Nicolas Samuelian Inrap

D C

Fanny Bocquentin C

U 

La mission Beisamoun : sur les traces des premiers

écouverte et prote ion du site de Beisamoun. Le site de Beisamoun est situé en Haute-Galilée (Israël), dans la partie la plus septentrionale de la vallée du Jourdain. Il faisait face à la zone marécageuse d'un grand lac, le Houleh, dont la rive la plus proche était située à - km plus au sud. L'occupation principale est datée du Néolithique précéramique B, à la fi n du  e millénaire avant notre ère. C'est à la faveur du creusement de grands bassins piscicoles que le site de Beisamoun a été découvert, dès , grâce aux pro e ions de surface menées par Amnon Assaf, conservateur du musée de Préhist oire du Houleh (Perrot, ). En ,  et , au gré des opérations de vidange et de nettoyage des bassins qui se succèdent, une équipe, dirigée par J. Perrot (C/C¹) puis M. Lechevallier (C/C), procède au relevé des st ru ures qui émergent au fond des bassins. Associé à la colle e de matériel en surface, ce travail de repérage syst ématique des st ru ures permet d'évaluer l'étendue du site néolithique à plus de  ha. Les bassins étant creusés en cuvette, c'est toujours sur leurs marges que les vest iges archéologiques en place ont été trouvés, ceux situés plus au centre ayant été généralement arasés. En , une courte opération de sauvetage dirigée par M. Lechevallier met au jour les vest iges d'une maison re angulaire à sol enduit (Lechevallier, ). Une surface d'environ  m² est exposée et fouillée sur  cm de profondeur où trois niveaux d'occupation sont identifi és. Ce se eur du site a livré des informations de grand intérêt à travers, notamment, des st ru ures variées et bien conservées, un agencement de l'e ace très st ru uré, un outillage exceptionnel, de nombreuses inhumations et un dépôt de deux crânes surmodelés. Ces découvertes, associées à l'importante superfi cie du village, ont fait de Beisamoun un site majeur de la fi n du Néolithique précéramique.

Par la suite, les a ivités agricole et piscicole ont continué à endommager certaines parties du site jusqu'à ce que l'Offi ce des antiquités israéliennes le protège en . En  et , notre équipe a mené deux courtes campagnes exploratoires, de  jours chacune, qui ont permis de localiser un vast e se eur bien préservé du site (Bocquentin et al., ) et de poser les jalons d'une recherche programmée pluridisciplinaire bénéfi ciant des savoir-faire écifi ques des diff érents membres de l'équipe. En  et  nous avons mené des campagnes de trois semaines qui ont permis d'élargir l'emprise de fouille à  m². Une quinzaine d'étudiants bénévoles participent

[Fig. 1] a. Sol enduit de chaux (environ  m²). De la base au sommet on dist ingue plusieurs couches : un radier fait de pierres de petit calibre (- cm) densément agencées, un niveau de gravier, et le niveau d'enduit à proprement parler (amalgame de matériaux d'une très grande dureté et qui peut atteindre  cm d'épaisseur). Une fosse contenant seulement une mandibule humaine vient percer le sol chaulé.

b et c. Deux calages de poteaux sont aménagés à chaque extrémité de la surface enduite préservée : il s'agit de petites st ru ures circulaires en creux pouvant maintenir un poteau d'une quinzaine de centimètres de diamètre. Les pierres de pourtour sont posées de chant ; à l'est , leur paroi est renforcée par des blocs de basalte. au travail de terrain chaque année. L'équipe scientifi que est composée désormais d'une vingtaine de écialist es français, israéliens et américains.

Le cadre chrono-culturel. Le Néolithique du Levant comporte deux étapes majeures : une première, qui succède immédiatement au Natoufi en (fi n de l'Épipaléolithique), et voit le développement d'une économie de produ ion ; et une deuxième, cara érisée par l'apparition de la céramique. Le Néolithique précéramique du Levant-sud a été divisé en trois phases à partir de la st ratigraphie des sites de Jéricho (Kenyon, ) et d ' Aïn Ghazal (Rollefson, ). On désigne ces trois phases par les abréviations PPNA, PPNB et PPNC (Pre Pottery Neolithic A, B et C).

Daté de  à  avant notre ère² , le PPNB, auquel appartient Beisamoun, représente un tournant fondamental de l'humanité, durant lequel les étapes majeures de la néolithisation, encore à leur balbutiement au Natoufi en et au PPNA, acquièrent toute leur maturité. Le passage des sociétés ancest rales de chasseurs-cueilleurs aux sociétés agro-past orales se fi nalise au cours de ces deux millénaires qui précèdent le Néolithique céramique. Ainsi, le mode de vie sédentaire s'intensifi e et l'organisation des groupes se modifi e. La culture des plantes se diff use et se diversifi e (seigle, blé, froment, orge nue, lentilles, lin, pois chiches, fèves), la domest ication des animaux débute avec la chèvre et connaît un développement rapide (mouton, porc, boeuf ). D'importantes innovations techniques sont faites, notamment dans le domaine archite ural où apparaissent les premières maisons re angulaires, les sols enduits et des édifi ces à vocation colle ive et écialisée. L'armement comporte désormais de grandes pointes de jet sur lames cara érist iques (Big Arrowheads Indust ries). Les pratiques funéraires, toujours très diversifi ées, se particularisent cependant par l'introdu ion d'un traitement complexe de certains crânes qui sont surmodelés, enduits et peints. Les grands sites de cette période sont situés dans le couloir levantin. Des sites plus modest es sont inst allés à la périphérie.

Les fouilles eff e uées à partir des années  en Jordanie ont enrichi considérablement nos connaissances sur la fi n du PPNB grâce à la découverte de « mégasites » (surface supérieure à  ha) (Rollefson, ). Ces premiers villages remplacent les hameaux des périodes précédentes. Leur population est est imée entre   et   habitants (Bar-Yosef, Meadow,  ; Kuijt, ). L'organisation des maisons, serrées les unes contre les autres et pouvant, pour la première fois, présenter un étage, suggère une occupation dense. Les cara érist iques de ces occupations (technologie lithique, archite ure, pratiques funéraires) sont proches de celles des hameaux du PPNB moyen. C'est un argument en faveur d'une continuité culturelle. Cependant, l'abandon, au même moment, de la plupart des sites situés à l'ouest de la vallée du Jourdain laisse supposer, au contraire, une rupture entre ces deux phases culturelles.

Le développement des mégasites sur les plateaux transjordaniens demeure énigmatique. Certains chercheurs est iment qu'ils témoignent de migrations de populations originaires de régions périphériques à la fi n du PPNB moyen, à la recherche d'un meilleur environnement (par exemple Simmons, ). Il s'agirait donc d'une agrégation des populations locales. On a également proposé l'hypothèse de l'arrivée de nouvelles populations et/ou d'une croissance démographique e aculaire des populations locales (Kuijt, ). Les causes de ces éventuelles migrations et/ou accroissement des populations sont en débat.

Le déclin rapide de ces grands sites n'est pas mieux compris. On a longtemps pensé que la plupart des sites du PPNB récent avaient été abandonnés à la fi n du  e millénaire (Perrot,  ; Bar-Yosef, Belfer-Cohen, ). En fait, les derniers travaux eff e ués en Jordanie indiquent que les mégasites décroissent considérablement, mais ne sont pas totalement abandonnés jusqu'à la fi n du Néolithique céramique (Simmons, , ). Une telle continuité de l'occupation avait été soulignée il y a déjà longtemps pour Beisamoun (Lechevallier, ). En outre, de nouveaux hameaux d'une superfi cie inférieure à , ha sont fondés. Pour expliquer le déclin brusque des mégasites, on a proposé des interprétations environnementales comme celle d'une aridifi cation du climat (Bar-Yosef,  ; Goring-Morris, Belfer-Cohen, ) ou bien celle d'une surexploitation des ressources alentours obligeant les populations à se déplacer (Rollefson,  ; Simmons,  ; Bar-Yosef, Bar-Yosef-Mayer, ). On a également évoqué un possible échec d'organisation politique et sociale pour faire face à des communautés de plus en plus grandes ainsi qu'une éventuelle détérioration des conditions d'hygiène et de santé liée à l'agrégation des populations (Kuijt, ). Ce n'est que trois millénaires plus tard, au début de l'âge du Bronze, que des villages aussi étendus seront à nouveau érigés au Proche-Orient.

La période charnière entre la fi n du PPNB et le début du PN est encore très mal cara érisée. Alors que, dans le Croissant fertile, la céramique fait son apparition dès la fi n du PPNB récent, il exist e au Levant-sud une phase de transition entre  et  avant notre ère dénommée PPNC (Rollefson, Kohler-Rollefson, ), ou PPNB fi nal (Goring-Morris, Belfer-Cohen, ) ou encore Ghazalien (Khalaily, ). Beaucoup de sites de la phase précédente sont abandonnés et de plus petites occupations s'établissent à proximité. Du point de vue de la technologie lithique, cette phase culturelle est en rupture par rapport au PPNB. Elle est très proche, en revanche, du Néolithique céramique du Levant-sud (Yarmoukien), bien qu'aucune céramique n'y soit associée.

Les enjeux scientifi ques. Les travaux menés dans les années  à Beisamoun ont révélé l'important potentiel de ce site qui est considéré comme majeur pour la fi n du PPNB. Toutefois, la fouille a concerné une très petite surface et, de surcroît, une st ru ure particulière, sans doute peu représentative du rest e du village. L'exploration de Beisamoun grâce à des opérations programmées off re l'opportunité de développer un programme pluridisciplinaire bénéfi ciant de toutes les avancées scientifi ques de ces dernières années. Elle off re aussi la possibilité de participer à une étude diachronique sur l'environnement et l'occupation du bassin du Houleh, passage privilégié entre le sud et le nord du Levant. Toutes les tentatives de datation radiocarbone ont échoué pour le moment, mais on e ère que la comparaison des assemblages lithiques de Beisamoun aux autres implantations PPNB de la région pourra permettre un calage techno-culturel plus précis de la fi n de l'occupation du site.

Savoir-faire, culture et héritage. Dans ce contexte de la fi n du Néolithique précéramique, c'est également la quest ion de la transmission culturelle et du remplacement éventuel des populations qui est posée. Y a-t-il une continuité des groupes qui occupent le bassin du Houleh ? Notre mission vise à mieux défi nir les écifi cités culturelles de cette région et les échanges que ces groupes pouvaient entretenir avec leurs voisins. L'approche de terrain écifi que (par décapages successifs et unités st ratigraphiques), ainsi que l'étude du matériel, sont dédiées, depuis le début du projet, à la recherche de l'identité du groupe, en l'occurrence les derniers habitants de Beisamoun. L'interprétation des données in situ et l'étude techno-économique des objets permettent d'approcher le quotidien comme des éléments plus pérennes qui font la écifi cité d'une communauté.

Les vest iges du village néolithique : premières découvertes. La parcelle fouillée se situe dans les limites d'un ancien bassin piscicole⁴ laissé en friche sur une surface de   m². En  nous avons ouvert trois tranchées de  et  m de long. Elles ont montré que les couches archéologiques sont présentes aux endroits sondés sur une épaisseur minimale de , m. Deux d'entre elles ont livré plusieurs st ru ures (murs et sols enduits). À présent, nous explorons deux se eurs situés à une vingtaine de mètres l'un de l'autre. Tous deux livrent une st ratigraphie similaire où deux occupations principales se succèdent. L'occupation la plus récente est érodée plus ou moins profondément par les aménagements modernes. Elle livre une indust rie sur silex dont les techniques de débitage annoncent celles du Néolithique céramique ancien, mais aucun tesson ne lui est associé. La phase de const ru ion qui la précède livre un assemblage classique de la fi n du PPNB.

