_REF].

The threefold definite article is traditionally considered in the literature as a "conservative" feature (Miletič, Koneski, Cyxun), following the analysis that the whole of South Slavic has gone through a phase of grammaticalization of three definite articles and then developed in some varieties to a single definite, while in others the three definites were maintained. Divergent analyses exist, arguing convincingly that the three-term definites are innovations of specific varieties but didn't develop to all the South Slavic languages (Mladenova 2007: 319-325 also citing Velčeva-Bojadžieva and Šaur).

In a synchronic perspective, the use of a one-term or three-term definite article requires a different dialectological analysis than the traditional dialectal classifications of South Slavic in terms of 'Eastern-Western' dialects following the phonetic distribution of the jat. Among the modern South Slavic varieties that have a grammaticalized definite article, the ternary deictic definite system is found in three geographic areas: this is the case of some Macedonian dialects and is the norm for Standard Macedonian (see Table 3); most of the varieties of the Rhodope Mountains (in Bulgaria and Greece), as well as some varieties that are spoken in Eastern Serbia (next to the Bulgarian border).

All the other South Slavic varieties that have developed a definite article are using a one-term system, usually based on the -t-form: this is the case of Standard Bulgarian and Bulgarian dialects 5 , the Eastern Macedonian dialects and the South-West Slavic varieties spoken in Greece [START_REF] Drettas | Le dialecte bulgaro-macédonien de Xr. (Edhessa, Grèce). Questions de typologie[END_REF], Adamou 2006). Some varieties are said to be in an unstable situation, evolving from the ternary to the single term system (Kanevska-Nikolova 2006, and fieldwork notes Adamou 2005Adamou -2006 for Evros Pomak2) 6 . The existence of two-fold definite systems is controversial (for a critical analysis see [START_REF] Mladenov | Ima li v bălgarskija ezik dvojno členuvane?[END_REF]).

Furthermore, it is important to note that even among the ternary systems, the parameters and the morphological distribution can differ. It would be interesting to know whether some differences observed are due to different analyses or to effective differences of the systems.

For example, Kanevska-Nikolova (2006: 79) describes the Rhodope definite system as what can be named after [START_REF] Anderson | Deixis. In Language, Typology and Syntactic Description[END_REF] terminology a distance oriented system (the speaker being its centre, in pragmatic or spatial terms), while in Mladenova (2007: 318) Macedonian also centre its system on the speaker (Minova-Gurkova 1997[START_REF] Friedman | Macedonian[END_REF] and for a different analysis Topolinjska 2006): The same split is found in the analyses of OCS demonstratives: for example, in Vaillant (1964: 140) the system appears to be distance-oriented while in Feuillet (1999: 148) it is described as person-oriented. According to [START_REF] Marchello-Nizia | From personal deixis to spatial deixis[END_REF] this sort of distinction could indicate different states of the language and is thus important to specify.

Definites in Pomak1

The definite article 7 is suffixed to the noun, the adjective or the tonic possessive pronoun and the possessive adjective. Three deictic suffixes are used in Pomak1 to form definite articles.

The choice of the deictic suffix depends on two types of parameters:

1/ For here and now situations the system depends on whether the entity is considered as being part of the speaker's sphere, the addressee's sphere, or none of them. According to [START_REF] Anderson | Deixis. In Language, Typology and Syntactic Description[END_REF], this type of system is named person-oriented.

Deictic systems are generally described in 'close vs. distant' terms: close to the speaker, close to the addressee, away form the speaker and the addressee. Still, it is broadly admitted that pragmatic parameters generally rule those systems [START_REF] Levinson | Deixis and pragmatics[END_REF]). In order to account for those uses, the term 'personal spheres' is preferred here [START_REF] Bally | L'expression des idées de la sphère personnelle et de solidarité dans les langues indo-européennes[END_REF][START_REF] Marchello-Nizia | From personal deixis to spatial deixis[END_REF].

In Pomak1, even though in most cases, the 'sphere' could appear to be spatial, other examples show that the system is determined by pragmatic and discursive criteria as well. The objective distance can thus be manipulated according to pragmatic criteria, but such uses will not be developed in detail here.

-s-: 'speaker's sphere'

( This definite is also used with a strong possessive meaning.

-t-: 'addressee's sphere'

( -n-: 'distal (not located in the participants' spheres)'

The suffix -n-is used for objects that are not included in the participant's spheres.

(3) j´ela nah kap´uje-ne come.IMP.2SG to doors-DEF.DIST 'Come to the entrance door (elsewhere).'

(The speaker is out of the house, the addressee is sitting at the balcony, and the entrance door is on the side of the house, away from both and not visible to any of them) (S, M3, 13, Pmk, Ell, Tur)

We should also note that in the variety described here, the deictic system does not seem obsolescent among younger speakers, contrary to other Pomak varieties which tend to reduce it.

2/ When the entities are situated in a different space and time, then the speaker adopts a temporal set of uses of the definite articles. In such cases, only the -t-and -n-articles are concerned, the -s-leading back to a spatial-temporal reading 'here and now'. Therefore, the addressee's -t-article no longer concerns the addressee's sphere but the past, while the -ndistal article is used for entities in the future, habitual in relation to the situation of utterance as well as for the situations with no anchoring to the situation of utterance (all 'non past' and 'non here an now' situations):

- In data available for South Slavic, demonstratives and definite markers do not seem to follow the same development within a variety and certainly not at the same speed. For example, even though the demonstrative system in Pomak1 is a three-term system, used in the same way as the definite system, in other close Pomak varieties (Kokkas 2004a: 22), demonstratives form a twofold system based on a 'close-distant' spatial contrast 10 , while the definite articles form a three-term person-oriented system. This difference in speed is also observed in Nashta, where a single definite article is in use, while demonstratives present traces of an old ternary system 'distal / proximal / unmarked'. In the same way, Standard Bulgarian has a two-term demonstrative system, the -t-suffix being used either as proximal (when contrasted) or as unmarked as far as distance is concerned, while the -n-suffix indicates distance. On the other hand, Standard Macedonian presents the same tripartite system in the demonstratives as in the definite articles. Mladenova (2007: 318) also signals this lack of symmetry in deixis between the articles and the pronominal and adverbial systems in her account of dialects (the so called mixed systems).

Demonstratives in Pomak1

11 hal-00398498, version 1 -24 Dec 2009

Demonstratives in Pomak1 also follow the double deictic system depending on the space and time. They have both exophoric and endophoric uses. The demonstratives have different forms according to the noun's gender and number: aisvak (m.), aisos (f.), aisva (n.) and aises (m/f.pl.) aiseva (n.pl) for close entities, ainvak (m.), ainos (f.), ainva (n.) and aines (m/f.pl.) aineva (n.pl) for distant entities and for some future, habitual time related to the situation of utterance, or for a moment with no anchoring to the situation of utterance; and aitvak (m.), aitos (f.), aitva (n.) and aites (m/f.pl.) aiteva (n.pl) for entities close to the addressee or for a past moment related to the situation of utterance.

