two : From Polynesia's 'Island of Women' to America's

initiated woman is worth..

1 published in S. Dunis (ed.) Sexual Snakes, Winged Maidens and Sky Gods: Myth in the Pacific, An Essay in Cultural Transparency, 2008, Editions Haere Po, Pape'ete : 141-154.

ONE INITIATED WOMAN IS WORTH… TWO : FROM POLYNESIA'S 'ISLAND OF WOMEN' TO AMERICA'S ALIEN 1

Marika Moisseeff, CNRS, Paris

Because in bygone times, it was the mother who had the power (the mana). …it was the mother who was the receptacle ('oto). … She was the receptacle in which the child is placed (vai'ia). How could it have been the father's power (mana)? He pulls out (tapae) and that's it! Vari'i, Recording by Henri Lavondès, 1966 -A-24 I will try to draw a parallel between two myths belonging to two very different cultural settings : traditional Polynesia and present-day USA. The first, presented in Henri Lavondès' PhD, Terre et mer, is based on the Isle of Women motif which, according to Lavondès who collected several versions of it in the Marquesas, is "very widespread in Polynesia and probably elsewhere in the Pacific. " (1975 : 411). 2 The second myth is composed of the four American movies dedicated to the battle between Lieutenant Ripley and her extraterrestrial enemy, Alien. Both these narratives describe a process of female initiation allowing women to assume and survive their reproductive role : pregnancy and childbirth. Their 'moral', however, is very different. In both cases childbirth is shown to be potentially lethal. Both narratives address the same scandalous human problem which is the asymmetry of men's and women's procreative roles. A comparative analysis of these two Marquesan and North-American myths reveal a number of structural inversions that shed light on the distinctive ways in which so-1 A first French version of this text was published in A. Babadzan (ed.), 2003 Insularités. Hommage à Henri Lavondès. Nanterre, Société d'Ethnologie : 79-107. And a shorter version was published in Nouvel Observateur Hors Série 'Lévi-Strauss et la pensée sauvage ', juillet 2003 : 82-85, with a new publication in Le siècle de Lévi-Strauss, CNRS Editions, Paris : 157-164. I wish to thank Serge Dunis, Naomi McPherson, Bambi Schieffelin, Heather Young Leslie, Helen Arnold and Michael Houseman for their support and comments on this version. 2 See for example the different versions of 'A'amu no Mahutari'i : a Maupiti warrior goes down to the beach at night to fish momote'a crabs with his torch. Mahutari'i inadvertently treads on a 'upa'i crab which sweeps him away to fenua Maaroaro (Rarotonga). The newcomer of the early morn waits all day in vain for a welcome and decides to go inland at sunset. He finds a pool shaded by a pandanus and upon hearing approaching voices, he ascends the tree. Young women are coming to bathe and enjoy their 'husband' : a young aerial root. Taken aback, Mahutari'i throws pandanus seeds into the pool and is then invited to have fun, thus becoming the women's husband on this island devoid of men. In another variant, Mahutari'i and nine other warriors set out to the same island on a pa'apa'a crab and found families. (cf. Llaona 2000).

called traditional and contemporary Western societies deal with maternity and, therefore, with male/female relationships.

In order to better appreciate the exotic character of the Alien planet, we shall first explore the richness of the Isle of Women by looking at the myth of Kae.

When women get a man

When Kae became a widower, he and another chief joined forces in a seafaring venture, in search of a wife. The other chief, an extremely ugly man, was jealous of Kae's beauty and marooned him on a wooden dish in the middle of the ocean.

The currents carried Kae to 'Pandanus Island', inhabited exclusively by women whose husbands were pandanus aerial roots. When they were pregnant with the children given to them by the pandanus roots, a magician, who was inhabited by a devil, a witch doctor, cut their womb open. The mother died and the child lived, but the children were always girls. That is why all of the island's inhabitants were women, and there were no men other than the magician, but he was a maleficent demon.

The chief of the island was a woman who did not make love with the pandanus aerial roots. "She did not become pregnant because she was the woman chief. The magician didn't go after her". One day, as she went to the beach with her companions, she found Kae there. She took him home with her and greatly appreciated the pleasure of having a human husband.

Kae asked her why there were no men in her country. She explained that the pandanus roots played that role, that the women only gave birth to girls, and that it was the magician who cut them open. "No more cutting open ! When it comes time to give birth, the women will deliver their children. When their pains come, we will help them to deliver, we will have them make efforts. When the magician comes, chase him away, bring the women to me, I will do the work."

From then on, when a woman was in pain, she was carried to Kae's house. The child was born without killing the mother. The magician was chased away. The women were very happy, because the children were born very well.

Kae's wife, the woman chief, became pregnant by her husband. One day, as she was delousing her husband's head, she came upon a white hair. Kae asked her to name their child-to-be 'Kae's white hair', and thereupon decided to return to his home country. And so he did, carried by a dolphin provided by the woman chief.

After her husband's departure, the woman chief gave birth to a boy, who, when he grew up, pestered his mother to find out who his father was. He then went off to find him, on the back of another dolphin belonging to his mother. Now Kae had a pig farm and plantations in his country, for his son. The entire property was surrounded by a stone wall that Kae's son caused to collapse as he ran away from people who tried to approach him. They finally caught the young man, tied him up and threw him into an earth oven, where he was stoned and was soon to be cooked. "My mother told me that Kae, my father, lives here," he said, in tears, "That is why my name is 'Kae's white hair". Kae was then able to recognize his son and could save him. (Lavondès, 1975, I : 412 and II : 227-246) The eponymous hero of this myth is a man, and it is as such that he meets his first opponent, in the person of another chief with whom he is allied at first and who later ejects him when at sea. Then it is he who ejects the magician from the Pandanus Island, so that its women inhabitants may survive the birth of their children. As a father-to-be, when the woman chief is pregnant with his child, he ejects himself, so to speak, upon learning from his wife of the growth of his first white hair. Kae's son, in turn, leaves his mother to find his father, the father being able to recognize his son as his alter ego once he has pronounced the term designating the filial ties between the two of them. The myth seems to indicate that to become a true man, accepted within the male community, one must first leave the maternal universe, confront the seas and strangers, and consent to become a father : that is, forego the status of child for that of parent. Kae was indeed clearly incapable of moving from one generation to another before he came to the island of women. His first wife had died before he could give her a child, and although a widower he was still as beautiful as an adolescent ; "Kae", says Lavondès, "is a man on the threshold of old age who persists in behaving like a Katie (a youngster)." (ibid. : 413).

In this respect, Kae's travels back and forth between land and sea suggest the different phases of men's initiation. Kae leaves his country to go to sea, where he is subjected to a first ordeal, since he finds himself on the flimsiest of vessels, a flat dish, out on the open sea. He spends a period living on the fringes of society, in a most unusual place -inhabited essentially by women -where he must save women from the grasp of a demon. He then becomes a father, and can return to his own country, again facing the seas, on a dolphin's back. At the end of these trials Kae is taken back into ordinary human society in which the male and female spheres coexist, albeit in spatially distinct areas. He then becomes the representative of the male group, a true chief capable of introducing his son to the bisexual world to which they both henceforth belong. But as he moves from the status of a childless man to that of a father, Kae must accept his passage from one generation to another ; that is, he must resign himself to the aging process, symbolized by his first white hair. The child therefore fully merits the name his father gave him, 'Kae's white hair', since the son causes the father to move up one step on the genealogical ladder. By doing so, Kae gains maturity at last, and can then go to the land of men where fathers work to produce riches and goods to be transmitted to their children.

The most remarkable element in this narrative is that male initiation, access to male maturity via fatherhood, is postulated as interdependent with female initiation. Before Kae becomes a father he initiates women to the specifically human process of "natural" childbirth, and to the suffering it implies. He embodies that sexual otherness so cruelly lacking on the pandanus island until his arrival. For indeed, on that exclusively female territory, the otherness with which the women must deal is not of a human, gendered nature : it pertains to plants, pandanus aerial roots, on the one hand, and to the demon, the magician, on the other. Reproduction therefore takes on a quasi vegetal-like, parthenogenetic character, with women only giving birth to women, the same generating the same. Mother and daughter form a single two-sided entity, with the mother becoming the slough, so to speak, of the daughter when they are split into two by the magician/gardener. The young sprout conceived with a pandanus root is then transformed into a cutting, put in the mother's stead and replacing her entirely.

There is a connection between the absence of specifically human, gendered otherness and women's inability to give birth without immediately dying : the coexistence of successive generations is impossible. To survive childbirth and thus move from one generation to the next, women must call upon the services of a real human male, not a demon/magician, who gives them access to the other sex. This allows them to achieve full, complete motherhood, thanks to which they can give birth not only to their own kind, girls, but also to the complementary sex, to boys. By introducing his wife to human sexuality, Kae changes her from a childless virgin to the mother of a son. Although Kae's wife's status is equivalent to her husband's -they are both chiefs -their respective sexual identities make them radically different. More exactly, they are perfectly complementary. As such, these two individuals who had no children before they came to know each other mutually initiate each other to their respective parental functions.

A real meeting of the sexes, that is, one capable of generating life and ensuring its continuity, requires that they be able to procreate together; the child born of this sexual encounter is at once the link between generations and between man and woman. Both Kae prior to his arrival and the demon/magician are incapable either of achieving an encounter of this type with the opposite sex or of situating themselves in a historical time frame. As such, they are responsible for the fact that the women around them do not survive. Kae's first wife dies childless, and the women, cut open by the magician, are replaced by their young sprouts ; their daughters simply replace them instead of integrating them into a genealogical sequence. Conversely, for both Kae and his wife, both chiefs and as such the symbolic representatives of their respective gender group, the effective encounter between the sexes through reproduction is tantamount to gaining access to the older generation. In other words, they gain access to the phase of growing old, that lying between youth and death. This integrates them in a historical time frame where different generations may coexist, a prerequisite for the process of intergenerational transmission and therefore any enduring social organization.

The theme of the island of women describes the transition from a timeless, homosexual, female world where the Other is of a different species -a plant (a pandanus aerial root) -to a division of the human species into two different genders, the Other of the female being the male. The introduction of maleness into the female universe brings humankind, threatened with being subsumed by the vegetal reign, to the genealogical time frame underlying any cultural order.

