UPR Pascale Dollfus

Peopling and migrations: some remarks based on research in Ladakh (India's western Himalayas) 1

Located in Northern India at the western end of the Tibetan plateau (Plate1) Ladakh is inhabited mainly by a population of Tibetan culture: in the eastern part (i.e. Leh District), a great majority of the inhabitants are followers of Tibetan Buddhism, while in the western part (i.e Kargil District) the inhabitants are mostly Muslims (Plate 2). The region covers some 60,000 square kilometres, except for Aksai Chin, which comes under Chinese administration. In 1901 it had a population of 60,000. Today its population amounts to 230,000 (as per the 2001 census). The density is only 4 persons per square kilometre.

Data on early peopling remain inadequate. No large-scale digs have been carried out and there has been very little systematic research to verify dates, locations and the stratification of settlements. In many parts of the country, rock art and petroglyphs have been found. These include illustrations of hunters with masses and bows, animals of various kinds (ibexes, felines, yaks, etc.), and many other objects (Plate 3). On the basis of stylistic comparisons with central Asian and Chinese glyphs, carvings and bronzes, Francfort et al. (1990) tentatively date the earliest petroglyphs to the Bronze Age 2 or even earlier to the Neolithic period. They attribute them to steppe groups linked to Chinese "Empires" and Persia. Yet this evidence is sketchy and requires more extensive study. Moreover, motifs may travel without their original populations. However, elements of Tibetan script associated with motifs such as stupas are sometimes found amongst these designs, so that we can be certain that these belong to the seventh or eighth century or later (Plate 4, Plate 5).

Just how, when, and where the people inhabiting Ladakh today originated remains a mystery. There are many varied and contradictory theories, and we still have a long way to go before settling these issues. Most scholars focus their attention on three main groups, who arrived from neighbouring regions, although the sequence and timing of inhabitation are still a subject of debate. 3 These peopling hypotheses, however, are largely worded in terms of the theoretical assumptions of past centuries. Some are based on naive western evolutionism while others reflect the desire of writers to link the origin of Himalayan peoples to those peoples mentioned in classical Greek or Indian sources. This later current is well illustrated 1 This paper is based on research carried out in Ladakh since 1980. I would like to thank Bernadette Sellers (CNRS, Villejuif) for revising the English. 2 Faced with the impossibility of distinguishing the Bronze Age from the Iron Age throughout the Tibetan world, some specialists consider it more advisable to use the expression "Metal Age", see Chayet, 1994. 3 See among others, Crook and Osmaston (1994), Francke (1980), Petech (1977), Rizvi (1983), Vitali (1996) and Vohra (1989a, 1989b). Using toponomy as a tool to trace the past may lead to erroneous conclusions. Indeed in Ladakh, like everywhere else, different places may bear the same name. For exemple, Sumdo, meaning "a junction where 3 [rivers, roads, etc] combine into one" is a very widespread name. Brang mkhar/Grang mkhar is found in Zangskar and in Spiti. The same is true for Rong ("valley") or even Rong chu brgyud. Therefore to claim along with Francke (1980: 28) that the so-called Dards once spread out to the East, a long way beyond Leh, because one of their hymns talked about Rong chu rgyud is just guess work.

by the instructive case of the so-called Dards who are commonly said to be among the earliest inhabitants of Ladakh. In fact no group refers to itself as "Dards". As Clarke (1977) demonstrated in an insightful paper, this name was largely coined by 19th-century travellers and philologists. In Western literature the Dards have been associated with the Daradrai of Ptolemy (Derdai, Dardae…) and the legends of "gold-digging ants" mentioned among other classical authors by Herodotus and Megasthenes who was Alexander's Ambassador to India. The term "Dard" itself appears to derive from a Persian root for the word "fierce", and is a generic term rather than a specific name for a population. It actually applies to various peoples speaking Indo-Iranian languages and inhabiting the north-western Himalayas.

In this paper, I attempt to illustrate the possible pitfalls of taking what is commonplace at face value. As an anthropologist, in order to create a narrative of the past, I look beyond the textual material for clues in the practices engaged in and in oral traditions. I have chosen 3 of them related to peopling and migrations to illustrate my sentiments: 1 -Mountains act as barriers. Isolated behind high ranges, mountain peoples are more archaic and mix less than those inhabiting plains and valley floors. 2 -Farming on barren land such as in Ladakh depends on large-scale waterworks calling not only for great hydraulic expertise, but a great workforce (or to use a Marxist vocabulary, requiring a high level of production forces), then government control for ensuring a proper distribution of the water, that leads to an absolutist managerial state (Karl Wittfogel's theory on "hydraulic civilizations"4 ) 3 -Nomadism is but primitive survival, a stage in evolution leading to agriculture and sedentarisation Mountains act as barriers... Embedded in the Karakoram range in the north-west, the Himalayas in the south-west, and the Trans-Himalayas at its core, Ladakh is a region characterized by high mountain ranges interspersed with high valleys, with no part of it at less than 2,600 metres in altitude (Plate 6). According to local traditions5 , it was formerly an uninhabited land, "an empty country", that was gradually "filled" with people from all the neighbouring countries. Most stories involve hunters who discovered the place by chance on hunting trips and really liked it. In many cases an arrow was thrown to find a spring and the best place to settle. Then grains and cuttings of wood were planted to test the soil's fertility. If they budded, the decision was taken to establish a village or a fortress on this spot.

The accounts generally concur to situate the founding of the first village and the beginning of barley cultivation in the western part of Ladakh. Stories come in various forms but may be summarized as follows:

Three brothers from Gilgit went hunting together. Exploring the mountains, they discovered a valley brimming with game. As it was already late, they decided to spend the night there. When they settled down to sleep, several grains of barley fell out of the straw stuffing of their shoes. Some months later, since their meat supplies were exhausted, they remembered this area with its abundant wildlife and decided to return there. At the place where they had previously stopped to rest, there was a field of barley ready to be harvested, which had sprung up from the grains they had dropped months before. Delighted by this fertile land where there was plenty of wildlife, they chose to settle there permanently. They started to cultivate the land and built houses. As time went by, a thousand people decided to move eastwards. They followed the caravan route, which from one pass to another, linked Kashmir to Leh, then proceeded to Nubra by another well-known route used by caravans because it remained open over longer periods than other routes.

A song describing the route travelled during their migration movement eastward towards Ladakh goes like this: One thousand of their members proceed to Dras. They arrive at Mulbek. At Kharbu the wind makes the sound gor gor. They arrive at Yuru Singé Gangs. They stop at a small field at Teya. They arrive at a rocky-stony place at Tinmosgang. They cross the Juniper grove at Hemis shukpa. They reach Basgo, and then Leh where there is a stone. 6 Going past Shey at not a very good place, they arrive at Sakti where food is taken. En route to Nubra past a deep gorge. They cross the water over a "hanging bridge". They arrive at Nubra, at a green-grassy place. Then reach a natural spring ... 7 Over the years, groups of people from all neighbouring countries followed: from Kashmir in the West, the cis-Himalayan valleys in the South, the oases of central Asia in the North and Tibet in the East. They founded many villages. The country was called Mang yul, a contracted form of mi mang po yul, "The Place of Many People".

All these migrants brought their own techniques and savoir-faire. The three brothers from Gilgit are said to have introduced agriculture, including the use of the irrigation system and especially wood gutters forming galleries round the cliff; Tibetan nomads, so it goes, brought tents made of yak hair as well as large-scale pastoralism involving yaks, goats and sheep; the Newars from Nepal are said to have introduced metalwork. (The same is true for gods, who are largely conceived as newcomers from afar, set up in the village by the enactment of appropriate rituals.)

A similar story is told about Zangskar. In the beginning, before there were any people, there were glaciers on all the high peaks. There used to be lakes and jungles. The region was covered in thick forest. There was not yet a single race of people, so it was said. And then slowly, in turn, the different sorts of men arrived. The first to arrive were hunters. They came and settled, each on a separate mountain. They lived in caves or built small huts. Then they began to live communally and built villages. 8 Throughout the accounts of settlements, emphasis is laid on the fact that: -Migration is not a rush large-scale movement. It is a to and fro pendulum process, rather than a unilinear process that never reverses itself or heads off in any other direction. As F. Jacquesson nicely points out, real peopling and population movements are not like arrows on maps. A migration flow generally produces at least one counter-flow. One wave after the other, migration takes centuries, and the presumed entity that has moved is rarely identical through the movement.

-High mountain ranges are not impenetrable barriers. Since the main rivers flow through impassable gorges or in alluvial plains subjected to floods, trade and pilgrimage routes do not generally follow riverbeds, but cross passes. Indeed Ladakh has been "a crossroads of High Asia" for centuries9 , and is inhabited by various populations that came from neighbouring countries across the mountains. Last but not least, remote does not mean isolate. People inhabiting the most distant areas are not necessarily the most isolated and "archaic" ones. On the Changthang plateau, nomadic pastoralists, for example, used to be long-distance traders. Men travelled to neighbouring regions with large flocks of sheep and goats loaded with salt and sheep wool, which they exchanged for food grain, but also tobacco and other necessities. While wool was a local product, they collected salt from lakes in western Tibet and traded it far away (Rizvi 1999: 122). Even today, these people move around more often and to farther away places -for business and/or religious reasons -than inhabitants of central Ladakh who are, however, not far from the town of Leh, the district capital, where buses, jeeps, trucks and aeroplanes are centred.

2 -Farming on barren land depends on large-scale waterworks ... Located behind the Great Himalayan range denying entry to monsoon clouds, Ladakh is a high-altitude cold desert with an extremely short growing season (Plate 7). Mean annual precipitation is very low, usually less than 100 millimetres per year. Farming is impossible without irrigation. Hence, despite Ladakh's large territorial expanse, less than 200 km 2 is farmed. The main source of water is the winter snow-fall in the mountains which runs off in summer and is diverted into built terraces where the soil is formed. Many observers have noticed the irrigation channels built over past centuries, which follow the contours of the hillsides high above the valley bottoms, and have underlined the contrast between the lush cultivated zones and the barren region outside these oases (Plate 8, Plate 9). Indeed the greater part of the population is settled on the edge of lateral valleys watered by snow-fed streams. According to one already old but still applied theory, these oases would be the work of the above mentioned "Dards" from neighbouring Baltistan and Gilgit who "especially exhibited an extraordinary skill in the construction of watercourses along almost inaccessible cliffs" (Francke 1980, 47) 10 . Later came the Tibetans from central Tibet who drove them out, but made the most of their hydraulic expertise.

In a recent piece of work Valérie Labbal (2001) challenged this view. Without denying the role played by these migrants in landscape patterns, especially in Lower Ladakh, she argues that practising agriculture in such an environment does not necessarily call for sophisticated knowledge. She suggests that people in central Ladakh first colonized the stream valleys and their alluvial fans by transforming natural gullies or ancient stream beds into distribution canals and by flooding pieces of land between them (Plate 10). These irrigation networks were progressively built without any collective planning or coercive political institutions. Moreover, cultivated terraces in Ladakh are not carved out of an abrupt stony mountainside as in Peru or in the Cévennes, but are shaped out of detritic material on valley floors and in alluvial fans. They are not the result of hard work, of earthmoving, but of a slow gradual process where man's contribution is partly indirect.

At the beginning of the 19 th century, W. Moorcroft and G. Trebeck (1993, 162) depicted the process as follows:

"In solitary spots, remote from human habitation, stone dykes may be observed crossing the sloping sides of mountains near their base: these are constructed by the peasants to assist the deposit of soil and gravel by the melting snows, and they are thus left for many years, perhaps for some generations, for the operation of natural agency to prepare for the labour of man, and the more ready conversion of an abrupt and sterile declivity into an accessible flight of terraces of cultivation."

This hypothesis which is backed up by a lexical analysis disputes the validity of Wittfogel's theory whereby the development of irrigation work in such a barren environment led to the use of mass labour, to an organizational hierarchy for coordinating and directing its activities, then to government control for ensuring a proper distribution of the water, resulting in an absolutist managerial state. According to V. Labbal, in Ladakh the royal authority would have established the collective order much later, in particular through the creation of irrigation networks in the Indus plain, but in no way would it have controlled the distribution of water. Throughout the country, one can see abandoned fields and the remains of dwellings at the high point of the valley near the glacier. Early settlements tended to be fortress villages built on a cliff, providing protection against marauders. The tendency for farming families to descend to habitations on the valley floor or alluvial plain developed later. In many places villages only moved down to the valley floor in the 19 th century (K. Gutschow 1998).

-Nomadism is but primitive survival, a stage in evolution leading thence to agriculture and sedentarisation

In literature, even recently, nomadism is commonly described as an evolutionary stage in human history, a phase following hunting-gathering and leading thence to agriculture and sedentarisation. However, archaeological and historical evidence has shown a more complex story. In most parts of the world, Africa excepted, agriculture seems to have come earlier than pastoralism. Furthermore, anthropological data demonstrate that taking up nomadism or leaving it in order to settle are processes both common and very widespread. All the same, as Salzman (2002) remarks, we do usually tend to think, in terms of evolutionary assumptions, of a "rise and fall" rather than an alternation or natural cycles. In thinking about nomadism, we are still influenced, consciously or unconsciously, by evolutionary schemes or by Marxist historical schemes, and we may assume a "natural development" from nomadic life to settled life.