Here

Deictic comparison markers, quantifiers and relatives in Pomak1

The three deictic suffixes -s-, -t-, -n-, are also used within comparison markers (kaks´o / kakt´o / kakn´o "as") and quantifiers (aisolk´os / aitolk´os / ainolk´os "this much", "that much") in a quite productive manner. Note that in other villages of the area, systems differ:

for instance kakn´o "as", is also used for 'simultaneous duration' (sharing those two usages with Greek), while it is not attested in my corpus of Pomak1 and ag´a is given in questionnaires. Deictic suffixes can also be encountered partially for manner (ais´i "this way" / ain´i "that way").

Relatives also follow the deictic system according to space and time: 'addressee's sphere'

Subordinate clauses

Subordinate clauses in the South Slavic languages

According to [START_REF] Cyxun | Tipologičeskie problemy balkanoslavjanskogo jazykovogo areala[END_REF] and [START_REF] Mladenova | Definiteness in Bulgarian[END_REF], the use of the particle -to in relative pronouns and conjunctions is an innovation, characterizing the Eastern South Slavic area: the Rhodopean dialects are the "centre of innovation in regards to the introduction of overt definiteness" (Mladenova 2007: 243). In this perspective, the uses of the deictics as described in this paper should also be viewed as innovative for South Slavic.

Indeed, conjunctions with -to, deriving from the demonstrative, are found relatively late in the written sources. Analyzing the damaskin 12 texts, Mladenova (2007: 242) signals a split between uses with or without -to for relative pronouns and conjunctions; the relative deto being the most frequent. Concerning the temporal conjunctions, she notes 96.88% uses of koga and a single use of kogato "when"; the latter being nowadays used in Standard Bulgarian. In Standard Macedonian, despite the use of a ternary definite and demonstrative system, there is no use of deictic suffixes in the temporals as the one observed in Pomak1; the speakers of the standard language use the temporal conjunction koga "when".

Subordinate clauses in Pomak1

Together with the Slavic subordinators, many loans from Greek and Turkish are used in Pomak1. Subordinating conjunctions are indeed high in the borrowability hierarchy in language contact settings [START_REF] Matras | Utterance modifiers and universals of grammatical borrowing[END_REF](Matras , 2007)). In Pomak, some result from old contact, and are well integrated in the language (as Greek oti "because"), while some result from recent contact (as Turkish tam "just as"), and thus are unstable and in competition with Slavic markers.

We will now focus on temporal subordinators based on these deictic suffixes.

Deixis in temporal subordinate clauses

In Pomak1 temporal subordinators combine reference to the situation of utterance and reference to the situation of the process through the use of the deictic suffixes as follows 13 :

1/ Presence of a deictic suffix indicates the anchoring of the utterance to the situation of utterance: ag´a-to, g´ato or kug´ato "when (past)", ag´a-no g´ano or kug´ano 14 "when (future), whenever". Absence of deictics indicates absence of anchoring to the situation of utterance (ag´a). Thus, the temporal marker with no deictic suffix is used for folk tales, jokes and other narratives, fictional or not. Ag´a seems to be taking over some conditional uses, too, while the form ga is specialized for causal usage.

2/ The choice of the deictic suffix indicates the relation between the situation of the process and the situation of utterance.

The 'addressee's' -t-suffix, in ag´a-to, is used for a 'past' event in relation to the situation of utterance. The 'distal' -n-suffix, in ag´a-no, refers to a generic or habitual timeframe and future reference. "whenever" future moment in relation to the situation of utterance (21,22,23) or habitual in relation to the situation of utterance (18,19,20) The interclausal relation is determined both by the subordinators and the TMA markers (on the importance of the complex TMA system in adverbial clauses in Bulgarian, see among others [START_REF] Fielder | Aspect and Modality in Bulgarian subordinate clauses[END_REF][START_REF] Guentchéva | L'imparfait perfectif en bulgare[END_REF].

Note that the three temporal markers presented above are not specialized in a specific type of temporal interclausal relation: for example ag´ato is used for anteriority, posteriority, simultaneity, terminus a quo, while all three markers can express anteriority (for this terminology see [START_REF] Kortmann | Adverbial Subordination: A typology and history of adverbial subordinators[END_REF].

To describe the system of Pomak1's temporal subordinators, I shall be using the terminology developed by [START_REF] Culioli | À propos d'opérations intervenant dans le traitement formel des langues naturelles[END_REF][START_REF] Culioli | À propos d'opérations intervenant dans le traitement formel des langues naturelles[END_REF]Culioli ( , and 1990) about the notion of situational anchoring, relating the situation of utterance (Sit0) to the situation of the process (Sit2), and applied by Robert (2006: 170) to the study of deixis and subordination in Wolof. Culioli has defined three types of relations between Sit0 and Sit2, often, but not uniquely expressed by TMA markers: the situation of the process may be different (≠) from the situation of utterance, towards being identical (=) to it, or have no relation to it (ω).

Pomak1 temporal conjunctions express those relations as follows:

Situation of utterance and situation of process are different: Sit2 ≠ Sit0

When the space-time frame of the denoted situation (Sit2) is different from the space-time of the utterance (Sit0), Pomak1 uses the deictic suffixes -n-and -t-to form the temporal subordinators. The -t-suffix, 'close to the addressee', is used for the 'past' in relation to the situation of utterance, while the 'distal' -n-suffix is used for 'habitual' and 'future' events in relation to the situation of utterance.

-to : 'past moment related to the moment of utterance' (Conversation about politicians and their politics about the villages) (Q, F2, 30, Pmk, Ell, Tur)

(
Below some examples of the use of ag´a-no/ kug´a-no for future events: (Instructions in order to locate a place in the village) (Q, F8, 21, Pmk, Ell, Tur)

The choice of the temporal subordinator in Pomak1 doesn't depend on the TMA choice, even though some TMA are more frequent for a given subordinator, since it relates to a past or future event.

The following examples illustrate that: in 24 and 25 the speaker gives two possible versions, changing the TMA markers while maintaining the temporal subordinator ag´ano.

(

No anchoring of the situation of process in the situation of utterance: Sit2 ω Sit0

When there is a break between the space-time of the process and the space-time of the The constant use of ag'a in tales is also confirmed by other corpus of the area, such as the tales published in [START_REF] Theoharidis | Pomakoi. Xanthi: Pakethra[END_REF] 16 and Kokkas (2004b).

Identification between the two situations: Sit2 = Sit0

When the space-time of the denoted situation is identified with the space-time of the utterance, Pomak1 could, in theory, use the 'proximal to the speaker' deictic suffix -s-, but in practice no such usages were attested during my fieldwork. When asked, the Pomak1 speakers say they cannot use ag´aso. An educated speaker from another village of the area thought this was a logical possibility, making a clear parallel with the threefold definite system, but still had difficulties in producing any such examples. Still, in the Rhodope dialects in Bulgaria, gasu seems to be still in use (p.c. Kanevska-Nikolova).

Deixis and immediate anteriority

Immediate anteriority is also marked differently according to the type of discursive anchoring.

Immediate anteriority is expressed by prefixing li to ag´a (becoming ga) for situations with no anchoring to speech time, to kug´ano for a future event related to the situation of utterance or for some habitual event, and to kug´ato for past events. As an adverb li in Pomak1 means:

1) 'immediately', e.g. li r´ipnala 'she hopped immediately';

2) 'all the time, constantly', e.g. li r´uka 'she was calling all the time'.

