Laetitia Zecchini

Postcolonial Little Magazines in India: 'Signatures of Dissent' and Worldliness

. While this essay focuses on little magazines which flourished in various Indian languages between the 1950s and the 1970s, I argue that these alternative publishing spaces were indeed a privileged site of postcolonial print cultures, and a privileged medium of modernisms in India. In these provisional pages, writers exercised and found their creative and critical voices; connected with each other and with the world outside; and honed a radicalism that was often inseparably aesthetical and political.

Many Indian writers who started writing at the turn of independence would probably identify with the writer Vilas Sarang when he acknowledged that he has always thought of himself as belonging to an international modernist tradition coterminous with 'the vigorous growth of a Marathi little magazine movement'. He defined this movement by its 'spirit of rebellion and experimentation', and its 'aliveness to diverse influences and cross-fertilization' (1994: 309-310). That is also what I identify as the 'worldliness' of these publications, which, following Edward Said, can be defined as the opposite of exclusivism and separatism, and as the awareness that works of literature are both situated in the political and secular world in which they emerge, and linked to the whole world of other literatures. Little magazines also represented worldly spaces in so far as they were open to the many voices and forms outside and beyond the established, guarded domains of 'art' and 'literature'.

Far from representing 'tiny, defensively constituted corner(s) of the world' or 'ethnographic specimens' suitable for the attention of area experts (Said 2000: 382), I suggest

Fig. 1. Some of the English little magazines published in the 60s and 70s. Author's collage and photographs, courtesy Arvind Krishna Mehrotra and Adil Jussawalla

I. Dialogical modernisms

Remembering the 'demotic openness' of the art world in the 1960s, Homi Bhabha also suggested that its limitations (there was no market for art at the time, no state support, few artists and galleries) and its absence of professionalization, could account for the 'tremendous translational vitality' and intensity of interaction that characterized the period (Bhabha 2013: 23-24). Artists and writers of Bombay's many creative worlds after independence in fact all recall the effervescent transactions between literary cultures and languages (especially Hindi, Gujarati, Marathi and English), and between literature, theatre, art, music or advertising in 1960s Bombay. 3 'The lower cases of modernism swept away the boundaries between languages that now seem so prominent' acknowledged Arvind Krishna Mehrotra. 4 Traffics between and across languages, in part fueled by the 'consumption', the practice, and the publishing of translation, were the norm. 5 Many writers involved in these little mags, such as Vilas Sarang, Dilip Chitre, Arun Kolatkar, Pavankumar Jain or Gauri Deshpande actually wrote and published in English and in another Indian language. What's more, little mags often published texts in different languages. Gujarati little magazines such as Kruti and Draooon Draooon, or Marathi little mags like Atta and Aso featured texts in English or Hindi; conversely, Vrishchik, whose name was chosen in part because it had the same meaning ('scorpio') across various Indian languages, carried texts in Gujarati.

Little publications also often aimed, like damn you, to be 'magazines of the arts', understood inclusively. Their contributors constantly crossed and bridged disciplinary, artistic or generic boundaries. Gulammohammed Sheikh, Gieve Patel, Arun Kolatkar, Prabodh Parikh and Bandu Waze, for instance, were visual artists and poets all at once. Several littles were hence magnificently illustrated by these writers-cum-artists, by Prabhakar Barwe, Jyoti Bhatt, Jeram Patel and others. That was especially the case of Vrishchik, started from Baroda by the two iconic painters Bhupen Khakhar and Gulammohammed Sheikh, that became a platform for Bombay poets and artists and published artworks or linocuts in each of its issues. 6 Pages of Vrishchik and Contra 66 (another little magazine founded by an artist, J.

Swaminathan, from Delhi), were filled with poems, translations and other literary texts, but also with critical pieces and reviews on literature, art or cinema, and with essays on the state of criticism in India, the role of the artist in society, or the implications of internationalism and indigenism. In these critical essays, writers often wrote about art; and artists about literature, once again highlighting the symbiotic milieu which sustained their creativity, and their conviction that they all shared similar questions and concerns, regardless of the medium used.

'Whatever I may have to say' about the role of painters or sculptors, 'has a lot to do with what I consider to be my own role as a writer' acknowledged poet Adil Jussawalla as a preamble to his damning critique of the 'artistic profession ' in Vrishchik (1972b: np).

Poet, translator, editor, critic, illustrator, distributor (of themselves or of each other): the role was often interchangeable, and writers could wear different hats, or many hats at once, depending on where they published. Putting their own creative, material and financial resources in these small publishing ventures, they relied exclusively on themselves to publishalso strengthening, as Jussawalla suggested again, the direct, conspirational link between poet and reader: 'We have been the only means by which poetry has been kept alive while the big publishers slept, welcome to the conspiracy' (Jussawalla 1978: 5).

II.

Anti-establishment conspiracies: 'everything justified except the margins'

It is to this anti-establishment conspirational ethos I would like to turn to now. 7 Often hand-crafted, cyclostyled or mimeographed, many little mags were defined by their irregularity, by their unstandardized (unperiodic and un-disciplined) character. Most were short-lived, without a fixed number of pages, and were not meant to endure, or to establish themselvesliterally. They were 'little' in terms of their genre and format, often composed of 'little forms' such as poems and reviews; 8 in terms of their limited readership and distribution; but also in terms of the defiant anti-mainstream stance they put forward; the de-professionalized space they cultivated.

'Everything justified except the margins', is a line featured on the cover of the inaugural issue ('numbuh one') of the American little mag manhattan review (1966)a journal in which several of Mehrotra's poems were published. This catchphrase could encapsulate the spirit of many littles. They were 'unjustified' typographically, with texts that were often visually 'uneven', that were also irregular or unconventional in typography, grammar, punctuation, spelling, and in format or layout as well. Many little magazines were hence worked out individually from the content itself, and refused to conform (or compromise) to the customary publishing or editorial requirements meant to apply to all kinds of texts indiscriminately.

But they were also unjustified in the sense that they dismissed powers of consecration (1967), also echoing the title of the little mag Shahane started in 1963: 'Aso', meaning 'let it be' or 'so be it'!

Asserting their independence from extraneous sources of endorsement, they refused to be 'justified' by an institution, a group, an ideology, even a specific agenda. Speaking for themselves only, and bestowing on themselves and on each other the only recognition they seemed to need, little mag editors and contributors acted in effect, as if they were their own creators of literary value. These journals therefore privileged 'untutored', 'untrained', 'raw' voices or 'unheard of' writers (words repeated in their pages). 'Preference will be given to poets who have not yet been published before ...' wrote Mehrotra in the closing statement of ezra 1, while Ashok Shahane suggested that Aso was born from the idea of collecting texts from writers who never imagined they were writers (Nerlekar 2017b: 113). Of course, if anyone could become a poet, editor, or a publisher and if, like art, writing could happen anywhere, it's also because using a mimeograph machine did not require specific skills, and little mags were cheap and easy to produce. They could even be printed, like the first issues of Vrishchik, on brown packing paper.

This was a leitmotiv of these not-for-profit publishing ventures: writing was not a means, used by a legitimate few, but a craft potentially available to anyone. In a sense, it was also 'life' itself, and explains why little magazines were often affective, non-professional affairs, born from and sustained by friendships. The Gujarati poet and little mag editor Pavankumar Jain suggested that a little mag like Tantrum (edited by Madhu Kothari in Gujarati)

was not for people with 'hard habits of writing and reading'. This defiantly inclusive and amateur ethos shows in the DIY appearance of several little magazines, which displayed their kinship with 'minor' arts or artefacts of popular culture by including songs, prospectus, postcards, stamps, advertisements, signboards, doodles, children texts or drawings, and sometimes draft-like, seemingly unfinalized texts. It also explains the experimentalism that characterized these publications. 'At the level of the creative imagination, all is possible' (italics in the text) acknowledged the editors of Dionysus (Pradhan and Benjamin 1965: 3). And little magazines were indeed often stretching the frontiers of literature to the limit, by publishing visual poems, concrete poems, poster poems, 'found poems' (in the form of shopping lists, overheard conversations, radio broadcasts, etc), and by rejecting formal diction and ornate style for colloquial, everyday speech.

Dilip Chitre equated one of Kolatkar's 'found poems', the 'Three Cups of Tea' sequence, first published in Aso and written in the Marathi/Hindi lingua franca of the Bombay streets, to the denial of poetry itself: the bourgeois-academic-formalist-poetry (1967: 19). 9

'Don't you know the rules?' is a line from the sequence. Precisely, in this anti-poem, the cultural, political and social etiquettes governing issues of hierarchy, diplomacy, discipline or decency are joyously overturned, along with linguistic and poetic conventions. The text remains 'unjustified', with an accordion-like layout; punctuation marks and capitals are omitted; crude vocabulary or slang (baby, wanna, guy, punk, man, cunt, fuck, etc.) is used.