In Tibet, as Matthew Kapstein (2006: 11) points out, "in the popular imagination, including not least the Tibetan imagination, nomads are generally regarded as the archetypical Tibetans." However, both Neolithic sites and the legendary and historical traditions which locate the birth of Tibetan society in the fertile valleys of Southern Tibet, suggest the primacy of agriculture in the development of Tibetan civilization. 11 The same is true in Ladakh. Nomadic pastoralists are said to have moved into Ladakh from the adjoining Tibetan Changthang or from the region known as Zhang-zhung lying on the western edge of the Tibetan plateau. Some scholars suggest that they migrated down the Indus valley on account of a shortage of food later on12 . The bulk of the population settled and became sedentary farmers and absorbed later migrations, while the residue formed the groups of nomads still living on the southeastern edge of Ladakh. Others have speculated that the descendants of the Tibetan nomadic tribes had been pushed eastwards, by Indo-Aryan farmers, to their current abode, the highlands of eastern Ladakh where the extreme climate and altitude precluded agriculture as an economic alternative. Whatever the chosen scenario, it is generally agreed today that present-day Ladakhi nomads trace their origins back to ancient Tibetan tribes and have always been nomads. Taking into account the case of the Kharnak people, I would like to challenge this view.

The Kharnakpa13 , "Those of the Black Castle", are one of the three nomadic pastoralist groups living in extreme conditions in the southeastern edge of Ladakh at an average height of 4,500 metres above sea level. The three groups share the same way of life raising goats, sheep and yaks on natural pastureland (Plate 11, Plate 12, Plate 13). They make full use of the animals, using wool, hair and tendons to make clothes, tents, blankets, carpets, ropes and pack bags; the hide and stomachs for boots and containers, dung for fuel; and of course, they gain sustenance from the meat and milk (Plate 14). They produce for their own subsistence but also for commercial purposes. In the past, men traded salt and wool, which they exchanged for barley. Nowadays they sell livestock products, such as pashmina wool14 -the winter undercoat of a variety of domestic goat -and culled animals for meat, mostly to Leh (primarily Muslim) traders, and they purchase foodstuffs, such as tea and food grains which are part of their staple diet, as well as many of the supplies and equipment they use (clothes, kitchen equipment, torches and radios ...).

They migrate from place to place throughout the year to gain access to resources, such as grass and water, which are sparse in any given location (Plate 15, Plate 16, Plate 17). They usually break and make camp about 6 to 10 times a year, making only short moves between a series of camping grounds (Plate 18, Plate 19, Plate 20, Plate 21). Nomadism does not mean wandering. Each group has well-established patterns of grazing migrations and claims exclusive access to grassland at a specific time of the year.

It is not known when Kharnak was first settled. The place is very much a backwater of mainstream events in Ladakh and does not figure in historical texts and records. For lack of archaeological and epigraphic evidence, or any reliable document, we have no choice but to stick to oral legends and local traditions. Though incapable of perfectly reflecting the reality, they give some interesting insight into the past as perceived by the Kharnakpa themselves. While the landscape is saturated with the past, most inhabitants could only go back as far as their grandparents' generation. Beyond the memories of those still alive, the past sinks rapidly into an undefined "before", situated "many years ago".

The name Kharnak is derived from the place name mkhar nag, "the Black Castle". The fortress in question stands in ruins high on a spur above the eponymous valley where shepherds sometimes graze their flocks and several families cultivate patches of land sown with barley (Plate 22). When questioned about its founder, people simply answer that it does not have anything to do with their own history, which goes like this:

At the very beginning, there were only two households -Togoche and Chukpo Lobde -in the happy land of Kharnak, both living within the Kharnak river watershed but at different locations. These two families did not like each other and did not co-operate at all. Once upon a time the country was hit by a winter of unheard-of harshness. For weeks on end the weather was freezing cold and there were abundant snowfalls. All the goats and sheep owned by the Chukpo Lobde family died of starvation. Bereft of animal herds, the head of the household and his son had no work. Eventually they decided to go to see what had happened to the Togoche family. They did not take "the low road" along the river, but walked along "the high road" through the deserted glen of Kugshel. When they arrived at the top of the pass, they looked down into the valley. Everywhere the snow was strewn with bones. Driven by hunger, wild dogs had greedily eaten all the small livestock. Not a single animal had survived. United in their sorrow, the two families decided to help each other. For days and nights they sang and danced with no respite. With all the dead animals lying everywhere, "dead meat" was plentiful. On the other hand it was difficult to find any barley to brew the beer essential to any feast. With each successive grain some success was achieved. Wedding celebrations were promptly organized, and a year later, children were born. They in turn had children and their children had children. One generation after the other, the number of households gradually increased. The herds grew accordingly. In search of more pastures to feed their flocks, the Kharnakpa went eastwards. At the time new land was conquered with no fighting or bloodshed. All these valleys were "empty". One simply had to go and take them. There was no frontier, no army, no Tibetan refugee, no gurkha, as local people called the migrants working for the Public Works Department, whatever country they belonged to. There were only a few rkyang (Tibetan wild ass), thus no forage competition with livestock. Grass was plentiful. But when war with the Chinese (Aksai Chin conflict in 1962) broke out, everything changed.

This story portrays the founders of Kharnak as shepherds living on goat and sheep farming, animals that still constitute more than 90% of the livestock population, but it does not tell us when and from where they came. Neither black-yak hair tents nor even yaks are mentioned. When questioned about the legendary and historical traditions of the other nomadic groups living nearby, who attribute to mythical heroes the creation of the first humans, the mountains around them, the sheep, goats, horses and yaks and everything people possess, the Kharnakpa claim that they have never heard of them.15 Just as they deny any link with their founding chief, said to be a Tibetan nobleman from Lhasa, who left the city because of a family rift and settled in the region approximately 600 years ago. 16 These divergent traditions are not the only clue throwing doubt on the common origin and path followed by the nomadic pastoralists of Ladakh. There is scattered evidence of another type, such as dwelling patterns, nomenclature, names of lineage, and so on.

Indeed, when I was in Kharnak, I was struck by the abundance of permanent structures -pens, storerooms for forage and fuel, sheds, watermills for grinding grains, one-room houses -that make the Kharnakpa's camps easily recognizable from any other (Plate 23, Plate 24). (Interestingly enough people constantly speaks of "the house", "house name" and "household" even when this involved tents.) The black tents are pitched over a 70-to 90-centimetre deep rectangular pit dug in the ground (Plate 25). A substantial stonewall built around its perimeter provides shelving and niches where food and other belongings are stored. In the centre, the fireplace lies on the sunken floor delimited by stones. At the rear, there is a small platform that serves as the family altar, and below is a storage place for butter (Plate 26, Plate 27). Next to its dwelling-place, each family sets up large stones around which they tie up their yak calves or to which the women fix their back-strap looms and the men their fixed-heddle looms. (Plate 28) Every household also possesses enclosures to pen sheep and goats during the night (Plate 29).

In some encampments, one may also see watermills for grinding grain on the stream running nearby (Plate 30, Plate 31). All these infrastructures are private. They belong to the people who built them, even if they moved away years ago and settled elsewhere. Indeed to camp, even temporarily, on a pitch, which is not one's own is punishable. Moreover, the possibility of moving away from people with whom you do not get along, either by pitching your tent at the opposite end of an encampment or moving to another encampment, described by some authors as one of the great psychological advantages of being a nomad, is not possible here. Furthermore, whatever the space available, the whole community occupies a deliberately small territory. In summer camps, people pitch tents side by side in a line, with the guy-lines of one literally overlapping the next (Plate 32). In winter quarters, tiny stone houses sharing a common wall are agglutinated in compact settlements (Plate 33, Plate 34, Plate 35). In fact, the first Westerners passing through on hunting trips in the late 19 th -early 20 th centuries preferred to call the settlements they came by villages rather than camps. Visiting Kharnak in 1931, Brig. Magan, for instance, noted in his diary: "the dwellings for the most part have walls of loose stone, built upon the other without any form of cement, and are roofed with stout sacking of closely woven wool which here and there is supplemented by the skins of animals. These houses are ornamented with poles from which are hung yak's tails and religious flags." 17 Building houses is not a recent phenomenon due to the development of roads and transportation. As far back as people remember, they have always built what they needed. Unlike the nomads from western Tibet who hire villagers in the summer to put up pens, prayer walls, storerooms and houses because they find these tasks unpleasant (Goldstein and Beall 1990, 62), the Kharnakpa build their dwellings themselves using only locally available construction material: earth, stones, bushes and wood. (Vegetation is scarce, yet there is some woodland -tamarisk, willow, poplar and juniper -in the lower part of the Kharnak valley, locally known as Rong, "the Gorge")(Plate 36, Plate 37).

Indeed in nearby Rusphu too, stone-dwellings have existed for decades, but contrary to Kharnak they were the prerogative of the clergy, who owned monasteries, and of the aristocracy. Thus the hereditary chief ruling over the region, not only had a home referred to as "a castle", but "crude houses" in main summer and winter encampments. 18 Besides herding, the Kharnakpa make a small part of their living from agriculture and haymaking. They grow barley and peas on patches of land in areas where irrigation is possible (Plate 38), and in September, just before harvesting, they cut grass in specific valleys that have been left ungrazed throughout the growing season (Plate 39, Plate 40, Plate 41). While recently, most fields were poorly tended and gave a low yield (2x to 3x), I was told that in former times, their forefathers used to get higher yields, up to 7-8x. At that time, so it is said, people worked hard in the fields: the irrigation systems were well-maintained, manure carefully spread on the soil and fields almost weed-free. However, since the closing of the borders with Tibet and the development of the lucrative pashmina wool trade, people prefer to comb out goats (Plate 42) than to grow barley on stony grounds (Plate 43). Consequently, the number of goats has increased many times over, while farming has reduced dramatically. Due to economic changes and government policy (forage supply, Public distribution system 19 ) none of the fields has been sown over the last three years.

It has been argued that winter quarters, haymaking and agriculture may be seen as the first step away from nomadic pastoralism towards sedentary farming. When dealing with the Kharnakpa, the process seems to me to be the opposite. Historical traditions together with dwelling patterns, agricultural and gathering practices, and clues in the form of "house names", worshipped deities and lineage belonging, rather suggest a shift from a form of mobile herding to one of a nomadic way of life, or in other words, a transformation from good shepherd to nomadic herder. My sentiment is that the earliest inhabitants of Kharnak were agro-pastoralists practising transhumance, with a permanent homestead and base. They perhaps began to cultivate some patches of land nearby to get grain, especially for making beer, but also winter fodder. (When the crop failed to ripen, the husks were harvested when still green and served as fodder for the cattle and horses.) When both the human and animal population increased, they crossed the watershed in search of more pastures to feed their flocks. Although the nomads occupy a well-defined territory, it is not a static and bounded entity. It changes over time depending on natural hazards, population increases, but also in response to economic or political events. Furthermore nomadism is not determinatively linked to a particular physio-biotic environment. The Kharnakpa occupy habitats that their forefathers previously exploited differently. Indeed a dozen Tibetan refugee families, who settled a few kilometres away, do not live as nomads. They are not involved in any agricultural work. Hence animal breeding is treated as a far nobler occupation, yet they live in two villages -a summer and a winter settlement -in modest accommodation made of mud bricks with two rooms or more.

Adopting the life of a nomadic pastoralist or settled agriculturist is not set in stone. In the case of Kharnak over the last fifteen years, dozens of families have moved from Changthang reducing the mobile community by 80 percent. In many instances, these migrants have sold up their livestock and settled on the outskirts of the district capital where they have established a permanent urban settlement known as Kharnakling ("Kharnak's island") and referred to as "a colony" by its inhabitants (Plate 44, Plate 45, Plate 46, Plate 47, Plate 48)20 . Sedentarization is not always a large-scale collective event. Quite frequently, individuals or families shift between a nomadic and sedentary lifestyle and back again depending upon circumstances. Nomads who have been forcibly settled often return to nomadism21 . Thus in Iran, the Yomut Turkmen, whose land had been taken as Crown land, returned to a nomadic, pastoral or farming life when Reza Shah abdicated. They tore down the houses in which they had been forced to live and again built felt-covered yurts (Salzman 2002). To conclude, being settled or nomadic, rather than being of two different types, are better thought of as opposite ends of a continuum with many gradations of stability and mobility. I do not think that this paper will conclusively identify the truth of the matter, but I hope to have demonstrated the benefit of cross-referencing data from different fields. Population genetics, linguistics, geography, archaeology, textual histories, oral legends and traditions, and current practices all afford different perspectives of the past. Taken together, they may actually clarify a history, which is itself uncertain.

According to Karl Wittfogel (Oriental Despotism, 1957), irrigation is the primary cause of the formation of coercive political institutions.