Une cinquantaine de locus ont été mis au jour dans les deux se eurs explorés. Il s'agit de st ru ures en creux (fosses dépotoirs, sépultures, calage de poteaux, bassins), de concentrations d'objets ou d'amoncellements de pierres calibrées, ou de segments de st ru ures de plus grande superfi cie, mais généralement incomplètes (murs, sols, radiers). Le soubassement des murs les plus massifs est composé de deux rangées de blocs parallèles entre lesquelles s'intercale un bourrage de cailloutis. D'autres murs ne sont matérialisés que par une seule rangée de blocs alignés. La hauteur conservée des diff érentes st ru ures ne dépasse pas deux assises et il faut imaginer qu'une paroi en pisé ou de briques crues venait compléter l'élévation. Des résidus de briques, parfois chauff ées, sont d'ailleurs régulièrement repérés sur le terrain. Les sols d'occupation découverts jusqu'à présent témoignent d'une grande diversité d'aménagement. Deux d'entre eux sont enduits de chaux, un procédé typique du PPNB mais pour lequel chaque site néolithique présente des techniques de fabrication écifi ques ; celles de Beisamoun sont en cours d'étude [Fig. 1]. Nous avons également mis au jour un radier de cailloutis ainsi qu'un niveau de galets sur lesquels le matériel qui reposait à plat suggère l'exist ence initiale de sols plus simplement aménagés, peutêtre en terre battue. Enfi n, nous avons découvert un sol d'occupation fait de gravillons, d'esquilles de silex, de granules d'argile et de coquillages compa és ensemble [Fig. 2] dont il exist e peu d'exemples comparables à notre connaissance. Les st ru ures annexes associées plus ou moins étroitement aux sols et aux segments de murs dégagés sont généralement des calages de poteaux [Fig. 1] ou des fosses. Certaines d'entre elles sont trouvées vides, d'autres sont remplies de pierres de moyen calibre (environ  cm), considérées parfois comme des fours à cuisson indire e (Cauvin, ). D'autres encore livrent un abondant matériel hétéroclite (faune, fragment d'outils et ust ensiles en basalte, outils en os, concentration de lamelles de silex, de petites perles en os et coquilles, blocs d'argile, et résidus de céréales) qui suggère une fon ion de dépotoir. On a pu const ater à plusieurs reprises une continuité des techniques archite urales au fi l du temps à Beisamoun. La const ru ion des murs, les sols enduits ou l'aménagement de bassins et d'aires empierrées montrent des similitudes avec les st ru ures du P moyen qui ont fait l'objet du sauvetage de . Cela concerne aussi les murs dont l'orientation se fait préférentiellement selon les points cardinaux et parfois l'aménagement interne des e aces fermés [Fig.2].

Six sépultures ont été découvertes jusqu'à présent qui viennent enrichir un corpus très rest reint dans tout le Levant-sud pour cette période. Une seule d'entre elles, celle d'un enfant décédé durant la période périnatale, appartient à la phase d'occupation la plus ancienne. Les autres témoignent d'une a ivité sépulcrale probablement post érieure à l'abandon du se eur en cours de fouille. On serait donc en présence d'une aire sépulcrale corre ondant à la fi n de l'occupation du village. Il ne rest e parfois que les fonds de fosses, le rest e ayant été érodé. La di osition de l'une de ces sépultures à l'intérieur du bourrage de pierres d'un mur appartenant à la phase d'occupation précédente témoigne d'une certaine volonté d'associer durablement les défunts aux ruines d'une phase plus ancienne de l'hist oire du village, nécessairement encore apparente [Fig. 3]. Le traitement funéraire apparaît varié : des inhumations, primaires et secondaires, peut-être une crémation, à confi rmer dans la mesure où il s'agit d'un cas unique pour la période. Un dépôt en fosse d'une calotte crânienne isolée a également été eff e ué dans la même zone. Les inhumations primaires témoignent de dépôts en pleine terre, le cadavre repose sur le dos, les membres inférieurs ramenés en fl exion sur le thorax [Fig. 3].

Le matériel exhumé est abondant et relativement bien conservé. La faune et l'indust rie en silex dominent. Contrairement aux autres sites du Néolithique précéramique où les e èces chassées sont remplacées au cours du temps par une consommation croissante des caprinés domest iques, le gibier favori des habitants de Beisamoun rest e le porc sauvage (d'après les travaux de L.K. Horwitz). Les objets façonnés sur os (outils et parure) sont de fa ure simple, issue de traditions régionales anciennes (d'après les travaux de G. Le Dosseur). L'indust rie en silex local est dominée par les pointes de proje iles, les lames de faucilles, les haches et les perforateurs. L'indust rie en basalte et en calcaire, relativement diversifi ée, témoigne également d'un approvisionnement local, alors que la présence de quelques pièces sur obsidienne témoigne d'échanges à longues dist ances, les sources les plus proches étant situées en Anatolie. Si la plupart des coquilles trouvées sur le site proviennent du lac Houleh ou de ses environs, quelques coquilles ont été ramenées de la Méditerranée ; elles présentent, en général, des modifi cations (perforation, incisions) qui montrent qu'elles étaient portées en parure.

La recherche intensive de rest es botaniques que nous avons menée dans les couches en cours de fouille n'a pas été concluante pour le moment, dans la mesure où graines et pollens sont mal conservés et que les bioturbations génèrent une pollution moderne. Toutefois, les phytolithes apportent des informations sur la composition végétale de certains matériaux de const ru ion ainsi que sur des phénomènes de concentrations atiales qui attest ent, par exemple, du st ockage de céréales (d'après les travaux de A. Emery-Barbier).

La collaboration C-I-Inrap. La convention de collaboration scientifi que signée en  entre l'Inrap et la mission archéologique de Beisamoun C-I s'inscrit dans le cadre d'une coopération franco-israélienne déjà très ancienne. Après les premières recherches menées par l'École biblique de Jérusalem dès la fi n du  e siècle, R. Neuville et J. Perrot marquent durablement les recherches préhist oriques en Israël. La création du Centre de recherche français à Jérusalem (C/ M ), il y a près de  ans, a permis de développer et de pérenniser cette collaboration fru ueuse.

Les méthodes de fouilles employées à Beisamoun sont in irées de l'ethnologie préhist orique préconisée par A. Leroi-Gourhan. Cette approche des couches archéologiques par décapages successifs en suivant les diff érents niveaux d'occupation, privilégie l'organisation atiale des vest iges afi n de faire une le ure ethnologique des rest es archéologiques qui nous sont parvenus. Il s'agit de faire ressortir les st ru ures latentes qui sont souvent négligées. La diversité des st ru ures découvertes à Beisamoun encourage à adopter ce type de méthode de fouille fi ne, qui seul permet la reconnaissance de st ru ures inconnues par ailleurs. Toutefois, la superfi cie des sites néolithiques du Levant, souvent fouillés en sauvetage, se prête mal à ce type d'approche onéreuse en temps. La collaboration C-I-Inrap sur le site de Beisamoun travaille à un ajust ement de méthodes fi nes issues de l'archéologie préhist orique à un site de grande superfi cie grâce à l'expérience acquise en France et en Israël dans le cadre d'opérations préventives.

[Fig. 2 ]

 2 Une const ru ion de tradition locale. Cette st ru ure quadrangulaire en cours de fouille a été dégagée pour le moment sur une largeur de  m et une longueur de  m. Il s'agit vraisemblablement d'une habitation, orientée selon un axe NE-SO. Sa partition interne divise l'e ace de façon asymétrique : un tiers à l'ouest , deux tiers à l'est , avec un passage entre ces deux aires situé au nord. Le sol de cette const ru ion est composé d'un mélange de gravillons, d'esquilles de silex, de granules d'argile et de coquillages compa és. Le plan de ce locus n'est pas sans rappeler la maison fouillée en  (Lechevallier,  : Fig.) orientée et subdivisée de la même manière (dessin dans le cadre). Il s'agit d'un témoignage particulièrement frappant d'une tradition archite urale forte qui s'est perpétuée du PPNB moyen au PPNB récent-fi nal à Beisamoun. [Fig.3] Le locus  est la sépulture d'un individu adulte dont la fosse a été creusée dans le bourrage du mur . Au nord, la sépulture est coupée par la fosse  à mi-hauteur de la cage thoracique. Le cadavre reposait sur le dos, la tête au N-NE, les membres inférieurs ramenés en fl exion forcée sur son côté gauche. Ses genoux et ses pieds reposaient dire ement sur les blocs du mur . Le coude gauche avait été di osé sous les genoux, l'avant-bras ramené sur l'abdomen. Du membre supérieur droit ne subsist ait que les os de la main posée à hauteur du coeur. Le cadavre s'est décomposé en pleine terre. La sépulture a connu une légère érosion de sa partie supérieure. Le défunt est associé aux ruines d'une phase plus ancienne de l'hist oire du village, nécessairement encore partiellement apparente au moment de l'inhumation.

  Une situation géographique clé. Situé au nord de la vallée du Jourdain, Beisamoun fait le lien entre les mégasites de la Jordanie et les sites contemporains de petite dimension en Damascène(Ramad, Ghoraifé, Aswad). La multiplication des fouilles en Jordanie ces dernières décennies avec la découverte de plusieurs mégasites, ainsi que la reprise des fouilles à Tell Aswad (Stordeur, ) en Damascène a totalement changé notre perception de la région au PPNB récent. À une plus grande échelle, Beisamoun apparaît comme un site incontournable pour appréhender les relations et les échanges entre le nord et le sud du Levant. Il a déjà montré des affi nités et des échanges de part et d'autre (Le Brun, in Lechevallier,  ; Kozlowski, Aurenche, ). Il off re la possibilité de discuter des phénomènes d'introdu ion ou de domest ication autochtone des e èces animales ainsi que de la circulation et des échanges de techniques et de matières premières. Bottema, ,  ; Meadows, ).Cette comparaison des séquences permettra d'appréhender le contexte environnemental local et la nature (anthropique ou non) de son éventuelle détérioration à la fi n du PPNB.Une occupation de longue durée qui couvre une période mal défi nie.Si Beisamoun est apparu depuis longtemps comme un site clé du PPNB récent, la reprise des fouilles par notre équipe a permis de mettre au jour des niveaux un peu plus récents, de la toute fi n de cette période ou bien du début du PN (Néolithique céramique) ainsi qu'une épaisse st ratigraphie préservée dans le se eur exploré, qui confère au site un st atut d'autant plus exceptionnel. La toute fi n du PPNB est en eff et très mal connue et particulièrement dans cette région du bassin du Houleh. Cette phase tardive d'occupation était su e ée d'exist er, auparavant, à Beisamoun, grâce au ramassage de surface mené syst ématiquement dans les années  et , mais aucun niveau d'occupation en place n'avait été mis au jour. En parallèle, la reprise de l'étude du matériel et des archives de fouille de ³ a permis d'attribuer les st ru ures fouillées lors du sauvetage de  au PPNB moyen (Bocquentin et al., sous presse ; Bocquentin, ). Ceci signifi e que, loin d'être brève, l'implantation humaine à Beisamoun a perduré probablement en continu du milieu du  e millénaire au début du  e millénaire.