'Set like a machinegun' (damn you 1968 6: np), the mimeo-machine was aimed at the Establishment, declined in different ways: 'established', precisely consecrated, award-winning writers; mainstream publishers and journals, such as the powerful periodical Satyakatha in Marathi, whose burning was staged publicly in 1969 by Raja Dhale, the Dalit poet, activist and editor of several little magazines; established formal or standardized language, which could take the form of the very 'correct', 'starch-like' and 'dreadfully moralist' English idiom A. K.

Ramanujan and Adil Jussawalla railed against; 10 or the ornamental and sanskritized linguistic registers in other languages, and of course other major gate-keepers of the literary field in the guise of the literary sammelans and the academies.

Vrishchik is perhaps the most striking illustration of little mags' anti-establishment struggle. In its pages, 'the establishment' took the form of the state and its institutions, namely A word here on Nissim Ezekiel, a central figure of Bombay modernisms, who nonetheless became a casualty of several little mags' 'blasts'. Along with other 'middle-aged intellects in terrylene', Ezekiel needed 'to be dynamited', according to Mehrotra, for more radical, avant-garde writers to be heard. 11 And yet, Ezekiel was in fact a prodigal mentor of many 'unheard of' writers at the time, and an instrumental presence in countless small ventures, for whom no magazine was 'too raw or unestablished to be given a poem'. 12 As editor of Quest and Poetry India, which were visually and formally much more conventional than magazines like damn you, he nevertheless published many of his more radical modernist contemporaries in English, Bengali, Marathi or Gujarati (including Mehrotra, Kolatkar, Chitre, Suresh Joshi, Buddhadev Bose or Benoy Mojumdar), and in other more mainstream magazines with which he was actively involved (such as The Illustrated Weekly of India) he never failed to advertise their innovative work and publishing ventures.

For the published final version of the essay see The Bloomsbury Handbook of Postcolonial Print Cultures, eds. Toral Gajarawala, Rajaswari Sunder Rajan, Neelam Srivastava, Jack Webb, Bloomsbury Academic, 2023, pp. 39-56. In the pages of little mags, metaphors flourished to describe this anti-establishment, antiprofessional and anti-academic 'signature' of dissent and marginality. The magazines presented themselves as platforms for outcasts, outlaws, drop-outs, or misfits. 'Call me a freak' is in fact the title of a short autobiographical text published by Pavankumar Jain in the special issue of Tantrum he guest-edited (undated. See Fig. 2). The trope of the tramp, the vagabond or the ecstatic 'bhakti beatnik' is a regular feature of these publications. 13 A quotation attributed to dissident Russian writer Yevgeny Zamyatin, and featured on the cover of Contra 66 could sum up their spirit: 'There can be real literature only when it is created not by executives and reliable civil servants but by madmen, hermits, heretics, dreamers and sceptics' (Swaminathan 1967: 5-6).

III. Transnational traffics, friendships and asymmetries

In many ways, these little magazines illustrate how 'minority' or 'littleness' can become less a predicament of literary production, and more a condition of worldliness and independence; a 'resource', even, to borrow from Leela Many of these publications thus relied on underground or informal circuits of communication, circulation and funding, on friendships and seemingly random discoveries. 16 'We just move around the place, and when there is no world to make us happy, we dish out DY. some goes places, some just hangs from the clouds, some is kept hidden' ('statement ', damn you 1967 4: np). And yet, as suggested, the 'carefree, unruly' attitude flaunted by many little mag contributors, their apparent indifference to consecration or publicization, did not mean that they were not also claiming to disrupt conventional cartographies of literature and asserting their place in the world geography of letters. The damn you statement quoted above is in fact immediately followed by the manifesto-like aphorism: 'damn you eating the corners of literature'.

It is finally important to keep in mind that these conspirational traffics were not necessarily exclusive of (or distinct from?) more institutional or politically-conditioned networks. The transnational circulation and affiliations of a little magazine like Quest, for instance, modelled on Encounter, and sponsored by the CIA-backed Congress for Cultural

Freedom, depended in part on a formidable publishing artillery. More generally, the immense broadening of literary horizons which also coincided with the 'paperback revolution' in the 1950s and 1960s, and a 'feverish activity in translation', was precipitated by the Cold War when both blocks were engaged in devising programs to translate and disseminate works of literature that were deemed useful in promoting an ideology and countering the opposing side's 'propaganda'. Yet, radical or dissenting Indian voices also used available Cold war funds and spaceslike the little magazine, partially coopted by American cold war authoritiesto bend them to their own projects and needs. [START_REF] Zecchini | That is also why I argue for a connected history of the anti-establishment little magazines like damn you or Contra 66, with magazines that appear more conventional, such as Quest[END_REF] Even Arvind Krishna Mehrotra, in his first letter to Howard McCord on October 16 1966, described the USIS (the United States Information Service) as the place to visit for aspiring Indian writers with an editorial project in mind and no cash! USIS, in a sense, was compatible with charas.

Here are the last enlightening lines of Arvind Krishna Mehrotra's opening statement in damn you 6: These little were also about claiming a space for what is usually considered too insignificant to get noticed: the 'little known little people', the 'corners' of literature and the world and all its 'forgotten territories' (an expression used in damn you 4). With its brash disregard of the consecrating 'capitals of the skyscraping earth', Mehrotra's text presents itself as a formidable declaration of emancipation and seems to corroborate Eric Bulson's comment that little mags created a decentered literary universe (2016). 18 Yet these dismissals, along with many of the East-West traffics of the time, were not devoid of ambivalence. Even though Mehrotra, in the above statement, brushes aside Eric Oatman's and Ken Geering's judgments about damn you being 'too yankee oriented' or 'too damn british', mentioning them was also signaling that he belonged to and was recognized by littles in the West. A desire to emancipate oneself from Western models often coincided with a desire to be recognized by Western modelshowever countercultural.

It is important to recall here that the 'curse' of belatedness, influence and mimicry has always plague Indian (and more generally postcolonial) modernisms. This was the time when many modernist writers and artists were being told that they were derivative, outdated, or inauthentic -both in India, where in the wake of India's freedom struggle, modernism was often suspiciously perceived as an offspring of the West and where cultural nationalism was virtually required of Indian artists, and in or vis-à-vis the West, where Indian modernists were often told to go back to their folk or classicali.e. authentically Indianart. Little mags and their contributors were struggling to shake off these different assignations, expectations and representations. They were also struggling to escape the 'Little Englands and Little Americas' to which some felt confined or doomed; [START_REF]The expression is Adil Jussawalla's[END_REF] and struggling to hone a voice of one's own, both distinct from and interconnected to world voices and modernisms.

Interestingly, Ginsberg himself had a particularly heated exchange with Abu Sayeed Ayyub (who headed the ICCF in Calcutta in the 1960s, and whom Ginsberg had solicited to help Malay Roy Choudhury) which illuminates some of the debates that were happening at the time on the provenance and idioms of modernism, and on the curse of influence mentioned above. If Ayyub actually denied the Hungryalists the epithet of 'writers' (only 'immature imitators of American beatnik poetry'), Ginsberg in turn refused to qualify their poetics as either imitative or obscene, instead pointing precisely to the transnational modernist syntax of opposition articulated simultaneously in various corners of the world. Quoting one of the lines with which Ayyub had taken umbrage ('Fuck the bastards of Gangshalik School of Poetry'),

Ginsberg wryly claimed that it was 'common parlance … from cafes of Paris or Calcutta to old manifestos by Tristan Tzara'. 20 If the interest shown by the Beats, and more generally by several American little magazines, for Indian writers, was not always free of clichés about India, they did acknowledge radical Indian little mag contributors as their own. 21

We therefore see an alternativeand synchronousgeography of letters taking shape through a shared modern idiom, linking, as Mehrotra suggested, Prévert in Paris to Lawrence

Ferlinghetti in San Francisco, to Kolatkar in Bombay: 'it's the same casual tone, the language stripped bare' (Zecchini 2017b: 202). Challenging the suspicion of belatedness or mimicry, many artists and writers were actuallyjust like other modernists the world overpoaching, cannibalizing and appropriating everything they had access to. [START_REF]The 'stealing metaphor' is abundantly used by Indian modernists, including Mehrotra and Kolatkar, but also Gieve Patel or Gulammohammed Sheikh (see their interviews in Nerlekar & Zecchini[END_REF] I've often discussed the voracious curiosity of all these writers and artists who staged themselves as indiscriminate gluttons or 'rag-pickers', and also talked about the formidable food-chain of reading-recycling-(re)writing where modernism originates. Both translations and quotations, which flourished on the covers of these little mags and in the intertextual fabric of many of these texts, are signs of this palimpsest-like assemblage or bricolage of world literature or world art.