See among others[START_REF] Francke | Ladakh: The mysterious land -History of western Himalayas[END_REF][START_REF] Francke | Ladakh: The mysterious land -History of western Himalayas[END_REF]);[START_REF] Kaplanian | Les Ladakhi du Cachemire : Montagnards du Tibet occidental[END_REF];Kloos (2004); Pathak (1996);Vohra (1989a & b).

Souvenir of a game played, which required lifting a stone.

Adapted fromVohra (1989a,

[84][85] 8 For a complete translation of this legendary history recited by Karsha Lonpo Sonam Angchug, see K.Gutschow (1998, 348).

In JanetRizvi's phrase (1983).

See alsoRibbach (1955: 30-32), and more recentlyVohra (2000)

See Chayet (1994),Kapstein (2006);[START_REF] Stein | La civilisation tibétaine. Paris : Le Sycomore-L'Asiathèque. VITALI, Roberto 1996 -The kingdoms of Gu.ge Pu[END_REF].

A. Cunnningham (1977) believes them to have extended westwards as far as Gilgit.

The suffix -pa is added to the end of place names to indicate a person who comes from that place.

Known locally as lena, this fine and luxurious fibre comes from the underbelly of the Capra hircus goat, which grows this extra layer each winter to protect itself against the rigours of this extreme climate.

Ahmed (2002, 31-34)

For more details, seeAhmed (2002, 34-38).

Magan c.1931. See also H. Z. Darrah

18 For a description of his "castle" and houses, see[START_REF] Francke | Ladakh: The mysterious land -History of western Himalayas[END_REF],[START_REF] Koelz | -Diary of the 1931 expedition to Western Tibet[END_REF] and, more recentlyAhmed 2002. 19 The Public Distribution System (PDS) refers to a network of retail outlets (popularly known as ration stores) through which the government sells subsidizied rice, wheat and kerosene at fixed prices lower than those of the market. The conditions govern the sale. First, the buyer must possess a ration card. Second, purchases are subject to quota.

On this rural-to-urban migration of Kharnak nomadic pastoralists in Leh, see[START_REF] Dollfus | Pascale 2004 -Vers une sédentarisation univoque ? Le cas des éleveurs nomades de Kharnak, Ladakh, Himalaya occidental indien[END_REF][START_REF] Goodall | -Rural-to-urban Migration and Urbanization in Leh, Ladakh. A case study of three nomadic pastoral communities[END_REF]

There are two processes of sedentarization. One is a spiral of failure. Pastoralists settle as propertyless tenants in villages or towns to become agricultural labourers on the land of others or daily wagers. The other process is the accumulation and conversion of animals into a more stable form of wealth, such as land. These well-to-do pastoralists become landowners and, as landlords, they receive rent from agriculturalists working the land.(See Salzmann 2002).


Véronique Maume-Deschamps
email: veronique.maume@univ-lyon1.fr

AND KHALIL SAID Didier Rullière
email: didier.rulliere@univ-lyon1.fr

EXTREMES FOR MULTIVARIATE EXPECTILES

Keywords: Mathematics Subject Classification. 62H00, 62P05, 91B30 Risk measures, multivariate expectiles, regular variations, extreme values, tail dependence functions

, a new family of vector-valued risk measures called multivariate expectiles is introduced. In this paper, we focus on the asymptotic behavior of these measures in a multivariate regular variations context. For models with equivalent tails, we propose an estimator of extreme multivariate expectiles, in the Fréchet attraction domain case, with asymptotic independence, or for comonotonic margins.

Introduction

In few years, expectiles became an important risk measure among more used ones, essentially, because it satisfies both coherence and elicitability properties. In dimension one, expectiles were introduced by [START_REF] Newey | Asymmetric Least Squares Estimation and Testing[END_REF] [START_REF] Newey | Asymmetric Least Squares Estimation and Testing[END_REF]. For a random variable X with finite order 2 moment, the expectile of level α is defined as

e α (X) = arg min x∈R E[α(X -x) 2 + + (1 -α)(x -X) 2 + ],
where (x) + = max(x, 0). Expectiles are the only risk measure satisfying both elicitability and coherence properties, according to Bellini and Bignozzi (2015) [START_REF] Bellini | On elicitable risk measures[END_REF].

In higher dimension, one of the proposed extensions of expectiles in [START_REF] Maume-Deschamps | Multivariate extensions of expectiles risk measures[END_REF] are Matrix Expectiles. Consider a random vector X = (X 1 , . . . , X d ) T ∈ R d having order 2 moments, and let Σ = (π ij ) 1≤i,j≤d be a d × d real matrix, symmetric and positive semi-definite such that i ∈ {1, . . . , d}, π ii = π i > 0. A Σ-expectile of X, is defined as e Σ α (X) ∈ arg min

x∈R d E[α(X -x) T + Σ(X -x) + + (1 -α)(X -x) T -Σ(X -x) -],
where (x) + = ((x 1 ) + , . . . , (x d ) + ) T and (x) -= (-x) + . We shall concentrate on the case where the above minimization has a unique solution. In [START_REF] Maume-Deschamps | Multivariate extensions of expectiles risk measures[END_REF], conditions on Σ ensuring the uniqueness of the argmin are given, it is sufficient that π ij ≥ 0, ∀i, j ∈ {1, . . . , d}. We shall make this assumption throughout this paper. Then, the vector expectile is unique, and it is solution of the following equations system (0.1)

α d i=1 π ki E[(X i -x i ) + 1 1 {X k >x k } ] = (1 -α) d i=1 π ki E[(x i -X i ) + 1 1 {x k >X k } ]
, ∀k ∈ {1, . . . , d}.

In case π ij = 1 for all (i, j) ∈ {1, . . . , d} 2 , the corresponding Σ-expectile is called a L 1 -expectile. It coincides with the L 1 -norm expectile defined in [START_REF] Maume-Deschamps | Multivariate extensions of expectiles risk measures[END_REF].

In [START_REF] Maume-Deschamps | Multivariate extensions of expectiles risk measures[END_REF] it is proved that, lim α-→1 e Σ α (X) = X F , and lim α-→0 e Σ α (X) = X I , where X F ∈ (R ∪ {+∞}) d is the right endpoint vector (x 1 F , . . . , x d F ) T , and by X I ∈ (R ∪ {-∞}) d is the left endpoint vector (x 1 I , . . . , x d I ) T of the support of the random vector X.

The multivariate expectiles can be estimated in the general case using stochastic optimization algorithms. The example of estimation by the [START_REF] Robbins | A stochastic approximation method[END_REF] [START_REF] Robbins | A stochastic approximation method[END_REF] algorithm, presented in [START_REF] Maume-Deschamps | Multivariate extensions of expectiles risk measures[END_REF], shows that for extreme levels, the obtained estimation is not satisfactory in term of convergence speed. This leads us to the theoretical analysis of the asymptotic behavior of multivariate expectiles. Asymptotic levels i.e. α → 1 or α → 0 represent extreme risks. Since the solvency thresholds in insurance are generally high (e.g. α = 0.995 for Solvency II directive), the study of asymptotic behavior of risk measures is of natural importance. The goal of this work is to establish the asymptotic behaviour of multivariate expectiles. The study of the extreme behaviour of risk measures in a multivariate regular variation framework is the subject of a balk of works, let us mention as examples, Embrechts

The MRV Framework

Regularly varying distributions are well suited to study extreme phenomenons. Lots of works have been devoted to the asymptotic behavior of usual risk measures for this class of distributions, and results are given for sums of risks belonging to this family. It is well known that the three domains of attraction of extreme value distributions can be defined using the concept of regular variations (see [11; 23; 8; 6]). This section is devoted to the classical characterization of multivariate regular variations, which will be used in the study of the asymptotic behavior of multivariate expectiles. We also recall some basic results on the univariate setting that we shall use.

1.1. Univariate regular variations. We begin by recalling basic definitions and results on univariate regular variations.

Definition 1.1 (Regularly varying functions). A measurable positive function f is regularly varying of index ρ at a ∈ {0, +∞}, if for all t > 0,

lim x→a f (tx) f (x) = t ρ ,
we denote f ∈ RV ρ (a).

A slowly varying function is a regularly varying function of index ρ = 0. Remark that f ∈ RV ρ (+∞) if and only if, there exists a slowly varying function at infinity, L ∈ RV 0 (+∞) such that f (x) = x ρ L(x). Theorem 1.2 (Karamata's representation, [START_REF] Resnick | REGULAR VARIATION AND POINT PROCESSES[END_REF]). For any slowly varying function L at +∞, there exist a positive measurable function c(•) that satisfies lim Throughout the paper, we shall consider generalized inverses of non-decreasing functions f :

f ←-(y) = inf{x ∈ R, f (x) ≥ y}.
Lemma 1.3 (Inverse of RV functions [START_REF] Resnick | PROBABILISTIC AND STATISTICAL MODELING[END_REF]). Let f be a measurable non-decreasing function defined on R + , such that lim

x→+∞ f (x) = +∞. Then f ∈ RV ρ (+∞) if and only if f ←-∈ RV 1 ρ (+∞)
, for all 0 ≤ ρ ≤ +∞, where we follow the convention 1/0 = ∞ and 1/∞ = 0. Lemma 1.4 (Integration of RV functions (Karamata's Theorem)), [START_REF] Mikosch | Modeling dependence and tails of financial time series[END_REF]). For a positive measurable function f , regularly varying of index ρ at +∞, locally bounded on [x 0 , +∞) with x 0 ≥ 0

• if ρ > -1, then lim x→+∞ x x0 f (t)dt xf (x) = 1 ρ + 1 , • if ρ < -1, then lim x→+∞ +∞ x f (t)dt xf (x) = - 1 ρ + 1 .
Lemma 1.5 (Potter's bounds [6]). For f ∈ RV ρ (a), with a ∈ {0, ∞} and ρ ∈ R. For any 0 < < 1 and all x and y sufficiently close to a, we have

(1 -) min x y ρ- , x y ρ+ ≤ f (x) f (y) ≤ (1 + ) max x y ρ- , x y ρ+ .
Many other properties of regularly varying functions are presented e.g. in [6].

1.2. Multivariate regular variations. The multivariate extension of regular variations is introduced in [START_REF] De Haan | Limit theory for multivariate sample extremes[END_REF]. We denote by µ n v -→ µ the vague convergence of Radon measures as presented in [START_REF] Kallenberg | Random measures, 3rd. Edition[END_REF]. The following definitions are given for non negative random variables.

(x) = +∞ such that (1.1) uP X b(u) ∈ • υ -→ µ X (•) as u -→ +∞.
There exist several equivalent definitions of multivariate regular variations which will be useful in what follows.

Definition 1.7 (MRV equivalent definitions). Let X be a random vector on R d , the following definitions are equivalent:

• The vector X has a regularly varying tail of index θ.

• There exist a finite measure µ on the unit sphere S d-1 , and a normalization function b

: (0, ∞) -→ (0, ∞) such that (1.2) lim t-→+∞ P X > xb(t), X X ∈ . = x -θ µ(.),
for all x > 0. The measure µ depends on the chosen norm, it is called the spectral measure of X. • There exist a finite measure µ on the unit sphere S d-1 , a slowly varying function L, and a positive real θ > 0 such that

(1.3) lim x-→+∞ x θ L(x) P X > x, X X ∈ B = µ(B),
for all B ∈ B(S d-1 ) with µ(∂B) = 0.

From now on, MRV denotes the set of multivariate regularly varying distributions, and MRV(θ, µ) denotes the set of random vectors with regularly varying tail, with index θ and spectral measure µ. From (1.3), we may assume that µ is normalized i.e. µ(S d-1 ) = 1, which implies that X has a regularly varying tail of index -θ. On another hand, lim

x-→+∞ P X X ∈ B X > x, = lim x-→+∞ P X > x, X X ∈ B P ( X > x) = lim x-→+∞ x θ L(x) µ(B)x -θ L(x) = µ(B),
for all B ∈ B(S d-1 ) with µ(∂B) = 0. That means that conditionally to { X > x}, X X converges weakly to µ. The different possible characterizations of the MRV concept are presented in [START_REF] Mikosch | Modeling dependence and tails of financial time series[END_REF].

1.3. Characterization using tail dependence functions. Let X = (X 1 , . . . , X d ) be a random vector. From now on, F Xi denotes the survival function of X i . In this paper, we use the definition of the upper tail dependence function, as introduced in [START_REF] Klüppelberg | Semi-parametric models for the multivariate tail dependence functionthe asymptotically dependent case[END_REF].

Definition 1.8 (The tail dependence function). Let X be a random vector on R d , with continuous marginal distributions. The tail dependence function is defined by

(1.4) λ X U (x 1 , . . . , x d ) = lim t-→0 t -1 P( FX1 (X 1 ) ≤ tx 1 , . . . , FX d (X d ) ≤ tx d ),
when the limit exists.

For k ≤ d, denote by X (k) a k dimensional sub-vector of X, C (k) its copula and C (k) its survival copula. The upper tail dependence function is

(1.5) λ k U (u 1 , . . . , u k ) = lim t-→0 + C(k) (tu 1 , . . . , tu k ) t ,
if this limit exists. The lower tail dependence function can be defined analogically by

λ k L (u 1 , . . . , u k ) = lim t-→0 + C (k) (tu 1 , . . . , tu k ) t ,
when the limit exists. In this paper, our study is limited to the upper version as defined in (1.5).