According to this myth, the passing of generations is dependent on sexual reproduction, that is, on the meeting of two complementary sexes participating jointly in the reproductive process. Furthermore, contends the myth, this union must be followed by a separation : in a second phase, Kae's wife must allow her husband and her son to leave her. In the everyday activities of this traditional Polynesian society, male and female spheres are kept separate from each other ; by providing dolphins to carry her husband and son to the land of men, the woman chief, representing the female community, proves her submission to this law. For a son to become an adult man capable of having children with women, he must first leave his mother and the woman's world in which he has lived. This separation of opposites, of mother and son, of men and women, is an essential prerequisite for their subsequent fertile reunion; the sexual complementarity underlying human reproduction requires the bringing together of what has previously been disjoined. 3 This to and fro movement between male and female is magnificently symbolized by the ebb and flow of the sea, which both maintains a distance between the islands it keeps apart, and makes it possible to overcome that distance (one must simply confront it) : the waters that carried Kae away from his wife are one and the same as those that took him to her.

Among other things, this myth restores the inescapable truth that men's procreative function is materialized in the body of women : men cannot give birth to their own kind and even less to their sexual opposites. Kae's son is born to a woman and that woman herself was born to a woman. However, while it recalls this naturally asymmetrical relationship favouring women and dictated by pregnancy, the myth contrasts this with the complementarity of men and women, along with a strictly converse asymmetry favoring men, who alone are in a position to introduce women to culture. Kae culturalizes nature, represented by the pandanus roots and the demon that inhabits the magician, by replacing the gardener's knife by obstetrical techniques more appropriate to human birthing. In this respect, Kae's initiation is much more complete than his wife's. Firstly, it is he who comes to her, and secondly, he goes from the status of novitiate to that of initiator. 4 The male otherness embodied by Kae is the underlying factor in men's responsibility for the fertility of women in general. Thus, through a symbolic manipulation often found in traditional societies (Moisseeff, 1995(Moisseeff, , 1998)), a shift occurs, from the natural encompassing of the male by the female to the cultural encompassing of the female by the male. Without men, women would remain shackled to a dangerously natural state, for women's powers are potentially lethal if they are not domesticated ; to give birth to those liberators who subordinate nature to culture, they must, precisely, have commerce with men.

The control and the appropriation of women's powers by men is a theme found in myths everywhere : for women's engendering of men to be tolerable, it must be subordinated to culture, of which men are made the guardians (cf. [START_REF] Godelier | La production des Grands Hommes[END_REF][START_REF] Dunis | Sans Tabou Ni Totem, inceste et pouvoir politique chez les Maori de Nouvelle-Zélande[END_REF]Moisseeff, 1995). In the Judeo-Christian mythology that pervades modern culture in the United States, as we shall see, this feat is accomplished by God. The bodily creation of Eve out of Adam's rib and that of Christ out of a virgin mother evidence men's efforts to culturalize the unavoidable transiting by the body of a woman. Eve's motherly function, her role in the process of engendering human kind is subordinated to that of Adam and of God, the universal father. Fatherhood precedes and encompasses motherhood, and the same is of course true for Virgin Mary. One noteworthy difference between these myths and the story of Kae is that, in them, the valued mode of reproduction is independent of any sexual intercourse : Adam, Eve, Christ and even his mother (remember the immaculate conception) are all conceived outside of any sexual relationship. This is also the case in another cycle of myths that I will now discuss : Alien is conceived without any need for sexual intercourse. This science fiction film reflects the evolution of representations and practices linked with reproduction in Western societies, where erotic activities -sex life -and procreation tend to be viewed as independent spheres : if individuals are to achieve full sexual pleasure, believed to be a human prerogative (Moisseeff, 2000a and b), they must be "protected" from any reproductive risk. Child-bearing, in turn, belongs to the preserve of the "medically assisted". The place and function of motherhood are under the authority of specialists, gynecologists and obstetricians in particular, who are more or less exclusively in charge of this domain. At the same time, having children is not connected with any particularly respected social status, and the parental function does not partake of the definition of adulthood, since adult sexual identity is assumed to derive directly from one's sexual identity at birth, and the transition from one to the other does not require any specific procedure, aside from the act of providing pubescent individuals with effective means of contraception aimed at avoiding pregnancy (Moisseeff, 1992).

In the Kae myth, the fertile, positive union of man and woman is connected with sexual intercourse between them and requires that they be separated previously and afterwards. In the present-day Western world, individuals of both sexes mix constantly, and a disjunction establishes a more or less radical break, not between the sexes but between sexuality and procreation (as shown by contraception, abortion and the new reproductive techniques). This disjunction should be seen as correlated with the determination to achieve symmetry between gender roles : once the issue of procreation is removed, men and women may be viewed as occupying status-equivalent positions.

In Alien and other similar films, pregnancy is likened to an inexorably fatal, parasitic occurrence affecting men as well as women : with the exception of the heroine, the male and female characters are equal in this respect and in all others (they perform similar activities together). Otherness is not represented by the difference between the sexes, as in Kae, but by reproduction, embodied by a monster. 5 It is presented as "a survivor unsullied by conscience, remorse or moral illusions" (Alien, the eighth passenger). And by a strange coincidence, in this science fiction world where men and women are equal and mix in all spheres, only a woman is able to combat the crazy, immoral 'relic' represented by pregnancy. This new type of myth therefore brings dramatically back to life what our 'egalitarian' societies tend to conceal; that is, the primordial asymmetry favouring women by virtue of their reproductive faculty, giving them specific, exclusive powers. For if there is one constant underlying symbolic elaboration on the difference between the sexes, it is the more or less systematic correlation between women's reproductive powers and the notion of danger, of some hidden, death-dealing force. The uterus is a Pandora's box, capable of letting loose a thousand devils. 6In the Kae myth, the demon takes the shape of a magician who kills the women he delivers. It is up to a man, a real man, to take responsibility for taming the function of motherhood by subordinating it to culture. The cultural elaboration of the difference between the sexes consists here in a reversal of the natural asymmetry favoring women that consolidates men's social and cultural superiority. In the Alien myth, the demon is pregnancy itself, and it is a woman who will succeed in suppressing its effects. Paradoxically, the cultural elaboration of the difference between the sexes, consisting here of creating symmetry between their social and cultural roles, tends to reinforce the asymmetrical nature of male and female positions with respect to reproduction. By inventing practices such as new reproductive techniques (IVF, test-tube babies, clones) corroborating the feasibility of dissociating sexuality and procreation -that is, by setting the field of obstetrics apart from the relations between the sexes -society simply recognizes what is specific about the female reproductive function, namely gestation. Pregnancy can then be reintegrated into cultural imagery as an independent entity, cut off from its usual sexual medium, the female body. Its appearance becomes that of a beast, whose monstrous mask hides those occult, lethal female powers. Thus, in movies such as Alien, we are led to witness a battle between a female and her procreative function, a means of connecting what has previously been disconnected. However, if this myth suggests the possibility of such a disjunction, it also provides a novel solution to this situation : a process of initiation, through which women may subjugate what have become the quasi autonomous powers inherent in their capacity from motherhood.

In order to examine this solution more closely, we will now leave the traditional insular world of the Marquesas Islands for the post-modern intergalactic space of an American utopia.

Where men become pregnant

The opening scene of the first episode of Alien's adventures with a woman lieutenant named Ripley is set in the closeted space of a spaceship containing glass cradles in which the protagonists, two women and five men, are in 'hypersleep'. "Mother", the computer in charge of the "mother vessel" wakes them up and gives them a mission : they are to explore a planet that has just sent out a message indicating the presence of a living organism. Their envoy, named Kane (Hawaiian for man)7 discovers thousands of egg-shaped objects inside a warm, dark cave. One of them hatches, releasing a thing -a sort of giant scorpion with fingers instead of legs -that jumps onto his face, thrusting a stinger down his oesophagus and coiling its tail around his neck. Shortly after Kane's return to the vessel, the thing drops off Kane's face like a sloughed skin. Some time later, Kane is suddenly assailed with terrible pain. A terrifying creature springs out of his chest and flees, shrieking, while he dies. The parasite baby grows at a phenomenal rate, and is soon huge. Ripley wants to blow the vessel up but the scientific officer on board, an android, refuses. He takes his orders from the Company that employs the crew and has just been told, via "Mother", the resident computer, to bring the specimen -Alien -back to Earth at all costs, even if he has to sacrifice everyone. Indeed, the extreme adaptability of this creature makes it an invaluable weapon in the eyes of the Company's officials and the military people with whom they are negotiating. As it turns out, the monster exterminates all of the passengers except Ripley and her kitten. The heroine finally blows up the ship containing "Mother" and after a last fight with Alien she throws it out of her rescue cabin into space.

Second episode, fifty-seven years later. The cabin is recovered by the Company which, in the meanwhile, has sent people to colonize the abominable planet. Ripley is forced to accompany an expedition of marines sent to the planet to find out why the colonists have ceased to communicate with Earth. These teenager-like military folk, muscle-bound, triggerhappy men and women, delight in discussing their feats of sexual prowess. They soon put Ripley in charge of their group, once their leader -a ridiculous macho -is eliminated by a beast that has wreaked havoc by establishing its hive in the middle of the colony. Indeed, prior to the military mission's arrival, the colonists have been captured and trapped in a web where they were changed into cocoons -chrysalides hanging from the ceiling -within which the contents of Alien eggs develop. The nest -the crèche of the huge egg-laying queen -is guarded by warriors of her species who are constantly attacking the members of the expedition, killing them off one by one. Ripley discovers a survivor, a little girl of about twelve, Newt, whom she looks after like her own daughter, and who ends up calling her "mommy". After all sorts of adventures, all particularly gory, Ripley succeeds in overcoming the queen, who is doing everything in her power to kidnap the little girl so as to fertilize her. She then discovers that the resident scientific officer has every intention of bringing the creature back to Earth once it has 'impregnated' both Newt and herself. For Alien must be born to a woman if it is to change into an egg-laying queen. Ripley, assisted by a new android, ends up overcoming her monstrous enemy and her enemy's ally, the scientific officer. The only people left on board are one adult woman, Ripley, one female child, Newt, an incapacitated android (he has been cut in half by the Alien queen) and a single man, the last remaining officer, unconscious and in very bad shape.