	(Baruch,
	Environnement et subsist ance à la fi n
	du Néolithique précéramique.
	Le bassin du Houleh, qui présente un climat
	de type méditerranéen, et plus particulièrement
	sa partie centrale irriguée en permanence
	par plusieurs sources importantes, a été propice
	à des inst allations préhist oriques de longue durée
	comme en attest e l'important site natoufi en
	de Mallaha dès  avant notre ère (Valla et al.,
	). Jusqu'au début des années , date qui
	marque le début de son drainage afi n d'étendre
	les terrains cultivables, le lac était peuplé de la plus
	grande diversité d'e èces aquatiques du Levant
	et était une importante halte pour les oiseaux
	migrateurs (Dimentman et al., ).
	L'emplacement de Beisamoun au coeur de cet
	écosyst ème particulièrement riche et la proximité
	immédiate du lac Houleh confèrent à l'étude des
	st ratégies de subsist ance au PPNB une dimension
	nouvelle. Cette vallée représente un cadre
	particulièrement favorable pour l'étude des
	relations de l'homme avec son environnement
	dans la durée. Les produits issus de la chasse
	et de la cueillette représenteraient encore
	une part importante de l'alimentation à Beisamoun.
	La pêche a pu ralentir l'élevage dans cette région
	privilégiée. D'autre part, les données propres
	à Beisamoun devraient permettre d'ajust er
	chronologiquement la séquence culturelle
	du PPNB aux données climatiques issues des
	séquences palynologiques du lac Houleh

 Toutes les datations qui sont données ici sont calibrées ; elles sont issues de Aurenche et al.,  et Kuijt, Goring-Morris, . Les datations absolues peuvent faire l'objet de certains écarts d'un auteur à l'autre, notamment pour le début du PPNB.


What Happened in the Final Natufian? The end of the Natufian culture and the rise of the PPNA horizon are still obscure. This is evidenced by the discussions prompted by the terms "Final Natufian" and "Khiamian" ( Bar-Yosef 1988[START_REF] Garfinkel | The Sultanian flint assemblage from Gesher and its implications for recognizing early Neolithic entities in the Levant[END_REF][START_REF] Nadel | The Khiamian as a case of Sultanian intersite variability[END_REF]). This situation is not new. Its origin can be traced in the pioneering works of René Neuville and Dorothy Garrod. The debate stems from the problematic stratigraphy at El Khiam in the Judean Desert where [START_REF] Neuville | Le préhistorique de Palestine[END_REF] claimed he could identify four phases (I to IV according to his terminology) in the Natufian culture. In his view, his phases I and II were equivalent to the Early and Late Natufian as recognized by [START_REF] Garrod | A New Mesolithic Industry: The Natufian of Palestine[END_REF] during her excavations at Shukba and El Wad. They were followed by the otherwise unknown phases III and IV. To Neuville's mind these two new episodes had the advantage to bridge the gap between what he considered to be a "Mesolithic" phenomenon -the Natufian -and the following "Eneolithic", when metal and pottery were in use. The confusion was made possible by the compacted chronology then accepted, according to which the Natufian culture could not possibly be dated earlier than the fifth millenium B.C. (Garrod ibidem). It was also fed by the assumption that there was no true Neolithic in the Levant. This perception, based on a narrow definition of the Neolithic as an "Age of the polished stone", prevented Neuville from correctly understanding the Tahunian industry described by [START_REF] Buzy | Une industrie mésolithique en Palestine[END_REF].

If this record is correct, three main difficulties contributed to confusing the situation. First: a basic conceptual misunderstanding about what the sequence should be, second: the lack of any good stratified site in the field, third: the poor chronological control on the available finds. These problems were partly cleared up by subsequent work. Neuville's theoretical framework became obsolete in the Near East due to the impact of the thinking of Childe and Braidwood who centered the identification of the "Neolithic" on economic aspects of human life. New observations at El Khiam by Perrot ( in Neuville 1952) and [START_REF] Gonzalez-Echegaray | Excavaciones en la Terraza de «El Khiam[END_REF], as well as a revision of the finds by [START_REF] Bar-Yosef | The Epi-Paleolithic Cultures of Palestine[END_REF], led to the conclusion that the sharply sloping deposits at El Khiam were probably not in-situ and should not be used as a type for a stratigraphic sequence. Meanwhile, works by Garstang, Kenyon and others accumulated new data. Finally, the chronological construct proved untenable when C 14 dating was introduced. After a period when confusion prevailed and any subdivision in the Natufian culture was rejected together with the term "Khiamian" (suggested by Echegaray for assemblages following the Natufian culture and including early arrowheads) a new terminology emerged: the term Khiamian was revived by J. [START_REF] Crowfoot-Payne | The terminology of the Aceramic Neolithic Period in the Levant[END_REF] and it was proposed to return to a partition of the Natufian culture in two phases, Early and Late, similar to those of Garrod with some modifications [START_REF] Bar-Yosef | L'évolution du Natoufien: nouvelles suggestions[END_REF].

As research continued, further observations at Eynan (Ain Mallaha) and analyses of lithic assemblages from Nahal Oren (Higgs and Noy excavations) and Fazael IV prompted one of us to identify a third phase in the Natufian. This stage -the Final Natufian --supposedly choronologically followed the Early and Late phases now currently accepted (Valla 1981a[START_REF] Valla | Les industries de silex de Mallaha et du Natoufien dans le Levant[END_REF][START_REF] Valla | Chronologie absolue et chronologies relatives dans le Natoufien[END_REF][START_REF] Valla | The first settled societies[END_REF]. The basis for this suggestion was the fact that, at Eynan, a stratigraphically clearly separate layer (labelled Ib) was comprised of a flint industry distinctive from Early and Late Natufian assemblages as identified at the site itself and elsewhere. Moreover, the very small lunates typical for this microlithic industry were not an isolated phenomenon. They could be observed in the aforementioned sites as well as at Mureybet IA (Cauvin 1977) and in the Harifian toolkit in the Negev [START_REF] Marks | Abu Salem: the Type site of the Harifian Industry of the Southern Levant[END_REF][START_REF] Scott | The Harifian of the Central Negev[END_REF], Goring-Morris 1987, 1991, Phillips 1977). Stratigraphic and chronological control, when possible, confirmed the terminal position in the Natufian sequence of the layers involved.

Nevertheless the Final Natufian suggestion was not received positively by many researchers. At least five important objections could be raised against it. First: there was no proof for Eynan layer Ib to be very late in the Natufian sequence. Helwan retouch on flint tools, a feature commonly associated with the Early phase of the Natufian, could still be found in that layer. Arguably, this could indicate that the layer was not so "final". Second: C 14 control on the chronological position of the Final Natufian was very poor. Not taking into account the well dated Harifian in the Negev, only Mureybet had a set of more or less acceptable dates. Third: the small size of the lunates, which was a shared feature of the proposed Final Natufian assemblages, was perhaps not significant. It was easy to show that there was no uniform decrease in the length of lunates during the Natufian era in the large area where this culture could be traced [START_REF] Henry | An analysis of settlement patterns and adaptive strategies of the Natufian[END_REF]. Fourth: no in-situ deposit could be assigned to the Final Natufian besides Mureybet IA. At Eynan, layer Ib was supposed to result from a "flash flood" (Perrot 1966: 439). Also, the analyzed sample from Nahal Oren was taken from deposits not less sloping than those at El Khiam and could not be confidently relied upon. Fifth: due to this situation, the Final Natufian, if accepted, stood as a cultural entity lacking any of the economic and other features prehistorians had become used to taking into account when analysing the rise of the Neolithic in the Near-East after the work of Childe and Braidwood. As such, it had but little value, if any, in picturing the Natufian phenomenon.

Some of these objections were clearly rooted in the experience gained in discussing Neuville's misconceptions. Nevertheless, as time passed, part of them tended to loose their virulence. For exemple, whenever a set of very small Natufian lunates was found, researchers classified them "at the end of the (Late Natufian) sequence" (Henry 1995: 326), implicitly acknowledging the chronological value of the length criteria (see also [START_REF] Grosman | A final Natufian site: Fazael IV[END_REF]. Moreover, laboratory excavation of a block of sediment cut at the base of the Natufian graveyard during the Stekelis and Yisraeli field work at Nahal Oren [START_REF] Stekelis | The excavations at Nahal Oren[END_REF] found a flint assemblage -admittedly small -, which fitted the decription of the Final Natufian [START_REF] Nadel | A Note on New Finds from the Natufian Graveyard at Nahal Oren[END_REF]. This finding was potentially very significant since it pointed to a Final Natufian age for the nearby fireplace apparently earlier than the graves. If so, an in-situ Final Natufian deposit was present, along with a set of data with cultural information beyond flint. Today, there are hints of Final Natufian deposits at six sites spread over a large area -from the Middle Euphrates valley (Mureybet IA) to Southern Jordan (Wadi Humeima) -including the Lower Jordan valley (Fazael IV), the Carmel mountains (El Wad and Nahal Oren) and the Galilee (Eynan). Among these sites, two, Mureybet and Eynan, and possibly three if Nahal Oren is added, have in-situ deposits. The aims of this paper is to report briefly about the architecture and burial customs recently uncovered at Eynan, which add to the Final Natufian the missing data, allowing comparisons with the former Early and Late Natufian episodes. We will then examine their originality compared to their earlier counterparts in order to validate, or not, the Final Natufian subdivision. A preliminary discussion will follow to introduce the data in the broader framework of changing Natufian behaviors.

FINAL NATUFIAN STRUCTURES AND GRAVES AT EYNAN

The discovery of architectural features in the layer associated with a Final Natufian industry is the most significant finding of the excavations renewed at Eynan since 1996. This finding allows us to study the transformations occurring in the architectural tradition during the Final Natufian in the "core area" of the Natufian culture. It also provides us with a new basis for analysis of possible sedentism during that period at the site. Moreover, the association of graves with some of the uncovered buildings provides us with an opportunity to observe the development of the relationship between buildings and graves, already known from the Early and Late phases.

In the newly excavated area, the Final Natufian layer, labelled Ib, is approximately 50 cm thick. This layer is densely packed with small stones, most of them angular limestone fragments, embedded in a matrix of red-brown "silt" that still awaits thorough geological analysis. Preliminarily, due to the archaeological remains it contains, this layer can be subdivided into two sub-layers: Ib2 (lower) and Ib1 (upper). As a further refinement, it is possible to suggest some subdivisions into 1b1 since this unit includes all the structures and graves associated with the surface of the layer. From a stratigraphic point of view, this means two different things. First, that the structures and associated graves were dug from the top of layer Ib during the unknown duration of its exposure. Except for cases of superposition, it seems to us impossible to establish detailed stratigraphic correlations (or subdivisions) between these structures, which are archaeologically synchroneous. Secondly, the graves were apparently dug from a subsequent deposit now supposedly eroded. For the purpose of this presentation we will tentatively (and we insist on this "tentatively") try to separate these later graves probably posterior to the structures, from the buildings and associated burials (figure 1) Two C 14 dates were obtained for the Ib2 building. These dates are 10530 ±100 bp (GifA. 99332) and 10540 ± 90 bp (GifA. 100400) (Valladas et Tisnérat-Laborde in [START_REF] Valla | Les fouilles à Mallaha en 2000 et 2001: 3ème rapport préliminaire[END_REF]. They fit well with the age assigned to the Final Natufian on the basis of comparisons with the flint assemblages and length of lunates from Mureybet IA and Harifian sites in the Negev altogether, five main buildings were excavated: one from Ib2 and four from Ib1. They share the same architectural characteristics. They are set in shallow pits lined on part of their circumference with blocks of limestone. These curvilinear walls never stand higher than the surface of the layer around. They are made of one row of unshaped blocks, end to end occasionally on only one course.When superstructures are probable, which is not always the case, they may have been made from light materials, possibly reeds from the nearby waterbodies (Eynan spring, the rivulet between the spring and lake Hula, the lake shore).