We assume that X has equivalent regularly varying marginal tails, which means: H1: FX1 ∈ RV -θ (+∞), with θ > 0. H2: The tails of X i , i = 1, . . . , d are equivalent. That is for all i ∈ {2, . . . , d}, there is a positive constant c i such that lim

x-→+∞ FXi (x) FX1 (x) = c i .
H1 and H2 imply that all marginal tails are regularly varying of index -θ at +∞.

The following two theorems show that, under H1 and H2, the MRV character of multivariate distributions is equivalent to the existence of the tail dependence functions. Theorem 1.9 (Theorem 2.3 in [START_REF] Li | Tail dependence for heavy-tailed scale mixtures of multivariate distributions[END_REF]). Let X = (X 1 , . . . , X d ) be a random vector in R d , with continuous marginal distributions F Xi , i = 1, . . . , d that satisfy H1 and H2. If X has a MRV distribution, the tail dependence function exists, and it is given by par

λ k U (u 1 , . . . , u k ) = lim x-→+∞ xP X 1 > b(x) u 1 c 1 -1/θ , . . . , X k > b(x) u d c d -1/θ
, for any k ∈ {1, . . . , d}.

Theorem 1.10 (Theorem 3.2 in [START_REF] Weng | Characterization of multivariate heavy-tailed distribution families via copula[END_REF]). Let X = (X 1 , . . . , X d ) be a random vector in R d , with continuous marginal distributions F Xi , i = 1, . . . , d that satisfies H1 and H2. If the tail dependence function λ k U exists for all k ∈ {1, . . . , d}, then X is MRV, its normalization function is given by b

(u) = 1 FX 1 ←- (u) and the spectral measure is µ([0, x] c ) = d i=1 c i x -θ i - 1≤i<j≤d λ 2 U (c i x -θ i , c j x -θ j ) + • • • + (-1) d+1 λ d U (c 1 x -θ 1 , . . . , c d x -θ d ).
By construction of the multivariate expectiles, only the bivariate dependence structures are taken into account. We shall use the functions λ (Xi,X k ) U , for all (i, k) ∈ {1, . . . , d} 2 . In order to simplify the notation, we denote it by λ ik U . If the vector X has an MRV distribution, the pairs (X i , X j ) have also MRV distributions, for any (i, j) ∈ {1, . . . , d} 2 . So, in the MRV framework, and under H1 and H2, the existence of functions λ ik is insured. In addition, we assume in all the rest of this paper that these functions are continuous.

Fréchet model with equivalent tails

In this section, we assume that X satisfies H1 and H2 with θ > 1. It implies that X 1 belongs to the extreme value domain of attraction of Fréchet MDA(Φ θ ). This domain contains distributions with infinite endpoint x F = sup{x : F (x) < 1} = +∞, so as α -→ 1 we get e i α (X) -→ +∞ ∀i. Also, from Karamata's Theorem (Theorem 1.4), we have for i = 1, . . . , d,

(2.1) lim x-→+∞ E[(X i -x) + ] x FXi (x) = 1 θ -1 ,
for all i ∈ {1, . . . , d}.

Proposition 2.1. Let Σ = (π ij ) i,j=1,...,d with π ij > 0 for all i, j ∈ {1, . . . , d}. Under H1 and H2, the components of the multivariate Σ-expectiles e α (X) = (e i α (X)) i=1,...,d satisfy

0 < lim α-→1 e i α (X) e 1 α (X) ≤ lim α-→1 e i α (X) e 1 α (X)
< +∞, ∀i ∈ {2, . . . , d}.

Proposition 2.1 implies that distributions with equivalent tails have asymptotically comparable multivariate expectile components.

Before we prove Proposition 2.1, we shall demonstrate some preliminary results. Firstly, let X = (X 1 , . . . , X d ) T satisfy H1 and H2, we denote x i = e i α (X) for all i ∈ {1, . . . , d}. We define the functions l α Xi,Xj for all (i, j) ∈ {1, . . . , d} 2 by (2.2) andl α Xi (x i ) = l α Xi,Xi (x i , x i ). The optimality system (0.1) rewrites

l α Xi,Xj (x i , x j ) = αE[(X i -x i ) + 1 1 {Xj >xj } ] -(1 -α)E[(X i -x i ) -1 1 {Xj <xj } ],
(2.3) l α X k (x k ) = - d i=1,i =k π ki π kk l α Xi,X k (x i , x k ) ∀k ∈ {1, . . . , d}.
We shall use the following sets:

J i 0 = {j ∈ {1, . . . , d} \ {i} | lim α-→1 x j x i = 0}, J i C = {j ∈ {1, . . . , d} \ {i} | 0 < lim α-→1 x j x i < lim α-→1 x j x i < +∞},
and

J i ∞ = {j ∈ {1, . . . , d} \ {i} | lim α-→1 x j x i = +∞}.
The proof of Proposition 2.1 is written for π ij = 1, for all (i, j) ∈ {1, . . . , d} 2 , ie for the L 1 -expectiles. The general case can be treated in the same way, provided that π ij > 0 for all (i, j) ∈ {1, . . . , d} (2) If t = Θ(s), 1 then for all (i, j) ∈ {1, . . . , d} 2 ,

F Xi (s) F Xj (t) ∼ c i c j s t -θ as t → ∞.
The proof is given in Appendix A.1.

1 Recall that t = Θ(s) means that there exist positive constants

C 1 and C 2 such that C 1 s ≤ t ≤ C 2 s
Proposition 2.3. Under H1 and H2, the components of the extreme multivariate expectile satisfy

0 < lim α-→1 1 -α FXi (e i α (X)) ≤ lim α-→1 1 -α FXi (e i α (X))
< +∞, ∀i ∈ {2, . . . , d}.

The proof is given in Appendix A.2. We may now prove Proposition 2.1.

Proof of Proposition 2.1. We shall prove that J 1 ∞ = ∅, the fact that J k ∞ = ∅ for all k ∈ {1, . . . , d} may be proven in the same way. This implies that J k 0 = J k ∞ = ∅ for all k ∈ {1, . . . , d}, hence the result. We suppose that J 1 ∞ = ∅, let i ∈ J 1 ∞ , taking if necessary a subsequence, we may assume that x i /x 1 → +∞ as α → 1. From Proposition 2.3, we have

0 < lim α-→1 1 -α FXi (e i α (X)) ≤ lim α-→1 1 -α FXi (e i α (X))
< +∞, ∀i ∈ {2, . . . , d}, so, taking if necessary a subsequence, we may assume that ∃ ∈ R * \{+∞} such that lim

α-→1 1 -α FX1 (x 1 ) = .
In this case,

lim α-→1 l α X1 (x 1 ) x 1 FX1 (x 1 ) = lim α-→1 (2α -1) E[(X 1 -x 1 ) + ] x 1 FX1 (x 1 ) - 1 -α F1 (x 1 ) (1 - E[X 1 ] x 1 ) = 1 θ -1 -< +∞.
Moreover,

E[(X i -x i ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) ≤ E[(X i -x i ) + ] x 1 FX1 (x 1 ) = E[(X i -x i ) + ] x i FXi (x i ) x i FXi (x i ) x 1 FX1 (x 1 ) -→ 0 using Lemma 2.2.
We get

lim α-→1 l α Xi,X1 (x i , x 1 ) x 1 FX1 (x 1 ) = lim α-→1 αE[(X i -x i ) + 1 1 {X1>x1} ] -(1 -α)E[(X i -x i ) -1 1 {X1<x1} ] x 1 FX1 (x 1 ) = lim α-→1 E[(X i -x i ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) - 1 -α FX1 (x 1 ) x i x 1 = -∞, ∀i ∈ J 1 ∞ .
Going through the limit (α -→ 1) in the first equation of the optimality System (2.3) divided by x 1 FX1 (x 1 ), leads to

(2.4) lim α-→1 k∈J 1 0 ∪J 1 C \J 1 ∞ l α X k ,X1 (x k , x 1 ) x 1 FX1 (x 1 ) = -∞. Now, let k ∈ J 1 0 E[(X k -x k ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) = x1 x k P (X k > t, X 1 > x 1 ) dt x 1 FX1 (x 1 ) + +∞ x1 P (X k > t, X 1 > x 1 ) dt x 1 FX1 (x 1 ) ≤ x1 x k P (X 1 > x 1 ) dt x 1 FX1 (x 1 ) + +∞ x1 P (X k > t) dt x 1 FX1 (x 1 ) , Karamata's Theorem (Theorem 1.4) leads to lim α-→1 x1 x k P (X 1 > x 1 ) dt x 1 FX1 (x 1 ) + +∞ x1 P (X k > t) dt x 1 FX1 (x 1 ) = 1 + c k θ -1 , ∀k ∈ J 1 0 . Consider k ∈ J 1 C E[(X k -x k ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) ≤ E[(X k -x k ) + ] x 1 FX1 (x 1 ) = E[(X k -x k ) + ] x k FX k (x k ) x i FX k (x k ) x 1 FX1 (x 1 ) , and E[(X k -x k ) + ] x i FX k (x k ) x k FX k (x k ) x 1 FX1 (x 1 ) ∼ α-→1 c k θ -1 x k x 1 -θ+1
.

Finally, we deduce that

- k∈J 1 0 ∪J 1 C \J 1 ∞ lim α-→1 x k x 1 ≤ lim α-→1 k∈J 1 0 ∪J 1 C \J 1 ∞ l α X k ,X1 (x k , x 1 ) x 1 FX1 (x 1 ) ≤ lim α-→1 k∈J 1 0 ∪J 1 C \J 1 ∞ l α X k ,X1 (x k , x 1 ) x 1 FX1 (x 1 ) ≤ k∈J 1 C c k θ -1 lim α-→1 x k x 1 -θ+1 -lim α-→1 x k x 1 + k∈J 1 0 \J 1 ∞ 1 + c k θ -1 .
This is contradictory with (2.4), and consequently J 1 ∞ is necessarily an empty set. The result follows.

Proposition 2.4 (Extreme multivariate expectile). Assume that H1 and H2 are satisfied and X has a regularly varying multivariate distribution in the sense of Definition 1.6. Consider the L 1 -expectiles e α (X) = (e i α (X)) i=1,...,d . Then any limit vector (η, β 2 , . . . , β d ) of

1-α FX 1 (e 1 α (X)) , e 2 α (X) e 1 α (X) , . . . , e d α (X) e 1 α (X)
satisfies the following equation system

(2.5) 1 θ -1 -η (β k ) θ c k = - d i=1,i =k +∞ β i β k λ ik U c i c k t -θ , 1 dt -η β θ-1 k c k β i , ∀k ∈ {1, . . . , d}.
By solving the system (2.5), we may obtain an equivalent of the extreme multivariate expectile, using the marginal quantiles.

Proof. The optimality system (2.3) can be written in the following form

(2α -1) E[(X k -x k ) + ] x k FX k (x k ) - 1 -α FX k (x k ) 1 - E[X k ] x k = d i=1,i =k (1 -α) E[(X i -x i ) -1 1 {X k <x k } ] x k FX k (x k ) - d i=1,i =k α E[(X i -x i ) + 1 1 {X k >x k } ] x k FX k (x k ) , ∀k ∈ {1, . . . , d}.
For all k ∈ {1, . . . , d}, we have (taking if necessary a subsequence)

lim α-→1 (2α -1) E[(X k -x k ) + ] x k FX k (x k ) - 1 -α FX k (x k ) 1 - E[X k ] x k = 1 θ -1 -η (β k ) θ c k ,
and for all i ∈ {1, . . . , d} \ {k}

lim α-→1 (1 -α) E[(X i -x i ) -1 1 {X k <x k } ] x k FX k (x k ) = lim α-→1 1 -α FX k (x k ) x i x k P(X i < x i , X k < x k ) - E[X i 1 1 {Xi<xi,X k <x k } ] x k = η β θ k c k β i β k = η β θ-1 k c k β i .
Moreover,

E[(X i -x i ) + 1 1 {X k >x k } ] x k FX k (x k ) = 1 x k FX k (x k ) +∞ xi P(X i > t, X k > x k )dt = +∞ x i x k P(X i > tx k , X k > x k ) FX k (x k ) dt = +∞ x i x k P FXi (X i ) < FXi (tx k ), FX k (X k ) < FX k (x k ) FX k (x k ) dt.
Firstly, we remark that

x i x k β i β k P FXi (X i ) < FXi (tx k ), FX k (X k ) < FX k (x k ) FX k (x k ) dt ≤ x i x k - β i β k .
Since the functions λ ik U are assumed to be continuous,

(2.6) lim α-→1 P FXi (X i ) < FXi (tx k ), FX k (X k ) < FX k (x k ) FX k (x k ) = λ ik U c i c k t -θ , 1 .
In order to show that lim

α-→1 α E[(X i -x i ) + 1 1 {X k >x k } ] x k FX k (x k ) = +∞ β i β k λ ik U c i c k t -θ , 1 dt,
we may use the Lebesgue's Dominated Convergence Theorem with Potter's bounds (1942) (Lemma 1.5) for regularly varying functions. First of all,

P FXi (X i ) < FXi (tx k ), FX k (X k ) < FX k (x k ) FX k (x k ) ≤ min 1, FXi (tx k ) FX k (x k ) , since FX i (tx k ) FX k (x k ) = FX i (tx k ) FX k (tx k ) FX k (tx k ) FX k (x k ) and lim α-→1 FX i (tx k ) FX k (tx k ) = ci c k , using Potter's bounds, for all ε 1 > 0 and 0 < ε 2 < θ -1, there exists x 0 k (ε 2 , ε 1 ) such that for min{x k , tx k } ≥ x 0 k (ε 2 , ε 1 ) FXi (tx k ) FX k (x k ) ≤ c i c k + 2ε 1 t -θ max(t ε2 , t -ε2 ).
Lebesgue's theorem gives lim α-→1

+∞ x i x k P FXi (X i ) < FXi (tx k ), FX k (X k ) < FX k (x k ) FX k (x k ) dt = +∞ β i β k λ ik U c i c k t -θ , 1 dt, so, for all (i = k) ∈ {1, . . . , d} 2 lim α-→1 E[(X i -x i ) + 1 1 {X k >x k } ] x k FX k (x k ) = +∞ β i β k λ ik U c i c k t -θ , 1 dt.
Hence the system announced in this proposition.