When the ship is recovered during the third episode, the male officer and Newt are dead. Ripley, who, like them, has travelled in a glass cradle that has kept her in hibernation, is again the only survivor. She is stranded on an intergalactic refinery-penitentiary with an exclusively male population : twenty-five prisoners possessing a double Y chromosome, all having murdered or raped women or young girls, as well as a ridiculously dictatorial director, his assistant with an abnormally low-IQ (whence his nickname, eighty-five) and last, a doctor, himself a former prisoner (he was responsible for the death of thirty-five people, years ago). The prisoners have vowed chastity and constitute a sort of religious brotherhood within which the presence of a woman is believed to bring misfortune. In the end, most of them are indeed massacred by the death-dealing power following on the heroine's heels. Alien, the giant proliferating insect has not only followed Ripley but has inseminated her. At the very end of the episode, the Company willingly kills the surviving outcasts in order to capture Ripley who has become the case enclosing the most precious jewel : an Alien queen. But before that, Ripley and the prisoners have the time to organize a hunt, pushing Alien towards the prison's foundry where its fate will be sealed in molten metal. The inventor of the android, sent by the Company, tries to convince Ripley to return to Earth with him. "Why give up a normal life and children ?" One of the prisoners makes a similar suggestion : "You should get married, have a baby." However, Ripley prefers to sacrifice herself rather than reintroduce a monstrous, lethal form of motherhood. She hurls herself, arms outstretched, Christ-like, into the molten metal. Her baby, the little queen mother, springs out of her chest; like a Madonna and child, she hugs it to her breast and takes it with her into death.

Two hundred years later, in the final episode, two doctors have succeeded in cloning Ripley using blood taken from the hellish cauldron. The operation results in the creation of two hybrid creatures : Ripley's genes and those of the Alien queen have been mixed, and each has acquired the aptitudes of the other. On the one hand, Ripley's growth and memory, her corrosive blood, her strength, agility and rapidity are the correlate of her lost emotions : she is much less concerned with the dangers her offspring will wreak on the human race than she was in her previous life. On the other hand, Thanks to her hybridization with Ripley, the Alien queen has acquired a complete reproductive system. She has a womb, and is delivered of a half-insect, half-human son, in the throes of terrible birth pangs. No sooner born, this son unfortunately decapitates her. Before this, however, the queen has had the time to lay eggs, from which other demons will spring. The future of the human race is again seriously threatened.

The sorcerer's apprentices in this great operation are men doing highly specialized research in new reproductive techniques under the auspices of the army. In this fourth episode -Resurrection -the action takes place on a spaceship given over to medical research, an aseptic world of men in white jackets flanked by soldiers armed to the teeth. The first scene introduces us to the alchemy of the process under way : from primordial DNA soup emerges a new Ripley, with her alien baby inside her. Some time later we attend the surgical extraction, by Caesarian section from her chest, of this queen, replete with a full set of teeth. The question arises : should they keep the mother alive? Won't she make another attempt to exterminate Alien, her eternal opponent ? In the hopes that she will help them pierce the mysteries of procreation, the doctors save her. This turns out to be a serious mistake, for Ripley again becomes the commander-in-chief of the group that will destroy the innumerable offspring the reconstituted Alien queen produces by laying her eggs in imprisoned human prey. These humans are impregnated by the parasitic seed and transformed into chrysalides carrying her prolific young. The army has procured these sacrificial victims by purchasing the services of a troop of mercenaries who kidnapped them in a state of hibernation from their spaceships. The pirates, however, were unaware of the fate to which their hostages were destined.

Acting concertedly and in total secrecy, doctors and soldiers devote themselves to exploring the mysteries of life and of its origins : procreation. These initiates, all in uniform (white jacket or military attire) conform to strict, puritanical rules of conduct : no drinking, no sex. The other group, the gun-slinging adolescent mercenaries, wearing eccentric clothing, are their exact opposite : they swear, fight, drink, talk continuously about sex, play cards and listen to rock and roll music. There are two women in the group : one is the leader's girlfriend, the other, Call, an excessively feminine "second-generation" robot (that is, generated by a robot herself). These two women differ from Ripley by their tendency to want to protect those around them at all costs. Through the influence of these feminine qualities, the heroine will retrieve her soul which was lost when she regressed, through motherhood, to a kind of animal state. Although she calls Call a "new model asshole " because Call is programmed to redeem decadent humankind and therefore is more sensitive than "real human beings", Ripley finally sides with her and once again saves the human race.

While Ripley and Call are in the ship's chapel, facing a crucifix, Ripley forces Call to plug into the central computer, "Father", thus replacing the evil, inhuman Father's program by the program of a nice, more-than-human robot. The count-down, leading to the explosion of the spaceship, can begin. Whereas the women stick together, the men are divided into two conflicting clans, the 'pro-sex' mercenaries and the 'pro-motherhood' doctors and servicemen, fighting to the death (the mercenaries want sex without motherhood; the doctors want motherhood without sex). Following a period of chasing each other around, during which an inseminated man gives birth to a monster the survivors are exterminated by Alien's offspring, who hang their trophies on the mesh of their queen's nest.

Ripley falls into a ditch filled with members of her teeming "little family" where she watches the queen giving birth to the first male alien. 8 The spectacle of the delivery is pitiful. Every effort is made to show that "natural", uterine delivery is monstrous. The queen, who has exchanged her nice, clean, detachable ootheca for a womb, lies on her back on her bed of suffering, made of sticky, quasi-excremental muck, "a pestilential mire" (Crispin and Whedon, 1997 : 261). Moreover, once born, the male, death-headed monster decapitates and shreds its mother the queen.

8 An explanation communicated on Internet tells us that Ripley acts as a catalyst, indispensable if her daughter, the queen, is to acquire a womb and if her offspring are to be gendered. Ripley and her daughter even had to have more or less sexual contact so as to set in motion the process by which the queen is transformed (that scene was cut out of the final scenario). Whereas Ripley is both the grandmother, the mother and the father of this half-insect, half-human creature, every effort is made to show us that she is in fact the true mother, and she defines herself as such with respect to all the aliens born on the spaceship. Spectators and critics made no mistake about that : "At the end of the film, we witness the inevitable confrontation between beauty and the beast, the difference being that in this case one is the mother, the other the son." (Thoret and Vérat, 1997 : 23) The only men to escape this quasi-obstetrical massacre are, in one case, physically disabled, and in the other, simple-minded. They will participate in the emergency landing on Earth of the mercenary vessel in which they flee along with Call and Ripley. In the last scene, we find the two women in an embrace, moved at the sight of their home planet as if they were witnessing the world coming into being for the first time, saved from disaster once again, while the two men kiss each other on the mouth. Before this final scene, Ripley has had to eliminate the monstrous son.

Comparison

Both the Kae myth and the Alien saga end with the main female protagonists' gaining access to their full-fledged procreative function : the woman chief of the pandanus island and the Alien queen give birth to sons, whereas they had previously been unable to do so. Ripley, the heroine of Alien, is clearly identified with the queen : the film and the subsequent book version make Ripley the 'mother' of this alien son graced with death's head features. This identification acts to portray natural childbirth as monstrous, animal-like and lethal. This, being further accentuated by scenes in which human beings give birth to parasites in the course of the different episodes. They are inseminated by the beast through an intermediate animal form -the kind of giant scorpion -and metamorphosed into chrysalides imprisoned in a web. From this standpoint, the two myths move in opposite directions. On the pandanus island, we start with a fatal operation, an artificial delivery, tending to turn the human race into plants, and end with natural delivery introducing culture. In intergalactic space, we go from natural delivery, which tends to animalize as well as to exterminate the human race, to the thoracic Caesarian section thanks to which the heroine is able to survive childbirth and reintegrate human society. Alien's pregnancy takes place outside her body, in the chest of her prey, and not in the lower part of the body. This transposition from bottom to top emphasizes both the asexual nature of this kind of reproduction and the equal status of men and women with respect to an externalized gravid womb in the form of an egg.

In the Kae myth, we move from timelessness, in which the same replaces the same, where daughters take their mother's place, to linear, historical, genealogical time, requiring the acceptance of both aging and heterosexuality, these being prerequisites for the coexistence of successive generations. In Alien, the heroine, Sleeping Beauty-like, never grows old. Her age is always the same, thanks to the hibernation into which she is plunged during the interval between the first three episodes, and thanks to cloning in the last episode. There is no coexistence of successive generations, since Ripley systematically kills her children (even the daughter she adopted during the second episode does not outlive her9 ). In intergalactic space, time is suspended. This difference between traditional and post-modern mythological time frames is correlated with a difference in spatial movements : respectively horizontal, between land and sea, and vertical, between sky and earth. However, this change of spatial axes actually points to much more fundamental structural differences.

As we have seen, the Marquesas island myth establishes a correlation between the succession of generations and sexually distinct activities and identities. To do so, it connotes natural birthing positively, as a process underlying the transmission, from parents to children, of a cultural order. While acknowledging the asymmetry between men and women with respect to reproduction, it adds a strictly converse asymmetry, in that it is men who initiate women to their role in the reproductive process. To secure men's responsibility in the reproductive process, the myth postulates male supremacy over women. By contrast, in the American myth, aging is rejected and children sacrificed, whereas childbearing is abhorred. The negative valence set on reproduction tends to cancel the difference between men and women; rather than separate male and female activities and spheres, men and women mix and homosexuality is valued. At the same time, female supremacy prevails.

Two forms of homosexuality are actually represented here. The first has to do with Ripley's physical relationship with her daughter, the Alien queen, with whom she has combined to the point where a mutual exchange of genetic material takes place between them. They represent the same, Janus-faced female entity, with the alien daughter representing the women's reproductive function. It is she who will be put to death. The other form of homosexuality has to do with the erotic relationship between Ripley and Call, two women who both epitomize femininity. Call replaces the childbearing female : the last episode begins with the union between Ripley and Alien and ends with a vision of her union with Call. Now, Call is a robot created by a robot (an immaculate conception if ever there was one), and her name is evocative of a 'religious vocation' (to save the world). It is thanks to this guardian angel -more than human and programmed to save the human race -that the heroine recovers the soul she had lost by animalizing herself through childbearing. In the end, the robot woman wins over Ripley's heart, taking the place of the creature who personifies her lost soul. This second form of homosexuality is purely erotic. By no means can Ripley bear the child of a robot woman, and the same is obviously true for the two surviving men of the final episode. In this myth, then, there is a transition from a universal mixing of the sexes, in which a form of reproductive homosexuality is shown as negative, to a non-reproductive, positive homosexuality. The moral of the story, then, is the superiority not of males over females, but of the erotic over the reproductive, and an appreciation of the disconnection between these two aptitudes. In the Kae myth, on the other hand, there is a transition from reproductive but lethal homosexuality to positive heterosexuality combining eroticism and childbearing (even before she becomes pregnant, the woman chief greatly appreciates having a human husband). But in that story it is a man who saves the human race. In Alien the primacy of womanhood is evidenced firstly by the heroine's sex, and secondly by the physical, moral and intellectual inferiority of the male characters throughout the narration.