Li is also a yes-no interrogative particle, as was already the case in Old Church Slavonic, and as commonly found in many other Slavic languages (there have been many studies on the meaning and placement of li in Slavic languages). Example in Pomak1:

h´ubeve li ste? 'Are you OK?' j´ala li si? 'Have you eaten?'

In Bulgarian, li is used as a subordinator with a meaning close to the conditional [START_REF] Guentchéva | L'imparfait perfectif en bulgare[END_REF].

Below some examples of immediate anteriority in Pomak1:

Polyfunctionality

Some of the adverbial subordinators presented above are polyfunctional.

Conditionals

Ag´a : from 'temporal' to 'hypothetical'

It is very common cross-linguistically for temporal subordinators to express conditional or hypothetical meanings (Traugott 1986). In Pomak1, ag´a, also has a hypothetical meaning, even though the most frequent subordinator is ak´u.

(

Conclusion

The This is an uncommon phenomenon among Slavic languages, even though a more common one cross-linguistically as shown in this volume. More research needs to be done, both in a synchronic and diachronic perspective, on this phenomenon that has received practically no attention in South Slavic studies.

older speakers, or younger ones with little education, seem to prefer ag´a-, although I have not conducted any precise sociolinguistic study on this topic. In any case, both forms are accepted by the speakers when asked.

  system of definite markers is attested in other South Slavic languages, such as Standard Macedonian, but the constraints of usage are very different from those of Pomak described in this paper (see §3.1.1). Standard Bulgarian makes use of one definite article, based on the -t-suffix, while practically all the other Slavic languages have no grammaticalized definite markers.

  the Rhodope system is clearly described as person-oriented. The most recent descriptions of 7 hal-00398498, version 1 -24 Dec 2009

  .HAB.FUT come.3SG MOD 3SG.F.DAT 3SG.M/N.ACC bring-1SG 'When she comes, I'll bring it to her.' (The speaker has a gift to give to a friend who is not at the village at the moment) (S, F5, 15, Pmk, Ell, Tur) (22) ag´a-no mi punarast´o-t m´ufka det´i-se when-LOC.HAB.FUT 1SG.DAT grow,up-3PL a,little children-DEF.S ʃe d´ojde-me nah par´isi MOD come-1PL to Paris 'When the (my) children grow up a little, we'll come to Paris.' .HAB.FUT arrive-2SG at square-DEF.DIST f´ati nah lj´avana str´ana seaze.IMP.2SG to left side 'When you arrive at the central square, take the left side.'

  study of the Pomak variety spoken in the Xanthi area (Greece) reveals an interesting use of deixis in the formation of temporal subordinators, indicating the anchoring of the event to the time of utterance while the choice of the subordinator indicates the type of event encoded in the clause. Absence of a deictic suffix indicates the break between the situation of utterance and the situation of the process.

Table 2 .

 2 Deictic suffixes in Pomak1

	hal-00398498, version 1 -24 Dec 2009
	-s-	speaker's sphere; here and now
	-t-	addressee's sphere; past moment related to the situation of utterance
	-n-	distal (not located in the participants' spheres); future, present moment related to the situation of
		utterance; no anchoring to the situation of utterance
		5

Table 3 .

 3 Deictic suffixes in Standard Macedonian's definite articles

	-f/v-	close to the speaker
	-t-	unmarked (no indication to the distance)
	-n-	away from the speaker

n-: 'future moment in relation to the moment of utterance'

	(4)	na sf´adba-na	ʃe n´adena-m tʃerv´en-en	fust´an
		at wedding-DEF.DIST

MOD wear-1SG red-DEF.DIST dress 'At the wedding I'll wear a red dress.' (Q, F2, 30, Pmk, Ell, Tur) -t-: '

past moment in relation to the moment of utterance'

	-s-: 'speaker's sphere'
	(6)	na sf´adba-sa	sam	/ ´ima mn´ogo ins´an
		at wedding-DEF.S am.1SG / is.3SG a,lot	people
		'I'm at the wedding. There is a lot of people.' (Q, F2, 30, Pmk, Ell, Tur)
	3.2. The demonstratives
	3.2.1. Demonstratives in the South Slavic languages
	hal-00398498, version 1 -24 Dec 2009
	(5)	na sf´adba-ta		bex	sas tʃerv´en-et	fust´an
		at wedding-DEF.LOC.PAS was.1SG with red-DEF.LOC.PAS dress
		'At the wedding I wore a red dress.' (Q, F2, 30, Pmk, Ell, Tur)
						10

past moment in relation to the situation of utterance'

  are some examples:

	(10)	dve gud´ini napr´eʃ mand´ili-te b´exa	tʃerv´eni
		two years to, the,	scarfs-	were.3SG red
				front	DEF.LOC.PAS
		´oti		b´eʃe	ait´os	m´oda-ta
		because	was.3SG	that.LOC.PAS	fashion-DEF.LOC.PAS
		'Two years ago, the scarfs were red because that was the fashion.'
	'speaker's sphere' (Q, F2, 30, Pmk, Ell, Tur)
	(7)	ais´os je	d´aed-va-sa	k´oʃta 11
	(8)	ts´alo aitv´a ʃe jad´e-ʃ
		all	DEM.A MOD eat-2SG
		'You'll eat all of this (your sphere).' (S, F8, 21, Pmk, Ell, Tur)
	'distal'
	(9)	ain´os		l´elka hi	z´oela	je	b´eʃ bin evr´o

DEM.S is.3SG grand,father-POSS-DEF.S house 'This (my sphere) is grand-father's house.' (Q, F2, 30, Pmk, Ell, Tur) 'addressee's sphere' DEM.DIST aunty POSS.3SG.F took.PRF AUX.3SG 5 000 euro 'That aunty of hers, she has got 5 000 euros.' (S, F5, 15, Pmk, Ell, Tur) '

Table 4 .

 4 The temporal subordinating conjunctions in Pomak1

	ag´a	"when"	no anchoring to the situation of utterance (24, 25, 26)
		"if"	(conditional) (34)

ag´ato/kug´ato "when" past moment in relation to the situation of utterance

(15, 16, 17)

ag´ano/kug´ano "when"

  utterance, Pomak1 does not use deictic suffixes, but the free subordinator ag´a (kug´a is not 20 hal-00398498, version 1 -24 Dec 2009 attested in such use). In this case, the process is not anchored in relation to the utterance. This subordinator is almost exclusively used in fictional narratives and tales (the habitual ag´ano

	subordinator is very seldom encountered):
	(26)	ag´a sa	v´ornal-i	dve gud´ini sj´etne ud	alam´anie
		when REFL return.EVID-3PL two years after from Germany
		utiʃl´il-i	da plat´ə -t jajts´a-na
		go.EVID-3PL to pay-3PL eggs-DEF.DIST
		'When they returned, two years later from Germany, they went to pay the
		eggs.'
		(Nasradin joke) (S, M3, 70, Pmk, Ell, Tur)
	(27)	ag'a	utiʃl'il-i	po	udv'ode sr'eʃtal-i	m'ajka	mu
		when	go.EVID-3PL more out		meet.EVID-3PL mother	POSS3SG.M
		'When they went a little further, they met his mother.'
		(Nasradin joke) (S, M3, 70, Pmk, Ell, Tur)
	(28)	ag'a	je	fprj'ala		je	razv'orzala	tʃuv'al-an
		when	AUX.3SG arrived.PRF	AUX.3SG untied.PRF	bag-DEF.DIST
		'When she arrived, she untied the bag.' (Folk tale) (S, F3, 7, Pmk, Ell)