In the general case of Σ-expectiles, with Σ = (π ij ) i,j=1,...,d ,

π ij ≥ 0, π ii = π i > 0, System (2.5) becomes 1 θ -1 -η (β k ) θ c k = - d i=1,i =k π ik π k +∞ β i β k λ ik U c i c k t -θ , 1 dt -η β θ-1 k c k β i , ∀k ∈ {1, . . . , d}.
Moreover, let us remark that System (2.5) is equivalent to the following system (2.7)

d i=1 +∞ β i β k λ ik U c i t -θ , c k β -θ k dt = d i=1 +∞ βi λ i1 U c i t -θ , 1 dt, ∀k ∈ {2, . . . , d}.
The limit points β i are thus completely determined by the asymptotic bivariate dependencies between the marginal components of the vector X.

Proposition 2.5. Assume that H1 and H2 are satisfied and the multivariate distribution of X is regularly varying in the sense of Definition 1.6, consider the L 1 -expectiles e α (X) = (e i α (X)) i=1,...,d . Then any limit vector (η, β 2 , . . . , β d ) of

1-α FX 1 (e 1 α (X)) , e 2 α (X) e 1 α (X) , . . . , e d α (X) e 1 α (X)
satisfies the following system of equations, ∀k ∈ {1, . . . , d}

(2.8) 1 θ -1 -η (β k ) θ c k = - d i=1,i =k    c i c k β i β k -θ+1 +∞ 1 λ ik U t -θ , c k c i β k β i -θ dt -η β θ-1 k c k β i    .
Proof. The proof is straightforward using a substitution in System (2.5) and the positive homogeneity property of the bivariate tail dependence functions λ ik U (see Proposition 2.2 in [START_REF] Joe | Tail dependence functions and vine copulas[END_REF]).

The main utility of writing the asymptotic optimality system in the form (2.8) is the possibility to give an explicit form to (η, β 2 , . . . , β d ) for some dependence structures.

Example: Consider that the dependence structure of X is given by an Archimedean copula with generator ψ. The survival copula is given by

C(x 1 , . . . , x d ) = ψ(ψ (x 1 ) + • • • + ψ (x d )),
where ψ (x) = inf{t ≥ 0|ψ(t) ≤ x} (see e.g. [START_REF] Mcneil | Multivariate archimedean copulas, d-monotone functions and l1-norm symmetric distributions[END_REF] for more details). Assume that, ψ is a regularly varying function with non-positive index ψ ∈ RV -θ ψ . According to [START_REF] Charpentier | Tails of multivariate archimedean copulas[END_REF], the right tail dependence functions exist, and one can get their explicit forms

λ k U (x 1 , . . . , x k ) = k i=1 x -1 θ ψ i -θ ψ .
Thus, the bivariate upper tail dependence functions are given by

λ ik U t -θ , c k c i β k β i -θ = t θ θ ψ + c i c k 1 θ ψ β k β i θ θ ψ -θ ψ .
In particular, if θ = θ ψ , we have

+∞ 1 λ ik U t -θ , c k c i β k β i -θ dt = 1 θ -1 1 + c i c k 1 θ β k β i -θ+1
, and System 2.8 becomes

1 θ -1 -η (β k ) θ c k = - d i=1,i =k   1 θ -1 c i c k β i β k + c i c k 1 θ -θ+1 -η β θ-1 k c k β i   .
Lemma 2.6 (The comonotonic Fréchet case). Under H1 and H2, consider the L 1 -expectiles e α (X) = (e i α (X)) i=1,...,d . If X = (X 1 , . . . , X d ) is a comonotonic random vector, then the limit

(η, β 2 , . . . , β d ) = lim α-→1 1 -α FX1 (e 1 α (X)) , e 2 α (X) e 1 α (X) , . . . , e d α (X) e 1 α (X) , satisfies lim α-→1 1 -α FX k (e k α (X)) = 1 θ -1 and β k = c 1/θ k , ∀k ∈ {1, . . . , d}.
Proof. Since the random vector X is comonotonic, its survival copula is

C X (u 1 , . . . , u d ) = min(u 1 , . . . , u d ), ∀(u 1 , . . . , u d ) ∈ [0, 1] d .
We deduce the expression of the functions

λ ij U λ ij U (x i , x j ) = min(x i , x j ), ∀(x i , x j ) ∈ R 2 + , ∀i, j ∈ {1, . . . , d}. So, +∞ 1 λ ik U t -θ , c k c i ( β k β i ) -θ dt = +∞ 1 min t -θ , c k c i β k β i -θ dt = β k β i c k c i -1 θ -1 + c k c i β k β i -θ + 1 θ -1 1 + β k β i c k c i -1 θ -1 + -θ+1
.

Under assumptions H1 and H2, and by Proposition 2.8, let (η, β 2 , . . . , β d ) be a solution of the following equation system.

η d i=1 β i - 1 θ -1 d i=1 c i β -θ+1 i = d i=1,i =k c k β -θ k β i β k β i c k c i -1 θ -1 + + 1 θ -1 d i=1,i =k c i β -θ+1 i   1 + β k β i c k c i -1 θ -1 + -θ+1 -1   , ∀k ∈ {1, . . . , d}. η = 1 θ-1 and β k = c 1 θ
k is the only solution to this system.

Proposition 2.7 (Asymptotic independence case). Under H1 and H2, consider the L 1 -expectiles e α (X) = (e i α (X)) i=1,...,d . If X = (X 1 , . . . , X d ) is such that the pairs (X i , X j ) are asymptotically independent, then the limit vector (η,

β 2 , . . . , β d ) of 1-α FX 1 (e 1 α (X)) , e 2 α (X) e 1 α (X) , . . . , e d α (X) e 1 α (X) satisfies η = 1 (θ -1)   1 + d j=2 c 1 θ-1 j   and β k = c 1 θ-1 k , for all k ∈ {1, . . . , d}.
Proof. The hypothesis of asymptotic bivariate independence means:

lim α-→1 P(X i > x i , X j > x j ) P(X j > x j ) = lim α-→1 P(X i > tx j , X j > x j ) P(X j > x j ) = 0,
for all (i, j) ∈ {1, . . . , d} 2 and for all t > 0, then, Lebesgue's Theorem used as in Proposition 2.4 gives

lim α-→1 E[(X i -x i ) + 1 1 {Xj >xj } ] x j FXj(xj) = lim α-→1 +∞ x i x j P(X i > tx j , X j > x j ) P(X j > x j ) dt = 0.
The extreme multivariate expectile verifies the following equation system

1 θ -1 - η c k β θ k = + d i=1,i =k η c k β θ-1 k β i , ∀k ∈ {1, . . . , d},
which can be rewritten as (2.9)

c k η(θ -1)β θ-1 k = d i=1 β i , ∀k ∈ {1, . . . , d}, hence β k = c 1 θ-1 k
for all k ∈ {1, . . . , d}, and

η = 1 (θ -1)   1 + d j=2 c 1 θ-1 j   .
In the general case of a matrix of positive coefficients π ij , i, j ∈ {1, . . . , d}, the limits β i , i = 2, . . . , d remain the same, but the limit η will change:

lim α-→1 e k α (X) e 1 α (X) = c 1 θ-1 k and lim α-→1 1 -α FX k (e k α (X)) = c 1 θ-1 k (θ -1)   1 + d j=2 π jk π k c 1 θ-1 j   , for all k ∈ {1, . . . , d}. We remark that lim α-→1 1 -α FXi (x i ) ≤ c 1 θ-1 i θ -1 ,
which allows a comparison between the marginal quantile and the corresponding component of the multivariate expectile, and since F -1 X k (1 -•) is a regularly varying function at 0 for all k ∈ {1, . . . , d} with index -1 θ (see Lemma 1.3), we get

e k α (X) ∼ α-→1 VaR α (X k ) (θ -1) -1 θ        1 + d i=2 c 1 θ-1 i c 1 θ-1 k        -1 θ ,
where VaR α (X k ) denotes the Value at Risk of X k at level α, ie the α-quantile F ← X k (α) of X k . These conclusions coincide with the results obtained in dimension 1, for distributions that belong to the domain of attraction of Fréchet, in [START_REF] Bellini | Risk management with expectiles[END_REF]. The values of constants c i determine the position of the marginal quantile compared to the corresponding component of the multivariate expectile for each risk.

Fréchet model with a dominant tail

This section is devoted to the case where X 1 has a dominant tail with respect to the X i 's. then

β i = lim α↑1 e i α (X) e 1 α (X) = 0, lim α↑1 1 -α FXi (e i α (X)) = 0, ∀i ∈ {2, . . . , d},
and

lim α↑1 1 -α FX1 (e 1 α (X)) = 1 θ -1 .
The proof of Proposition 3.1 follows from the following lemmas.

Lemma 3.2. Under H1, consider the L 1 -expectiles e α (X) = (e i α (X)) i=1,...,d . If

lim x↑+∞ FXi (x) FX1 (x) = 0, ∀i ∈ {2, . . . , d}, then lim α↑1 E[(X i -x i ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) = 0, ∀ i ∈ {2, . . . , d}. Lemma 3.3. Under H1, consider the L 1 -expectiles e α (X) = (e i α (X)) i=1,...,d . If lim x↑+∞ FXi (x) FX1 (x) = 0, ∀i ∈ {2, . . . , d}, then 0 < lim α↑1 1 -α FX1 (x 1 ) ≤ lim α↑1 1 -α FX1 (x 1 ) < +∞.
The poofs of Lemmas 3.2 and 3.3 are given in Appendix A.3 and A.4 respectively. Now, we have all necessary tools to prove Proposition 3.1.

Proof of Proposition 3.1. From Lemma 3.3, we have Taking if necessary a subsequence (α n ) n∈N α n -→ 1, we suppose that 1-α FX 1 (x1) is converging to a limit denoted 0 < η < +∞ and that the limits lim α→1 xi x1 = β i exist. Going through the limit (α → 1) in the 1 st equation of System 0.1 divided by x 1 FX1 (x 1 ), leads using Lemma 3.2 to

(3.1) lim α↑1 1 -α FX1 (x 1 ) i∈J C ∪J∞ x i x 1 = lim α↑1 η i∈J C ∪J∞ x i x 1 = 1 θ -1 -η,
we deduce that J ∞ = ∅. We suppose that J C = ∅, so there exists at least one i ∈ {2, . . . , d} such that i ∈ J C , and for all j ∈ {1, . . . , d}\{i}, we have

lim α↑1 E[(X j -x j ) + 1 1 {Xi>xi} ] x 1 FX1 (x 1 ) = 0. indeed, if j ∈ J C \{i} lim α↑1 E[(X j -x j ) + 1 1 {Xi>xi} ] x 1 FX1 (x 1 ) = lim α↑1 +∞ βj P(X j > tx 1 , X i > x i ) P(X 1 > x 1 ) dt, because P(X j > tx j , X i > x i ) P(X 1 > x 1 ) = P(X j > tx j |X i > x i ) P(X i > x i ) P(X 1 > x 1 )
,

and
P(X j > tx 1 , X i > x i ) P(X 1 > x 1 ) ≤ min P(X j > tx 1 ) P(X 1 > x 1 ) , P(X i > x i ) P(X 1 > x 1 )
, and using Potter's Bounds associated to FX1 as regularly varying function in order to apply the dominated convergence Theorem, we get

lim α↑1 E[(X j -x j ) + 1 1 {Xi>xi} ] x 1 FX1 (x 1 ) = +∞ βj lim α↑1 P(X j > tx 1 , X i > x i ) P(X 1 > x 1 ) dt = 0, ∀j ∈ J C \{i}. Now, if j ∈ J 0 then E[(X j -x j ) + 1 1 {Xi>xi} ] x 1 FX1 (x 1 ) = x1 xj P(X j > t, X i > x i ) x 1 P(X 1 > x 1 ) dt + +∞ 1 P(X j > tx 1 , X i > x i ) P(X 1 > x 1 ) dt ≤ 1 - x j x 1 P(X i > x i ) P(X 1 > x 1 ) + +∞ 1 P(X j > tx 1 , X i > x i ) P(X 1 > x 1 )
dt, Going through the limit (α → 1) in the i th equation of System 0.1 divided by x 1 FX1 (x 1 ), leads to

thus lim α↑1 E[(X j -x j ) + 1 1 {Xi>xi} ] x 1 FX1 (x 1 ) = 0, ∀j ∈ J 0 ,
-ηβ i = η(1 + j∈J C \{i} β j ),
which is absurd, and consequently J C = ∅.