These buildings can be subdivided into two main groups. One of these groups is comprised of buildings we interpret as "houses". The other group appears to be less homogeneous: it includes a variety of buildings, which share as a main common feature a lack of obvious housing facilities. In both groups, each building is associated with minor features, which may be found inside the buildings or outside and adjacent to them. These minor structures display a range of variation. Among those we are able to interpret are post holes (associated only with houses according to current evidence) and different kinds of hearth. But there is, as well, a number of minor structures that we don't know how to interpret. Most of them we call "basins" because they are hollow and made of stones, but they lack any uniformity, either in shape (some of them are open, others are closed) or in size. (1996)(1997)(1998)(1999)(2000)(2001). Structure 215 (Ib2) is older than structures 200,202,203,218,230 (Ib1). Graves H.150,H.151,H.154,H.155,as well as H.168 and H.170, belong to the surface of the layer; they were excavated later than the structures mentioned above Each of the buildings passed through a history that is recorded in a succession of floors or stages. These are evidenced by superimposed features, sometimes with interstratified graves. For example, the main house under excavation (labelled 200-208) exhibits at least three successive floors, each of which is associated with a reorganisation of the same hearth. In general, floors are not built with any coating of plaster or clay, but some of them have a kind of pavement in limited areas. They are separated with deposits only a few centimeters thick, which are never devoid of artefacts. This situation makes it difficult to expose their entire surface with absolute security but the successive arrangements of the small structures testify for the multiplicity of stages. The interstratified graves, in some cases, demonstrate that there were periods of abandonment between the reorganizations of the floors. Unfortunately, we have little control of the duration of these sequences of occupations and abandonments. From the little we know, it is apparent that some changes in the use of the buildings may have occured during their lifetime. In one case, if our inferences are correct, a building ( 203) first aimed at other purposes was then turned into a house. Arguably, this may indicate that the sequences involved are more of the amplitude of months or years than days or even weeks.

According to current understanding, two of the exposed buildings (200-208 and 203) were houses. One of them (200-208) seems to keep this function for the three levels of floor exposed up to now. The other was turned into a house after a stage when the space was crowded with hearths. It then worked as a house for two successive floors. The general organizations of the houses are variations on the same schematic outline. They are set in shallow pits dug from a sloping surface; oval in general shape and lined with a stone wall only upslope, at most on half of their circumference. The associated minor structures, mainly post holes and hearths, are set on two axis: one is a line approximately joining the extremities of the stone wall and the other is perpendicular to that line in its middle point. They have several superimposed floors which do not reach 10 m 2 in surface.

Besides the basic characteristics shared by all of the main constructions, the buildings which are not understood as houses do not follow a single pattern. As opposed to houses, the walls of most of those we excavated are not open downslope but have a different orientation. The limits of the floors where it is not made by the wall are not so clearcut as in the case of houses. Whenever they were observed, they seem to gradually vanish. These buildings seem to have a wider size range than the houses. Finally, the surface they include is often densely occupied by minor structures which allow no place for living. Their particular history seems to be not less complicated than that of the houses, with superimposed inner structures which sometimes suggest changes in function. But, again, we have no control on the length of time involved and we believe it should be premature to enter in discussions or interpretations of these buildings as long as we don't have a full record of the sequences of their successive floors.

The equivocacity of the data can best be illustrated with two examples. In building 215 (Ib2) the latest stages involve a large hearth on top of which two big standing blocks were then erected. Clearly, our understanding will be dependant on the suggested time beween these events. Another building (202, from Ib1) encompasses an elaborate hearth (an oven ?) associated with a thick layer of ash and burnt material. This hearth was later capped with a pile of cobbles in which an open "basin" was arranged. Then, a corpse (H.157) was deposited on the cobbles, between the "basin" and the outer wall, and covered with undifferentiated sediment and stones. Again, the meaning of this sequence would be very different if we understand it as a short succession of events aimed at the burial of the corpse, or if we see it as a series of non related actions ending with the opportunistic use of a pre-existing excavation to bury a dead body (the favored interpretation at the time of writing).

Final Natufian burial customs at Eynan were not totally unknown before our excavations, since some graves had been previously found on the surface of layer Ib [START_REF] Perrot | Les Hommes de Mallaha (Eynan), Israël[END_REF]). But for the first time graves are excavated into the thickness of the deposit and in association with the architectural remains. Not taking into account isolated human bones scattered in the layer, the remains of seventeen individuals were recovered in the Final Natufian deposit since 1996. Of these, only one can be tentatively associated with Ib2. It should be emphisized that this figure is not definitive since it includes neither a partially excavated grave under building 200 with at least five individuals, nor a probable burial recognized at the top of layer Ib.

Each sample taken separately is relatively small, so it is not feasible to draw general conclusions. One can only note that, according to available observations, graves are not evenly dispersed. In the thickness of the layer, they are clearly related to buildings -mainly to houses. On the surface of the layer they are found in two groups of six and two individuals respectively. For all the sequence, individual burials seem to be the rule but some exceptions were recorded. [START_REF] Valla | Natufian seasonality: a guess[END_REF][START_REF] Valla | Les fouilles à Mallaha en 2000 et 2001: 3ème rapport préliminaire[END_REF].

Graves related to buildings may often have been shallow pits, but the original depth is often difficult to ascertain because of later disturbances. At least one individual (H.156) was deposited on the floor of a house in a box of some kind without any digging. Relatively deep holes (few tens of centimeters) were observed at the surface of the layer. Primary burials are overwhelmingly predominant. One skull (H.169: probably not in a grave) has a problematic status (see below) and one newborn is still to be studied in detail (H.175). It should be mentionned as well that there are hints that the partially excavated grave under building 200 is still problematic. More analyses are needed in order to reach a conclusion about its status.

Bodies in the graves are most often (but not always) in an hypercontracted position and the space they occupy is minimal. They were sometimes bound with ties or inserted in bags. None of them wears any item of personal ornament. "Grave goods" are not totally excluded but they are very difficult to identify due to undiscriminate packing of objects in the fills. Disturbances of the bones are not exceptional. They sometimes result from human activities as can be observed in the buildings. There, bones were cut, removed or rearranged according to needs. It is worth noting that no manipulation of the skull is evidenced. On the other hand, removal of large long bones (femur, etc.) that made problems for the builders is well attested. In this context, the finding of a large part of an isolated skull of a child on what is perhaps a floor outside of the Ib2 building, is difficult to interpret. At the moment, a conservative hypothesis would prefer to see it as the result of a mere "accident" preferably to a further testimony of the "skull removal practice" [START_REF] Belfer-Cohen | The Natufian Graveyard in Hayonim Cave[END_REF].

The small sample at hand provides no evidence for the criteria of selection of the part of the population buried at the site. As far as we can determine, newborns, children, adolescents and adults are present. Young people are slightly under-represented, but this may be due to the small sample size. Males and females are in equal number among adults. Caries were noted on four out of six individuals, all with their set of teeth.

COMPARISONS WITH EARLY AND LATE NATUFIAN FINDS

A superficial comparison of these Final Natufian findings with the Early and Late Natufian architectures and graves as they are known from the site will suffice to emphasize some of the main general similarities and differences by which the Final Natufian appears well in the continuation of earlier traditions and yet displays its own originality. Final Natufian buildings are clearly rooted in the practice which produced the Early and Late Natufian architectures (figure 2). The basic concepts behind it (preparing a curvilinear excavation in the ground and lining the cut with a wall, at least on part of its circumference) are the same. Past analyses suggested that houses tend to become smaller from the Early to the Late Natufian. Final Natufian houses (or understood as such) seem to fall in the same size range, or are somewhat smaller, that their Late Natufian counterparts. Another difference is seen in a tendency for the pits to be more and more superficially dug. Some Early Natufian walls are more than 1 m high. In other words, houses seem to move out from the ground and become more protruding on the surface. But the time where there will be no preliminary digging has yet to come.

What is known from the way Early Natufian houses underwent successive flooring is strikingly similar to what is observed in our Final Natufian examples. In Early Natufian house 131-151, a series of unplastered floors associated with inner structures were found closely stratified on top of each other and are occasionally linked to re-building of the wall. No sterile layer can be isolated between the floors [START_REF] Valla | Aspect du sol de l'abri 131 de Mallaha (Eynan)[END_REF]. A circumstancial parallel of the planning of the house can hardly be developed because of later destructions but a few observations are appealing. We will never know whether house 131 was fully oval in shape as are houses 200-208 and 203 from the Final Natufian. It is worth noting, however, that the reconstruction suggested for house 131, with a semi-circular wall and a row of structures (post holes, hearths) on a line joining the two ends of the wall, finds close parallels in Final Figure 2: Schematic examples of selected houses from the Natufian at Eynan: 1, House 131 (Early Natufian); 2, House 29 (Late Natufian); 3, House 200 (Final Natufian). Note that the dotted lines do not have the same meaning in 1 and 3: in 1 the dots intend to suggest a probable reconstruction of a destroyed part of the wall; in 3 they suggest the full extension of the house where no wall was built. Redrawn from [START_REF] Valla | Aspect du sol de l'abri 131 de Mallaha (Eynan)[END_REF][START_REF] Perrot | Le gisement natoufien de Mallaha (Eynan)[END_REF][START_REF] Valla | Les fouilles à Mallaha en 2000 et 2001: 3ème rapport préliminaire[END_REF] Natufian houses. It now appears that this outline may have been part of the architectural tradition at the site. This can also be understood as further illustration for a tendency to view space as a geometric construct, which was already recognizable through the general shape of the houses. But this tendency doesn't reach a point where a strict regularity would be imposed on buildings. None of the examples at hand exhibit a full geometric perfection and there is a range of variability accepted in the basic scheme.

At the more trivial level of everyday life, it is perhaps of interest to note that in both houses 131 and 200, more than one hearth was simultaneously in use. In their shape, these hearths are very different from each other. It is not excessively risky to suggest that they may have served different purposes, which indicates some kind of specialization at least among structures and space, if not among people themself.

We do not have the same information for the houses of the Late Natufian, like their counterparts from the Early and Final Natufian excavated more recently. Moreover, some stratigraphic uncertainties remain, which may hamper comparisons. Nevertheless, the well preserved houses from this stage appear as more or less rounded pit-dwellings lined with a stone wall along their full circumference. Exceptions are, perhaps, the last stage of a house (26) and the building dug from its fill ( 22), which are reported to be semi-circular by Perrot (1966, figure 12). It is noticeable as well that no organization of small structures along one diameter can be observed in any of the Late Natufian buildings.