Another parallel can be drawn between the myth of Kae and the Alien saga. Both describe an initiation through which their respective heroes are led to fulfil their sexually specific reproductive roles. The interstellar epic indeed provides an allegoric account of a genuine initiation to womanhood. A young woman is taken out of her usual setting -Earthand immersed in a wild, untamed environment -intergalactic space -where she must face a great many dreadful ordeals through which she is confronted with that facet of her femininity to which she did not yet have access : motherhood, shown to be terrifying and hideous. During this transitional period she takes a number of steps leading her to gradually accept this other side of herself. Consecutively, she plays mother to a kitten, becomes a foster mother for a little girl, and mothers a female and then a male, both of which are non human. The most perilous ordeal for Ripley, enacted through her submission to Alien, thus involves integrating the animal side of her womanly nature, her maternal alter ego, in her body. In this way, her animal nature will be silenced once and for all : she must die, to be born again totally metamorphosed, stronger and more feminine than ever. She can then be reintegrated into human society with a new status, that of a full-fledged woman, entitled to indulge in a sex life freed from any reproductive constraint. This perspective reveals two further structural oppositions between the two myths. In the Marquesan myth, the main hero is a male, whereas in the American saga, the main hero is a female ; and while in the former, initiation leads to an assumption of the full-fledged status of parent, in the latter, it leads to a definitive rejection of this status.

With this comparison in mind, let us now consider the bases -the preconditions -for the transformation of gender roles in our contemporary Western world.

What's born in the bone comes out in the flesh

During all four episodes of the Alien saga, the eponymous character to be combated by the heroine -the foreigner -has the physical appearance of an extraordinary predator, a kind of giant insect, half ant, half spider, which transforms its human prey into cocoons into which to discharge the content of its eggs. The monster's only aim is reproduction. The beast doesn't devour its victims, it takes hold of their bodies only so as to procreate. It is a cancer-like invader, exterminating from within by penetrating the host organisms, where it develops lightning-fast. Alien's weaponry is its reproductive system, and its supreme weapon is pregnancy. Thus, the contents of its eggs are introduced into the victim's chest and, following a relatively short period of thoracic gestation, the carrier explodes, delivering the baby. This repugnant 'thing' with its many Hollywood avatars (The Thing, Species I and II, etc.) is no other than the hideous mask by which the modern Western world, patterned after the United States, designates the inhuman, beastly, invasive -in short, parasitic -side of procreation (Moisseeff, 2000a). To corroborate this view, it is useful to point out that one of the major goals of contemporary medical research, essentially unknown to the wider public, is to pierce the mystery of how a foreign body (an alien) -the baby -can develop in its mother's body -a modern version, dare we say, of the mystery of incarnation. Genetic research and the subsequent manipulations of reproduction are indeed part of an attempt to unveil the mystery of the origin of life. Given the amount of publicity such efforts have received in the media, it is not surprising that they have inspired symbolic constructions such as works of science fiction, echoing the concerns of the public at large. It is my belief that such works should be viewed as a new form of mythology. Considered from this angle, they may enlighten us as to our own ideology.

We observe, then, that pregnancy -that exclusively female phase of sexual reproduction -is close to representing an almost insufferable asymmetry in a cultural universe whose proclaimed ideal is to achieve perfect symmetry in gender roles. Medical science has successfully taken over this problem : making babies outside their mother's womb is apparently a particularly adequate solution. It is as though medical agencies have appropriated the female reproductive function, thanks to their increasingly sophisticated technology and the growing power it gives them in this sphere (Moisseeff, 1990). After all, the idea of men appropriating female reproductive powers is a rather commonplace theme in anthropology. It is easily found in the rituals and myths of more traditional cultural contexts, where it serves to justify the superiority of men as a group, notably with respect to fertility.

In the Alien myth too, it is men (scientific officers working for a mercantile Company, and soldiers who attempt to monopolize the reproductive function by capturing the creature. However, the difference is that they are systematically and unambiguously depicted as the bad guys. Indeed, one of the paradoxes of Western culture resides in its association of reproduction -represented here by Alien -not so much with the perpetuation of the species as with its destruction. A connection should be made between the fictional beast's extraordinary ability to proliferate, ability which underlies its phenomenal destructiveness, and the constantly hammered theme, in real life, of the threat of annihilation by overpopulation. Generally speaking, the degree of cultural evolution of a given group is deemed inversely proportional to its rate of reproduction. As opposed to the general assumption that holds sway in traditional societies, in the Western world, misfortune, even affliction, tends to be associated with excessive fertility10 rather than with sterility. This idea is anything but new. Remember Jonathan Swift's famous lampoon, A Modest Proposal for Preventing the Children of Poor People from Being a Burthen to their Parents, published in 1729, in which the author, with stark, devastating humour, suggests anthropophagy as a solution to the miseries of overpopulation.

Controlling the access of the poor, the 'under-civilized', to reproduction by controlling the wombs of their women is indeed one of the primary objectives of the 'civilized' world (consider, for example, the 1995 United Nations Cairo Conference). With sterilization viewed as an efficient means of warding off the supposedly dramatic risks entailed by the global overpopulation to which unbridled natural reproduction would allegedly expose the human race, one has the foreboding, at the turn of the third millennium, that Huxley's brave new world, foreseen in 193211 , will soon be upon us. In that dystopian world, only a few savages surviving in faraway reservations still resort to the womb, to the great displeasure of the "civilized" world in whose favourite motto is 'civilization is sterilization'. The corollary of this is access to sexual pleasure. Thus in Brave New World, the civilized indulge in unlimited erotic enjoyment, free of any risk of pregnancy (Moisseeff 2000b), with babies being produced in test tubes or cloned in "incubation" and "conditioning" centers. This shows how prophetic science fiction can be, and demands that we recognize it as an anthropological object in its own right.

Alien may be viewed as the science-fiction follow-up to the logic underlying Brave New World. The saga's plot seems to suggest that Mother Nature -procreation depicted as a monster -has returned to haunt the human race which has rejected her by cutting sexuality off from childbearing. The bad guys, naturally, are the people who try to recuperate the beast, perceived as the supreme weapon for subjugating the population, for their own profit. As we have seen, they are men. The hero, who may succeed in thwarting the terrible doings of the monster and its allies, is a woman. If the threat preying on the human race has to do with the transmission from mother to daughter of woman's ability to procreate, then only a woman possessing the capacity to bear children but refusing to do so is in a position to embody the force that triumphs over the procreating dragon. 12 This heroine, Lieutenant Ripley, is obliged to follow a very special path along which she endures increasingly perilous ordeals, the last of which is death from which she emerges ever stronger and more feminine. This symbolism of death and rebirth, so characteristic of male initiation rituals under other skies and in other times [START_REF] Moisseeff | Entre maternité et procréation : l'inceste[END_REF](Moisseeff, , 1995(Moisseeff, , 1998)), is applied here to the initiation of females. This transposition is highly revealing of the modification of women's role in modern western societies. They may gain access to a highly valued social status on a par with men, under the explicit provision that they set aside their role as mothers.

I hope I have convinced you of the relevance of comparing these two myths from such different regions and eras. The exercise, I suggest, sheds light on how reproduction and the relations between men and women are being transformed in our society. Moreover, you will have noticed, I expect, that in both cases, one initiated woman is, in fact, worth two. In the Marquesas Islands, a mother destined to die if giving birth, acquires a daughter and the possibility of passing along the torch of motherhood. In America, solitary Ripley acquires a brand-new, very pretty female super-robot as a life companion. The similar yet quite different morals of these two stories should perhaps be thought about together.

Bourdieu, in his analysis of Kabyle rites and practices, identifies the same complementary principles with regard to relations between men and women : "on the one hand, the joining of separated opposites… generating life, … and on the other hand, the separation of reunified opposites …"(1980 : 366- 367)

Another form of "ageing" from learner to learned. Thanks to Naomi McPherson for this comment.

The creature's name, Alien, with its various implications, clearly indicates the terrible Otherness facing humanity : stranger ; different or set apart ; supposedly linked to creatures from other worlds ; a specimen coming from elsewhere, entering this world and having accustomed itself to its new environment ; hostile ; unacceptable or repellent(Allen, 1990).

Brian de Palma's film,Carrie (1972), is a masterful, more direct illustration of how terrifying women's powers are, to this day. Carrie is an innocent teenage girl frightened to death by her first menstrual blood. The rest of the story will corroborate her fears : with puberty, her innocence turns into a devilish force destroying everyone around her.

One cannot but help be struck by the connection long established by Ben Finney between space exploration and Polynesian ocean voyaging.Commentaire [MSOffice1] :

In the book based on this fourth episode (seeCrispin and Whedon, ibid.), and contrary to what is hinted in the films, Ripley is supposed to have had a daughter on earth, but she practically never knew her, and that daughter did not survive her either, of course.

The French have a word for" breeding like rabbits, lapinisme, defined in Le Nouveau Petit Robert dictionary (1995) as :"Excessive fertility (in a couple or a nation)". In the exotic societies studied by anthropologists, fertility is hardly ever and possibly never spoken of as excessive. The existence of this notion in modern western societies, and definitely in our society, as indicated by this French term and the English equivalent, is associated not only with a form of malady, but also with animality. One occasionally hears people say : "Those people act like animals". The fact that rabbits are extremely prolific qualify them as metaphors for both the affliction of excessive fertility and the animality on which such behaviour confines. A very peculiar affliction indeed, since it is supposed to affect not only couples but also whole peoples, incidentally those who have not reached a par for civilization : the poor, the savages, the 'developing'.

It should be recalled that Huxley believed that the risk of having a "dictatorship of scientists" set up to impose widespread sterilization would be an unavoidable consequence of overpopulation, as he explicitly showed in Brave New World Revisited, published in 1958.

Crispin's enigmatic dedication of his book (1997) to Sigourney Weaver, the actress who interprets the title role of the combatant in Alien, is none the less significant in this respect: "Thank you for


Frdric Lavancier

Anne Philippe

Some convergence results on quadratic forms for random fields and application to empirical covariances

Keywords: Gaussian Random field, Long memory, Non central limit theorems, Quadratic mean convergence, Stochastic integral, Triangular array

Limit theorems are proved for quadratic forms of Gaussian random fields in presence of long memory. We obtain a non central limit theorem under a minimal integrability condition, which allows isotropic and anisotropic models. We apply our limit theorems and those of Ginovian (1999) to obtain the asymptotic behavior of the empirical covariances of Gaussian fields, which is a particular example of quadratic forms. We show that it is possible to obtain a Gaussian limit when the spectral density is not in L 2 . Therefore the dichotomy observed in dimension d = 1 between central and non central limit theorems cannot be stated so easily due to possible anisotropic strong dependence in d > 1.