We have thus proved that J 0 = {2, . . . , d} which means

lim α↑1 e i α (X) e 1 α (X) = β i = 0, ∀i ∈ {2, . . . , d}.
And from Equation 3.1 we deduce also that

η = lim α↑1 1 -α FX1 (e 1 α (X)) = 1 θ -1 ,
and by Lemma 2.2 that

lim α↑1 1 -α FX k (e i α (X))
= 0, ∀i ∈ {2, . . . , d}.

Proposition 3.1 shows that the dominant risk behaves asymptotically as in the univariate case, and its component in the extreme multivariate expectile satisfies

e 1 α (X) α↑1 ∼ (θ -1) -1 θ VaR α (X 1 )
α↑1 ∼ e α (X 1 ), the right equivalence is proved, in the univariate case, in [START_REF] Bellini | Risk management with expectiles[END_REF], Proposition 2.3.

Example: Consider Pareto distributions, X i ∼ P a(a i , b), i = 1, . . . , d, such that a i > a 1 for all i ∈ {1, . . . , d}. The tail of X 1 dominates that of the X i 's and Proposition 3.1 applies.

Estimation of the extreme expectiles

In this section, we propose some estimators of the extreme multivariate expectile. We focus on the cases of asymptotic independence and comonotonicity, for which the equation system is more tractable. We begin with the main ideas of our approach, then, we construct the estimators using the extreme values statistical tools and prove its consistency. We terminate this section with a simulation study. Proposition 4.1 (Estimation's idea). Using notations of previous sections, consider the L 1 -expectiles e α (X) = (e i α (X)) i=1,...,d . Under H1, H2 and the assumption that the vector

1-α FX 1 (e 1 α (X)) , e 2 α (X) e 1 α (X) , . . . , e d α (X) e 1 α (X) has a unique limit point (η, β 2 , . . . , β d ), e α (X) ∼ α-→1 VaR α (X 1 )η 1 θ (1, β 2 , . . . , β d ) T . Proof. Let (η, β 2 , . . . , β d ) = lim α→1 1-α FX 1 (e 1 α (X)) , e 2 α (X) e 1 α (X) , . . . , e d α (X) e 1 α (X) , we have e α (X) ∼ α-→1 e 1 α (X)(1, β 2 , . . . , β d ) T .

Moreover, lim α-→1

1-α FX 1 (e 1 α (X)) = η, and Theorem 1.5.12 in [6] states that F ← X1 (1 -.) is regularly varying at 0, with index -1 θ . This leads to

e 1 α (X) ∼ α-→1 F ← X1 (α) 1 η -1 θ ,
and the result follows.

Proposition 4.1 gives a way to estimate the extreme multivariate expectile. Let X = (X 1 , . . . , X d ) T be an independent sample of size n of X, with X i = (X 1,i , . . . , X d,i ) T for all i ∈ {1, . . . , n}. We denote by

X i,1,n ≤ X i,2,n ≤ • • • ≤ X i,n,n the ordered sample corresponding to X i .
4.1. Estimator's construction. We begin with the case of asymptotic independence. Propositions 2.7 and 4.1 are the key tools in the construction of the estimator. We have for all i ∈ {1, . . . , d}

β i = c 1 θ-1 i , and lim α-→1 1 -α FXi (e i α (X)) = c 1 θ-1 i (θ -1)   1 + d j=2 c 1 θ-1 j   . Proposition 4.1 gives e α (X) α↑1 ∼ VaR α (X 1 ) (θ -1) -1 θ 1 + d i=2 c 1 θ-1 i -1 θ 1, c 1 θ-1 2 , . . . , c 1 θ-1 d T .
So, in order to estimate the extreme multivariate expectile, we need an estimator of the univariate quantile of X 1 , of the tail equivalence parameters. and of θ. In the same way, and for the case of comonotonic risks, we may use Proposition 2.6 lim

α-→1 1 -α FXi (e i α (X)) = 1 θ -1
and

β i = c 1/θ i , ∀i ∈ {1, . . . , d},
and by Proposition 4.1 we obtain e α (X) T α↑1 ∼ VaR α (X 1 ) (θ -1)

-1 θ (1, c 1 θ 2 , . . . , c 1 θ d ) T .
The X i 's have all the same index θ of regular variation, which is also the same as the index of regular variation of X . We propose to estimate θ by using the Hill estimator γ. We shall denote θ = 1 γ . See [START_REF] Hill | A simple general approach to inference about the tail of a distribution[END_REF] for details on the Hill estimator. In order to estimate the c i 's, we shall use the GPD approximation: for u a large threshold, and

x ≥ u, F (x) ∼ F (u) x u -θ
.

Let k ∈ N be fixed and consider the thresholds u i :

FXi (u i ) = FX1 (u 1 ) = k n , ∀i ∈ {1, . . . , d}.
The u i are estimated by X i,n-k+1,n with k → ∞ and k/n → 0 as n → ∞. Using Lemma 2.2, we get

c i = lim n→∞ u i u 1 θ .
We shall consider (4.1) ĉi = X i,n-k+1,n X 1,n-k+1,n Proof. The results in [START_REF] Beirlant | STATISTICS OF EXTREMES: THEORY AND APPLI-CATIONS[END_REF] page 86 imply that for any i = 1, . . . , d X i,n-k+1,n

u i P -→ 1.
Moreover, it is well known (see [START_REF] Hill | A simple general approach to inference about the tail of a distribution[END_REF]) that the Hill estimator is consistent. Using (4.1), and the fact that

X i,n-k+1,n X 1,n-k+1,n ∼ u i u 1
in probability and thus is bounded in probability, we get the result.

To estimate the extreme quantile, we will use Weissman's estimator (1978) [START_REF] Weissman | Estimation of parameters and large quantiles based on the k largest observations[END_REF]:

VaR α (X 1 ) = X 1,n-k(n)+1,n k(n) (1 -α)n γ .
The properties of Weissman's estimator are presented in [START_REF] Embrechts | MODELLING EXTREMAL EVENTS[END_REF] [START_REF] Embrechts | MODELLING EXTREMAL EVENTS[END_REF] and also in [START_REF] Beirlant | STATISTICS OF EXTREMES: THEORY AND APPLI-CATIONS[END_REF] page 119. In order to prove the consistency of our estimators of extreme multivariate expectiles, we shall need the following second order condition (see [START_REF] Beirlant | STATISTICS OF EXTREMES: THEORY AND APPLI-CATIONS[END_REF] Section 4.4).

Definition 4.3.

A random variable X satisfying H1 with θ = 1 γ > 0 will be said to verify the second order condition SOC -β (b) with β > 0 and b ∈ RV -β (+∞) if the function U : y F ← (1 -1 y ) satisfies for u > 0:

U (ux) U (x) = u γ (1 + h -β (u)b(x) + o(b(x))
) , as x goes to infinity.

where

h -β (u) = 1-u -β β .
Now, we can deduce some estimators of the extreme multivariate expectile, using the previous ones, in the cases of asymptotic independence and perfect dependence. Definition 4.4 (Multivariate expectile estimator, Asymptotic independence). Under H1 and H2, in the case of bivariate asymptotic independence of the random vector X = (X 1 , . . . , X d ) T , we define the estimator of the L 1expectile as follows

ê⊥ α (X) = X 1,n-k(n)+1,n k(n) (1 -α)n γ γ 1 -γ γ 1 1 + d =2 ĉ γ 1-γ γ 1, ĉ γ 1-γ 2 , . . . , ĉ γ 1-γ d T = VaR α (X 1 ) γ 1 -γ γ 1 1 + d =2 ĉ γ 1-γ γ 1, ĉ γ 1-γ 2 , . . . , ĉ γ 1-γ d T .
Definition 4.5 (Multivariate expectile estimator, comonotonic risks). Under the assumptions of the Fréchet model with equivalent tails, for a comonotonic random vector X = (X 1 , . . . , X d ) T , we define the estimator of L 1 -expectile as follows

ê+ α (X) = X 1,n-k(n)+1,n k(n) (1 -α)n γ γ 1 -γ γ 1, ĉγ 2 , . . . , ĉγ d T = VaR α (X 1 ) γ 1 -γ γ 1, ĉγ 2 , . . . , ĉγ d T .
We prove below that if the second order condition SOC -β (b) is satisfied, then the term by term ratio ê⊥ α (X)/e α (X) goes to 1 in probability in the asymptotically independent case and ê+ α (X)/e α (X) goes to 1 in probability in the comonotonic case. More work is required to get the asymptotic normality.

Theorem 4.6. Assume that H1, H2 and SOC

-β (b) are satisfied. Choose k = k(n) such that • k(n) → ∞ as n → ∞, • k(n)/n → 0 as n → ∞, • k(n) 1 + log 2 k(n) n(1-α) -1 2 → ∞ as n → ∞.
Then, if each pair of the random vector X is asymptotically independent, ê⊥ α (X)/e α (X) -→ 1 in probability, as n → ∞.

If the random vector X is comonotonic, then

ê+ α (X)/e α (X) -→ 1 in probability, as n → ∞.
Proof. With the SOC -β (b) hypothesis and the choice of k, we get by using (4.18) p.120 in [START_REF] Beirlant | STATISTICS OF EXTREMES: THEORY AND APPLI-CATIONS[END_REF] that

VaR α (X 1 ) VaR α (X 1 ) -→ 1 in probability as n → ∞.
Then the announced results follow from Propositions 2.7 and 4.2.

Numerical illustration.

The attraction domain of Fréchet contains the usual distributions of Pareto, Student, Burr and Cauchy. In order to illustrate the convergence of the proposed estimators, we study numerically, the cases of Pareto, Burr and Student distributions.

In the independence case which is a special case of asymptotic independence, the functions l α Xi,Xj defined in (2.2) have the following expression

l α Xi,Xj (x i , x j ) = α FXj (x j )E (X i -x i ) + -(1 -α) F Xj (x j )E (X i -x i ) -.
In the comonotonic case we have

l α Xi,Xj (x i , x j ) = α FXj (x j )(µ i,j -x i ) + + E (X i -max(x i , µ i,j )) + -(1 -α) F Xj (x j )(x i -µ i,j ) + + E (X i -min(x i , µ i,j )) -,
where µ i,j = F ←- Xi (F Xj (x j )). From these expressions, the exact value of the extreme multivariate expectile is obtained using numerical optimization, and we can confront it to the estimated values. The choice of k(n) is function of the distributions parameters, and it is done in our simulations using graphical illustrations. We present the estimators for different values of k(n) that belong to the common convergence range of the estimators of tail equivalence coefficients, in order to verify the stability of the expectile estimator?s convergence. In what follows, we consider two models for which the exact values of the L 1 -expectiles are computable. In the first model, the X i 's are independent. In the second one, the X i 's are comonotonic and for Pareto distributions µ i,j = bi bj x j . In the simulations below, we have taken the same k = k(n) to get γ and êα (X). For X i ∼ P a(2, 5•(i+1)) i=1,2 , and n = 100000, Figure 1 illustrates the convergence of estimator ĉ2 . On the left, the shaded area indicates suitable values of k(n) for n = 100000. The boxplots are obtained for different values of n and a fixed k ∈ k(n), the data size is 1000. 4.2.2. Burr distributions. We consider a multivariate Burr model X i ∼ Burr(a, b i , τ ), i ∈ {1, . . . , d}. In this case, the tails are equivalents with equivalence parameter

c i = lim x→+∞ FXi (x) FX1 (x) = lim x→+∞ bi bi+x τ a b1 b1+x τ a = b i b 1 a , ∀i ∈ {2, . . . , d},
and FXi ∈ RV a * τ (+∞) for all i in {1, . . . , d}. In the Burr comonotonic case µ i,j = bi bj 1 τ x j . The model is asymptotically equivalent to the Pareto one, but the margins are different, which helps to test the pertinence of the estimation processes compared to the theoretical resultants. Figures 5 and6

Student distributions.