We do not want to say that the tradition of house building passed through a major break during the Late Natufian at Eynan. The data are too slender to sustain such a drastic conclusion. Neither Early, Late or Final Natufian houses are known from samples large enough to allow full comparisons between buildings, which certainly allowed some variability in outline during each of these periods, the shortest of which covered several hundred years. Fully lined rounded or oval pit-dwellings are possible (probable ?) during the Early Natufian at Eynan. We just want to emphasize, taking into account the record presently available, some striking structural similarities between some Early and Final Natufian houses and to note that the sample at hand does not provide so close parallels among Late Natufian houses. This does not prevent us thinking of continuity in building conception as well as in building technique.

Another important fact needs to be emphasized. This is the relatively large number of structures which do not look like houses among Final Natufian buildings. Again, comparisons with the Early and Late Natufian data are difficult, no detailed description being available. Buildings which are not houses possibly existed in the Early Natufian (see structure 61 in Perrot 1966, fig. 2). In the Late Natufian, structure 41 provides the most obvious case (Perrot,ibidem fig.14). At this low level of knowledge, there is no basis for a lengthy discussion. But some observations should be made. If our interpretations are correct, these structures are not less numerous than houses among the excavated Final Natufian buildings. Clearly this proportion needs confirmation through further work. It is necessary as well to get a better understanding of their functions before suggesting any conclusion. Nevertheless their presence should be underlined as a possibly significant development in Natufian building habits.

Since burial customs seem to be one of the Natufian traditions subjected to the most obvious changes from one phase to the other at Eynan, a synthetic overview will suffice in order to compare Final Natufian habits to former ones (figure 3). With very few exceptions, Early Natufian people buried individual corpses in primary deposition. Many of these graves are in groups which are, at least by virtue of superposition, related to house floors, but detailed stratigraphic successions often elude us (see cemetery A and B of [START_REF] Perrot | Les Hommes de Mallaha (Eynan), Israël[END_REF]. Later, isolated graves can also be found in the upper fill of the houses (H.96, H.80, H.82) (See [START_REF] Perrot | Les Hommes de Mallaha (Eynan), Israël[END_REF]. Even when arms and legs are tightly folded, the body does not appear reduced in as narrow as possible a space. People were sometimes buried with personal adornments.

Individual graves in primary deposition eventually still existed during the Late Natufian, but associations between graves and house floors become less close. A striking change is the presence of several pits up to 50 cm deep where two to eleven individuals were buried. Most of these pits were probably kept empty for further inhumations and the old bones were rearranged progressively to make place for new corpses (Bocquentin 2003: 264). Secondary burials were probably exceptional (contrary to Perrot and Ladiray's views). None of the individuals in these graves has any personal adornment, except for two people who may have been buried with horn cores of gazelle attached to their head (Perrot et al.: 59). Originally, corpses were probably folded or seated, but later disturbances prevent detailed observation of their initial position.

One cannot ignore that the changes in burial habits as seen in the Natufian sequence at Eynan are only variations in the tradition of burying at least part of the dead in association with buildings in the perimeter of the village. There is a good probability for a continuous trend in grouping the dead either in graves closely dug near each other, or in pits planed to receive more than one individual, to be at work (as suggested by [START_REF] Byrd | Death, Mortuary Ritual and Natufian Social Structure[END_REF]. But outstanding variations occured. Among the most noteworthy are the use of grave pits in the Late Natufian only, the disappearance of personal adornments from the Late Natufian on, and a growing tendancy to reduce the space devoted to individual corpses. Despite these trends, it is between the Early and the Final Natufian that the closest affinities can be observed. In both Periods, graves are usually individual. Their relation to architecture seems to be closer than during the Late Natufian episode and results in a series of interstratifications which are difficult to elucidate. The tendancy to group individual graves, as observed at the top of the Final Natufian layer, is also more reminiscent of the Early Natufian than of the Late Natufian.

DISCUSSION

Ongoing excavations at Eynan (Ain Mallaha) uncovered in-situ deposits including buildings and graves associated with a flint assemblage previously defined as Final Natufian. A few C.14 dates seem reliable enough to confirm the very late occurence of these discoveries in the Natufian phenomenon. We now need to discuss whether the data are significant enough, and show enough specificity, to deserve the distinctive denomination "Final Natufian" or whether they simply can be lumped with the Late Natufian. If the finds in the suggested Final Natufian appear to differ significantly from those in the Late Natufian, then the term "Final Natufian" will be justified. From that point of view, the discoveries at Eynan are of particular interest because they testify to behavioral changes that took place in the same geographic setting and were subjected to minimal pressures other than internal cultural development.

Architectural features and graves, despite some hesitation, are currently the finds best associated with stratigraphic determination at Eynan from a global point of view. Comparisons demonstrate that there are continuous trends from the Early to the Final Natufian, both in architectural traditions and in burial customs. Buildings are in pits lined with a wall of some kind. These walls were not standing above the ground from which the pits were dug. Possible superstructures, when they existed, were probably made of light material. Beside these basic features, houses tend to be smaller and set in lesser deep pits. This results in a less impressive aspect of the buildings from one period to the other. As for graves, two main salient habits can be followed for the overall sequence; one is a tendency to bury at least some of the dead in the village. The other, a tendency to group graves together.

These trends allow for a range of variations through which the specificity of each period is manifested. What is perhaps significant for our discussion is that parallels are closer between the Early and Final Natufian than between the Final Natufian and the Late Natufian. The Late Natufian, with its mostly fully walled round houses and its group burials in pits kept empty for further interments, appears as relatively original. These particularities accentuate the apparent changes associated with the Final Natufian. Thus, the small houses, numerous buildings not dedicated to housing and sometimes related to adjacent minor dependant structures, and the individual graves in hypercontracted position, stand as even more innovative than they really are. Everything taken into consideration thus seems to validate the Final Natufian subdivision.

To go one step further, it should be necessary to widen the comparison and to take into account other aspects of prehistoric life, among them technical production besides the flint industry, subsistence and symbolism. Unfortunately, data bases are problematic in most of these fields. One can only note a few points that will deserve verification when more securely stratified material will become available. The fact that obsidian pieces first reach the site during the Final Natufian should be mentioned, because of the possible increase in relations with the northern cultures. It is also probable that more grinding slabs and more handstones are produced instead of mortars and pestles. This may indicate some changes in grinding and pounding techniques and perhaps in the processed materials. On the other hand, game seem not to be very different from the earlier phases. Cervids (Dama mesopotamica, Cervus elaphus, Capreolus capreolus) still contribute a large part of the hunted mammals, together with suids (Sus scrofa) and, mainly, Gazelles (Gazella gazella). Waterfowl and fish are important parts of the diet. In the area of symbolism, to restrict oneself to decorative items, an increase in the use of green stone probably occured. Dentalium shells are still numerous but they seemingly tend to be used as shorter beads than in earlier levels (Bar-Yosef Mayer in [START_REF] Valla | Les fouilles à Mallaha en 2000 et 2001: 3ème rapport préliminaire[END_REF]).

These observations are the result of punctual rather than systematic comparisons. They are somewhat impressionistic and should not be relied on without reservations. From what we can guess, continuity is warranted. In the field of food procurement, game suggest a relative stability. It is not clear how the shift in grinding techniques should be understood. Even if it is related to food processing, more work is needed to understand whether it means a change in the ground material, or in the quantities ground, or whether it is mainly aimed at producing tools easier to make and to use. Besides that, one notices that most of the other changes listed above are related to exotic items: obsidian, green stone and even dentalium. This may mean an increase in permeability to objects and ideas foreign to the Carmel and Galilee area. This should be understood in relation to indications for a general weakening of the influence of the Carmel-Galilee "core area" in peripheries like the Negev and the Middle Euphrates valley, where original flint tools are developed independantly [START_REF] Valla | Natufian seasonality: a guess[END_REF].

Besides the fairly well known Harifian in the Negev, Final Natufian evidence away from Eynan is still meagre but not without existence. Due to very small exposures, no architecture was found in Mureybet but hearths and floors made of brought in and packed clay were reported. If we rely on Nadel's observations, Final Natufian Nahal Oren comprised a floor with a large hearth and possible post holes. Graves are numerous. The ones we know are not similar to those at Eynan. They are relatively large pits, some of them lined with stones. Bodies are in primary position. Sometimes more than one person were put in the same pit. Hypercontraction of all the body is not common. No personal adornment were found, but a mortar was, in a few cases, buried with the corpse and some graves were marked with a standing "stone pipe" (elongated large limestone mortar). Skull removal was unusual but not totally absent [START_REF] Noy | Some aspects of Natufian mortuary behaviors at Nahal Oren[END_REF][START_REF] Bocquentin | Pratiques funéraires, paramètres biologiques et identités culturelles au Natoufien : une analyse archéo-anthropologique[END_REF]. Scanty as these data are, their diversity may look surprising. As a matter of fact, they hint at a situation not very different from what is known from the Early and Late Natufian, when each site appears with its own particularities.

Data related to subsistence activities are available for Eynan, Mureybet and Nahal Oren. But direct comparisons are problematic because of the extreme diversity in the environmental setting of the sites and because of the difference in nature of the data relative to vegetable food. Eynan is reputed to be in an exceptionally favorable place as far as food procurement is concerned [START_REF] Perrot | Le gisement natoufien de Mallaha (Eynan)[END_REF]. Nahal Oren is far from being so valuable (Noy et al.1973: 96). These discrepancies are probably hightly significant, as were the cultural habits of Natufian people themselves. Eynan hunters exploited the lake (for fish and birds), the swamps and the valley (for wild boar, etc.), as well as the slopes and hills (for gazelles). Their counterparts from Mureybet neglected the bottom of the valley to concentrate on the Euphrates river (for fish and birds) and the steppes (for gazelles and asses) [START_REF] Helmer | Etude de la faune de la phase IA (Natoufien final) de Tell Mureybet (Syrie), fouilles[END_REF].

Nahal Oren hunters caught mainly gazelles, probably in the foothills of the Carmel mountain along the coastal plain. As for vegetal food, phytoliths analysis suggests that wheat was collected for food in the hills above Eynan, but in small quantities (Miller-Rosen in [START_REF] Valla | Les fouilles à Mallaha en 2000 et 2001: 3ème rapport préliminaire[END_REF]). On the other hand, gatherers at Mureybet showed little interest for cereals and legumes but collected a lot of Polygonum from the Euphrates valley (Van Zeist and Bakker-Heerres 1974). Again, this diversity appears as the result of cultural habits with some adaptative behavior. It demonstrates that no behavioral uniformity has been reached by the turn to the PPNA (see also [START_REF] Belfer-Cohen | Early Sedentism in the Near-East[END_REF].

To conclude, fieldwork during the last 20 years substanciated the Final Natufian as a particular episode in the Natufian sequence. Excavations at Eynan since 1996 provided the opportunity for a large exposure of relevant in-situ deposits with buildings and associated graves. Other in-situ structures at Nahal Oren were assigned to this phase with a relatively good probability. Together with the data obtained earlier at Mureybet, this provides a firm basis for a preliminary approach to the period, which is also known from a few other places spread from the Middle Euphrates valley to the Southern Jordan desert.