They were also a means, for these writers, to choose both their ancestors and their contemporaries. You are less born into a world (as lineage and as cartography), than make it new, and make it for you. Remember the line: 'when there is no world to make us happy, we dish out DIY', or the 'homemade worlds' many little mag writers were claiming to invent.

These citations, then, also had a 'world-making' function. Original, national or 'natural' filiations are converted into affiliations, connections that are invented rather than inherited (Said 1983: 23), and that were either transnational or corresponded to a remaking of what an 'Indian poet' or an 'Indian artist' was supposed to be.

IV. The 'street-fighting' years: Little Magazines and the struggles outside

This worldliness has other (more political) resonances as well. New poets, wrote Anil Saari 'will write about dejecta and diseasemore appropriate to India than DDT-ed verse' (1966). Obviously, here, 'DDT-ed' verse is made to correspond to the unchallenging, appropriate verse of the establishment, the opposite of the avant-garde. But Adil Jussawalla also hints at another dimension of this 'purgation' process in 'Boys and Girls in Purdah' (1972a). 'Art with its balls removed' is the art that is curtain-filtered and state-sanctioned ('what filters through the curtain is only fit for nothing but the international shit-pot'), especially the material disseminated by international agencies like USIS or by the ICCF. As Serge Guilbaut (1983) and Greg Barnhisel (2015) have shown, modernism was indeed turned into a culturaldiplomatic weapon by American cultural authorities during the Cold War, which contributed to defanging modernism and taming its privileged medium, the little mag.

That is also how we must understand Geeta Kapur's indictment of the commodified and standardized (i.e. 'stylized') form of mainstream International avant-garde art (that must be 'new' and 'advanced', but that should neither be obscure nor disturbing) promoted by institutions in India, and abroad (Kapur 1971: np). In a 'continuingly Anglo-American colony like India', Adil Jussawalla noted again, art is not just selected, but trivialized or, indeed, castrated: the violence and the morally painful, the inventive and the demotic, the aesthetically and politically radical are kept at bay (1972a). In the same text, Jussawalla called for the 'living acid' of literature and language to dissolve the curtain. In many ways, the turn towards the Beats, towards Central and East European poets, towards alternative Third-World, African or Latin American writers, and towards bhakti are different facets of this quest for 'untrivialized', more challenging literatures and countercultures. [START_REF]Jussawalla also organized the 'Dangerous Animals' series of readings at St Xavier's in Bombay in the 70s[END_REF] A whole insurgent counter-library was surfacing in the 60s, both in India and the world, and Jussawalla's influential New Writing in India anthology (1974) must be read in that light.

Destined to challenge the neglect of Indian writers on the global literary map, and the clichés about 'Indian literature', Jussawalla offered a categorical dismissal of the paradigm of 'influence' or 'parasitism'. 24 He also chose to showcase works of literature that were characterized by their thematic and formal boldness and did not shy away from the 'social evils' and political struggles of the world.

That is another defining feature of the worldliness of these magazines. Like Geeta Kapur, Adil Jussawalla attacked the elitism and the escapism of the art establishment in India, its bourgeois defensiveness and abdication from 'social responsibility ' (1972b). He satirized those who promoted 'decorative' art featuring fishermen and tribals, while refusing to acknowledge the revolts of the landless, Dalits or Naxalites: 'We can safely hang them up in our living rooms with no fear that they might ever want to hang us'. 'Eating the corners' of literature, then is as much about salvaging art where it's not supposed to be, in the seemingly trivial, un-DDTed, even obscene or filthy (the many 'waste papers' of literature and culture) ;

than it is about opening literature to the 'troubles and turmoil of the world' outsidean expression used by J. Swaminathan in an article where he likewise criticizes the mannerism of Indian art (1966: 9-14).

It would therefore be erroneous to take the seemingly non-committal attitude of some little mags at face value. 'Fuque politics' (a line from the statement of damn you 6 quoted above) is less the refusal of politics, than the refusal to be pigeonholed into neat, predictable political or ideological alignments, including Cold War ones, and the refusalagain -to straightjacket literature, or submit it to extraneous sources of evaluation.

In the 50s, 60s and 70s political and social turbulences, both in and outside India, were of course pressing upon art and texts in myriads of ways. In a global context, this was the height of the Cold War which subjected many postcolonial countries to the 'competitive courtship' of the United States and the Soviet Union. It was also the years of an international Counter Culture, youth movements, student protests and anti-Vietnam rallies. In India, more specifically, the 1950s-70s are often described as a period of widespread disenchantment born from the betrayed promises of Independence, and of social and political unrest, illustrated by the Naxalite insurgency in the late 60s, and by the Emergency (1975-1977).

These 'street-fighting years' (Ali 2018) In the pages of many little mags, local and global political struggles were constantly correlated. In Vrishchik, an editorial published after anti-Muslim riots, and entitled 'Against

Communalism' called for artists to mobilize against communalism (which, as the editors noted, also plagued art and literature), and connected the horrors happening inside and outside India:

'horror is the word … whether it is an entire marriage party burnt alive during the communal riots or South Vietnamese prisoners held like animals in "tiger-caves", starved, choked and forced to drinking their own urine' (Khakhar and Sheikh 1970 7-8: np).

Two globalized local struggles were in fact mobilizing countercultural energies worldwide, and figured prominently in the pages of Indian little mags: Vietnam, and Bangladesh. A huge 'concert for Bangladesh', organized by George Harrison and Ravi Shankar was held in New York in August 1971, and Ginsberg wrote his poignant long anti-war poem, 'September on Jessore Road', after receiving photos of the atrocities sent to him by Ashok Shahane, who was the first to publish the poem as a broadside from India. 28 In many little mags, both struggles were framed as the resistance of the little peoples, or young nations of the world, to the 'Big Nations' or 'great powers'. Vrishchik, for instance, published letters sent by an American GI in Vietnam that were introduced by a few lines suggesting these documents 'mock at the fake capsules of humanity presented to us by the glamorous propaganda of the Big Nations', while advocating an anti-imperialist call to arms (Khakhar and Sheikh 1970 6: np). The war in Bangladesh is likewise presented as evidence that 'so called giant nations' have slept over the 'mass-scale butchery of unarmed innocents' (Khakhar and Sheikh 1971 6-7: np).

Conclusion

In postcolonial little magazines many of the most innovative or radical Indian writers and artists found their voices, while clearing spaces (that would not have existed otherwise) for themselves. Ezekiel remembered the disdain of editors when he went from publisher to publisher with his manuscripts or 'looked in vain for review space' in the 60s (Ezekiel in Anklaseria 2008: 222). Little magazines provided him and countless others with that space, and in their pages other non-mainstream genres, forms, voices and modes found a home: critical, at times provocative essays, translations from Indian and world languages, Indian poetry in English, and many of the radical, 'raw and unestablished' writers or painters who used the freedom these publications gave them to challenge the status quo, and craft a modernism they could call their own.

Their 'space-clearing gestures' also illustrate a conception of literature which was not walled in exclusive frontiers (be they linguistic, generic or geographical), and not blunted or isolated from the world outside -the 'everyday' and the 'street' in its many forms, including the street-fighting struggles of all those who, in the 60s, stood up against forces of repression and discrimination.

These little magazines were 'space-clearing' or worldly in another sense as well: their anti-establishment 'littleness' paved the way for the future consecration of many of those who

showcased their work or wrote (sometimes exclusively) in them. [START_REF]That is why Anjali Nerlekar talks about these littles as the 'originary spaces of the canon[END_REF] Writers like Arun Kolatkar, Bhalchandra Nemade, Arvind Krishna Mehrotra or Namdeo Dhasal, and artists like Ghulammohammed Sheikh and Bhupen Khakhar did not remain in the 'rain shadow region' of literature and art, but are today recognized as the most significant modernists of their time. 31 The fugitive or minor lives and works that seemed to dwell 'at the other end of permanence' (Mehrotra, ibid), are in fact remarkable, memorableand worldly.

the

  Lalit Kala Akademi (the National Akademy of Art, established in 1954), with which the magazine spelt out its 'unequivocal dissent', and the Triennale India, launched in 1968. The Akademi is attacked as a hotbed of mediocrity, and as an apathetic, complacent, selfrepresenting and self-perpetuating body. In 1970, Vrishchik formed an 'All India Committee of Protest', and spearheaded the pan-Indian signature campaign (among whom featured Adil Jussawalla, Gieve Patel and Arvind Krishna Mehrotra) to boycott the Akademi and the second Indian Triennale.

Fig. 2 .