In order to illustrate the convergence of the two estimators for other distributions nature, we close this subsection by a Student model. We consider a risk vector (X 1 , . . . , X d ) such that X i = a i T i for all i ∈ {1, . . . , d} and (T i ) i are identically distributed following a t-distribution of parameter z. Using L'Hôpital's rule, the tails are equivalent since

lim x→+∞ FXi (x) FX1 (x) = lim x→+∞ FT1 (x/a i ) FX1 (x) = lim x→+∞ a 1 f T1 (x/a i ) a i f T1 (x/a 1 ) = a i a 1 z = c i , ∀i ∈ {2, . . . , d}.
The marginal tails are all RV -(z+1) (+∞). For the Student comonotonic model µ i,j = ai aj x j . For the numerical illustration the parameters are a i = 2 i-1 for i = 1, . . . , d and z = 2. In the case of the independence (T i ) i are supposed independent, and they are comonotonic in the comonotonic case. Figures 7 and8 present an illustration of the obtained results in the two cases. For the three Fréchet models, Pareto, Weibull, and Student, the different illustrations show that the convergence is better for values of α close to 1. This is natural since we are approaching the extreme level and therefore the estimate value converges towards the theoretical value. The convergence seems to be stable for values of k(n) in the convergence zone. When α moves away from 1, the difference with the theoretical value is apparently a function of the marginal risk level represented by the coefficients of tails' equivalence c i .

Conclusion

We have studied properties of extreme multivariate expectiles in a regular variations framework. We have seen that the asymptotic behavior of expectiles vectors strongly depends on the marginal tails behavior and on the nature of the asymptotic dependence. The main conclusion of this analysis, is that the equivalence of marginal tails leads to equivalence of the extreme expectile components.

The statistical estimation of the integrals of the tail dependence functions would allow to construct estimators of the extreme expectile vectors. This paper's estimations are limited to the cases of asymptotic independence and comonotonicity which do not require the estimation of the tail dependence functions. The asymptotic normality of the estimators proposed in the last section of this paper requires a careful technical analysis which is not considered in this paper.

A natural perspective of this work, is to study the asymptotic behavior of Σ-expectiles in the case of equivalent tails of marginal distributions in the domains of attraction of Weibull and Gumbel. The Gumbel's domain contains most of the usual distributions, especially the family of Weibull tail-distributions, which makes the analysis of its case an interesting task.

Appendix A. Proofs A.1. Lemma 2.2.
Proof. We give some details on the proof for the first item, the second one may be obtained in the same way. Under H1 and H2, for all i ∈ {1, . . . , d} FXi ∈ RV -θ (+∞), there exists for all i a positive measurable function

L i ∈ RV 0 (+∞) such that FXi (x) = x -θ L i (x), ∀x > 0,
then for all (i, j) ∈ {1, . . . , d} 2 and all t, s > 0 1 -α FXi (e i α (X))

(A.1) s FXi (s) t FXj (t) = s t -θ+1 L i (s) L j (t) = s t -θ+1 L i (s) L i (t) L i (t) L j (t
< +∞, ∀i ∈ {2, . . . , d}.

Using H2, it is sufficient to show that lim α-→1

1 -α FX1 (e 1 α (X))

< +∞.

Assume that lim α-→1

1-α FX 1 (e 1 α (X)) = +∞, we shall prove that, in that case, (2.3) cannot be satisfied. Taking if necessary a subsequence (α n → 1), we may assume that lim

α-→1 1-α FX 1 (e 1 α (X)) = +∞. We have l α X1 (x 1 ) (1 -α)x 1 = αE[(X 1 -x 1 ) + ] -(1 -α)E[(X 1 -x 1 ) -] (1 -α)x 1 = (2α -1) E[(X 1 -x 1 ) + ] (1 -α)x 1 - (1 -α)(x 1 -E[X 1 ]) (1 -α)x 1 = (2α -1) E[(X 1 -x 1 ) + ] x 1 FX1 (x 1 ) FX1 (x 1 ) 1 -α -1 + E[X 1 ] x 1 α↑1 -→ -1 recall (2.1) .
Furthermore, for all i ∈ {2, . . . , d}

l α Xi,X1 (x i , x 1 ) (1 -α)x 1 = αE[(X i -x i ) + 1 1 {X1>x1} ] -(1 -α)E[(X i -x i ) -1 1 {X1<x1} ] (1 -α)x 1 = αE[(X i -x i ) + 1 1 {X1>x1} ] (1 -α)x 1 - E[(X i -x i ) + 1 1 {X1<x1} ] x 1 - x i P(X 1 < x 1 ) x 1 + E[X i 1 1 {X1<x1} ] x 1 .
On one side,

E[(X i -x i ) + 1 1 {X1>x1} ] (1 -α)x 1 ≤ E[(X i -x i ) + ] (1 -α)x 1 = FX1 (x 1 ) 1 -α E[(X i -x i ) + ] x i FXi (x i ) x i FXi (x i ) x 1 FX1 (x 1 )
, ∀i ∈ {2, . . . , d}.

So that, Lemma 2.2 implies

(A.3) lim α-→1 E[(X i -x i ) + 1 1 {X1>x1} ] (1 -α)x 1 = 0, ∀k ∈ J 1 C ∪ J 1 ∞ .
Let i ∈ J 1 0 , taking if necessary a subsequence, we may assume that xi x1 → 0.

(A.4) E[(X i -x i ) + 1 1 {X1>x1} ] (1 -α)x 1 = x1 xi P (X i > t, X 1 > x 1 ) dt (1 -α)x 1 + +∞ x1 P (X i > t, X 1 > x 1 ) dt (1 -α)x 1 . Now, x1 xi P (X i > t, X 1 > x 1 ) dt (1 -α)x 1 ≤ x1 xi P (X 1 > x 1 ) dt (1 -α)x 1 = FX1 (x 1 ) 1 -α 1 - x i x 1 .
Thus,

(A.5) lim α-→1 x1 xi P (X i > t, X 1 > x 1 ) dt (1 -α)x 1 = 0.
Consider the second term of (A.4)

+∞ x1 P (X i > t, X 1 > x 1 ) dt (1 -α)x 1 ≤ +∞ x1 P (X i > t) dt (1 -α)x 1 ,
Karamata's Theorem (Theorem 1.4) gives

+∞ x1 P (X k > t) dt (1 -α)x 1 α↑1 ∼ 1 θ -1 FX k (x 1 ) 1 -α , which leads to (A.6) lim α-→1 +∞ x1 P (X i > t, X 1 > x 1 ) dt (1 -α)x 1 = 0.
Finally, we get

(A.7) lim α-→1 E[(X i -x i ) + 1 1 {X1>x1} ] (1 -α)x 1 = 0, ∀i ∈ J 1 0 .
We have shown that lim

α-→1 E[(X k -x k ) + 1 1 {X1>x1} ] (1 -α)x 1 = 0, ∀k ∈ {2, . . . , d},
so, the first equation of optimality system (2.3) implies that

-lim α→1   k∈J 1 0 \J 1 ∞ l α X k ,X1 (x k , x 1 ) (1 -α)x 1 + k∈J 1 C l α X k ,X1 (x k , x 1 ) (1 -α)x 1 + k∈J 1 ∞ l α X k ,X1 (x k , x 1 ) (1 -α)x 1   = lim α-→1 d k=2 x k x 1 = -1,
this is absurd since the x k 's are non negative, and consequently lim α-→1

1 -α FX1 (x 1 ) < +∞. Now, we prove that the components of the extreme multivariate expectile satisfy also

0 < lim α-→1 1 -α FXi (e i α (X))
, ∀i ∈ {2, . . . , d}.

Using H2, it is sufficient to show that

0 < lim α-→1 1 -α FX1 (e 1 α (X))
.

Let us assume that lim α-→1

1-α FX 1 (e 1 α (X)) = 0, we shall see that in that case, (2.3) cannot be satisfied. Taking if necessary a convergent subsequence, we may assume that lim

α-→1 1-α FX 1 (e 1 α (X)) = 0. In this case, l α X1 (x 1 ) x 1 FX1 (x 1 ) = (2α -1) E[(X 1 -x 1 ) + ] x 1 FX1 (x 1 ) - 1 -α F1 (x 1 ) (1 - E[X 1 ] x 1 ) α↑1 -→ 1 θ -1 > 0.
On another side, let i ∈ J 1 ∞ , taking if necessary a subsequence, we may assume that x 1 = o(x i ). Lemma 2.2 and Proposition 2.3 give:

1 -α F1 (x 1 )

x i x 1 = 1 -α F Xi (x i ) • x i F Xi (x i ) x 1 F X1 (x 1 ) -→ 0 as α → 1. Moreover, E[(X i -x i ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) ≤ E[(X i -x i ) + ] x 1 FX1 (x 1 ) = E[(X i -x i ) + ] x i FXi (x i ) x i FXi (x i ) x 1 FX1 (x 1 ) , We deduce l α Xi,X1 (x i , x 1 ) x 1 FX1 (x 1 ) = αE[(X i -x i ) + 1 1 {X1>x1} ] -(1 -α)E[(X i -x i ) -1 1 {X1<x1} ] x 1 FX1 (x 1 )
-→ 0, ∀i ∈ J 1 ∞ . Going through the limit (α -→ 1) in the first equation of the optimality system (2.3) divided by x 1 FX1 (x 1 ), leads to lim

α-→1 k∈J 1 0 ∪J 1 C \J 1 ∞ l α X k ,X1 (x k , x 1 ) x 1 FX1 (x 1 ) = - 1 θ -1 , which is absurd because lim α-→1 k∈J 1 0 ∪J 1 C \J 1 ∞ l α X k ,X1 (x k , x 1 ) x 1 FX1 (x 1 ) = lim α-→1 k∈J 1 0 ∪J 1 C \J 1 ∞ αE[(X k -x k ) + 1 1 {X1>x1} ] -(1 -α)E[(X k -x k ) -1 1 {X1<x1} ] x 1 FX1 (x 1 ) = lim α-→1 k∈J 1 0 ∪J 1 C \J 1 ∞ E[(X k -x k ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) - 1 -α FX1 (x 1 ) x k x 1 = lim α-→1 k∈J 1 0 ∪J 1 C \J 1 ∞ E[(X k -x k ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) ≥ 0.
We can finally conclude that lim α-→1

1 -α FX1 (x 1 ) > 0.

A.3. Lemma 3.2.

Proof. Taking if necessary a convergent subsequence (α n ) n∈N α n -→ 1, we consider that the limits lim

α→1 xi x1 = β i exist. Using the notation J C = {i ∈ {2, . . . , d}|0 < β i < +∞}, for all i ∈ J C lim α↑1 E[(X i -x i ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) = lim α↑1 +∞ βi P(X i > tx 1 , X 1 > x 1 ) P(X 1 > x 1 ) dt, because P(X i > tx 1 , X 1 > x 1 ) P(X 1 > x 1 ) = P(X i > tx 1 |X 1 > x 1 ) ≤ 1.
On another hand,

P(X i > tx 1 , X 1 > x 1 ) P(X 1 > x 1 ) ≤ min{1, P(X i > tx 1 ) P(X 1 > x 1 ) }, and
P(X i > tx 1 ) P(X 1 > x 1 ) = FXi (tx 1 ) FX1 (tx 1 )
FX1 (tx 1 ) FX1 (x 1 ) , then, using lim α↑1 FX i (tx1) FX 1 (tx1) = 0 and the Potter's bounds (1.5) associated to FX1 , we deduce that for all 1 > 0 and 0 < 2 < 1, there exists x 0 1 ( 1 , 2 ) such that for x 1 ≥

x

0 1 ( 1, 2) min{1,βi} P(X i > tx 1 ) P(X 1 > x 1 ) ≤ 1 (1 + 2 ) max t -θ+ 2 , t -θ-2 .
And the application of the Dominated Convergence Theorem leads to lim

α↑1 E[(X i -x i ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) = +∞ βi lim α↑1 P(X i > tx 1 , X 1 > x 1 ) P(X 1 > x 1 ) dt = 0, ∀ i ∈ J C .
We denote by J ∞ the set J ∞ = {i ∈ {2, . . . , d}|β i = +∞}. So, for all i ∈ J ∞ , x 1 = o(x i ) and

E[(X i -x i ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) = +∞ xi P(X i > t, X 1 > x 1 ) x 1 P(X 1 > x 1 ) dt ≤ +∞ x1 P(X i > t, X 1 > x 1 ) x 1 P(X 1 > x 1 ) dt = +∞ 1 P(X i > tx 1 , X 1 > x 1 ) P(X 1 > x 1 )
dt.