Introduction

Many statistical methods proposed for long memory processes are based on statistics, which can be expressed using quadratic forms (see [START_REF] Beran | Statistics for long memory processes[END_REF], for a review). The empirical covariance sequence (see [START_REF] Hosking | Asmyptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series[END_REF][START_REF] Hannan | The asymptotic distribution of serial covariances[END_REF], the estimation of the long memory parameter using Whittle contrast (see [START_REF] Fox | Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series[END_REF][START_REF] Giraitis | A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittles's estimate[END_REF]Giraitis and Taqqu, 1999, for example) or using the integrated periodogram (see [START_REF] Lobato | Averaged periodogram estimation of long memory[END_REF], some change point detection procedure (see [START_REF] Beran | Testing for a change of the long-memory parameter[END_REF] rely on quadratic forms.

It would be interesting to validate the same statistical tools for the study of random fields having long memory. As a first step, we study in this paper the convergence of the quadratic forms in the d-dimensional (d > 1) case.

Let us introduce our framework. Let X = (X n ) n∈Z d be a stationary L 2 random field having long memory, i.e. its covariance sequence (r(n)) n∈Z d is not a summable series. We assume that X is a Gaussian process and admits the 1 linear representation

X n = E a(x)e i<n,x> dW (x), n ∈ Z d , E = [-π, π] d , ( 1
)
where W is the Gaussian white noise spectral measure and where the function a is in L 2 (E). < •, • > denotes the inner product in R d . Hereafter, we denote by f the spectral density of X, f is proportional to |a(x)| 2 , and we have r(n) = fn := E e i<n,λ> f (λ)dλ.

(2)

The quadratic forms associated to X and (g i ) i∈N d ∈ ℓ 1 (N d ) are defined as

Q n = 1 n d i∈An j∈An g i-j X i X j (3)
where A n = {1, . . . , n} d . The statistics Q n can be rewritten as a functional of the periodogram I n of X. Indeed let

I n (t) = (2πn) -d k,l∈A 2 n X k X l e i<k-l,t> ,
we have

Q n = (2π) d E g(t)I n (t)dt, (4)
where g(t) = (2π) -d j∈Z d

g j e -i<j,t> .

In dimension one (d = 1), many papers deal with the asymptotic behavior of Q n for long memory processes. When the intensity of the memory is not too strong or when specific conditions are imposed to g in order to kill the effect of the long memory involved by f , the convergence rate and the asymptotic normality obtained in short memory are preserved. The asymptotic normality of the quadratic forms is proved by [START_REF] Avram | On bilinear forms in Gaussian random variables and toeplitz matrices[END_REF], [START_REF] Fox | Central limit theorem for quadratic forms in random variables having long-range dependence[END_REF] and [START_REF] Ginovian | On the central limit theorem for Toeplitz quadratic forms of stationary sequences[END_REF] in the Gaussian case and extended by [START_REF] Giraitis | A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittles's estimate[END_REF] to include the non-Gaussian linear case. Avram's result is restricted to f and g in L 2 , but requires only integrability conditions. On the opposite, [START_REF] Fox | Central limit theorem for quadratic forms in random variables having long-range dependence[END_REF], [START_REF] Ginovian | On the central limit theorem for Toeplitz quadratic forms of stationary sequences[END_REF] specify the behavior of f and g around zero. The extension to linear models obtained by [START_REF] Giraitis | A central limit theorem for quadratic forms in strongly dependent linear variables and its application to asymptotic normality of Whittles's estimate[END_REF] holds under the assumptions of [START_REF] Avram | On bilinear forms in Gaussian random variables and toeplitz matrices[END_REF] or [START_REF] Fox | Central limit theorem for quadratic forms in random variables having long-range dependence[END_REF].

Under other conditions on f and g fixing the behavior of both functions around zero, [START_REF] Rosenblatt | Weak convergence to the rosenblatt process[END_REF], [START_REF] Fox | Non central limit theorem for quadratic forms in random variables having long-range dependence[END_REF], [START_REF] Terrin | A noncentral limit theorem for quadratic forms of Gaussian stationary sequences[END_REF] prove non central limit theorems for Gaussian processes. The normalization is not standard (i.e. it is not equal to n -d/2 ) and the limiting process is not Gaussian. The extension to linear models is proved in [START_REF] Giraitis | Limit theorems for bivariate Appell polynomials. Part II: Non-central limit theorems[END_REF] in the general context of bivariate Appell polynomials.

Fixing the behavior of f and g around zero, the dichotomy in d = 1 can be summarize as follows. Assume that

f (x) ∼ |x| 2α when x → 0, g(x) ∼ |x| 2β when x → 0.
The limiting distribution depends on the value of (α + β).

• If α + β > - 1 4
then the limit is Gaussian and the normalization is n -1/2

(Q n satisfies a central limit theorem).

• If α + β < - 1 4
then the limiting distribution is not Gaussian and the normalization in this case is n 2(α+β) , which is lower than n -1/2 (Q n satisfies a non central limit theorem).

The case α + β = 1 4 is investigated in [START_REF] Ginovian | On the central limit theorem for Toeplitz quadratic forms of stationary sequences[END_REF] (up to a slowly varying function in f and g which vanishes at zero). In this case, the limit remains gaussian.

In the d-dimensional case (d > 1), a central limit theorem has been proved by [START_REF] Ginovian | Nonparametric estimation of the spectrum of homogeneous Gaussian Fields[END_REF], in the same spirit as [START_REF] Avram | On bilinear forms in Gaussian random variables and toeplitz matrices[END_REF] (i.e. f and g are in L 2 and satisfy an integrability condition). This result is recalled in Section 2.1. When f or g are not in L 2 , some central limit theorems have been proved under some specific conditions on these functions, see Theorem 3.1 in [START_REF] Doukhan | Central and non central limit theorems for quadratic forms of a strongly dependent gaussian field[END_REF] and Lemma 3.3 in [START_REF] Boissy | Parameter estimates for fractional autoregressive spatial processes[END_REF].

On the other hand, few results are available on non-central limit theorems in dimension d > 1. In [START_REF] Doukhan | Central and non central limit theorems for quadratic forms of a strongly dependent gaussian field[END_REF], the asymptotic of Q n is investigated when the memory of X is "isotropic" (see [START_REF] Lavancier | Long memory random fields[END_REF] Definition 1), i.e. the memory of X is due to |x| 2α in the expression of f where | • | denotes the Euclidian norm. The authors obtain a non-central limit theorem very similar to [START_REF] Fox | Non central limit theorem for quadratic forms in random variables having long-range dependence[END_REF]. However the investigation of "anisotropic" memory and more general functions g (including the particular case of the empirical covariance) is of great interest for statistical applications. In [START_REF] Lavancier | Long memory random fields[END_REF], different classes of random fields having long memory are described: The important role of anisotropic processes in terms of modeling is illustrated.

Our paper is organized as follows. In Section 2, after a reminder of the central limit theorem obtained by [START_REF] Ginovian | Nonparametric estimation of the spectrum of homogeneous Gaussian Fields[END_REF], we present a non-central limit theorem under a general integrability condition on f and g. We show that this condition is satisfied by a large class of isotropic and anisotropic models (including all previous results). Since random fields in d > 1 may exhibit much more different types of long memory than processes in d = 1, the dichotomy between central and non central limit theorems is not so simple to formulate. As already highlighted in Remark 4.2 of [START_REF] Dobrushin | Non central limit theorems for non-linear functionals of gaussian fields[END_REF], a spectral singularity outside zero may lead to new limiting results. This fact is illustrated through examples in Section 2.3.

In Section 3, we apply limit theorems for quadratic forms to the convergence of the empirical covariance series. Contrary to the classical setting in dimension one, we show that it is possible to obtain a Gaussian limit in dimension d > 1 when f is not in L 2 .

2 Convergence of quadratic forms

Central limit theorem

We recall in this section the central limit theorem obtained by [START_REF] Avram | On bilinear forms in Gaussian random variables and toeplitz matrices[END_REF] in dimension d = 1 and extended by [START_REF] Ginovian | Nonparametric estimation of the spectrum of homogeneous Gaussian Fields[END_REF] to d > 1. This result will be useful in section 3, where the asymptotic behavior of the covariance series is investigated.

Theorem 1 [START_REF] Avram | On bilinear forms in Gaussian random variables and toeplitz matrices[END_REF], [START_REF] Ginovian | Nonparametric estimation of the spectrum of homogeneous Gaussian Fields[END_REF]). Let X be a random field as in (1) and denote f its spectral density. Let Q n , defined in (3), be the quadratic form associated with g and X. If f ∈ L p and g ∈ L q with p ≥ 2, q ≥ 2 and 1/p + 1/q ≤ 1/2, then

n d/2 (Q n -E(Q n )) L -→ N 0, 2(2π) 3d E f 2 (t)g 2 (t)dt , (5)
where L -→ denotes the convergence in law.

This result provides a large setting since no specific form for functions f and g is required, but only integrability conditions. In return, f and g are restricted to L 2 . A similar result is obtained in [START_REF] Doukhan | Central and non central limit theorems for quadratic forms of a strongly dependent gaussian field[END_REF], Th. 6.2., but under semi-parametric assumptions on f and g.

When f or g are not in L 2 but a compensation between them occurs, a central limit theorem may still arise as in [START_REF] Fox | Non central limit theorem for quadratic forms in random variables having long-range dependence[END_REF] for d = 1. To our knowledge, two situations in dimension d > 1 have been investigated so far. In dimension d = 2, assuming f and g have a tensorial product form around zero, [START_REF] Boissy | Parameter estimates for fractional autoregressive spatial processes[END_REF] obtain a central limit theorem, which can be straightforwardly extended to d > 2. In [START_REF] Doukhan | Central and non central limit theorems for quadratic forms of a strongly dependent gaussian field[END_REF], Th. 6.1, the authors prove the same result when f has an isotropic form around zero and when g is regular enough and g(0) = 0.

It is known that the lonely condition 0 < E f 2 (t)g 2 (t)dt < ∞ is not sufficient to obtain a central limit theorem, as proved in [START_REF] Ginovian | On the central limit theorem for Toeplitz quadratic forms of stationary sequences[END_REF], Prop. 2.2. A general setting leading to a central limit theorem (included f or g not in L 2 ) is still missing. This issue is not investigated in this paper.

Main result : non-central limit theorem

The following theorem gives the convergence of the quadratic forms under the general condition (H) below on f and g. Contrary to the previous studies in the one dimension case or in dimension d > 1, this condition does not require to specify the behavior of f around its singularities. Such a condition is satisfied by many examples as shown in Section 2.3.