 2 Fig. 2. Tantrum special issue guest-edited by Pavan Kumar Jain. Author's photograph and collage,

Fig. 3 :

 3 Fig. 3: 'statement', damn you 6. Courtesy Arvind Krishna Mehrotra

  reverberated in the pages of little mags, many of which identified with the Counterculture of the times, with anti-imperialist, Third World struggles, and with those of the landless, Dalits or Naxalites of Jussawalla's article. As F. N. Souza wrote, again in Vrishchik, the struggle against the Lalit Kala Akademi is only one among many battles: 'Now Vrishchik should fight the barriers on temple entry and let loose a nest of scorpions'.[START_REF]How I Hum the Song of Bhagwad![END_REF] In this damning indictment of casteist-Brahmin segregation, we hear echoes of the Dalit struggle. Dalit activism and Dalit literature in fact also emerged from the defiant oppositional ethos of the little magazine movement. All four founders of the 'Dalit Panthers', an anti-caste political and cultural movement formed in 1972 in Bombay, were writers and two of them, the poets Namdeo Dhasal and Raja Dhale (whose burning of Satyakatha in 1968 was also meant to be a revolt against the upper-caste establishment) were active little magazine editors and contributors. Raja Dhale was a close friend and collaborator of Ashok Shahane, and both Dhale and Dhasal have acknowledged their debts to Marathi little magazines such as Shahane's Aso, or the pioneering cyclostyled Shabda launched in 1954 by Arun Kolatkar and Dilip Chitre, among others.[START_REF]See Nerlekar and Zecchini 2017a, especially Raja Dhale's interview[END_REF] Internationalism was an important feature of these street-fighting times and the Dalit Panthers in India echoed the 'Black Panthers' in the United States, which also took root in the United Kingdom with the formation of the British Black Panthers in 1968. The BBP included Indian writers and artists connected to the little magazine movement in Bombay, such as Farrukh Dhondy, Mala Sen or Vivan Sundaram (celebrated modern Indian painter and Geeta Kapur's husband). Adil Jussawalla was very close to all three, and dates the beginning of his political awakening to the time he spent in London in the 60s, like other writers and artists who later contributed to the little mag scene in India, such as Eunice de Souza (who lived for some time with Farrukh Dhondy and Mala Sen), Gulammohammed Sheikh (who remembers starting his collection of little magazines in London), or F. N. Souza. Jussawalla's Marxist education was consolidated upon his return to Bombay, where he joined a progressive Marxist youth movement (PROYOM), that brought out a journal called Lalkaar (meaning Challenge) to which he contributed. Several PROYOM were actually jailed and tortured during the Emergency.[START_REF]PROYOM and the political ferment of the 1960s and 70s, but also of the reading and print cultures of Bombay see: 'We All Have A Genuine Fear of Violence[END_REF]

Fig. 4 .

 4 Fig. 4. Some of the images sent by Ashok Shahane to Ginsberg. Courtesy of the Department of Special Collections, Stanford University Libraries, 29 and Peter Hale/Ginsberg Estate.

  refuse alignment along the secure axes of filiation to seek expression outside, if not against, possessive communities of belonging' (2006: 10). But little magazines were not only platforms for like-minded Indian artists to connect or affiliate with each other against inherited, 'possessive communities of belonging' and institutions. They could become places of dialogue and friendship with magazines outside India. Some littles in fact advertised both Indian and non-Indian magazines, and many staged their interconnection between world subcultures by publishing reviews, translations, quotations, excerpts of correspondence and occasionally lists of little publications with which they corresponded. An issue of Poetry India, for instance, published an ad for damn you announcing an 'editorial policy / open as hell'. The ad included a long list of national and Octavio Paz, to whom he became close when Paz was the Mexican ambassador to India until his resignation in 1968, in protest of the Mexican government's bloody repression of the students' movement.14 Mehrotra maintained a close correspondence with Howard McCord and Douglas Blazek, and they published each other in their respective little mags. Allen Ginsberg corresponded with Malay Roy Chaudhury and befriended the Bengali Hungryalists, making sure their manifestoes, their poems, and later their arrest received worldwide attention. In Bombay, Ginsberg also became friends with Ashok Shahane and Arun Kolatkar, whose partial translation of 'Kaddish' was published in Aso in 1963. In his letters to Ginsberg in the early 60s, Shahane in fact talks about those who 'joined the army' of little magazines, and about himself, Kolatkar and Ginsberg as 'spy-poets'. The anti-establishment 'conspiracy' was also formed transnationally.Hence if texts by Ginsberg, André Breton, Octavio Paz, Antonin Artaud, Carl Rakosi, Vasco Popa, Cesare Pavese, Hans Arp, Jean Genet, or John Berger, as well as excerpts from letters by Eric Oatman, James Laughlin or Lawrence Ferlinghetti, appeared in these little mags, it wasn't just the 'world', world literature or world magazines coming to India, but Bombay, Calcutta and several other places in India materializing on the global map of 'little mags'. Poets such as Mehrotra, Kolatkar, Malay Roy Choudhury, Pritish Nandy, SrinivasRayaprol and others published texts in Evergreen Review, Manhattan Review, Kulchur, City Lights Journal,

	A few months later McCord wrote to say that this being 'a bad, broke month', he could not send
	along 'bucks' with his letter, but suggested that Mehrotra follow in Malay Roy Choudhury's

Gandhi in another context. Minority, she suggests, involves a protest of what is dominant or normative, and is connected to the trope (and the politics, after Derrida) of friendship, which gestures toward all those invisible affective gestures 'that international contributors and closed on the tagline: 'support "little mags" everywhere'. But it also advertised klactoveedsedsteen, the beat magazine published from Heidelberg by Carl Weissner, asking the 'new generation' of modern Indian poets to send him 'strong stuff', including drawings suitable for 'photo-offset-repro', and promising distribution 'thru the major avantgarde bookshops of Europe and America'

(Ezekiel 1967 2: 2)

. Some of these small publications in fact testify to the transnational communities of friendship established at the time. In Contra 66, for instance, J. Swaminathan, published Salted Feathers, Intrepid, measure 3, Ole', Delos, and other littles, like the iconic English-Spanish countercultural magazine, El Corno Emplumado, published from Mexico.

15

Some of these publications, then, did travel the world, or at least were staged as destined to do so, with prices featured in rupees, pounds or dollars, and readers of damn you encouraged to 'smuggle' the mag in all the countries of the world. A passage from Mehrotra's correspondence with Howard McCord offers a humorous snapshot of these clandestine modes of circulationand funding! Although little mag editors often claimed that money was not an issue, little magazines were constantly struggling financially. And if Mehrotra regularly sent poems and issues of his own magazines for McCord to sell in the United States, other less literary artefacts were 'trafficked' between borders. In December 1966, Mehrotra sent McCord 100 greeting cards with Indian gods destined to be sold and help the ezra-fakir press stay afloat.

footsteps. The latter had apparently devised a stratagem to fool the post-office and include charas (the local name for cannabis, presumably more marketable than Indian gods!) between the pages of his little magazines (7 September 1967).

See the extraordinary bhakti special issue of Vrishchik (1: 11-12, September-October 1970), illustrations of which are reproduced in 'The Worlds of Bombay Poetry'.

I discuss the image of 'conspiracy', used both by Adil Jussawalla and Octavio Paz in 'Translation as Literary Activism: On Invisibility and Exposure, Arun Kolatkar and the Little Magazine "Conspiracy"'(Zecchini 2017c).

See my interview with Sheikh (Zecchini 2017a) for details on Octavio Paz in India.

After the project of editing a special klacto issue fizzled out, Intrepid brought out a special Indian poetry issue guest-edited by Carl Weissner (Spring 1968), and measure 3, edited by Howard Mc Cord published a 'Young Poets of India' issue in 1972. Rayaprol started the little magazine East and West in 1956 and regularly corresponded with William Carlos Williams (See Rayaprol 2020).

In the course of a conversation in Bombay, Ashok Shahane also remembered how the only available copy of Burroughs' Naked Lunch was circulated clandestinely among Bombay writers and artists in the 1960s.

, 'What Filters Through the Curtain': Reconsidering Indian Modernisms, Travelling Literatures and Little Magazines in a Cold War Context, Interventions: International Journal of Postcolonial Studies, 22:2, 2019: 172-194. Zecchini, L. (2022), Struggles for 'Cultural Freedom' in Cold War India: Fostering a Critical Spirit in the Liberal Journals of the 1950s-70s, in The Bloomsbury Handbook to Cold War Literary Cultures, ed. Greg Barnhisel, Bloomsbury Academic, forthcoming.