In the same way as in the previous case, and using the Potter's bounds, we show that

lim α↑1 +∞ 1 P(X i > tx 1 , X 1 > x 1 ) P(X 1 > x 1 ) dt = +∞ 1 lim α↑1 P(X i > tx 1 , X 1 > x 1 ) P(X 1 > x 1 ) dt = 0, from which we deduce that lim α↑1 E[(X i -x i ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) = 0, ∀ i ∈ J ∞ .
Let J 0 be the set J 0 = {i ∈ {2, . . . , d}|β i = 0}. For all i ∈ J 0 we have In addition, for all > 0, we have lim

x i = o(x 1 ), then lim α↑1 E[(X i -x i ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) = lim α↑1 +∞ xi P(X i > t, X 1 > x 1 ) x 1 P(X 1 > x 1 ) dt = lim α↑1 +∞ 0 P(X i > tx 1 , X 1 > x 1 ) P(X 1 > x 1 ) dt
α↑1 +∞ P(X i > tx 1 , X 1 > x 1 ) P(X 1 > x 1 ) dt = 0,
because the Dominated Convergence Theorem is applicable using the Potter's bounds, and lim

α↑1 P(Xi>tx1,X1>x1) P(X1>x1) = 0 for all t > 0 since c i = 0. Let κ > 0, ∀ > 0 ∃α 0 such that ∀α > α 0 +∞ P(X i > tx 1 , X 1 > x 1 ) P(X 1 > x 1 ) dt < κ, then +∞ 0 P(X i > tx 1 , X 1 > x 1 ) P(X 1 > x 1 ) dt = 0 P(X i > tx 1 , X 1 > x 1 ) P(X 1 > x 1 ) dt + +∞ P(X i > tx 1 , X 1 > x 1 ) P(X 1 > x 1 ) dt < + κ, we deduce that lim α↑1 +∞ 0 P(X i > tx 1 , X 1 > x 1 ) P(X 1 > x 1 ) dt = 0, so, lim α↑1 E[(X i -x i ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) = 0, ∀i ∈ J 0 .
We have therefore shown that lim

α↑1 E[(X i -x i ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) = 0, ∀i ∈ {2, . . . , d}.
A.4. Lemma 3.3.

Proof. We suppose that lim α↑1 1-α FX 1 (e 1 α (X)) = +∞. Taking if necessary a convergent subsequence (α n ) n∈N with α n -→ 1, we consider that the limits lim α→1 xi x1 = β i exist and that lim α↑1 1-α FX 1 (x1) = +∞. We use the notations J C = {i ∈ {2, . . . , d}| 0 < β i < +∞}, J 0 = {i ∈ {2, . . . , d}| β i = 0}, and J ∞ = {i ∈ {2, . . . , d}| β i = +∞}. The first equation of the optimality system (0.1) divided by x 1 FX1 (x 1 ) can be written (2α -1)E[(X 1 -x 1 ) + ]

x 1 FX1 (x 1 )

+ d i=2 αE[(X i -x i ) + 1 1 {X1>x1} ] x 1 FX1 (x 1 ) = 1 -α FX1 (x 1 ) x 1 -E[X 1 ] x 1 + 1 -α FX1 (x 1 ) d i=2 E[(X i -x i ) -1 1 {X1<x1} ] x 1 .
By (2. x 1 FX1 (x 1 ) = 0, so, going through the limit (α → 1) in the previous equation leads to

lim α↑1 1 -α FX1 (x 1 ) x 1 -E[X 1 ] x 1 + d i=2 E[(X i -x i ) -1 1 {X1<x1} ] x 1 = 1 θ -1 , nevertheless, lim α↑1 1 -α FX1 (x 1 ) x 1 -E[X 1 ] x 1 + d i=2 E[(X i -x i ) -1 1 {X1<x1} ] x 1 = lim α↑1 1 -α FX1 (x 1 ) 1 + d i=2 x i x 1 = +∞.
From this contradiction, we deduce that the case lim We denote J C = {i ∈ {2, . . . , d}| 0 < β i < +∞}, J 0 = {i ∈ {2, . . . , d}| β i = 0}, and J ∞ = {i ∈ {2, . . . , d}| β i = +∞}. Going through the limit (α → 1) in the first equation of System 0.1 divided by x 1 FX1 (x 1 ), and using Lemma 3.2 and Equation 2.1, leads to (A.8) lim α↑1 1 -α FX1 (x 1 ) i∈J∞

x i x 1 = 1 θ -1 .
If J ∞ = ∅, so, there exists i ∈ {2, . . . , d} such that i ∈ J ∞ . In this case,

lim α↑1 E[(X i -x i ) + ] x 1 FX1 (x 1 ) = lim α↑1 FXi (x i ) FX1 (x i ) x i FX1 (x i ) x 1 FX1 (x 1 ) E[(X i -x i ) + ] x i FXi (x i ) = 0, because lim α↑1 E[(Xi-xi)+] xi FX i (xi) = 1 θ-1 , lim α↑1 FX i (xi)
FX 1 (xi) = 0, and by Lemma 2.2 (X i = X j = X 1 ) lim

α↑1 xi FX 1 (xi)
x1 FX 1 (x1) = 0. On another hand, for all j ∈ {1, . . . , d}\{i}, E[(X j -x j ) + 1 1 {Xi>xi} ]

x 1 FX1 (x 1 ) = +∞ xj P(X j > t, X i > x i ) x 1 P(X P(X j > t, X i > x i )

x 1 P(X 1 > x 1 ) dt = +∞ 1 P(X j > tx 1 , X i > x i ) P(X 1 > x 1 ) dt,
we show in the same way as in the previous case that x 1 FX1 (x 1 ) = 0, ∀j ∈ J ∞ \{i}.

If j ∈ J 0 , then

E[(X j -x j ) + 1 1 {Xi>xi} ] x 1 FX1 (x 1 ) = x1 xj P(X j > tx 1 , X i > x i ) x 1 P(X 1 > x 1 ) dt + ∞ x1 P(X j > tx 1 , X i > x i ) x 1 P(X 1 > x 1 ) dt ≤ x 1 -x j x 1 FXi (x i ) FX1 (x 1 ) + +∞ 1 P(X j > tx 1 , X i > x i ) P(X 1 > x 1 )
dt, since lim FXi (x i ) FX1 (x i )

x i FX1 (x i ) x 1 FX1 (x 1 )
x 1

x i = 0, we obtain from that lim α↑1 E[(X j -x j ) + 1 1 {Xi>xi} ] x 1 FX1 (x 1 ) = 0, ∀j ∈ J 0 , and consequently lim α↑1 E[(X j -x j ) + 1 1 {Xi>xi} ] x 1 FX1 (x 1 ) = 0, ∀j ∈ {1, . . . , d}\{i}.

The i th equation of System 0.1 divided by x 1 FX1 (x 1 ) can be written in the form

(2α -1)E[(X i -x i ) + ] x 1 FX1 (x 1 ) - 1 -α FX1 (x 1 ) x i -E[X i ] x 1 = d j=1 j =i (1 -α)E[(X j -x j ) -1 1 {Xi<xi} ] x 1 FX1 (x 1 ) - d j=1 j =i αE[(X j -x j ) + 1 1 {Xi>xi} ]
x 1 FX1 (x 1 ) , going through the limit (α → 1) in this equation leads to

1 ε 1 ρ

 11 x→+∞ c(x) = c ∈]0, +∞[, and a measurable function ε(•) with lim x→+∞ ε(x) = 0, such that L(x) = c(x) exp x s representation is generalized to RV functions. Indeed, f ∈ RV ρ (+∞) if and only if it can written in the form f (x) = c(x) x (t) t dt, where ρ(t) t→∞ = ρ and c(t) t→∞ = c ∈]0, +∞[.

Definition 1 . 6 (

 16 Multivariate regular variations). The distribution of a random vector X on [0, ∞] d is said to be regularly varying if there exist a non-null Radon measure µ X on the Borel σ-algebra B d on [0, ∞] d \0, and a normalization function b : R -→ R which satisfies lim x-→+∞ b

Proposition 3 . 1 (

 31 Asymptotic dominance). Under H1, consider the L 1 -expectiles e α (X) = (e i α (X)) i=1,...,d . Iflim x↑+∞ FXi (x) FX1 (x)= 0, ∀i ∈ {2, . . . , d}, (dominant tail hypothesis)

Proposition 4 . 2 .

 42 k) is the Hill's estimator of the extreme values index constructed using the k largest observations of X . Let θ = 1 γ(k) . Let k = k(n) be such that k → ∞ and k/n -→ 0 as n → ∞. Under H1 and H2, for any i = 2, . . . , d, c i P -→ c i .

4. 2 . 1 .FX2

 21 Pareto distributions. We consider a bivariate Pareto model X i ∼ P a(a, b i ), i ∈ {1, 2}. Both distributions have the same scale parameter a, so they have equivalent tails with equivalence parameter c 2 = lim x→+∞

Figure 1 .

 1 Figure 1. Convergence of ĉ2 . X i ∼ P a(2, 5 • (i + 1)) i=1,2 .

Figure 2

 2 Figure 2 presents the obtained results for different k(n) values in the independence case where n = 100000. A multivariate illustration in dimension 4 is given in Figure 3. The comonotonic case is illustrated in Figure 4. The simulations parameters are a = 2, b 1 = 10 and b 2 = 15.

Figure 2 .

 2 Figure 2. Convergence of ê⊥ α (X) (asymptotic independence case). On the left, the first coordinate of e α (X) and ê⊥ α (X) for various values of k = k(n) are plotted. The right figure concerns the second coordinate.

Figure 3 .

 3 Figure 3. Convergence of ê⊥ α (X) (asymptotic independence case). The coordinates of e α (X) and ê⊥ α (X) in dimension d = 4, n = 100000 and k(n) = 100.

Figure 4 .

 4 Figure 4. Convergence of ê+ α (X) (comonotonic case). On the left, the first coordinate of e α (X) and ê⊥ α (X) for various values of k = k(n) are plotted. The right figure concerns the second coordinate.

  present the obtained results for different k(n) values in the independence and the comonotonic cases respectively. The simulations parameters are a = 4, b 1 = 10, b 2 = 15, τ = 0.75 and n = 10000.

Figure 5 .

 5 Figure 5. Convergence of ê⊥ α (X) (asymptotic independence case). On the left, the first coordinate of e α (X) and ê⊥ α (X) for various values of k = k(n) are plotted. The right figure concerns the second coordinate.

Figure 6 .

 6 Figure 6. Convergence of ê+ α (X) (comonotonic case). On the left, the first coordinate of e α (X) and ê⊥ α (X) for various values of k = k(n) are plotted. The right figure concerns the second coordinate.

Figure 7 .

 7 Figure 7. Convergence of ê⊥ α (X) (asymptotic independence case). On the left, the first coordinate of e α (X) and ê⊥ α (X) for various values of k = k(n) are plotted. The right figure concerns the second coordinate.

Figure 8 .

 8 Figure 8. Convergence of ê+ α (X) (comonotonic case). On the left, the first coordinate of e α (X) and ê⊥ α (X) for various values of k = k(n) are plotted. The right figure concerns the second coordinate.

1 FX1

 1 1)E[(X 1 -x 1 ) + ] x X i -x i ) + 1 1 {X1>x1} ]

1 α

 1 (X)) = 0. Taking if necessary a subsequence (α n ) n∈N with α n -→ 1, we consider that the limits lim α→1 xi x1 = β i exist and that lim α↑1 1-α FX 1 (x1) = 0.

lim α↑1 +∞ 1 P

 1 (X j > tx 1 , X i > x i ) P(X 1 > x 1 ) j > tx 1 , X i > x i ) P(X 1 > x 1 ) X j -x j ) + 1 1 {Xi>xi} ]

x 1 -x j x 1 FXi

 11

  and that is contradictory with Equation A.8.

  = 0 for all t > 0. We apply the dominated convergence Theorem, using Potter's bounds associated to FX1 , to getlim α↑1 E[(X j -x j ) + 1 1 {Xi>xi} ] x 1 FX1 (x 1 ) = > tx 1 , X i > x i ) P(X 1 > x 1 ) -x j ) + 1 1 {Xi>xi} ] x 1 FX1 (x 1 ) = 0, ∀j ∈ J C . If j ∈ J ∞ \{i}, then E[(X j -x j ) + 1 1 {Xi>xi} ] x 1 FX1 (x 1 ) =

	because	P(Xj >tx1,Xi>xi) P(X1>x1)	≤	P(Xj >tx1) P(X1>x1) and lim α↑1	P(Xj >tx1) P(X1>x1) +∞
									βj	lim α↑1	P(X dt,
	and since
			P(X j > tx 1 , X i > x i ) P(X 1 > x 1 )	≤	P(X i > x i ) P(X 1 > x 1 )	=	x 1 x i	x i FX1 (x i ) x 1 FX1 (x 1 )	FXi (x i ) FX1 (x i )	,
	so, by Lemma 2.2
				lim α↑1	P(X j > tx 1 , X i > x i ) P(X 1 > x 1 )	= lim α↑1	P(X i > x i ) P(X 1 > x 1 )	= 0,
	we deduce finally that
					α↑1 lim	E[(X
									dt,

1 > x 1 ) dt, so if j ∈ J C , then lim α↑1 E[(X j -x j ) + 1 1 {Xi>xi} ] x 1 FX1 (x 1 ) = lim α↑1 +∞ x j x 1 P(X j > tx 1 , X i > x i ) P(X 1 > x 1 ) dt = lim α↑1 +∞ βj P(X j > tx 1 , X i > x i ) P(X 1 > x 1 ) j j +∞ xj P(X j > t, X i > x i ) x 1 P(X 1 > x 1 ) dt ≤ +∞ x1

Acknowledgment

We are grateful to the editor and to the reviewers. We thank them for their valuable comments which helped to improve the quality of our manuscript.