The Natufian tradition seems fully preserved during the Final Natufian. The basic scheme is that of "villages" including buildings closely associated with graves, and scattered small ephemeral sites, all of them sharing a similar microlithic flint industry. In each of the areas of human activity still available for archaeological observation, the sites involved share the same cultural tradition. But in each of these areas as well, no two sites are exactly similar. The Final Natufian is not an exception to this situation and it appears that there is no standardization reached by the end of the Natufian, either in building, burial or hunting and foraging habits. At Eynan, where we are able to follow these habits in the same place for a long period of time, the observations taken into consideration in this paper underline two characteristics of the Final Natufian. One is a tendency to revive some Early Natufian behaviors. The other could be a tendency to "think smaller", if we can borrow an expression from Belfer-Cohen and Goring-Morris.

Together, the sum of these observations raises the question of Final Natufian sedentism. In the past, sedentism has been questioned for Nahal Oren (Higgs in [START_REF] Noy | Recent excavations at Nahal Oren[END_REF]) and for the Final Natufian of Eynan [START_REF] Valla | Les établissements natoufiens dans le nord d'Israël[END_REF]. Since most sites in the Carmel and Galilee area either had been deserted or were thought to have no architecture by the Final Natufian, it has even been suggested that people had given up sedentism for a while in this area before settling in new villages during the PPNA [START_REF] Valla | The first settled societies[END_REF][START_REF] Valla | Natufian seasonality: a guess[END_REF]. This interpretation has to be revised in view of the new data. The Final Natufian occupation near the spring of Eynan shares the same characteristics with the prior Early and Late Natufian. It is not less ambiguous in term of year round or seasonal stays in the houses. It seems to indicate that some kind of sedentism or semi-sedentism not very different from the way of life of the former inhabitants was maintained on the site during part of the Final Natufian. Nevertheless, if it is true that the graves at the top of the layer were dug from a now eroded deposit, then they probably mean that the site was still used as a cemetary by Final Natufian people after being abandoned as a living place. This reevaluation simply makes the general trend toward mobility noticed by the end of the Natufian less abrupt and the passage to the PPNA somewhat softer than was previously understood.

FRANÇOIS

  UMR 7041 (ArScAn): 21, Allée l'Université, 92023 Nanterre, France 2 Israel Antiquities Authority, P.O.B 586, Jerusalem 91004 3 CNRS. UMR 5809 (Anthropologie des populations du passé): 1, Avenue des Facultés, 33405 Talence, France

Figure 1 :

 1 Figure 1: General map of main Final Natufian structures and graves excavated at Eynan (1996-2001). Structure 215 (Ib2) is older than structures

Figure 3 :

 3 Figure 3: Schematic examples ofselected graves from the Natufian at Eynanan: 1, H.91 (with a dentalium belt), Early Natufian; 2, locus 9 (stones are in black), Late Natufian; 3, H.156, Final Natufian.After[START_REF] Perrot | Les Hommes de Mallaha (Eynan), Israël[END_REF] (1 and 2),Valla et al. 2001 (3)

ACKNOWLEDGEMENTS

The authors are grateful to J.Phillips and A.Belfer-Cohen who invited them to contribute in honor of O.Bar-Yosef. Some of the ideas expressed in the paper were discussed at length with D.Stordeur during fruitful exchanges for many years. Ongoing excavations at Eynan were made possible by grants from the DGRCST of the Ministère des Affaires Etrangères (Paris), the Irene-Levi-Sala CARE Archaeological Foundation, The Wenner Gren Foundation for Anthropological Reasearch and the National Geographic Society. Illustrations were completed by Marjolaine Barazani (Centre de Recherche Français de Jérusalem) and Maurice Hardy (ArScAn).


Thomas Taimre
email: t.taimre@uq.edu.au

Patrick J Laub
email: patrick.laub@univ-lyon1.fr

Rare tail approximation using asymptotics and L 1 polar coordinates

In this work, we propose a class of importance sampling (IS) estimators for estimating the right tail probability of a sum of continuous random variables based on a change of variables to L 1 polar coordinates in which the radial and angular components of the IS distribution are considered separately. The estimation of this quantity is of particular interest in application areas ranging from finance to wireless communications as well as appearing as a classical problem in the applied probability literature (notably in queueing theory). Particular models which appear time and again are Pareto, Lognormal, and Weibull summands, and we choose to focus on these here. Due to the unavailability of closed-form expressions, the study of this quantity has been the subject of intense research, specifically in uncovering asymptotic expressions for various summands on the one hand and in devising computationally effective Monte Carlo estimation schemes on the other. In the latter case, much of the literature has been devoted either to the case of light-tailed summands or to heavy-tailed summands (and in particular, sub-exponential summands). Here, we draw on the

growing base of knowledge of asymptotic expressions in order to devise a broadly-applicable computationally effective Monte Carlo estimation procedure which is not restricted to either light-or heavy-tailed summands.

When the asymptotic behaviour of the sum is known we exploit it for the radial change of measure, and the resulting estimator has the appealing form of the (known) asymptotic multiplied by a random multiplicative correction factor. Given we assume knowledge of the asymptotic behaviour of the sum in this framework, traditional notions of efficiency that appear in the rare-event literature hold little practical meaning here. Indeed, the estimator by design enjoys bounded relative error provided the likelihood ratio of the angular component remains bounded as the rarity parameter increases. Instead, we focus on the practical behaviour of the proposed estimator in the pre-asymptotic regime for right tail probabilities.

The proposed estimator and procedure are applicable in both the heavy-and light-tailed settings, as well as for independent and dependent summands. In the case of independent summands, we find that our estimator compares favourably with exponential tilting (iid light-tailed summands) and the Asmussen-Kroese method (independent subexponential summands).

However, for dependent subexponential summands using the same simple angular distribution as for the independent case, the performance of our estimator rapidly degenerates with increasing dimension, suggesting an open avenue for further research.

Introduction

A typical problem in the field of rare-event estimation is to determine the probability (γ) := P(S > γ) [START_REF] Alink | Diversification of aggregate dependent risks[END_REF] where S := X 1 + • • • + X d for a random vector X X X := (X 1 , . . . , X d ) with fixed d ∈ N + having joint probability density function (pdf) f X X X and where the γ ∈ R is large or increasing. In applications, we often wish to understand the behaviour of a combination of random factors, and hence the random variable (r.v.) S is ubiquitous in real-world modeling problems. It can model, for example: aggregate risk or portfolio value for holding d risky assets [START_REF] Mcneil | Quantitative Risk Management: Concepts, Techniques and Tools[END_REF][START_REF] Rüschendorf | Mathematical Risk Analysis[END_REF], the aggregate losses for d insurance policy claims [START_REF] Asmussen | Ruin probabilities[END_REF][START_REF] Klugman | Loss models: from data to decisions[END_REF], or the combined signal interference from d wireless transmission sources [START_REF] Rached | Importance sampling estimator of outage probability under generalized selection combining model[END_REF][START_REF] Rached | On the sum of order statistics and applications to wireless communication systems performances[END_REF]. Probabilities of the form (1) are used to understand how a system would behave under extreme scenarios such as a market crash, a power surge, or a natural disaster. One is typically interested not just in the quantity (γ) but also in the behaviour of the summands when the extreme event {S > γ} occurs.

This probability is available in closed-form for only a few basic cases, when the density of S (which is a d-fold convolution) has a known solution, c.f. [START_REF] Nadarajah | A review of results on sums of random variables[END_REF]. For example, when the summands are independent and identically distributed (iid) then it is sometimes simple to calculate (for example exponential, gamma, or normal summands, and in the discrete case binomial, geometric, or negative binomial summands) and sometimes it is still intractable (for example lognormal, Weibull, Laplace, or Beta summands). However, requiring the assumption of independence (let alone iid-ness) of the summands is a stifling restriction when modeling realworld events; a notorious example would be the partial blame of the 2008-9 global financial crisis on mathematicians' inappropriate use of a simplistic dependence model (the Gaussian copula) [START_REF] Salmon | Recipe for disaster: The formula that killed Wall Street[END_REF].

When analytical solutions are unavailable, the next best option is numerical integration, and after that Monte Carlo integration (or quasi-Monte Carlo). Numerical integration algorithms applied to

(γ) = R d I{x 1 + • • • + x d > γ}f X X X (x x x) dx x x ,
where I{A} denotes the indicator of an event A (taking value 1 if A occurs and 0 otherwise), are typically slow, inaccurate, and misleading. This is because the indicator is rarely 1, floating-point errors accumulate, and the curse of dimensionality applies for d larger than about 2 or 3. Some of these algorithms attempt to estimate the error in their result, but there are few (if any) theoretical guarantees that these estimates are reliable.

Rare-event problems also cause difficulties for the crude Monte Carlo (CMC) estimator. This is obvious as the CMC estimator's relative error explodes for large

γ -that is, the CMC estimator CMC (γ) := I{S > γ} has lim γ→∞ RelativeError{ CMC (γ)} = lim γ→∞ Var[ CMC (γ)] (γ) 2 = lim γ→∞ (γ)[1 -(γ)] (γ) 2 = ∞ .
Intuitively, the problem is because the indicator I{S > γ} is eventually always 0 when γ gets very large. In response, various variance reduction techniques have been applied so that there are now a large collection of estimators with better performance in this setting, c.f. 'rare-event estimation' in [START_REF] Kroese | Handbook of Monte Carlo Methods[END_REF][START_REF] Asmussen | Stochastic Simulation: Algorithms and Analysis[END_REF]19].

There is, of course, no silver bullet for the problem. Some estimators only apply to specific distributions (e.g. [START_REF] Botev | Fast and accurate computation of the distribution of sums of dependent lognormals[END_REF] for sums of lognormals, [START_REF] Yao | Estimating tail probabilities of random sums of infinite mixtures of phasetype distributions[END_REF] for sums of phase-type mixtures) or to certain classes of distributions (exponential tilting for light-tailed summands [START_REF] Kroese | Handbook of Monte Carlo Methods[END_REF][START_REF] Asmussen | Stochastic Simulation: Algorithms and Analysis[END_REF], hazard-rate twisting or the Asmussen-Kroese method [START_REF] Asmussen | Improved algorithms for rare event simulation with heavy tails[END_REF] for heavy-tailed summands). Other estimators are general but require specifying either some extra information (e.g. availability of conditional distributions for conditional Monte Carlo [START_REF] Asmussen | Conditional Monte Carlo for sums, with applications to insurance and finance[END_REF], or an appropriate sampling distribution for use in importance sampling).

The most general estimators -such as the generalised splitting method [START_REF] Botev | Efficient Monte Carlo simulation via the generalized splitting method[END_REF], cross-entropy method [START_REF] De Boer | A tutorial on the cross-entropy method[END_REF], or Markov Chain Monte Carlo (MCMC) methods such as [START_REF] Chan | Improved cross-entropy method for estimation[END_REF] are usually computationally demanding, they often depend upon an intelligent selection of input parameters to perform efficiently, and are somewhat complicated.