1 Bombay was a breeding ground for small collaborative publishing ventures and for modernisms in India, but the story of postcolonial little mags could be told from other vantage points. As suggested below, the Bombay little magazine scene was connected to other cities, including Calcutta, Baroda, Delhi, or Allahabad. Some of the material used in this essay draws on 'The Worlds of Bombay Poetry' double special issue (Nerlekar and Zecchini 2017a), in which many visual documents are included, and on my ongoing work on Indian modernisms, such as my monograph on Kolatkar (2014). 2 Kapur 2001. This essay is in many ways also a tribute to Geeta Kapur's trailblazing work. 3 See Nerlekar & Zecchini 2017a, especially interviews with Gulammohammed Sheikh (69-82) and Gieve Patel (25-32). 4 Personal email to the author, July 24 th 2014. 5 Many little mags, such as Poetry India; Vagartha, started in 1973 from Delhi by Sujit and Meenakshi Mukherjee, or Kavi (later Kavi-India) founded by Santan Rodrigues in 1976 from Bombay, aimed at forming a bridge between different Indian languages by publishing translations.


Laetitia Miguel

Juliette Gervais

Gaël Nicolas

Magalie Lecourtois
email: magalie.lecourtois@univ-rouen.fr

SorLA protective function is restored by improving SorLA protein maturation in a subset of Alzheimer's disease-associated SORL1 missense variants

Keywords:

SORL1 loss of function is associated with Alzheimer disease (AD) risk through increased Aβ peptide secretion. We expressed 10 maturation-defective rare missense SORL1 variants in HEK cells and showed that decreasing growing temperature led to a significant increase in the maturation of the encoded protein SorLA for 6/10. In edited hiPSC carrying two of these variants, maturation of the protein was restored partially by decreasing the culture temperature and was associated with concomitant decrease in Aβ secretion. Correcting SorLA maturation in the context of maturation-defective missense variants could thus be a relevant strategy to improve SorLA protective function against AD.

Introduction

The SORL1 gene (Sortilin-related receptor 1) encodes the SorLA protein, which acts as an intracellular sorting receptor for numerous cargo proteins, including the Amyloid β Precursor Protein (AβPP) and the Amyloid β (Aβ) peptide [START_REF] Schmidt | Sorting receptor SORLA: cellular mechanisms and implications for disease[END_REF][START_REF] Andersen | Risk factor SORL1: from genetic association to functional validation in Alzheimer's disease[END_REF]. The Aβ peptide is the major component of senile plaques in the brain of Alzheimer's Disease (AD) patients, and genetic studies show that it plays a central role in triggering AD pathophysiology. Functional studies demonstrated that SorLA plays a key protective role against the production and the secretion of the Aβ peptide by neurons [START_REF] Andersen | Neuronal sorting 11 protein-related receptor sorLA/LR11 regulates processing of the amyloid precursor protein[END_REF][START_REF] Dodson | Loss of LR11/SORLA enhances early pathology in a mouse model of amyloidosis: evidence for a proximal role in Alzheimer's disease[END_REF][START_REF] Rohe | Sortilin-related receptor with A-type repeats (SORLA) affects the amyloid precursor protein-dependent stimulation of ERK signaling and adult neurogenesis[END_REF][START_REF] Knupp | Depletion of the AD Risk Gene SORL1 Selectively Impairs Neuronal Endosomal Traffic Independent of Amyloidogenic APP Processing[END_REF]. First, by binding to the retromer complex, SorLA redirects AβPP from the early endosomes back to the Golgi apparatus through the recycling pathway. This in turn results to reduced amyloidogenic processing of AβPP, by preventing it from being targeted to the late endosome compartments, where AβPP is cleaved into Aβ. Second, SorLA may directly bind nascent Aβ peptides and target them to lysosomes, which also contributes to the reduction of Aβ secretion [START_REF] Caglayan | Lysosomal sorting of amyloid-β by the SORLA receptor is impaired by a familial Alzheimer's disease mutation[END_REF]. Accordingly, rare coding genetic variants hitting SORL1 have been found in ~3% of AD patients and are considered as a major AD risk factor [START_REF] Nicolas | SORL1 rare variants: a major risk factor for familial early-onset Alzheimer's disease[END_REF][START_REF] Bellenguez | Contribution to Alzheimer's disease risk of rare variants in TREM2, SORL1, and ABCA7 in 1779 cases and 1273 controls[END_REF][START_REF] Pottier | High frequency of potentially pathogenic SORL1 mutations in autosomal dominant early-onset Alzheimer disease[END_REF].

Newly synthesized SorLA molecules follow the constitutive secretory pathway from the endoplasmic reticulum (ER) to the cell surface, through the Golgi. During their transport, SorLA proteins are subjected to several post-translational modifications, resulting in the presence of immature and mature species. Recently, we demonstrated that a subset of rare missense AD-associated SORL1 variants was associated with impaired maturation of the SorLA protein, resulting in its retention in ER. These trafficking defects were associated with a loss of SorLA protective function leading to an increase in Aβ peptide secretion [START_REF] Rovelet-Lecrux | Impaired SorLA maturation and trafficking as a new mechanism for SORL1 missense variants in Alzheimer disease[END_REF].

Defects in protein trafficking and ER retention have been shown to be a common cellular mechanism involved in the pathogenesis of a wide range of inherited human diseases, such as cystic fibrosis or kidney diseases [START_REF] Schaeffer | Protein trafficking defects in inherited kidney diseases[END_REF][START_REF] Welsh | Molecular mechanisms of CFTR chloride channel dysfunction in cystic fibrosis[END_REF]. Mutations act by impacting protein folding and conformational stability, thus interfering with efficient post-translational modification and subsequent protein trafficking. Conformational protein stability depends on environmental factors including temperature, pH and salt concentration [START_REF] Uversky | Intrinsically Disordered Proteins[END_REF]. In agreement, the processing defects associated with various missense mutations can be reverted when temperature is reduced [START_REF] Zhou | Correction of defective protein trafficking of a mutant HERG potassium channel in human long QT syndrome. Pharmacological and temperature effects[END_REF][START_REF] Anderson | Most LQT2 mutations reduce Kv11.1 (hERG) current by a class 2 (trafficking-deficient) mechanism[END_REF][START_REF] Denning | Processing of mutant cystic fibrosis transmembrane conductance regulator is temperature-sensitive[END_REF]. In this report, we tested the effect of low temperature on maturation and function of maturation-defective SorLA proteins.

Material and methods

Cell culture

All cDNA constructs and human induced pluripotent stem cell (hiPSC) clones used in this report were previously described in ref [START_REF] Rovelet-Lecrux | Impaired SorLA maturation and trafficking as a new mechanism for SORL1 missense variants in Alzheimer disease[END_REF]. HEK293 and hiPSCs were cultured as described in [START_REF] Rovelet-Lecrux | Impaired SorLA maturation and trafficking as a new mechanism for SORL1 missense variants in Alzheimer disease[END_REF] with slight modifications. HEK293 cells were transfected using the Transporter 5 reagent (Polysciences Europe GmbH, Eppelheim, Germany) according to the manufacturer's protocol. Six hours after transfection for HEK293 cells, or 2 days after plating for hiPSCs, cells were shifted or not at 29°C for 24h, before being collected.

Protein extraction and Western blotting

Protein extraction and western blotting were performed as previously described [START_REF] Rovelet-Lecrux | Impaired SorLA maturation and trafficking as a new mechanism for SORL1 missense variants in Alzheimer disease[END_REF]. Note that due to reduced expression of CMV promoter-dependent transgene at lower temperature, 4 times more proteins were loaded for HEK samples incubated at 29°C, to obtain similar signal intensities and consequently accurate quantification of bands between both temperatures.

Proteins were resolved by Tris-acetate NOVEX NuPAGE 3-8% (Invitrogen, Carlsbad, CA, USA). Nitrocellulose membrane were blotted with monoclonal anti-SorLA antibody (1:5,000; mAb# 611860, BD Bioscience, Franklin lakes, NJ, USA) and with recombinant anti-vinculin antibody (1:10,000; ab129002, Abcam, Cambridge, UK). The signal intensity corresponding to mature and immature forms of SorLA proteins in each lane was quantified using the Genetools software (Syngene, Cambridge, UK).

Aβ40 secretion assay

HiPSCs were plated in triplicate on 12-well plates. Two days later, cells were shifted or not at 29°C and 4 hours later, medium was replaced and supplemented with phosphoramidon (10µM, Sigma-Aldrich, Saint-Louis, MI, USA). Twenty-four hours later, conditioned medium was collected and cleared by centrifugation. The quantification of Aβ40 peptide was performed using the V-PLEX Aβ40 Peptide (6E10) kit (Meso Scale Diagnostics, Rockville, MD, USA).

Statistics

Statistical analyses were performed using one-way Analysis of Variance (ANOVA) using Sidak's post hoc test, allowing correction for multiple comparisons. Statistical analyses were performed with GraphPad Prism software (GraphPad, San Diego, CA, USA). All n reported are biological replicates. All data are presented as mean ± standard error of the mean (SEM).