Table 5 :

 5 Immediate anteriority subordinating conjunctions in Pomak1

	li ga	"as soon as" immediate anteriority in a moment
		with no anchoring to the situation of utterance
	li kug´ato "as soon as" immediate anteriority in a past moment
		in relation to the situation of utterance
	li	"as soon as" immediate anteriority in future moment
	kug´ano	or habitual event with relation to the situation of utterance

li kug´ato 'immediate anteriority-past'

	(31) li-kug´a-no		sa	je	r´adalo	d´ete-no
		immediately-when-LOC.HAB.FUT REFL	AUX.3SG was,born.PRF child-DEF.DIST
		d´avali	mu	sa		ʃik´er
		were,giving.PRF 3SG.M/N.DAT AUX.3PL	sugar
		'As soon as the child was born, they were giving him sugar.' (Q, F2, 30, Pmk,
		Ell, Tur)
	(29) li-kug´a-to		gu		r´ekah-	stan´a
						me
		immediately-when-	3SG.M/N.ACC said-1PL became.3SG
		LOC.PAS
		'As soon as we said it, it happened'. (S, F2, 30, Pmk, Ell, Tur)
	li kug´ano 'immediate anteriority-future/habitual'
	(30)	li-kug´a-no		sa	razbud´i-ʃ z´oe[mi]	ma	telef´on
		immediately-when-	REFL wake,up-	take.IMP.2SG 1SG.ACC telephone
		LOC.HAB.FUT			2SG
		'As soon as you wake up, call me.' (Q, F2, 30, Pmk, Ell, Tur)

tam 'immediate anteriority, past moment related to the moment of utterance' tam

  (from Turkish) 'just as, as soon as' 17 , is external to the native system relative to the time of utterance. Thus, language contact, first through loan, but then finally through conceptual pressure in case of stable contact situation, might trigger the change of the Pomak system of

	temporal conjunctions.
	(32) tam	fpraeh	faf	isk´etʃe zoe	ma	telef´on
	just,as	arrived.1SG in	Isketshe took.3SG	1SG.ACC telephone
	'Just as I arrived in Isketshe (Gr. Xanthi), s-he called me.' (Q, F2, 30, Pmk,
	Ell, Tur)

Ga: from 'temporal' to 'causal'

  ) ,birth.P.PRF another-DEF.DIST child more to,the,front 'If I had known, I would have had another child earlier.' (Q, F2, 30, Pmk, Ell, It is also common cross-linguistically for temporal subordinators to express causality. In Pomak1, ga also has some causal uses, most probably derived from the temporal ag´a after phonetic erosion. The use of ga presents the cause-effect relation as obvious and has a strong discursive component. The causal clause with ga usually precedes the main clause. Since you don't want, we won't go.'(Q, F2, 30, Pmk, Ell, Tur) The other causal subordinator ´oti, is probably a loan from New Testament Greek ´oti 'because'. In the causal clauses with ´oti, the main clause generally precedes the subordinate wanted.3SG because say.3SG supposedly is.3SG ugly 'She didn't want him because, she says, supposedly he's ugly.' (S, F5, 15, The boy offered her a necklace because it was her birthday.'(S, F5, 15, Pmk,

		(38)	ʃe da d´ojda ga ni ʃte-ʃ AUX.3SG had.PRF birthday nah s´elo n´ema	da	h´ode-me
		MOD MOD come.3SG to village since NEG want-2SG not,have.3SG because	to	go-1PL
		'If my husband doesn't work, he will come to the village.' (Q, F2, 30, Pmk, Ell, Tur) '´oti: causal 'Ell, Tur)
		ni (36) ak´u beh	p´oje-m	n´ema zn´ala	da	´ida-m	faf panepist´imjo
			ag´a if	AUX.1SG	knew.P.PRF
		clause.	if NEG study-1SG not,have.3SG to	go-1SG in university
		'If I don't study, I won't go to the university.' (Q, F3, 7, Pmk, Ell) ʃe beh rud´ila dr´ugo-no d´ete po (39) ait´us tr´aeva da je let´alo	napr´eʃ
	hal-00398498, version 1 -24 Dec 2009	ak´u 'conditional' This is the most frequent conditional subordinator in Pomak1, used for both realis and irrealis clauses. (34) ak´u ´iʃta-t da ´ida-t da sa utʃ´o-t faf panepist´imjo-no if want-3PL to go-3PL to REFL study-3PL in university-DEF.DIST MOD AUX.1SG Tur) 5.2. Causal here must.3SG to AUX.3SG rained.PRF ´oti sab´alahin be j´atse m´okro p´ot-en because in,the,morning was.3SG a,lot wet street-DEF.DIST 'Here, it must have rained because this morning the street was wet.' (S, F8, 21, Pmk, Ell, Tur) (40) gu ni iʃt´ae ´oti v´ika gjo je par´atik
			tr´aeva 3SG.M/N.ACC	da	´ima-t	h´ubavi	kit´ape	ut	ðimotik´o-no
			must.3SG		to	have-3PL	good	books	from	primary-DEF.DIST
			'If they want to go to study to the University, they must have good books since Pmk, Ell, Tur)
			the primary school.' (Q, F2, 30, Pmk, Ell, Tur)
		(37) (41) k´opel-an ga n´ema-me hi	tumaf´il ni je	dal	h´ode-me nah sindʒ´ir	den´iz-en
		(35) ak´u since not,have-1PL car ne r´aboti boy-DEF.DIST 3SG.F.DAT AUX.3SG gave.PRF necklace tʃylj´aek-an mi NEG go-1PL to	sea-DEF.DIST
			if ´oti	j	NEG work.3SG im´ala	husband-DEF.DIST POSS.1SG du´omgyny
								27

gave'Since we don't have a car, we won't go to the sea.' (Q, F2, 30, Pmk, Ell, Tur) NEG

The distinction between 'Bulgarian' and 'Macedonian' dialects depends on the authors and their ideological background. Bulgarian linguists have long considered the whole of the South Slavic varieties as 'Bulgarian' dialects, despite variation and important structural differences; the demarcation line being with Serbian. Since the standardization of Macedonian, and reinforced by the creation of an independent Republic, the term 'Macedonian' has slowly made its way to the international bibliography. Within this context, where linguistics are linked with politics, we choose to apply the term 'Bulgarian dialects' for the varieties situated

Variation between kug´a-no/to (based on the interrogative kug´a) and ag´a-no/to (based on the temporal subordinator ag´a) depends on the speaker's age and possibly his education;

This term refers to a special social event that takes place in the afternoon, when unmarried girls walk in the main village street in front of the boys in order to make a potential match. This walk takes place in two villages of the area and gathers young people from the surrounding villages.