Let ζ be a function defined on R d , hereafter we denote by ζ (p) (p ∈ N) the periodic function with period 2πp (with respect to each component), that coincides with ζ on the set pE d ,

ζ (p) (x) = ζ(x) ∀x ∈ [-pπ, pπ] d . (6)
Theorem 2. Let X be the linear field defined by ( 1) and Q n the quadratic form defined by (3).

Assume that for all x ∈ E a(-x) = a(x), and g(-x) = g(x). Moreover, assume that

a(x) = ã(1) (x)L 1 (x), g(x) = g(1) (x)L 2 (x),
where ã is a homogeneous function of degree

α (i.e. ∀c > 0, ∀x ∈ E, ã(cx) = c α ã(x))
, where g is a homogeneous function of degree 2β and where L 1 and L 2 are bounded functions, continuous at zero and

L i (0) = 0 (i = 1, 2). If the following assumption (H) is satisfied (H): R 2d ã2 (x)ã 2 (y) R d |g(t)| d k=1 1 (1 + |x k + t k |)(1 + |y k -t k |) dt 2 dxdy < ∞, then n d+2α+2β (Q n -E(Q n )) L 2 -→ L 2 1 (0)L 2 (0) R 2d ã(x)ã(y) R d g(t)H(x + t)H(y -t)dtdW (x)dW (y), ( 7
)
where

H(z) = d j=1 e iz j -1 izj .
It seems that (H) is almost necessary for the convergence, since it guarantees that the integral in (7) exists.

The proof is relegated to Section 2.4. The main tool consists in rewriting Q n as a double stochastic integral. Then the convergence in quadratic mean is deduced from a simple convergence in L 2 (R 2d ). Note that contrary to the proofs for non central limit theorems in [START_REF] Terrin | A noncentral limit theorem for quadratic forms of Gaussian stationary sequences[END_REF] and [START_REF] Doukhan | Central and non central limit theorems for quadratic forms of a strongly dependent gaussian field[END_REF], we do not apply the scheme of convergence of [START_REF] Dobrushin | Non central limit theorems for non-linear functionals of gaussian fields[END_REF], especially their Lemma 3 which involves a spectral measure convergence assumption.

Some class of models satisfying the hypothesis (H)

The setting of random fields allows many kind of dependencies (see [START_REF] Lavancier | Long memory random fields[END_REF]). Apart from its intensity, the dependence can be isotropic or can occur all over several particular directions, depending on the form of the spectral density. Since we did not want to restrict ourselves to one particular case, Theorem 2 involves a general integrability condition, i.e. assumption (H). We check in this section that this hypothesis is not too restrictive. Indeed, we prove that in all the preceding studies about quadratic forms of Gaussian fields, this condition is fulfilled. Moreover, it allows to extend to new cases, mainly when the dependence is not isotropic.

As a consequence of the following Lemmas and according to Theorem 2, a non-central theorem for Q n arises for all the cases treated in this section.

Lemma 3. Assume that there exists some positive constants c and c ′ such that

|ã(x)| ≤ c d i=1 |x i | α/d |g(x)| ≤ c ′ d i=1 |x i | 2β/d . ( 8
)
If moreover α > -d/2, β > -d/2 and α + β < -d/4, then (H) is fulfilled.
Remark 1. The result of our Theorem 2 in the setting of Lemma 3 is the same as the convergence stated in dimension 1 by [START_REF] Terrin | A noncentral limit theorem for quadratic forms of Gaussian stationary sequences[END_REF] (Th. 1). The assumptions in Lemma 3 allow the filter a(x) to be isotropic, i.e. equivalent at zero, up to a constant, to |x| α . This is the hypothesis done in Theorem 5.1 of [START_REF] Doukhan | Central and non central limit theorems for quadratic forms of a strongly dependent gaussian field[END_REF] where, moreover, it is assumed β = 0.

Proof. We have to check that the function defined on (R d ) 4 by

(x, y, s, t) → ã2 (x)ã 2 (y)|g(t)||g(s)| d k=1 (1 + |x k + t k |)(1 + |y k -t k |)(1 + |x k + s k |)(1 + |y k -s k |) is integrable on (R d ) 4 .
From the hypothesis of Lemma 3, it is enough to prove that for all k = 1, . . . , d

(t k , s k , x k , y k ) → |x k | 2α/d |y k | 2α/d |t k | 2β/d |s k | 2β/d (1 + |x k + t k |)(1 + |y k -t k |)(1 + |x k + s k |)(1 + |y k -s k |)
is integrable on R 4 . Such integrals are studied in Lemma 1 of [START_REF] Terrin | A noncentral limit theorem for quadratic forms of Gaussian stationary sequences[END_REF]: Under the assumptions of Lemma 3 on α and β, this integral is finite.

When the functions a and g involved in Theorem 2 do not satisfy (8), condition (H) may be investigated thanks to power counting theorems (cf. Theorem 2 in [START_REF] Terrin | A noncentral limit theorem for quadratic forms of Gaussian stationary sequences[END_REF]). The following lemma focus on a particular situation in dimension d = 2: When ã admits one or two independent lines of singularities.

Lemma 4. Assume that d = 2 and that g follows the same conditions as in (8).

If, for p = q, ã(

x 1 , x 2 ) = |x 1 + px 2 | αp |x 1 + qx 2 | αq , and if α p > -1/2, α q > -1/2, β > -1 and α p + α q + β < -1/2, then (H) is fulfilled. Proof. Checking (H) is equivalent to prove that R 8 |x 1 + px 2 | 2αp |x 1 + qx 2 | 2αq |y 1 + py 2 | 2αp |y 1 + qy 2 | 2αq |t 1 | β |t 2 | β |s 1 | β |s 2 | β 2 k=1 (1 + |x k + t k |)(1 + |y k -t k |)(1 + |x k + s k |)(1 + |y k -s k |) dtdsdxdy (9)
is finite. We apply Theorem 2 of [START_REF] Terrin | A noncentral limit theorem for quadratic forms of Gaussian stationary sequences[END_REF]. Let us introduce some notations. The integral above can be written :

R 8 |L 1 (u)| 2αp |L 2 (u)| 2αq |L 3 (u)| 2αp |L 4 (u)| 2αq × |L 5 (u)| β |L 6 (u)| β |L 7 (u)| β |L 8 (u)| β 16 k=9 (1 + |L k (u)|) -1 du, ( 10
)
where u = (x 1 , x 2 , y 1 , y 2 , t 1 , t 2 , s 1 , s 2 ) and the L k 's are the linear functionals involved in (9). For instance

L 1 (u) = x 1 + px 2 , L 5 (u) = t 1 , L 9 (u) = x 1 + t 1 .
Let T = {L 1 , . . . , L 16 } and let γ k be the exponent associated to L k in (10). For instance,

γ 1 = 2α p , γ 5 = β, γ 9 = -1. Consider now the subsets W ⊂ T such that span(W ) ∩ T = W . A subset W is said padded if any L k ∈ W is a linear combination of the L i 's in W -{L k }.
The integrability of (9) near 0 is obvious since 2α p > -1, 2α q > -1 and β > -1. According to Theorem 2 in [START_REF] Terrin | A noncentral limit theorem for quadratic forms of Gaussian stationary sequences[END_REF], the integrability at infinity is achieved if for every padded

W considered above but T , d ∞ (W ) := rank(T ) -rank(W ) + T -W γ k < 0.
The maximum value for d ∞ (W ) is obtained with W = {L 9 , . . . , L 16 }. In this case d ∞ (W ) = 8 -6 + 4α p + 4α q + 4β. This leads to α p + α q + β < -1/2.

Proof of Theorem 2

Proof. Let Z be a Gaussian random spectral measure associated to the measure µ. [START_REF] Major | Multiple Wiener-It Integrals[END_REF] defines

R 2d f (x, y)dZ(x)dZ(y)
for all f ∈ H µ where H µ denotes the space of functions f :

R d × R d → C such that f (-x, -y) = f (x, y) and |f (x, y)| 2 dµ(x)dµ(y) < ∞. The second order moment of this integral satisfies, for all f in H µ , E f (x, y)dZ(x)dZ(y) 2 ≤ 2 R 2d |f (x, y)| 2 dµ(x)dµ(y). (11)
And the so-called Ito formula follows: For all f 1 and f 2 in H µ ,

f 1 (x)f 2 (y)dZ(x)dZ(y) = f 1 (x)dZ(x) f 2 (y)dZ(y)-f 1 (x)f 2 (x)dµ(x). (12)
Now, let us rewrite Q n as a double stochastic integral. According to (1),

Q n -E(Q n ) = 1 n d k∈An l∈An g k-l (X k X l -r(l -k)) = E g(t) 1 n d k∈An l∈An e i<k-l,t> (X k X l -r(l -k)) dt. ( 13
)
From the Ito formula ( 12) <k,x>+<l,y>) dW (x)dW (y),

X k X l -r(l -k) = E a(x)e i<k,x> dW (x) E a(y)e i<l,y> dW (y) - E e i<l-k,x> a 2 (x)dµ(x) = E 2 a(x)a(y)e i(
where µ, the spectral measure of ǫ, is proportional to the Lebesgue measure λ.

Assume that g(-x) = g(x) and a(-x) = a(x), then the function

(x, y) → a(x)a(y) 1 n d E g(t) k∈An e i<k,x+t> l∈An e i<l,y-t> dt (14)
belongs to H λ . Therefore we can rewrite (13) as

Q n -E(Q n ) = E 2 a(x)a(y) 1 n d E g(t)H n (x + t) H n (y -t) dt dW (x)dW (y), (15)
where H n (t) = k∈An e i<k,t> .

In (15), we make the change of variables x → x/n, y → y/n, t → t/n. Since the Gaussian measure W satisfies, for all Borelian set A, W (n -1 A) = n -1/2 W (A) and since ã and g are homogeneous, we get

n d+2α+2β (Q n -E(Q n )) = nE 2 ã(n) (x)ã (n) (y)L 1 x n L 1 y n ψ n (x, y)dW (x)dW (y),
where

Ψ n (x, y) = nE g(n) (t)L 2 t n 1 n d H n x + t n 1 n d H n y -t n dt.
As a consequence, according to (11), it suffices, for proving (7), to show that the following integral tends to zero as n → ∞:

nE 2 ã(n) (x)ã (n) (y)L 1 x n L 1 y n Ψ n (x, y)-ã (n) (x)ã (n) (y)L 2 1 (0)Ψ(x, y) 2 dxdy where Ψ(x, y) = L 2 (0) R d g(t)H(x + t)H(y -t)dt.
From the decomposition A n B -CD = (A n -C)B + (B -D)C, the L 2 -norm above is lower than the sum 2(I 1 + I 2 ) where

I 1 = nE 2 ã(n) (x)ã (n) (y)Ψ n (x, y) -ã(n) (x)ã (n) (y)Ψ(x, y) 2 × L 1 x n L 1 y n 2 dxdy
and

I 2 = nE 2 L 1 x n L 1 y n -L 2 1 (0) 2 ã(n) (x)ã (n) (y)Ψ(x, y) 2 dxdy.
The following lemma will be useful.