Statistical significance was defined as ns (not significant) p > 0.05, * p < 0.05, ** p < 0.01, *** p < 0.001, **** p < 0.0001.

Results

Reduced temperature improves SorLA protein processing for a subset of maturation-defective SORL1 variants in HEK cells.

Over the 15 maturation-defective SORL1 variants identified by our group [START_REF] Rovelet-Lecrux | Impaired SorLA maturation and trafficking as a new mechanism for SORL1 missense variants in Alzheimer disease[END_REF], 10 were selected based on their differential impact on SorLA maturation profile when expressed in HEK cells: R332W, S564G, S577P, R654W, R729W, D806N, Y934C, D1535N, D1545E and W1862C.

To evaluate the impact of lower growing temperature on SorLA protein maturation, we transiently transfected HEK cells with cDNA constructs encoding the full-length wild-type (WT)-SorLA or SorLA variants. As previously described [START_REF] Rovelet-Lecrux | Impaired SorLA maturation and trafficking as a new mechanism for SORL1 missense variants in Alzheimer disease[END_REF], at 37°C, WT-SorLA protein consisted of two distinct forms on immunoblot analysis (Figure 1A), corresponding to mature (upper band) and immature species (lower band). In contrast, all 10 missense variants showed an alteration of the maturation profile associated with lower levels of mature SorLA protein.

Following culture at 29°C, WT-SorLA protein showed a maturation pattern similar to that observed at 37°C, indicating that WT-SorLA protein maturation was not affected by decreased temperature. On the other hand, 6 of the 10 variants (R332W, S564G, S577P, R654W, D806N, and W1862C) showed a significant increase in the proportion of mature SorLA, compared to 37°C (Figure 1B, Supplementary table 1). Although there was a slight increase in the proportion of mature forms for the R729W and the Y934C variants, this trend did not reach significance.

Decreasing temperature worsened the maturation profile for D1535N and D1545E variations.

Taken together, these results indicate that decreasing the growing temperature of HEK cells improves the processing of a subset of maturation-defective SORL1 variants.

Reduced temperature improves SorLA protein maturation defects linked to missense SORL1 variants in hiPSC.

We then sought to validate these results on endogenous SORL1 levels, using CRISPR/Cas9edited hiPSC lines carrying homozygous R332W and R654W variants [START_REF] Rovelet-Lecrux | Impaired SorLA maturation and trafficking as a new mechanism for SORL1 missense variants in Alzheimer disease[END_REF]. As reported before, at 37°C, endogenous WT-SorLA migrates mostly as mature species (Figure 2A,B) in hiPSCs.

The R332W variant is associated with a drastic reduction of the mature forms with a concomitant increase of the immature species of the protein, and the overwhelming majority corresponds to immature forms for the R654W variant. Growing cells at 29°C did not affect the WT-SorLA protein maturation, but resulted in a significant increase in the proportion of mature forms of SorLA at 29°C for both variants. These data confirmed the beneficial effect of reducing growing temperature on SorLA maturation defects linked to missense SORL1 variants in an endogenous SORL1 expressing model.

Improvement in SorLA protein maturation results in decreased Aβ secretion in hiPSCs.

To evaluate if this improvement in SorLA maturation was associated with a rescue in SorLA function regarding Aβ secretion, we quantified Aβ40 peptides in the conditioned media of WT, R332W, R654W or KO SorLA hiPSC lines cultured at 37 or 29°C. As previously reported [START_REF] Rovelet-Lecrux | Impaired SorLA maturation and trafficking as a new mechanism for SORL1 missense variants in Alzheimer disease[END_REF], at 37°C, KO and mutants SorLA hiPSCs showed a significant increase in Aβ secretion, both variants leading to similar levels of Aβ as the SORL1 KO hiPSC lines (Figure 2C). At 29°C, significant lower levels of Aβ peptides were detected in conditioned media of R332W and R654W hiPSCs, compared to the KO SorLA hiPSCs. Similar levels of Aβ were detected in WT and R332W lines. Note that KO SorLA hiPSCs still showed a significant increase in Aβ secretion compared to the WT lines, indicating that at lower temperature WT-SorLA protective function is maintained. Altogether, these results indicate that reducing growing temperature of hiPSCs is associated with both an improvement in SorLA protein maturation and a rescue of its protective function against Aβ secretion.

Discussion

Recently, we demonstrated that a subset of rare missense AD-associated SORL1 variants was associated with impaired maturation/trafficking of the SorLA protein, resulting in a loss of SorLA protective function against the secretion of Aβ peptides [START_REF] Rovelet-Lecrux | Impaired SorLA maturation and trafficking as a new mechanism for SORL1 missense variants in Alzheimer disease[END_REF]. In this report, we demonstrate that reducing HEK cell culture temperature at 29°C leads to a significant improvement of SorLA protein maturation for 6 (R332W, S564G, S577P, R654W, D806N, and W1862C) of the 10 maturation-defective variants tested. These results were confirmed for the R332W and R654W variants, when expressed in CRISPR/Cas9-modified hiPSCs at endogenous levels. In both cell lines, SorLA protein maturation rescue was not complete. It is conceivable that the arbitrary temperature of 29°C used here is not optimal, and that changes in temperature and/or culture time could improve the efficiency of the treatment. In agreement, it has been demonstrated for the SUR1 mutant protein that decreasing the temperature to 30°C allowed only a partial exit from the ER, whereas full exit was observed at 18°C [START_REF] Yang | Low temperature completely rescues the function of two misfolded K ATP channel disease-mutants[END_REF]. It is also possible that several misfolded states coexist. Pending the severity of the conformational defect, partially folded proteins can be chaperoned to attain their native conformation or not, with only some being temperature-sensitive. Instead of being improved, the maturation profile of D1535N and D1545E was even worsened. Differential effect of temperature on maturation-defective variants has already been reported for other misfolded proteins. For instance, maturation defects associated with the CFTR G85E and R560T mutations remain unchanged and even worsen with the E92K mutant [START_REF] Lopes-Pacheco | Combination of Correctors Rescues CFTR Transmembrane-Domain Mutants by Mitigating their Interactions with Proteostasis[END_REF][START_REF] Farinha | Revertants, low temperature, and correctors reveal the mechanism of F508del-CFTR rescue by VX-809 and suggest multiple agents for full correction[END_REF].

Several mechanisms can explain how decreased temperature may act on SorLA protein maturation. First, slowing down protein translation increases time for proteins to get to their native state through the help of cellular chaperones. Post-translational modifications, particularly glycosylation, are maintained or even improved at low temperature [START_REF] Yoon | Effect of low culture temperature on specific productivity, transcription level, and heterogeneity of erythropoietin in Chinese hamster ovary cells[END_REF][START_REF] Bollati-Fogolín | Temperature reduction in cultures of hGM-CSF-expressing CHO cells: effect on productivity and product quality[END_REF]. Finally, decreasing temperature may have a more global effect on cells. It has been shown that low temperature treatment causes a profound alteration in cell transcriptome [START_REF] Sondo | Rescue of the mutant CFTR chloride channel by pharmacological correctors and low temperature analyzed by gene expression profiling[END_REF] and proteome [START_REF] Gomes-Alves | Low temperature restoring effect on F508del-CFTR misprocessing: A proteomic approach[END_REF]. It is therefore possible that the corresponding change in the proteostasis environment, and not the activity/expression of a single protein, is globally responsible for SorLA rescue by low temperature. Understanding the mechanism of low temperature and mimicking it with small molecules appear as an important priority to design novel therapeutics for SorLA rescue.

The study in hiPSCs also allowed us to highlight that reducing culture temperature is associated with a rescue of the protective function of the R332W and R654W SorLA variants against Aβ secretion. The significant increase in Aβ secretion observed in R332W hiPSCs is no longer detected at lower temperature, suggesting a nearly complete rescue of the SorLA protein function. These results are in full agreement with the great improvement of SorLA maturation in these lines at lower temperature. In contrast, the R654W hiPSCs displayed a significant increase in Aβ secretion compared to the WT lines at both temperatures, but to a lesser extent at lower temperature. Importantly, whereas at 37°C R654W mutant lines displayed similar Aβ secretion as compared to the KO lines, at lower temperature, Aβ secretion is significantly reduced indicating that this mutant does not behave anymore as a complete loss of function.

Accordingly, the improvement in SorLA maturation of this variant observed at lower temperature is by far incomplete, suggesting that the proportion of mature SorLA is insufficient to maintain the protein function against Aβ accumulation. On the other hand, we cannot exclude that, in addition to having a deleterious effect on SorLA maturation, the mutation impacts protein function by other mechanisms.