In this corpus, segmentation is not always accurate, sometimes including the auxiliary 'be' or the reflexive.

tamam 'at the same time' and li tamam 'as soon as', are known to be used in Pomak but did not figure in our corpus. Tamam is also used in Macedonian and Bulgarian.

3 Given the political pressure on Pomak speakers, within a context of linguistic shift to Turkish, I decided not to mention the villages' names, despite the obvious interest this would present in a dialectological perspective. Pomak1 stands for the village of the Xanthi area and Pomak2 for the village of the Evros area. I sincerely thank the speakers who accepted to participate in this study. 4 Old Church Slavonic [OCS] refers to the eldest available Slavic documents, mostly ecclesiastical. OCS extends from the 9th to the 11th century and is classified as South Slavic.

in Bulgaria, 'Macedonian dialects' for those in FYROMacedonia -even though for Macedonian linguists, the 'Macedonian' dialects go beyond these borders (in Albania, Bulgaria, Greece, Kosovo). For the denomination of Slavic in Greece see [START_REF] Adamou | Slave. In Le patrimoine plurilingue de la Grèce[END_REF].

Moreover, the terms 'language' and 'dialect' are in this context highly connected to political and ideological factors. Therefore, we use 'varieties' as a linguistic term for oral tradition languages seen in their diversity. Nevertheless, it is obvious that from a linguistic point of view those South Slavic standard languages and non-standard varieties are closely related and are part of the same linguistic diasystem. 6 In Greece, in a village of the Evros area, the system has been reduced to the -t-definite, while possessive uses of the -s-article occur occasionally. I consider this change to be a loss; the speakers in their metalinguistic comments mention three articles, which are not found in their oral productions.

7 Indefinite nouns have no articles; they may be preceded by the numeral 'one' which is not a fully grammaticalized indefinite article. 8 There are many loan words from both Greek and Turkish in Pomak1. I do not signal their origin, the Pomak morphology and syntax being respected in most cases. 9 Each example is followed by a description of the context. I systematically signal if the example is spontaneous (S), for data recorded or not, or extracted by questionnaire (Q). The questionnaire consists in translating from Greek a sentence that was usually produced in Pomak by a speaker, in natural speech, but that I didn't have the chance to record. The speaker's gender (male M, female F) is followed by an identification number valid for this paper. Are also noted the age and the languages spoken by the consultant: Pmk stands for Pomak, Ell for Greek and Tur for Turkish. All data were collected between 2005-2007 by the author.

10 izí (m.), isáy (f.), izí, isáy (n.) and iséy (pl.) for close entities, inazí (m.), ináy (f.), inazí, ináy (n.) and inéy (pl.) for distant entities.

11 Note that the vowels /o/, /e/, /a/ when accentuated are diphthongized [u´o], [j´e], [j´a], and, when not accentuated, are reduced.

12 Translated literary texts of a religious and didactic nature, in what is referred to as Modern and for some of them Middle Bulgarian (17th-18th century). 13 This is not a unified analysis for all the Rhodope varieties using deixis in the temporal conjunctions. Observations based on short term fieldwork in a Pomak variety in the Evros department in Greece, show a different system whose understanding requires further research (fieldwork notes Adamou 2005Adamou -2006)). Moreover, a quick look on data from Siroka Laka (Bulgaria) in [START_REF] Sobolev | Bolgarskij shirokolykskij govor: Sintaksis, Leksika duchovnoj kul'tury, Teksty[END_REF], doesn't seem to corroborate the uses observed in Pomak1, even though deictic suffixes are very frequently used with temporal subordinators. It would be interesting, among others, to compare with Kabasanov's description of Tixomir (1963), a Rhodope variety in Bulgaria, very close to the Greek border, spoken by Muslims.


William Kengne
email: william.kengne@u-cergy.fr

Strongly consistent model selection for general causal time series

Keywords: Model selection, strong consistency, causal processes, quasi-maximum likelihood estimation, penalized contrast

We consider the issue of strong consistency for model selection in a large class of causal time series models, including AR(∞), ARCH(∞), TARCH(∞), ARMA-GARCH and many other classical processes. We propose a penalized criterion based on the quasi likelihood of the model. We provide sufficient conditions that ensure the strong consistency of the proposed procedure. Also, the estimator of the parameter of the selected model obeys the law of iterated logarithm. It appears that, unlike the result of weak consistency obtained by Bardet et al. [2], dependence between the regularization parameter and the model structure is not needed.

Introduction

We consider a general class of autoregressive time series in a semiparametric framework. Let M, f : R N → R be two measurable functions and (ξ t ) t∈Z a sequence of centered independent and identically distributed (iid) random variables satisfying var(ξ 0 ) = 1. Consider the class of affine causal models, Class AC(M, f ) : A process X = (X t ) t∈Z belongs to AC(M, f ) if it satisfies:

X t = M (X t-i ) i∈N * ξ t + f (X t-i ) i∈N * for any t ∈ Z. (1)
The existence of a stationary and ergodic solution of the class (1) has been studied by [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF] as a particular case of models considered in [START_REF] Doukhan | Weakly dependent chains with infinite memory[END_REF]. [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF] and [START_REF] Bardet | Asymptotic behavior of the laplacian quasimaximum likelihood estimator of affine causal processes[END_REF] carried out the inference question in the semiparametric setting in the class AC(M, f ), whereas [START_REF] Bardet | Detecting multiple change-points in general causal time series using penalized quasi-likelihood[END_REF], [START_REF] Kengne | Testing for parameter constancy in general causal time-series models[END_REF], [START_REF] Bardet | Monitoring procedure for parameter change in causal time series[END_REF] focussed on the change-point problem in this class. Numerous classical time series models belong to the class [START_REF] Bardet | Asymptotic behavior of the laplacian quasimaximum likelihood estimator of affine causal processes[END_REF], for instance AR(∞), ARCH(∞), TARCH(∞), ARMA-GARCH or APARCH processes.

Consider a trajectory (X 1 , . . . , X n ) of a process X = (X t ) t∈Z that belongs to AC(M * , f * ) where M * and f * are unknown. We consider a finite collection M of affine causal models, where the true model m * ∈ M corresponds to M * and f * . Our main aim is to select a model m (among the collection M) which is "close" to m * for large n.

We focus on the semiparametric framework and assume that the distribution of ξ 0 is unknown and that the functions f and M are known up to a parameter θ ∈ Θ, where Θ is a compact subset of R d (d ∈ N). That is, the model m * corresponds to the true parameter θ * ∈ Θ and the process X belongs to AC(M θ * , f θ * ). In the sequel :

• each model m ∈ M is considered as a subset of {1, . . . , d} and denote by |m| the dimension of the model, typically, |m| = #(m);

• for m ∈ M, the parameter space of m is Θ(m) = (θ i ) 1≤i≤d ∈ Θ, θ i = 0 if i / ∈ m ; θ(m)
is the parameter vector associated to m;

• the collection M is considered as a subset of the power set of {1, . . . , d}, i.e. M ⊂ P {1, . . . , d} .

Therefore, for any model m ∈ M, m ∈ AC(M θ , f θ ) when θ ∈ Θ(m). Also, we could consider hierarchical as well as exhaustive families of models.