Lemma 5.

(i) ∀ z = (z 1 , . . . , z d ) ∈ nE, 1 n d H n z n ≤ d j=1 π 1 ∧ 1 |z j | . (ii) ∀ z = (z 1 , . . . , z d ) ∈ R d , |H(z)| ≤ d j=1 2 1 ∧ 1 |z j | . (iii) For a.e. z ∈ R d , lim n→∞ 1I nE (z) 1 n d H n z n -H(z) = 0.
Proof of Lemma 5. Since for all j, |z j | ≤ nπ,

1 n d H n z n = d j=1 1 n e i z j n e izj -1 e i z j n -1 = d j=1 sin(z j /2) z j /2 z j /2n sin(z j /2n) ≤ d j=1 π 1 ∧ 1 |z j | .
Similarly, for all z ∈ R d ,

|H(z)| = d j=1 e izj -1 iz j = d j=1 sin(z j /2) z j /2 ≤ d j=1 2 1 ∧ 1 |z j | .
Finally, for proving (iii), suppose first that d = 1. If z = 0,

1 n d H n z n -H(z) = 1 n e i z n e iz -1 e i z n -1 - e iz -1 iz ≤ 2 |z| e i z n iz n(e i z n -1) -1 .
The norm in the right hand side term is equivalent to |z|/n when n goes to infinity, hence it tends to 0. In dimension d, (iii) is proved by induction thanks to the decomposition

AB -CD = (A -C)B + (B -D)C.
Let us first prove that I 1 asymptotically vanishes. Since L 1 is bounded, it suffices to prove the convergence to 0 of

I 11 = nE 2 ã2 (n) (x)ã 2 (n) (y) nE Φ (11) n (t, x, y)dt 2 dxdy, (16)
with

Φ (11) n (t, x, y) = g(n) (t) L 2 t n -L 2 (0) 1 n d H n x + t n 1 n d H n y -t n
and

I 12 = nE 2 ã2 (n) (x)ã 2 (n) (y)L 2 (0) 2 nE Φ (12) n (t, x, y)dt 2 dxdy, (17)
with

Φ (12) n (t, x, y) = g(n) (t) 1 n d H n x + t n 1 n d H n y -t n -H(x + t)H(y -t)
In both I 11 and I 12 , the 2π-periodicity of g, H n and H allows us to reduce the domain of integration nE (with respect to t) to

nD x,y = {|x -t| < nπ} ∩ {|y + t| < nπ} ∩ nE.
Therefore, (i) of Lemma 5 can be applied and since L 2 is bounded,

nE Φ (11) n (t, x, y)dt ≤ c nDx,y |g (n) (t)| d k=1 1 ∧ 1 |x k + t k | 1 ∧ 1 |y k -t k | dt,
where c is a positive constant. Hence,

I 11 ≤ c ′ R 2d ã2 (x)ã 2 (y) R d |g(t)| d k=1 1 (1 + |x k + t k |)(1 + |y k -t k |) dt 2 dxdy.
Besides, according to (i) of Lemma 5,

1I nE 2 (x, y)1I nDx,y (t 1 )1I nDx,y (t 2 )ã 2 (n) (x)ã 2 (n) (y)Φ (11) n (t 1 , x, y)Φ (11) n (t 2 , x, y) ≤ c ã2 (x)ã 2 (y)|g(t 1 )||g(t 2 )| L 2 t 1 n -L 2 (0) L 2 t 2 n -L 2 (0) .
From the continuity of L 2 at 0, this term tends to zero for any fixed (x, y, t 1 , t 2 ) ∈ R 4d . Therefore, thanks to assumption (H), the Lebesgue's dominated convergence theorem applies and lim n→∞ I 11 = 0.

The convergence of I 12 is proved similarly. From (i) and (ii) of Lemma 5, we have nE Φ (12) n (t, x, y)dt ≤ c nDx,y

|g (n) (t)| d j=1 1 ∧ 1 |x k + t k | 1 ∧ 1 |y k -t k | dt and I 12 ≤ c ′ R 2d ã2 (x)ã 2 (y) R d |g(t)| d k=1 1 (1 + |x k + t k |)(1 + |y k -t k |) dt 2 dxdy,
where c and c ′ are positive constants.

Besides, for almost every (x, y, t 1 , t 2 ) ∈ R 4d , according to (iii) of Lemma 5,

1I nE 2 (x, y)1I nDx,y (t 1 )1I nDx,y (t 2 )ã (n) (x)ã (n) (y)Φ (12) n (t 1 , x, y)Φ (12) n (t 2 , x, y) n→∞ ----→ 0.
The Lebesgue's dominated convergence theorem applies thanks to (H) and I 12 tends to zero when n → ∞.

It is easy to see that I 2 tends similarly to zero.

Convergence of the empirical auto-covariance function

We present an application of the preceding theorems to the asymptotic law of the empirical covariance function in

Z d . Indeed, for h ∈ Z d , r(h) = 1 n d i∈An X i X i+h
is a particular case of (3) with g(t) = (2π) -d e i<h,t> . We will consider further

r(h) = 1 n d i∈An X i -Xn X i+h -Xn
where Xn = n -d i∈An X i and we denote in the following r(h) = E(X i X i+h ). We are dealing with linear fields as (1). In dimension 1, this framework provides a dichotomy in the asymptotic behavior of r(h) depending on whether the spectral density of X belongs to L 2 or not (see [START_REF] Hosking | Asmyptotic distributions of the sample mean, autocovariances, and autocorrelations of long-memory time series[END_REF]). We prove later on the same kind of division in dimension d, although some kind of intermediate behavior can arise. Let us note that this dichotomy does no longer hold in d = 1 when X is not linear, see for instance [START_REF] Giraitis | A model for long memory conditional heteroscedasticity[END_REF] in the case of a LARCH process or [START_REF] Giraitis | Whittle estimator for finite-variance non-Gaussian time series with long memory[END_REF] for the consequences on the Whittle estimator.

When f ∈ L 2 , we prove that r(h) and r(h) follow the same central limit theorem. This is the object of Proposition 6.

When f / ∈ L 2 , the asymptotic behavior of r(h) and r(h) may differ and we focus on r(h) which is a proper quadratic form. In this setting, the asymptotic law comes from Theorem 2 where β = 0, provided condition (H) is satisfied. Therefore, the normalization and the limit in law depend on the filter a. We summarize in Proposition 7 and in the beginning of section 3.2 the two situations already studied in Lemma 3 and Lemma 4 before.

But Theorem 2 does not apply in the example of Lemma 4 when α p α q = 0 and β = 0. This corresponds to the particular situation when the long memory occurs only along one direction in dimension 2. In this case, we study the asymptotic behavior of r(h) in Proposition 9. It appears that a non-central limit theorem holds in the sense that the normalization is not n d/2 . Yet, contrary to the classical non central limit results for r(h) (see references therein), the limiting law is Gaussian. This shows that the covariance series of a linear field may be asymptotically gaussian even if its spectral density does not belong to L 2 .

General results in dimension d

Let us first present the central limit theorem for r(h) and r(h) when f ∈ L 2 . Proposition 6. Let X be the linear field defined by (1).

If f ∈ L 2 , then, for all h ∈ Z d , n d/2 (r(h)r(h)) and n d/2 ( r(h)r(h)) converge both in law to a Gaussian random variable with zero mean and variance 2(2π) d f 2 2h , where f 2 2h stands for the (2h)-th Fourier coefficients of f 2 as defined in (2).

Proof. The central limit theorem for r(h) follows from Theorem 1 where p = 2, q = +∞ and g(t) = (2π) -d e i<h,t> .

For r(h), we prove that almost surely, n d/2 (r(h) -r(h)) = o(1). Indeed,

n d/2 (r(h) -r(h)) = 1 n 3d/2 k1∈An k2∈An X k1 X k2+h .
Using representation (1) of X, this term is equal to

1 n 3d/2 E f (x)e i<h,x> k∈An e i<k,x> 2 dx = 1 n d/2 E f (x)e i<h,x> d j=1 F n (x j ) dx
where F n denotes the Fejer kernel on [-π, π]. Now, from the Cauchy-Schwarz inequality, this last term is lower than

1 n d/2 E f 2 (x) d j=1 F n (x j ) dx.
Since f 2 ∈ L 1 , the Lebesgue's Theorem implies that the integral above is a o(n d ). Therefore, n d/2 (r(h) -r(h)) = o(1) and the central limit theorem for r(h) is inherited from the one for r(h).

Let us recall that the notation ã( 1) is defined at the beginning of Section 2.2. The following proposition provides central and non central limit for r(h), depending on the parameter α, when condition (18) below is fulfilled. This framework includes isotropic and anisotropic models. The result shows the same kind of dichotomy than in dimension d = 1. In section 4.2, some anisotropic models that do not follow this dichotomy are presented.

Proposition 7. Let X be the linear field defined by (1).

Assume that for all x ∈ E a(-x) = a(x) and that a(x) = ã(1) (x)L 1 (x), where ã is a homogeneous function of degree α > -d/2 and L 1 is a bounded function, continuous at zero and non-null at 0. If

|ã(x)| ≤ c d i=1 |x i | α/d (18)
for some positive constant c, then for all h ∈ Z d ,

• if α > -d/4, n d/2 (r(h) -r(h)) L -→ N 0, 2(2π) d f 2 2h , (19)
where f 2 2h is the (2h)-th Fourier coefficients of f 2 as defined in (2).

• if α < -d/4, n d+2α (r(h) -r(h)) L -→ L 2 1 (0) R 2d ã(x)ã(y)H(x + y)dW (x)dW (y), ( 20
)
where

H(z) = d j=1 e iz j -1 izj .
Proof. In the case α > -d/4, the convergence result is a consequence of Proposition 6. For α < -d/4, Lemma 3 applies since β = 0 and condition (H) in Theorem 2 is fulfilled. Let us justify the simplification of the limit in (20). Here g(t) = (2π) -d e i<h,t> , so g(t) = 1 and L 2 (0) = (2π) -d . The simplification comes from the main term I 12 in the proof of theorem 2, where we use the identity

nE 1 n d H n x + t n 1 n d H n y -t n dt = (2π) d 1 n d H n x + y n .
The pointwise convergence of this last term relies on (iii) of Lemma 5 and an application of the Lebesgue's theorem concludes the proof of (20).