Whilst one rarely has an exact expression for (γ), it is somewhat common to know an asymptotic approximation to it, and this forms the basis for our proposed estimator. For example, if X X X ∼ Lognormal(µ µ µ, Σ Σ Σ) where and covariance matrix Σ Σ Σ), then it has been shown that [START_REF] Asmussen | Asymptotics of sums of lognormal random variables with Gaussian copula[END_REF] (γ) = P(S > γ) ∼

µ µ µ ∈ R d and Σ Σ Σ ∈ R d×d
d i=1 P(X i > γ) as γ → ∞ (2)
where f (x) ∼ g(x) denotes lim x→∞ f (x)/g(x) = 1. Thus, one is tempted to label the right-hand side (RHS) of [START_REF] Alink | Diversification for general copula dependence[END_REF] as Asym (γ) and use it as an approximation for (γ). For certain values of (µ µ µ, Σ Σ Σ) this asymptotic approximation can be accurate, in others it can be wildly inaccurate, depending on how fast the asymptotic approximation converges to the true value; see Figure 1 for an illustration where it is only when (γ) 10 -10 that the asymptotic form begins to give accurate estimates (i.e., Asym (γ)/ (γ) > 0.99). A discussion of this phenomenon is in [START_REF] Botev | Fast and accurate computation of the distribution of sums of dependent lognormals[END_REF].

One Term Two Terms

Fig. 1: A comparison of (γ) and Asym (γ) for X 1 + X 2 where X 1 ∼ Lognormal(0, 1) is independent to X 2 ∼ Lognormal(0, 3 4 ). The y axis plots Asym (γ)/ (γ), and the x axis showslog 10 (γ). The two curves describe two possible asymptotics: the yellow "Two terms" describes Asym (γ) as given in (2), whereas the blue "One Term" uses just the (eventually dominant) first term of this sum.

We propose an importance sampling (IS) estimator which incorporates the asymptotic approximation and uses Monte Carlo sampling to estimate a correction to Asym (γ) in order to construct an unbiased estimator (γ) of (γ). The main drawback to IS is likelihood degeneration, where one can face statistical errors if γ or d is extremely large. The degeneration caused by a large d is only partially compensated by our approach, so we take d ≤ 100. Our goal is to provide an estimator which is practically useful when (γ) is in the pre-asymptotic regime -that is, for the values of the rarity parameter γ that are sufficiently large so that direct CMC estimation is no longer practical and yet the asymptotic form does not yet yield accurate estimates (i.e.,

Asym (γ)/ (γ) ≤ 0.99)
The range of probabilities that we consider is explicitly less rare the standard rare-event literature. The orthodox approach is to construct an estimator (γ)

and analyse the limit lim γ→∞ Var( (γ))/ (γ) 2 ;

if the limit is small (i.e., zero, bounded, or at least grows only at a polynomial rate) then the estimator is branded as a success (it has 'vanishing relative error', 'bounded relative error', or is 'logarithmically efficient' respectively) regardless of its behaviour in the finite γ situation. It can happen that these desirable limiting properties are only discernible in cases when the probabilities are truly minuscule (e.g. of order 10 -10 or smaller); in a situation such as this, the model error would surely dominate any estimation error.

The remainder of this paper is structured as follows.

The estimator is introduced in Section 2, the results from numerical comparisons are in Section 3, and Section 4 concludes the discussion.

2 The L 1 polar estimator

The general form

We construct an estimator of the quantity (γ) := P(S > γ), where S = X 1 + • • • + X d for large γ by applying IS. Standard IS theory says to construct an estimator which samples from a distribution close to f X X X | S>γ (that is, the distribution of X X X conditioned on S > γ), rather than the original f X X X .

To do this, we perform a change of variables to Pickand's coordinates [START_REF] Falk | On Pickands coordinates in arbitrary dimensions[END_REF] so

X X X -→ (S, Θ Θ Θ) := (X 1 + • • • + X d , X X X -d /[X 1 + • • • + X d ]) ,
where the notation X X X -i refers to the vector X X X after its i-th element has been removed. Here Θ Θ Θ is defined to be of length d -1, though we will often abuse notation and refer to Θ Θ Θ as d-dimensional, in which case the element

Θ d should be read as a shorthand for 1 -d-1 i=1 Θ i . The new density f (S,Θ Θ Θ) is available (if f X X X is known), and is f (S,Θ Θ Θ) (s, θ θ θ) = f X X X (sθ θ θ) × |s| d-1 .
Consider IS in this new form. Imagine that we have a density g (S,Θ Θ Θ) which is in some way similar to f (S,Θ Θ Θ) , for which we also know the marginal density g S (s) := g (S,Θ Θ Θ) (s, θ θ θ) dθ θ θ and the conditional density g Θ Θ Θ|S := g (S,Θ Θ Θ) /g S . If we truncate g (S,Θ Θ Θ) so that S > γ a.s., and use this as the IS distribution, we obtain

l IS (γ) := G S (γ) R R r=1 f (S,Θ Θ Θ) (S [r] , Θ Θ Θ [r] ) g S (S [r] )g Θ Θ Θ|S (Θ Θ Θ [r] | S [r] ) (3)
for

S [r] iid ∼ g S|S>γ , Θ Θ Θ [r] ind ∼ g Θ Θ Θ|S ( • |S [r]
), where we define G S (γ) := ∞ γ g S (s) ds and g S|S>γ := g S I{S > γ}/G S (γ). We investigate estimators of the general form of (3) which we call (L 1 ) polar estimators. These are accurate when

g (S,Θ Θ Θ) = g S × g Θ Θ Θ|S closely resembles f (S,Θ Θ Θ) = f S × f Θ Θ Θ|S
. This is carried out in two steps: (i) by finding a radial approximation g S which approximates f S , and (ii) an angular approximation g Θ Θ Θ|S similar to f Θ Θ Θ|S , which we discuss in the following sections.

The radial approximation

As mentioned in the introduction, we consider utilising an asymptotic form of the sum in our estimator -they form our radial approximation. To clarify the notation, we precisely define the relevant asymptotic forms:

Definition 1 (Asymptotic form) If for some func- tion f S ∈ L 1 (R), with tail F S (s) = ∞ s f S (x) dx, and con- stant c S ∈ R + , we have that f S (s) ∼ c S f S (s) , for s → ∞ (4)
then we say f S is an asymptotic form of f S .

Thus, in the general form (3) we will use g S = f S when it is available and is a proper pdf. There are some technicalities for the cases when f S does not form a proper pdf which we do not discuss in this work. The estimator resulting from this radial approximation is

l IS2 (γ) := c S F S (γ) R R r=1 f (S,Θ Θ Θ) (S [r] , Θ Θ Θ [r] ) c S f S (S [r] )g Θ Θ Θ|S (Θ Θ Θ [r] | S [r] ) (5)
for

S [r] iid ∼ f S|S>γ and Θ Θ Θ [r] ind ∼ g Θ Θ Θ|S ( • |S [r] ).
Remark 1 The estimator in ( 5) has natural interpretation which we find appealing. By defining a "correction factor" to the asymptotic form, R(γ), via (γ) = Asym (γ)R(γ), and noting that Asym (γ) := c S F S (γ), we see that

IS2 (γ) = Asym (γ) × R(γ) ,
where R(γ) is an unbiased Monte Carlo estimate of the factor R(γ), which by definition has expected value equal to 1.

The recent applied probability literature has found the f S for a staggering array of distributions of X X X. Perhaps the simplest case is when the X i are iid subexponential random variables. By definition (cf. [START_REF] Foss | An Introduction to Heavy-tailed and Subexponential Distributions[END_REF]), they satisfy

f S (s) ∼ d f 1 (s) , for s → ∞ . ( 6
)
For sums of independent non-identically distributed (ind) subexponential variables (or for sums containing some subexponential and some lighter-tailed variables) we have

f S (s) ∼ d i=1 f i (s) ∼ i∈I f i (s) , for s → ∞ ( 7
)
where I is the set of indices of slowest tail decay. The asymptotics in [START_REF] Asmussen | Improved algorithms for rare event simulation with heavy tails[END_REF] The Weibull distribution is interesting as it is a family which can be heavy-tailed, light-tailed (the Rayleigh distribution is a special case), or on the boundary between these (i.e. the exponential distribution). The asymptotic form for the heavy-tailed Weibull sum is covered by ( 6) and ( 7) as the summands are subexponential.

The difficulty in finding the asymptotics for the lighttailed case led the authors to investigate it in detail, leading to the paper [START_REF] Asmussen | Tail asymptotics for light-tailed Weibull-like sums[END_REF] which uses results originally from [START_REF] Balkema | Densities with Gaussian tails[END_REF].

λ ∈ R + , d ≥ 2. Then (γ) ∼ 2βπ β -1 (d-1)/2 d -1/2 γ λd β(d-1)/2 F γ d d as γ → ∞, where F is the cdf of the Weibull(β, λ) dis- tribution.
The exposition in [START_REF] Asmussen | Tail asymptotics for light-tailed Weibull-like sums[END_REF] details this and more general asymptotics (i.e. the independent but non-identically distributed case, and when the variables are not exactly

Weibull but are 'Weibull-like').

By its very construction, one would expect the estimator utilising an asymptotic form for the right-tail, [START_REF] Asmussen | Stochastic Simulation: Algorithms and Analysis[END_REF], to enjoy good efficiency properties as γ → ∞. As mentioned in the introduction, our goal is to provide a practically useful estimator for 'moderately' rare problems, in the range of γ before the asymptotic regime takes hold. As such, it is our view that the orthodox notions of efficiency have little meaning in our setting.

Nevertheless, we note that it is straightforward to verify that if the ratio f Θ Θ Θ|γ /g Θ Θ Θ|γ remains bounded by some finite constant K ≥ 1 for all θ θ θ as γ → ∞, then the estimator (5) enjoys bounded relative error, and if K = 1 then the estimator enjoys vanishing relative error.

The angular approximation

The choice of angular approximation is not as obvious as was the choice of radial approximation. Moreover, we have a computational benefit to finding a g Θ Θ Θ|S>γ which is similar to f Θ Θ Θ|S>γ as this distribution will be constant across all Monte Carlo iterates, in contrast to g Θ Θ Θ|S [r] and f Θ Θ Θ|S [r] .

Nevertheless, when it is possible, we follow the same approach as the radial approximation and utilise some asymptotic information. However, we note that if one simply re-uses the previous asymptotic form, that is

f Θ Θ Θ|S (θ θ θ|s) = f S,Θ Θ Θ (s, θ θ θ) f S (s) ∼ f X X X (sθ θ θ)|s| d-1 f S (s) =: g Θ Θ Θ|S (θ θ θ|s) ,
which may appear natural, then the estimator (5) degenerates to the deterministic

l IS2 (γ) := c S F S (γ) R R r=1 1 = c S F S (γ) .
Moreover, if it is known that f Θ Θ Θ|S>γ degenerates in This is just a re-casting of the principle of the single big jump (cf. [START_REF] Foss | An Introduction to Heavy-tailed and Subexponential Distributions[END_REF]). Unfortunately, for finite γ we cannot use this directly as in this case the likelihood ratio appearing in (3) is not well defined. One density, which we call the optimistic density (see the algorithm below), that is not degenerate (and therefore will have welldefined likelihood ratio) but is asymptotically equivalent to (Θ Θ Θ | S = s) for the case of ind subexponential summands is

g Θ Θ Θ|S (θ θ θ | s) = |s| d-1 d i=1 p i (s)f X X X -i (sθ θ θ -i )I{θ i = 1 -1 1 1 • θ θ θ -i } (8)
where the p i functions are defined by

p i (s) = F i (s) d j=1 F j (s) . ( 9
)
Algorithm 1 shows a method for sampling from this g Θ Θ Θ|S (θ θ θ | s), and Proposition 2 shows that it has limiting distribution consistent with [START_REF] Asmussen | Improved algorithms for rare event simulation with heavy tails[END_REF] as s → ∞.