For instance, assume that (X 1 , . . . , X n ) is generated from a GARCH(p * , q * ) process. The collection M could be a family of ARMA(p, q)-GARCH(p , q ) with (p, q, p , q

) ∈ {0, 1, • • • , p max } × {0, 1, • • • , q max } × {0, 1, • • • , p max } × {0, 1, • • • , p max }
where p max , q max , p max , p max are the fixed upper bounds of the orders, assumed to satisfy p * ≤ p max and q * ≤ q max . Therefore, consider Θ as a compact subset of R pmax+qmax × (0, ∞) × [0, ∞) p max +q max . Thus, a model m is a subset of {1, • • • , p max + q max + p max + q max + 1} and its (i) Under the assumptions that E|ξ 0 | r < ∞ with r > 4, the functions θ → f θ , M θ are twice continuously differentiable on Θ and satisfy some Lipschitz-type properties, we establish the strong consistency of the proposed procedure.

parameter space is Θ(m) = (θ i ) 1≤i≤d ∈ Θ, θ i = 0 if i / ∈ m .
(ii) We show that the quasi maximum likelihood estimator (QMLE) of the selected model obeys the law of iterated logarithm.

The rest of the paper is structured as follows. In Section 2, we set some notations, assumptions and define the model selection criterion. The main results are provided in Section 3 whereas Section 4 is devoted to a concluding remarks. Section 5 focuses on the proofs of the main results.

2 Assumptions and the model selection criterion

2.1 Assumptions on the class of models AC(M θ , f θ )

In the sequel, we will use the norms :

1. • applied to a vector denotes the Euclidean norm of the vector;

2. for any compact set K ⊆ R d and for any g : Θ -→ R p , g K = sup θ∈Θ g(θ) .

Throughout the sequel, we assume that the functions θ → M θ and θ → f θ are twice times continuously differentiable on Θ. Also, we will use H θ = M 2 θ and for any function g θ which is i times differentiable on Θ, we set ∂ θ i g θ = ∂ i g θ /∂θ i . Let us consider the following assumptions for any compact set K ⊆ Θ, i = 0, 1, 2 and

Ψ θ = ∂ θ i f θ or ∂ θ i M θ : Assumption A(Ψ θ , K): for any x ∈ R N , the function θ → Ψ θ (x) is continuous on Θ with Ψ θ (0) Θ < ∞ and
there exists a sequence of non-negative real numbers

α k (Ψ θ , K) k≥1 satisfying ∞ k=1 α k (Ψ θ , K) < ∞ such that: Ψ θ (x) -Ψ θ (y) K ≤ ∞ k=1 α k (Ψ θ , K)|x k -y k | f or all x, y ∈ R N .
In the sequel we refer to the particular case of the non linear ARCH(∞) (NLARCH(∞), see [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]) processes define when f θ = 0. In this case, we consider the following assumption for i = 0, 1, 2 :

Assumption A(∂ θ i H θ , K): for any x ∈ R ∞ , the function θ → ∂ θ i H θ (x) is continuous on Θ with ∂ θ i H θ (0) Θ <
∞ and there exists a sequence of non-negative real numbers

α k (∂ θ i H θ , K) k≥1 satisfying ∞ k=1 α k (∂ θ i H θ , K) < ∞ such that : ∂ θ i H θ (x) -∂ θ i H θ (y) K ≤ ∞ k=1 α k (∂ θ i H θ , K)|x 2 k -y 2 k | f or all x, y ∈ R N .
Then define the set:

Θ(r) = θ ∈ R d , A(f θ , {θ}) and A(M θ , {θ}) hold with ∞ k=1 α k (f θ , {θ}) + ξ 0 r ∞ k=1 α k (M θ , {θ}) < 1 ∪ θ ∈ R d , f θ = 0 and A(H θ , {θ}) holds with ξ 0 2 r ∞ k=1 α k (H θ , {θ}) < 1 (2)
The above Lipschitz-type conditions are classical when studying the existence of a stationary and ergodic solution of such model, see [START_REF] Doukhan | Weakly dependent chains with infinite memory[END_REF]. In the case of the class

AC(M θ , f θ ), if θ ∈ Θ(r),
then there exists a unique causal, stationary and ergodic solution X = (X t ) t∈Z ∈ AC(M θ , f θ ) with a finite moment of order r, see [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF].

The following assumptions are useful in the study of the asymptotic behavior of the QLME .

Assumption D(Θ): ∃h > 0 such that inf θ∈Θ (H θ (x)) ≥ h for all x ∈ R N . Assumption Id(Θ): For all (θ, θ ) ∈ Θ 2 , f θ (X 0 , X -1 , • • • ) = f θ (X 0 , X -1 , • • • ) and M θ (X 0 , X -1 , • • • ) = M θ (X 0 , X -1 , • • • ) a.s. ⇒ θ = θ . Assumption Var(Θ): For all θ ∈ Θ, one of the families ∂f θ ∂θ i (X 0 , X -1 , • • • ) 1≤i≤d or ∂h θ ∂θ i (X 0 , X -1 , • • • ) 1≤i≤d is a.s. linearly independent.
In the following assumption, we make the convention that if

A(∂ θ i M θ , Θ) holds then α k (∂ θ i H θ , Θ) = 0 for all k ∈ N and if A(∂ θ i H θ , Θ) holds, then α k (∂ θ i M θ , Θ) = 0 for all k ∈ N. Set for = 0, 1, 2,
Assumption K (Θ): there exists r > 4 such that θ * ∈ Θ(r)∩Θ and for i = 0,

• • • , , A(∂ θ i f θ , Θ), A(∂ θ i M θ , Θ) (or A(∂ θ i H θ , Θ)) hold with k≥1 1 √ k log log k j≥k i=0 α j (∂ θ i f θ , Θ) + α j (∂ θ i M θ , Θ) + α j (∂ θ i H θ , Θ) < ∞.
These aforementioned assumptions hold for many classical models, including AR(∞), ARCH(∞), TARCH(∞) type processes , see for instance [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF], [START_REF] Bardet | Detecting multiple change-points in general causal time series using penalized quasi-likelihood[END_REF], [START_REF] Kengne | Testing for parameter constancy in general causal time-series models[END_REF]. In the case of assumption K(Θ), let us consider for = 0, 1, 2 :

1. The geometric case:

i=0 α j (∂ θ i f θ , Θ) + α j (∂ θ i M θ , Θ) + α j (∂ θ i H θ , Θ) = O(a j
) for some a ∈ [0, 1). In this case, assumption K (Θ) holds.

The Riemanian case:

i=0 α j (∂ θ i f θ , Θ)+α j (∂ θ i M θ , Θ)+α j (∂ θ i H θ , Θ) = O(j -γ ) with γ > 0. If γ > 3/2, then K (Θ) holds.