Obviously, lowering temperature is not a transposable strategy to prevent AD or any misfolded protein disease. However, after successful correction of folding defects by decreased temperature, extensive studies were performed to identify small molecules acting as pharmacological chaperones able to rescue trafficking defects of proteins involved in misfolded protein disease, some of were later approved for clinical use [START_REF] Gámez | Protein misfolding diseases: Prospects of pharmacological treatment[END_REF]. Our results may encourage similar strategies for the treatment of SORL1 maturation-defective variants in AD.

Legends

Figure 1 -

 1 Figure 1 -Black and White -Decrease of growing temperature facilitates SorLA protein

Figure 2 -

 2 Figure 2 -Black and White -Decrease of growing temperature facilitates SorLA protein

Figure 1 -

 1 Figure 1 -Color -Decrease of growing temperature facilitates SorLA protein maturation

Figure 2 -

 2 Figure 2 -Color -Decrease of growing temperature facilitates SorLA protein maturation

Acknowledgments

We thank Anne Rovelet-Lecrux for her help in proof-reading the manuscript.

Funding

This work was co-supported by a grant from France Alzheimer association (AAPSM2019grant n°1957) and the European Union and the Région Normandy (RIN-NICE). Europe gets involved in Normandy through the European Regional Development Fund (ERDF). This work was performed in the framework of FHU-G4 Génomique.

Data availability

The data supporting the findings of this study are available within the article and/or its supplementary material.

Conflict of interest

The authors have no conflict of interest to report.


Ousmane Boly
email: ousmane.boly@cyu.fr

Felix Cheysson
email: felix.cheysson@univ-eiel.fr

Thi Hien Nguyen
email: thi-hien.nguyen5@cyu.fr

Mixing properties for multivariate Hawkes processes

Keywords: multivariate Hawkes process, strong mixing (α-mixing), functional central limit theorem

Properties of strong mixing have been established for the stationary linear Hawkes process in the univariate case, and can serve as a basis for statistical applications. In this paper, we provide the technical arguments needed to extend the proof to the multivariate case. We illustrate these properties by establishing a functional central limit theorem for multivariate Hawkes processes.

Introduction

Linear Hawkes processes form a versatile family of models for point processes that exhibit selfexcitation properties: in addition to a constant rate of arrivals, each new arrival increases the rate at which future arrivals occur. This additive dependence structure can also be decomposed such that the process can be viewed as a branching process with immigration, thus admitting a Poisson cluster structure where each cluster is itself a branching process without immigration [START_REF] Hawkes | A cluster process representation of a self-exciting process[END_REF]. While they were initially introduced by [START_REF] Hawkes | Point spectra of some mutually exciting point processes[END_REF] in order to model seismic aftershocks [START_REF] Ogata | Statistical models for earthquake occurrences and residual analysis for point processes[END_REF], Hawkes processes have become increasingly popular due to their wide applicability to various other elds including but not limited to nance [START_REF] Embrechts | Multivariate hawkes processes: an application to nancial data[END_REF], ecology [START_REF] Gupta | Lasso and probabilistic inequalities for multivariate point processes[END_REF], neurophysiology [START_REF] Chornoboy | Maximum likelihood identication of neural point process systems[END_REF], and social network analysis [START_REF] Zannettou | On the origins of memes by means of fringe web communities[END_REF]. The study of Hawkes processes in itself is also of interest, as theoretical advances in the past decades have paved the way to the development for statistical inference methods [START_REF] Gupta | Lasso and probabilistic inequalities for multivariate point processes[END_REF][START_REF] Bacry | Sparse and low-rank multivariate Hawkes processes[END_REF] and thus to the popularity of the model.

In particular, works interested in the ergodic and mixing properties of the Hawkes process play an important role, for they can provide strong moment inequalities and coupling methods useful to prove asymptotic properties. It is well known that the linear Hawkes process is ergodic due to its Poisson cluster structure [START_REF] Westcott | On Existence and Mixing Results for Cluster Point Processes[END_REF]. When the reproduction function has compact support, [START_REF] Reynaud-Bouret | Some non asymptotic tail estimates for Hawkes processes[END_REF] used this structure to derive exponential inequalities which specify the rate of convergence in the ergodic theorem. Recently, [START_REF] Graham | Regenerative properties of the linear Hawkes process with unbounded memory[END_REF] extended these results by proving regenerative properties for the linear Hawkes process, even when the reproduction function has unbounded support. Issues arise for the nonlinear Hawkes process in which the rate at which future arrivals occur is non-linearly dependent on the past arrivals since the Poisson cluster structure of the process no longer exists. While the existence and stationarity of the process can be proven under general sets of assumptions [START_REF] Bremaud | Stability of nonlinear hawkes processes[END_REF], studies of the mixing properties of the process have only been established under specic conditions. In [START_REF] Costa | Renewal in Hawkes processes with selfexcitation and inhibition[END_REF], by assuming a bounded support for the reproduction function (which may take negative values), the authors obtained exponential concentration inequalities using renewal techniques, further extending the results from [START_REF] Reynaud-Bouret | Some non asymptotic tail estimates for Hawkes processes[END_REF]. Without the bounded support assumption, but supposing the reproduction function is exponential, [START_REF] Dion | Exponential ergodicity for diusions with jumps driven by a Hawkes process[END_REF] showed that the non-linear Hawkes process is exponentially β-mixing, relying on the general theory of Markov processes.

In this work, we focus on establishing rates of convergence for the α-mixing of the linear Hawkes process, using the Poisson cluster decomposition of the process. Initially used to prove asymptotic properties for time series and random elds (see e.g. [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF], the αmixing property can be straightforwardly extended to the eld of point processes [START_REF] Westcott | The probability generating functional[END_REF][START_REF] Poinas | Mixing properties and central limit theorem for associated point processes[END_REF]. In the univariate context, [START_REF] Cheysson | Spectral estimation of Hawkes processes from count data[END_REF] already derived a polynomial rate of α-mixing under tail assumptions for the reproduction function of the Hawkes process. However, their proof cannot be directly extended to the multivariate case, as a critical argument of independence within each Poisson cluster, that is between the arrivals' times and the number of arrivals in any given generation of the branching representation of the cluster, does not hold when multiple components can mutually excit each other. This is solved by establishing an exponential inequality in the multitype branching process, which allows us to bound the covariance measure of the branching process.

In turn, the α-mixing of the process can serve as a basis for statistical applications, for example by using central limit theorems (CLTs) that already exist in the literature [START_REF] Doukhan | The functional central limit theorem for strongly mixing processes[END_REF]. To provide useful tools for statistical applications related to the multivariate Hawkes process, we enunciate both a classical and a functional CLTs in this context, and show that the parameters appearing in these CLTs can be easily calculated in practice, using the spectral representation of the process. Indeed, while the covariance structure of the Hawkes process is often non-explicit, except for exponential reproduction functions (Da Fonseca andZaatour, 2014, 2015), its Bartlett spectrum (i.e. the Fourier transform of the autocovariance of the process) has an explicit and simple form [START_REF] Daley | An introduction to the theory of point processes[END_REF]. Specically, this yields a simple estimator of the long run variance of the process which appears naturally in the CLTs.

Section 2 introduces the notation we use throughout and the Hawkes model. In Section 3, we establish the main results of our contribution, namely an α-mixing property and a functional CLT for the multivariate Hawkes process. Proofs are postponed to subsequent sections, respectively in Section 4 and 5.

Main denitions

Notation

Throughout this paper, we consider a probability space (Ω, F, P) and the measurable space (N, N ) of locally nite counting measures on R with values in N d , with d ≥ 1. We denote by B(A) the Borel σ-algebra generated over a subset A ⊂ R, and for a multivariate point process N = (N 1 , . . . , N d ) i.e. a measurable map from (Ω, F, P) to (N, N ), by E(A) the cylindrical σ-algebra generated by N on A and dened by

E(A) := σ({N ∈ N : N(B) = m}, B ∈ B(A), m ∈ N d ).
Finally, denote by N |A the point process restricted to A, that is the process dened by N |A (B) = N(A ∩ B) for all B ∈ B(R).

We also introduce some notation for function and vector norms. For a function f : R → R,

denote its L 1 -norm by ∥f ∥ L 1 , ∥f ∥ L 1 = R |f (x)|dx.
For a vector u = (u 1 , . . . , u n ) ∈ R n , denote its ℓ 1 -norm by ∥u∥ 1 ,

∥u∥ 1 = n k=1 |u k |,
and its supremum norm by ∥u∥ ∞ ,

∥u∥ ∞ = sup 1≤k≤n |u k |.