Algorithm 1 Sampling from the optimistic angular density

1: procedure Optimistic(s, F 1 , . . . , F d )

2:

Simulate index I in {1, . . . , d} by P(I = i) = p i (s) from (9).

3:

for i = 1 to d except I do 4:

X * i ← Random sample from F i 5:
end for 6:

X * I ← s -i =I X i 7:
return Θ Θ Θ ← X X X * /s 8: end procedure Our somewhat-odd name for this density derives from the fact that the value of X * I in Algorithm 1 can sometimes be negative, but we are optimistic that it usually won't. If this value becomes negative then the polar estimator's likelihood ratio will simply weight this sample by zero, so no great loss is incurred, and this situation is increasingly rare as s becomes larger.

When the subexponential summands are only independent in the extreme, a simple generalisation of Algorithm 1 is to replace Lines 3 to 5 with taking a random sample X X X * from f X X X .

Proposition 2

The optimistic density (8) converges as s → ∞ to the singular density

g ∞ (θ θ θ) := d i=1 p i I{θ θ θ = e e e i } , ( 10
)
where p i = lim s→∞ p i (s) for i = 1, . . . , d.

Proof For some

t t t = (t 1 , . . . , t d ) ∈ R d , the characteristic function of g Θ Θ Θ|S is ϕ g Θ Θ Θ|S (t t t | s) = E exp i t t t Θ Θ Θ = E exp i t t t s X X X *
where X X X * = sΘ Θ Θ as in Algorithm 1.

So, with I as the discrete variable defined in Algorithm 1, we have

ϕ g Θ Θ Θ|S (t t t | s) = d j=1 p j (s) E e i t t t s X X X * I = j = d j=1 p j (s) E   e i t t t -j s X X X * -j + t j s (s-1 1 1 X X X * -j ) I = j   = d j=1 p j (s)e it j E e i (t t t -j -t j 1 1 1) s X X X -j = d j=1 p j (s)e it j ϕ X X X -j (t t t -j -t j 1 1 1) s .
Therefore,

lim s→∞ ϕ g Θ Θ Θ|S (t t t; s) = d j=1 p j e it j = d j=1
p j e it t t e e e j which corresponds to the singular density as in [START_REF] Bingham | Regular Variation[END_REF].

Remark 2

The polar estimator for ind subexponential summands with the optimistic angular approximation (8) simplifies to

l IS2 (γ) = c S F S (γ) R R r=1 HM(f X 1 (S [r] Θ [r] 1 ), . . . , f X d (S [r] Θ [r] d )) c S f S (S [r] )
where S [r] iid ∼ f S|S>γ , Θ Θ Θ [r] ind ∼ g Θ Θ Θ|S ( • |S [r] ), and HM(. . . )

is the harmonic mean of the inputs.

The conditional angular asymptotic distribution is more challenging to obtain in the case of light-tailed summands. The following example shows these distributions differ qualitatively when different copulas are considered.

Example 1 Consider X 1 and X 2 to be Exp(1) variables which are: i) independent, ii) Clayton( 1) dependent, or iii) Ali-Mikhail-Haq(-1) dependent. The sum densities can be calculated explicitly and are given by

f Ind S (s) = se -s for s > 0 , f Cla S (s) = 2 -2 cosh(s) + s sinh(s) (cosh(s) -1) 2 for s > 0 , f AMH S (s) = 8 csch(s) 3 sinh(s/2) 4 for s > 0 ,
respectively. Hence, for s > 0 and θ ∈ (0, 1), we have angular densities

f Ind Θ 1 |S (θ|s) = 1 , f Cla Θ 1 |S (θ|s) = se -sθ (e s -e sθ )(e sθ -1) 2 + s -2e s + se s , f AMH Θ 1 |S (θ|s) =
se -sθ (e s + e 2sθ ) 2(e s -1) , respectively. It is interesting to note that the asymptotic independence of the Clayton copula would indi-

cate that f Cla Θ 1 |S (θ|s)/f Ind Θ 1 |S (θ|s) → 1 as s → ∞ which is indeed the case. In contrast, f AMH Θ 1 |S (θ|s)
degenerates to a pair of atoms at 0 and 1 as s → ∞.

One (light-tailed) case where we can determine an asymptotic angular distribution is for light-tailed Weibull sums. The angular asymptotic can be deduced from the results in [START_REF] Asmussen | Tail asymptotics for light-tailed Weibull-like sums[END_REF], and appears as follows.

W i (x) = ω(x)(X i -x/d) , for i = 1, . . . , d , where ω(x) := 2β(β -1)(x/d) β-2 . Then as γ → ∞ we have (W W W (γ) | S > γ) D --→ Normal (0 0 0, (1 -)I I I + ) , where = -1/(d -1).
Note that the d-dimensional multivariate normal distribution appearing in Proposition 3 is supported on a (d -1)-dimensional subspace.

When the asymptotic angular approximation is unavailable, there are several conceivable alternatives, however we haven't yet found a method worth mentioning here that is successful in a diverse range of problems.

Results

In this section we give illustrative results of numerical experiments. For subexponential summands, we compare to the most competitive alternative, the Asmussen-Kroese estimator, and for light-tailed summands we compare to the standard IS approach of exponential tilting. In what follows, we adopt Mathematica's parameterisations for the lognormal, Pareto, and Weibull distributions. The code we used is available online [START_REF] Taimre | Online accompaniment for[END_REF].

Subexponential Summands

Below we present the estimates and the estimated relative errors for the polar estimator and the Asmussen-Kroese estimator for various distributions of X X X. Each estimator is given R = 10 5 iid samples of X X X.

The first test (Figs. 2 and3) takes the sum of d = 12 independent lognormal random variables, with marginal

distributions X i ∼ Lognormal(-i d , i d ).
Here, the sum behaves asymptotically as the dominant term X 12 ∼ Lognormal(-1, 1), and the optimistic angular distribution is used.

The second test (Figs. 4 and5) considers the sum of d = 16 independent Pareto random variables, where X i ∼ Pareto(1, i, 0). The sum behaves asymptotically as the dominant term X 1 ∼ Pareto(1, 1, 0), and the optimistic angular distribution is used. The marginal distributions are X i ∼ Weibull( 1 4 , i d ). The sum behaves asymptotically as the last summand X 8 ∼ Weibull( 14 , 1), and the optimistic angular distribution is used.

• • • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆
• • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆
• • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 10

Dependent Summands

Next we reproduce the three subexponential tests above with dependence added by Archimedean copulas. We use the Asmussen-Kroese estimator as outlined in Section 3.2.2.2 of [START_REF] Nandayapa | Risk probabilities: asymptotics and simulation[END_REF] as the traditional form of this esti- • dom variables where X i ∼ Pareto(1, i, 0), except that the summands exhibit dependence via a Clayton( 9 10 ) copula. •

• • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆
X i ∼ Lognormal(-i d , i d ), except dependence is added via a Frank( 1 2 ) copula. • • • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 10
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 10
• • • • • • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆
■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 10

•

• • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 1
• • • • • • • • ■ ■ ■ ■ ■ ■ ■ ■ ■ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ 1

  is positive definite (by which we mean that X X X = exp(Y Y Y ) component-wise, where Y Y Y has a multivariate normal distribution with mean vector µ µ µ

Proposition 1 (

 1 Corollary 3.1 of [6]) Assume that X 1 , . . . , X d are iid light-tailed Weibull(β, λ) where β > 1,

  Finding a conditional density g Θ Θ Θ|S which is similar to f Θ Θ Θ|S has little precedent in the literature. Instead of taking an S which is larger than γ and asking 'what is the distribution of Θ Θ Θ given this S?', we can instead ask 'what is the distribution of Θ Θ Θ given S > γ?'. This is the same situation that is studied in multivariate extreme value theory, where the spectral density characterises the behaviour of f Θ Θ Θ|S>γ in the limit as γ → ∞ [15]. This second conditional distribution will resemble the first in cases that E[S -γ | S > γ] converges quickly to zero when γ becomes large.

  the limit [e.g. to a point mass at 1/d in each coordinate (perfect extremal dependence) or is degenerate on axes (independence in the extreme)], this does not tell us what we should do for finite γ. Indeed, when the summands are iid subexponentials, then the distribution of (Θ Θ Θ | S = s) as s → ∞ degenerates to a discrete distribution over the d-dimensional unit vectors e e e 1 , . . . , e e e d (with e e e i having a single 1 in the i-th coordinate and zeros in all other coordinates).

Proposition 3

 3 Say X 1 , . . . , X d are iid and are distributed as light-tailed Weibull(β, 1) with survival function F (x) = e -x β where β > 1, d ≥ 2. Define the vector function W W W (x) component-wise by

Fig. 2 :

 2 Fig. 2: Estimates of P(S > γ) from each estimator.

Fig. 3 :

 3 Fig. 3: Estimated relative errors for each estimator.

Fig. 4 :Fig. 5 :

 45 Fig. 4: Estimates of P(S > γ) from each estimator.

Fig. 6 :

 6 Fig. 6: Estimates of P(S > γ) from each estimator.

Fig. 7 :

 7 Fig. 7: Estimated relative errors for each estimator.

Fig. 8 :

 8 Fig. 8: Estimates of P(S > γ) from each estimator.

Fig. 9 :

 9 Fig. 9: Estimated relative errors for each estimator.

Fig. 10 :

 10 Fig. 10: Estimates of P(S > γ) from each estimator.

Fig. 11 :

 11 Fig. 11: Estimated relative errors for each estimator.

Fig. 12 :

 12 Fig. 12: Estimates of P(S > γ) from each estimator.

Fig. 13 :

 13 Fig. 13: Estimated relative errors for each estimator.

Fig. 14 :Fig. 15 :

 1415 Fig. 14: Estimates of P(S > γ) from each estimator.

Fig. 16 :Fig. 17 :

 1617 Fig. 16: Estimates of P(S > γ) from each estimator.

Fig. 18 :Fig. 19 :Fig. 20 :Fig. 21 : 4 Conclusion

 181920214 Fig. 18: Estimates of P(S > γ) from each estimator.

  also hold in many regimes where dependence has been introduced, cf.[START_REF] Foss | On sums of conditionally independent subexponential random variables[END_REF][START_REF] Wüthrich | Asymptotic value-at-risk estimates for sums of dependent random variables[END_REF][START_REF] Alink | Diversification of aggregate dependent risks[END_REF][START_REF] Alink | Diversification for general copula dependence[END_REF].

	A distribution can satisfy a stronger property called
	regular variation which implies subexponentiality and
	hence the asymptotics above. Examples of regularly
	varying distributions are Cauchy, Fréchet, and Pareto
	distributions [10]. The lognormal and heavy-tailed Weibull
	distributions are subexponential but not regularly vary-
	ing.

Acknowledgements

This work was supported under the Australian Research Council's Discovery Projects funding scheme (DP180101602). PJL was previously supported by an Australian Government Research Training Program Scholarship and by the Australian Research Council Centre of Excellence for Mathematical & Statistical Frontiers (ACEMS), under grant number CE140100049. Now PJL is supported by the DAMI -Data Analytics and Models for Insurance -Chair under the aegis of the Fondation du Risque, a joint initiative by UCBL and BNP Paribas Cardif.