The model selection criterion

Consider a model m ∈ M and the class

AC(M θ , f θ ) for θ ∈ Θ(m) ⊂ Θ ⊂ R d . Assume that a trajectory (X 1 , . . . , X n ) is observed. The conditional Gaussian quasi (log)likelihood (up to a constant) L n is defined for all θ ∈ Θ(m) by, L n (θ) := - 1 2 n t=1 q t (θ) , with q t (θ) := (X t -f t θ ) 2 H t θ + log(H t θ ) (3)
where

f t θ := f θ (X t-1 , X t-2 , • • • ), M t θ := M θ (X t-1 , X t-2 , • • • ) and H t θ = M t θ 2 .
Since L n (θ) depends on (X t ) t≤0 which are not observed, it is common practice (see [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF], [START_REF] Bardet | Detecting multiple change-points in general causal time series using penalized quasi-likelihood[END_REF], [START_REF] Kengne | Testing for parameter constancy in general causal time-series models[END_REF]) to consider the approximated quasi (log)likelihood given (up to a constant) for all θ ∈ Θ(m) by

L n (θ) := - 1 2 n t=1 q t (θ) , with q t (θ) := (X t -f t θ ) 2 H t θ + log( H t θ ) (4)
where

f t θ = f θ (X t-1 , X t-2 , • • • , X 1 , 0, • • • ), M t θ = M θ (X t-1 , X t-2 , • • • , X 1 , 0, • • • ), H t θ = ( M t θ ) 2 .
Note that, the "best" parameter associated to the model m is defined by,

θ * (m) = argmin E[q 0 (θ)] θ∈Θ(m)
.

According to [START_REF] Bardet | Consistent model selection criteria and goodness-of-fit test for common time series models[END_REF], θ * (m) exists and it is unique under Id(Θ). When m = m * , we have θ * (m * ) = θ * . For any m ∈ M, the QMLE of θ * (m) is given by

θ(m) = argmax θ∈Θ(m)
L n (θ).

(

) 5
The selection of the "best" model m among the collection M is performed by minimizing the penalized contrast

C(m) = -2 L n θ(m) + |m|κ n , (6)
that is

m = argmin m∈M C(m), (7)
where

• (κ n ) n
is the sequence of the regularization parameter (possibly data-dependent) that will be used to calibrate the penalty term;

• |m| is the dimension of the model m, typically, the cardinal of m (considered as a subset of {1, . . . , d}), which is also the number of the estimated components of θ (the others are fixed to zero).

Asymptotic results

Recall that, when the model is correctly specified, [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF] have established the consistency and the asymptotic normality of θ(m * ). The following proposition shows that the estimator θ(m * ) obeys the law of iterated logarithm.

Proposition 3.1 Let (X 1 , . . . , X n ) be a trajectory of a process X belonging to AC(M θ * , f θ * ) where θ * ∈ Θ(r) ∩ o Θ ⊂ R d with r > 4. Assume that D(Θ), Id(Θ), V ar(Θ), K 2 (Θ) hold. Then, θ(m * ) -θ * = O … log log n n a.s. . (8)
The following theorem provides sufficient conditions that ensure the strong consistency of the model selection procedure. -→

n→∞ m * . ( 9
)
Remark 3.1 1. This result, in addition to being stronger than those obtained by [START_REF] Bardet | Consistent model selection criteria and goodness-of-fit test for common time series models[END_REF], does not impose any condition on the dependence between the regularization parameter κ n and the Lipschitz-

type coefficients α j (∂ θ i f θ , Θ), α j (∂ θ i M θ , Θ), α j (∂ θ i H θ , Θ).
For instance, in the Riemanian case with

i=0 α j (∂ θ i f θ , Θ) + α j (∂ θ i M θ , Θ) + α j (∂ θ i H θ , Θ) = O(j -γ
) for some 3/2 < γ < 2, the BIC (κ n = log n) is strongly consistent as a consequence of this theorem, whereas the result of [START_REF] Bardet | Consistent model selection criteria and goodness-of-fit test for common time series models[END_REF] does not yield the weak consistency of the BIC. The next corollary show that the estimator of the parameter of the selected model θ( m) obeys the law of iterated logarithm.

Corollary 3.1 Let (X 1 , . . . , X n ) be a trajectory of a process X belonging to AC(M θ * , f θ * ). Under the assump- tions of Theorem 3.1, θ( m) -θ * = O … log log n n a.s. . (10)

Concluding remarks

This paper focuses on the model selection in a large class of causal time series models in a semiparametric framework. The strong consistency of an estimator based on a penalized quasi likelihood contrast is established, under some classical conditions on the regularization parameters κ n .

For the estimation of the order of an ARMAX model, Hannan and Deistler (2012) have established that, there exists a constant c 0 such that, if lim sup κ n /(2 log log n) < c 0 then the strong consistency of the estimator of the order fails. A topic of a future works could be to investigate if such result is applied to the general class of model considered here or to derive an upper bound of κ n for which the strong consistency fails.

Another extension of this works is to carry out the model selection problem in the class AC(M θ , f θ ) with a procedure based on a non Gaussian (for instance Laplacian, see [START_REF] Bardet | Asymptotic behavior of the laplacian quasimaximum likelihood estimator of affine causal processes[END_REF]) quasi likelihood, for the purpose of reducing the number of moments required.

Proofs of the main results

The following lemma will be useful in the sequel. The proof is carried out by going along similar lines as in Lemma 2 of [START_REF] Bardet | Consistent model selection criteria and goodness-of-fit test for common time series models[END_REF] by using Corollary 1 of [START_REF] Kounias | An inequality and almost sure convergence[END_REF]; so, it is then omitted.

Lemma 5.1 Let X ∈ AC(M θ , f θ ) and Θ ⊆ Θ(r) with r > 4. Assume that the assumptions D(Θ) and K 1 (Θ)

hold. Then, 1 √ n log log n ∂ L n (θ) ∂θ - ∂L n (θ) ∂θ Θ a.s. -→ n→∞ 0. ( 11
)
Proof of Proposition 3.1

According to [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF], it holds that θ(m * ) a.s.

-

∂θ i = ∂ L n (θ * ) ∂θ i + ∂ 2 L n θi (m * ) ∂θ∂θ i θ(m * ) -θ * ,
where θi (m * ) lies between θ(m * ) and θ * . Therefore,

… n log log n θ(m * ) -θ * = 2 √ n log log n F -1 n (m * ) ∂ L n (θ * ) ∂θ where F n (m * ) = -2 ∂ 2 L n θi (m * ) ∂θ∂θ i i∈m * . (12)
Note that, by dealing with the first (stationary) regime in the Corollary 6.1 of [START_REF] Bardet | Detecting multiple change-points in general causal time series using penalized quasi-likelihood[END_REF] and since θi (m * ) a.s.

-→

n→∞ θ * for i = 1, • • • , |m * |, we get F n (m * ) a.s. -→ n→∞ F (θ * , m * ) where F (θ * , m * ) = E ∂ 2 q 0 (θ * ) ∂θ i ∂θ j i,j∈m * . ( 13
)
Since F (θ * , m * ) is invertible (see [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]), then for n large enough and with a sufficiently large probability, the matrix F n (m * ) is invertible. We have from Lemme 5.1, ( 12) and ( 13),

… n log log n θ(m * ) -θ * = 2 √ n log log n F -1 n (m * ) ∂L n (θ * ) ∂θ + o(1) a.s. . (14)
We have,

∂L n (θ * ) ∂θ = n t=1 ∂q t (θ * ) ∂θ .
Denote for all t ∈ Z, F t = σ(X t , X t-1 , • • • ) the σ-field generated by the whole past at time t. Then,

∂qt(θ * )
∂θ , F t is a stationary ergodic square integrable martingale difference process (see [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF]). Therefore, from the law of iterative logarithm for martingales (see [START_REF] Stout | The Hartman-Wintner law of the iterated logarithm for martingales[END_REF][START_REF] Stout | Almost sure convergence[END_REF]), we get,

1 √ n log log n ∂L n (θ * ) ∂θ = O(1) a.s. . (15)
Thus, the proposition follows from ( 13), ( 14) and [START_REF] Resende | Model identification using the efficient determination criterion[END_REF].