Some anisotropic examples in dimension d = 2

Starting with the anisotropic case studied in Lemma 4, we confirm that new limiting results can be obtained as suggested by [START_REF] Dobrushin | Non central limit theorems for non-linear functionals of gaussian fields[END_REF] (Remark 4.2). If we suppose that a(x) = ã(1) (x)L(x), where L is a bounded function, continuous and non-null at 0 and

ã(x 1 , x 2 ) = |x 1 + px 2 | αp |x 1 + qx 2 | αq ,
with α p > -1 2 , α q > -1 2 and (p, q) ∈ R 2 , then, under the assumptions of Lemma 4 n 2+2αp+2αq (r(h)r(h))

L -→ Z
where Z is defined as the limit in ( 20). This convergence is a simple application of Theorem 2. Condition (H) is fulfilled thanks to Lemma 4 and the simplification of the limit holds for the same reasons as for Proposition 7.

The assumptions of Lemma 4, in the case when β = 0, imply the existence of two lines where the spectral density is unbounded. When α p = 0 or α q = 0 (or p = q), that is when the long memory occurs along only one direction, Lemma 4 does not imply (H) when β = 0. This case provides a new limiting behavior for r(h) as stated in Proposition 9.

Before stating this result, we deduce in the following lemma the covariance structure of X when the spectral density is unbounded along a line that goes through the origin. We say that the process has a long memory along one direction if its covariance function is not summable along this direction.

Lemma 8. Let p ∈ R and for all (

x 1 , x 2 ) ∈ R 2 , f (x 1 , x 2 ) = 1 2π f (x 1 + px 2 ),
where f is an even, non-negative, and 2π-periodic function on R.

Let us denote, for all

(h 1 , h 2 ) ∈ Z 2 , σ(h 1 , h 2 ) = [-π,π] 2 e i(h1x1+h2x2) f (x 1 , x 2 )dx 1 dx 2 and σ(h 1 ) = π -π e ih1x f (x)dx.
Then, for all (h 1 , h 2 ) ∈ Z 2 , we have

σ(h 1 , h 2 ) = σ(h 1 ) if h 2 = ph 1 , sin((h2-ph1)π) (h2-ph1)π σ(h 1 ) otherwise.
The proof of this lemma is given at the end of this section.

Proposition 9. Let X be a stationary Gaussian process in dimension d = 2.

Let us suppose that its spectral density is

f (x 1 , x 2 ) = 1 2π f (x 1 + px 2 ),
where p ∈ Z and where f , defined on [-π, π], is a spectral density in dimension d = 1. Assume moreover that for -1/2 < α < 0 and for all x ∈ [-π, π], f (x) = L(x)|x| 2α where L is a bounded function, continuous at zero and nonnull at 0. Then,

• if α > -1/4, n(r(h) -r(h)) L -→ N 0, 2(2π) 2 f 2 2h , (21)
where f 2 2h is the (2h)-th Fourier coefficients of f 2 as defined in (2).

• if α < -1/4 and if h 2 = ph 1 , n (4α+3)/2 (r(h) -r(h)) L -→ N 0, σ 2 α,p , (22)
where σ 2 α,p = lim n→∞ n 4α+3 V ar(r(h)). Proof. In the case α > -1/4, the result is a consequence of Proposition 6. Let us focus on α < -1/4. We restrict the proof to the case p ≥ 0 since p ≤ 0 can be treated in the same way.

We prove the result thanks to a central limit theorem for triangular arrays stated in [START_REF] Romano | A more general central limit theorem for m-dependent random variables with unbounded m[END_REF].

An alternative proof could be to start from the representation of r(h) in terms of a double stochastic integral as in (15), then to use the necessary and sufficient conditions for the convergence of such an integral given in [START_REF] Nualart | Central limit theorems for sequences of multiple stochastic integrals[END_REF] and [START_REF] Nualart | Central limit theorems ofr multiple stochastic integrals and Malliavin calculus[END_REF]. But checking these conditions leads to hard computations, which would not simplify or shorten the present proof.

So, let us take advantage of the natural triangular array structure that arises in our setting. Indeed, let Y i1,i2 = X i1,pi1+i2 and let i

= i 1 , j = i 2 -pi 1 , we have r(h 1 , h 2 ) = 1 n 2 n i1=1 n i2=1 X i1,i2 X i1+h1,i2+h2 = 1 n 2 n-p j=-pn+1 i∈Bj Y i,j Y i+h1,j-ph1+h2 , where B j = {i | 1 ∧ 1-j p ≤ i ≤ n-j p ∨ n}. Therefore r(h 1 , h 2 ) is the triangular array r(h 1 , h 2 ) = (p+1)n-p j=1 Ỹn,j where Ỹn,j = n -2 i∈Bj-pn Y i,j-pn Y i+h1,j-pn-ph1+h2 .
Let us summarize the properties of Ỹn,j .

From Lemma 8, we have

r(h 1 , h 2 ) = r(h 1 ) if h 2 = ph 1 , 0 otherwise. ( 23
)
where r is the covariance function associated with f . Consequently (Y i,j ) is a zero mean Gaussian process such that

E(Y i,j Y i+h1,j+h2 ) = r(h 1 ) if h 2 = 0, 0 otherwise.
The process Ỹn,j , viewed as a function of ((Y i,j-pn ) i∈Z , (Y i+h,j-pn-ph1+h2 ) i∈Z ), is thus a (h 2ph 1 )-dependent process. Moreover, we can compute the moments of Ỹn,j thanks to the representation of the moments of Gaussian variables in terms of Wick's product. Since we have assumed h 2 = ph 1 , we obtain

E( Ỹn,j ) = 0, E( Ỹn,j1 Ỹn,j2 ) = n -4 i1,i2∈Bj-pn r2 (i 2 -i 1 ) if j 1 = j 2 = j, 0 otherwise and E( Ỹ 4 n,j ) = 3n -8   i1,i2∈Bj-pn r2 (i 2 -i 1 )   2 + + 6n -8 i1,i2,i3,i4∈Bj-pn r(i 2 -i 1 )r(i 3 -i 2 )r(i 4 -i 3 )r(i 4 -i 1 ).
We are now in position to apply Theorem 2.1 in [START_REF] Romano | A more general central limit theorem for m-dependent random variables with unbounded m[END_REF] which gives sufficient condition for the convergence in law of a triangular array of m-dependent random variables to a normal distribution. Following the same notations as in this theorem, we choose δ = 2 and γ = 0 and we look for ∆ n , K n and L n such that

• E( Ỹ 4 n,j ) ≤ ∆ n for all j,
• V ar a+k-1 j=a Ỹn,j ≤ k K n for all a and for all k ≥ (h 2ph 1 ),

• V ar

(p+1)n-p j=1
Ỹn,j ≥ ((p + 1)np) L n .

According to [START_REF] Romano | A more general central limit theorem for m-dependent random variables with unbounded m[END_REF], the convergence in law holds whenever K n /L n = O(1) and ∆ n /L 2 n = O(1). From Theorem 2.24 of [START_REF] Zygmund | Trigonometric series[END_REF], we have r(h) ∼ c α h -2α-1 when h → ∞ where c α is a constant depending on α and L(0).

Let us first suppose that h 2 = ph 1 . When p ≥ 2, the above domain of integration can be cut up as follows (the case 1 ≤ p ≤ 2 is not detailed but can be treated similarly):

σ(h 1 , h 2 ) = 1 2π(h 2 -ph 1 ) (I 1 + I 2 ),
where The domain of integration in I 1 can be split into -π < s < π and s ∈ [-π, -π + eπ] ∪ [π, π + eπ]. From the 2π-periodicity of f , this is easy to check that, when summing up I 1 and I 2 , all the integrals involving e in their range of integration sum up to zero. Hence I 1 + I 2 reduces to and it is handled according to the parity of ⌊p⌋ and j. Since f is a 2π-periodic function, f (u + jπ) = f (u) when j is even and f (u + jπ) = f (uπ) when j is odd.

I 1 = -i -(p-1)π - (
When ⌊p⌋ is even, the sum (25) above is then This proves the result of the lemma for h 2 = ph 1 in the case ⌊p⌋ even. When ⌊p⌋ is odd, the sum (25) is Now, in (24), we split the domain of the first integral into -π < s < 0 where f (s + ⌊p⌋π) = f (s + π) and 0 < s < π where f (s + ⌊p⌋π) = f (sπ). We apply respectively the change of variables s = s + π and s = sπ. This allows to exhibit the integral 2 π -π f (s) sin(h 1 s + (h 2ph 1 )π)ds. With the help of (26), some trigonometric computations show that the remaining terms coming from this change of variables simplify with the remaining term in (24).

Therefore, when h 2 = ph 1 , the result of the lemma is proved for all p.

The proof when h 2 = ph 1 is simpler and it can be conducted in the same way.

e

  ih1u f (u) e i(h2-ph1)πe i(h2-ph1) u+π p du andI 2 = -i (p-1)π -(p-1)π e ih1u f (u) e i(h2-ph1) u+π p e i(h2-ph1) sin(h 1 u + (h 2ph 1 )π) s = u -⌊p⌋ and e = p -⌊p⌋, ⌊p⌋π) (sin(h 1 s + h 2 ⌊p⌋π + (h 2ph 1 )π)

  ⌊p⌋π) (sin(h 1 s + h 2 ⌊p⌋π + (h 2ph 1 )π)

  When plugging in this latest result in (24), it simplifies and I 1 + I sin(h 1 s + (h 2ph 1 )π)ds = 2 sin((h 2ph 1 )π) π -π f (s) cos(h 1 s)ds.

Since (i ∈ B j ) ⇒ (1 ≤ i ≤ n), an integral test leads to ∆ n = O(n -8α-8 ) and K n = O(n -4α-4 ). For L n , let us compute directly the equivalent of the variance. From the decomposition in terms of Wick's product, we obtain

In view of ( 23) and since h 2 = ph 1 , most of the terms in the sum above vanish. We finally obtain

The latest sum involves positive terms which are equivalent to c 2 α k -4α-2 (n-pk) when k → ∞. As a consequence, V ar (p+1)n-p j=1 Ỹn,j ∼ σ 2 α,p n -4α-3 , where σ 2 α,p is a positive constant. This leads in particular to L n = O(n -4α-4 ). The conditions in Theorem 2.1 in [START_REF] Romano | A more general central limit theorem for m-dependent random variables with unbounded m[END_REF] are fulfilled and the convergence in law holds.

Proof of Lemma 8. When p = 0, the result is obvious. Let us assume, without loss of generality, that p ≥ 1.

Let the change of variables u = x 1 + px 2 and v = x 2 :