Proof of Theorem 3.1

1. Let m ∈ M such as m m * . We have,

1 log log n C(m * ) -C(m) = 2 log log n L n θ(m) -L n θ(m * ) - κ n log log n (|m| -|m * |). (16)
Let us establish that

1 log log n L n θ(m) -L n θ(m * ) = O(1) a.s. . (17
L n θ(m) -L n θ * ) = 1 2 θ(m) -θ * ∂ 2 L n θ(m) ∂θ 2 θ(m) -θ * . (18)
Also, for any i = 1, • • • , |m|, we can find θi (m) between θ(m) and θ * such that, for n large enough,

0 = ∂ L n ( θ(m)) ∂θ i = ∂ L n (θ * ) ∂θ i + ∂ 2 L n θi (m) ∂θ∂θ i θ(m) -θ * .
Hence,

θ(m) -θ * = 2 n F -1 n (m) ∂ L n (θ * ) ∂θ where F n (m) = -2 ∂ 2 L n θi (m) ∂θ∂θ i i∈m . (19)
Since θ(m), θ(m), θi (m) a.s.

-→ n→∞ θ * for i = 1, • • • , |m|, in this case of overfitting, the same arguments as in the proof of Proposition 3.1 lead to

F n (m) a.s. -→ n→∞ F (θ * , m) and -2 n ∂ 2 L n θ(m) ∂θ 2 a.s. -→ n→∞ F (θ * , m) where F (θ * , m) = E ∂ 2 q 0 (θ * ) ∂θ i ∂θ j i,j∈m . (20)
For the overfitted model m, on can deduce from [START_REF] Bardet | Asymptotic normality of the quasi-maximum likelihood estimator for multidimensional causal processes[END_REF] that F (θ * , m) is invertible, thus for n large enough and with a sufficiently large probability, the matrix F n is invertible. From ( 18), ( 19), (20) and Lemma 5.1, it holds Proof of Corollary 3.1

According to the proof of Theorem 3.1 (equations ( 23) and ( 25)) it holds that m = m * a.s. for large n. Thus, the corollary follows from Proposition 3.1.

  The model selection problem for time series has already been considered by several authors; we refer to the book of McQuarrie and Tsai (1998), the monograph of Rao and Wu (2001), the recent review paper of Ding et al. (2018), the recent works of Hsu et al. (2019) and the references therein for an overview on this topic. Hannan (1980) and Hannan and Deistler (2012) provided general conditions for strong consistency of the order estimator of an ARMA and ARMAX model. Resende and Dorea (2016) proposed the efficient determination criterion (introduced by Zhao et al. (2001) for the strongly consistent estimation of the order of multiple Markov chains) for model selection in a general class of multivariate time series. They established the strong consistency of the procedure under some conditions which may seem a bit strong for some applications, for instance, the existence of the third order derivative of the contrast function (likelihood), the existence of moments of order 16 for the BEKK-GARCH model. Recently, Bardet et al. (2020) addressed the model selection question in the class of model AC(M θ , f θ ). They proposed a procedure based on the quasi likelihood of the model and provided sufficient conditions that ensure the weak consistency of the selected model. In this new contribution, we focus on the model selection in the class of model AC(M θ , f θ ) with a penalized contrast which is based on the Gaussian quasi likelihood of the model.

Theorem 3 . 1

 31 Let (X 1 , . . . , X n ) be a trajectory of a process X belonging to AC(M θ * , f θ * )). Under the assumptions of Proposition 3.1, and if κ n / log log n -→ n→∞ ∞ and κ n /n -→ n→∞ 0, then m a.s.

2 .

 2 [START_REF] Hannan | The statistical theory of linear systems[END_REF] have considered the estimation of the order of an ARMAX (including ARMA), where the contrast C is based on the Gaussian likelihood of the model. Under the condition κ n /n -→ n→∞ 0, they have established that there exists a constant c 1 > 0 such that, if lim inf κ n /(2 log log n) > c 1 , then the estimator of the order is strongly consistent. From the proof of Theorem 3.1, one can see that such result holds for the general class of model considered here; that is, we can find a constant c 2 > 0 such that if lim inf κ n / log log n > c 2 , then m a.s.-→ n→∞ m * .

  → n→∞ θ * . Also, since θ * ∈ Θ(m * ) ∩ o Θ, we get ∂ L n ( θ(m * )) ∂θ = 0 for n large enough. Thus, for any i = 1, • • • , |m * |, the Taylor expansion of ∂ L n ∂θ i implies 0 = ∂ L n ( θ(m * ))

)

  Since θ * ∈ Θ(m) ∩ o Θ and θ(m) a.s. -→ n→∞ θ * , then ∂ L n ( θ(m)) ∂θ = 0 for n large enough. Therefore, from the Taylor expansion of L n , we can find θ(m) between θ(m) and θ * such that

) 2 .

 2 that 1 log log n L n θ(m) -L n θ(m * ) to (21) and (15), (17) follows. Therefore, since κ n / log log n -→ n→∞ ∞ and |m| > |m * |, then (16) and (17) lead to lim n→∞ 1 log log n C(m * ) -C(m) = -∞ a.s. .(22)This implies,C(m) -C(m * ) > 0 a.s. for large n. (23Let m ∈ M such as m m * . We have, 1 n C(m * ) -C(m) = 2 n L n θ(m) -L n θ(m * ) -κ n n (|m| -|m * |).(24)For all θ ∈ Θ, denoteL(θ) = -1 2 E[q 0 (θ)].According to the proof of Theorem 3.1 of [2], we get 1 n L n θ(m) -L n θ(m * ) = L(θ * (m)) -L(θ * ) + o(1) a.s. . Note that, from [5], the function L : Θ → R has a unique maximum at θ * . Since m m * , it holds that θ * / ∈ Θ(m). Hence, L(θ * (m)) -L(θ * ) < 0 a.s. . Thus, according to (24) and since κ n /n -→ n→∞ 0, we get lim n→∞ 1 n C(m * ) -C(m) < 0 a.s. and C(m) -C(m * ) > 0 a.s. for large n. (25) Thus, the strong consistency of m = argmin m∈M C(m) = argmin m∈M C(m) -C(m * ) follows from (23) and (25).

Acknowledgements

The author is grateful to the referee for many relevant suggestions and comments that helped to improve the contents of the paper.

 [START_REF] Bardet | Asymptotic behavior of the laplacian quasimaximum likelihood estimator of affine causal processes[END_REF]Developed within the ANR BREAKRISK (ANR-17-CE26-0001-01) 2 This work was funded by CY Initiative of Excellence (grant "Investissements d'Avenir" ANR-16-IDEX-0008), Project "EcoDep" PSI-AAP2020-0000000013 1