2.2

The multivariate linear Hawkes process

The multivariate Hawkes process N = (N 1 , . . . , N d ) can be dened as a point process such that For all 1 ≤ j ≤ d, the conditional intensity λ j (t) of the component N j satisfying

P N j (t, t + h) = 1 | F t -= λ j (t)h + o(h)
, with F t the natural ltration of the process N is a left-continuous stochastic process given by

λ j (t) = η j + d i=1 t -∞ h ij (t -u)N i (du) = η j + d i=1 {n:T n i <t} h ij (t -T n i ),
where η j > 0 is the baseline intensity of N j , h ij : R + → R + the reproduction function from component i to j, and {T n i } n are the atoms of N i . The point process is orderly: for all 1 ≤ i ̸ = j ≤ d, N i and N j never jump simultaneously,

P ∥N (t, t + h) ∥ 1 ≥ 2 | F t -= o(h).
The multivariate Hawkes process N can also be seen as a Poisson cluster process (see [START_REF] Hawkes | A cluster process representation of a self-exciting process[END_REF], where clusters are mutually independent branching processes generated recursively as follows:

Cluster centres, also called immigrants of the process, form the generation 0 of the branching processes. Immigrants of type j arrive according to a homogeneous Poisson process with rate η j .

Then, an individual T n ki of generation k and type i generates osprings of generation k + 1 and type j according to an inhomogeneous Poisson process with intensity h ij (• -T n ki ). For a single branching process, denoting by Z ki the number of individuals of generation k and type i, we have that (Z ki ) k∈N,1≤i≤d is a multivariate GaltonWatson process. We will make the following assumption, which ensures the existence and stationarity of the multivariate Hawkes process [START_REF] Bremaud | Stability of nonlinear hawkes processes[END_REF].

Assumption 2.1. The spectral radius of the reproduction matrix, dened by M := (∥h ij ∥ L 1 ) 1≤i,j≤d , is strictly less than 1.

3 Main results

3.1

Strong mixing of the multivariate Hawkes process

For a probability space (Ω, F, P) and A, B two sub σ-algebras of F, the strong mixing coecient is dened as the measure of dependence between A and B [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF]:

α(A, B) := sup{|P(A ∩ B) -P(A) P(B)| , A ∈ A, B ∈ B.}
The denition of the strong mixing coecient can be adapted to a multivariate point process N, by dening (see [START_REF] Poinas | Mixing properties and central limit theorem for associated point processes[END_REF] α

N (τ ) = α(E t -∞ , E +∞ t+τ ) = sup t∈R sup A∈E t -∞ B∈E ∞ t+τ Cov 1 A (N), 1 B (N)
where E b a stands for E((a, b]), i.e the σ-algebra generated by the cylinder sets on the interval (a, b], and 1 A (N) is the indicator function of the cylinder set A, i.e for any elementary cylinder set

A B,n = {N ∈ N : N(B) = n}, 1 A (N) = 1 if N(B)
= n and 0 otherwise. As in the univariate case, the process is said to be strongly mixing if α N (τ ) → 0 as τ → ∞.

In the context of multivariate Hawkes processes, the main result of this contribution is the following theorem. Theorem 3.1 can serve as a basis for various statistical applications, as examplied by the next subsection.

Theorem 3.1. Let N be a multivariate Hawkes process with reproduction functions (h ij ) 1≤i,j≤d such that Assumption 2.1 holds true. Further assume that there exists β > 0 such that the reproduction kernels have nite moment of order 1 + β:

sup 1≤i,j≤d R t 1+β h ij (t)dt < ∞.
Then, for any 0 < γ < β, as τ → ∞,

α N (τ ) = O τ -γ .
(1)

While the proof (postponed to Section 4) mostly extends that of Cheysson and Lang (2022) to the multivariate case, it also circumvents a lemma in the proof of the aforementioned article: in the univariate Galton-Watson process, the number of individuals in any given generation is independent from the arrival times of all individuals, and vice versa, whereas this is only valid conditionally on the genealogy of individuals in the multivariate case. This is the heart of the proof and the focus of Lemmas 4.3 and 4.4, which aim to prove that we can make do with exponential inequalities instead of independence when bounding the strong mixing coecient, thanks to the asymptotic contraction property of the reproduction matrix M, ensured by Assumption 2.1.

3.2

An application to statistics

A functional central limit theorem

Suppose that we observe a point process N = (N 1 , . . . , N d ) from a stationary multivariate Hawkes model with intensities m i ≥ 0, and that we are interested in the asymptotic behaviour of statistics of the form

d i=1 N i (f i [0,T ] ) = d i=1 [0,T ] f i (x)N i (dx),
or rather their centered version,

S T = d i=1 [0,T ] f i (u) N i (du) -m i du , (2)
where

f i : R → R denote bounded measurable functions (1 ≤ i ≤ d) and f i [0,T ] : x → f i (x)1 [0,T ] (x) their restriction to [0, T ].
The variance of S T can be related to the covariance measure of N, whose elements are denoted C ij , through the use of Campbell's formula:

σ 2 T = Var S T = d i=1 d j=1 [0,T ] 2 f i (u)f j (v)C ij (du × dv).
Finally, for 0

≤ u ≤ 1, dene v T (u) = inf{t ∈ (0, T ) : σ 2 t /σ 2 T ≥ u} and W T (u) = σ -1 T S v T (u) .
The following proposition is a consequence of the functional central limit theorem for nonstationary strongly mixing processes in Merlevède and Peligrad (2020, Corollary 2.2), in conjunction with the mixing property derived in Theorem 3.1. Its proof is deferred until Section 5. Proposition 3.1. Let N be a multivariate Hawkes process satisfying the conditions of Theorem 3.1. Assume also that σ 2

T = T σ 2 + o(T ) as T → ∞, and that each f i is locally (2 + δ)-integrable, with β, δ > 0, such that (β -1)δ > 2.
Then, {W T (u), u ∈ (0, 1)} converges in distribution in the Skorokhod space D([0, 1]) (equipped with the uniform topology) to the standard Brownian motion.

Note also that this result implies that S T satises the classical central limit theorem (see Merlevède and Peligrad, 2020, Section 2.1.3).

Corollary 3.1. Let N be a multivariate Hawkes process as in Proposition 3.1. Then σ -1 T S T converges in distribution to the standard normal distribution.

However, in practice, the covariance measure of the Hawkes process has no explicit form (except in the case of exponential kernels, see Da [START_REF] Fonseca | Clustering and Mean Reversion in a Hawkes Microstructure Model[END_REF], so we can turn to the spectral properties of the process. Recall that the Bartlett spectrum of a stationary multivariate point process N on R is dened as the unique, positive-denite, Hermitian measure Γ of auto-and cross-spectral measures (Γ ij (•)) such that, for any rapidly decaying function f and g on R (Daley and Vere-Jones, 2003, chapter 8),

Cov N i (f ), N j (g) = R Ff (ξ)Fg * (ξ)Γ ij (dξ), (3)
where F• denotes the Fourier transform,

Ff (ξ) = R e -2iπξu f (u)du, (4)
and g * (u) = g(-u), so that Fg * is the complex conjugate of Fg.

It turns out that the Bartlett spectrum for the multivariate Hawkes process is known, and has density given by

γ(ξ) = I -[ H(2πξ)] ⊺ -1 diag(m 1 , . . . , m d ) I -H(-2πξ) -1 , with diag(m 1 , . . . , m d ) = (I -[ H(0)] ⊺ ) -1 η;
discrepancies with the expression in Daley and Vere-Jones (2003, example 8.3(c)) can be attributed to the denition of the reproduction matrix H and our choice of convention for the Fourier transform (4). Then the variance of S T can be calculated by the relation

σ 2 T = Var S T = d i=1 d j=1 R Ff i [0,T ] (ξ)Ff j * [0,T ] (ξ)γ ij (ξ)dξ.
(5)

Two examples

Though the integral appearing in (5) can be dicult to compute analytically, we highlight two cases where the computation is explicit, so that it may be used in applications.

Asymptotically constant functions.

Assume each

f i is of the form f i (•) = k i + g i (•
) with k i constants such that ∥k∥ 1 > 0 and g i bounded and integrable functions. Then, we nd that

σ 2 T = T d i=1 d j=1 k i k j γ ij (0) + o(T ) = T k ⊺ γ(0)k + o(T ),
yielding that the process (T k ⊺ γ(0)k) -1/2 S T u , u ∈ (0, 1) converges in distribution to the stan- dard Brownian motion, and (T k ⊺ γ(0)k) -1/2 S T to the standard normal distribution.

Proof. Since F1 [0,T ] (ξ) = T e -iπξT sinc(πξT ), the leading term in the variance ( 5) is given by

d i=1 d j=1 k i k j R F1 [0,T ] (ξ)F1 * [0,T ] (ξ)γ ij (ξ)dξ = d i=1 d j=1 k i k j T 2 R sinc 2 (πξT )γ ij (ξ)dξ = d i=1 d j=1 k i k j T R sinc 2 (πω)γ ij (ω/T )dω.
The dominated convergence theorem then yields that

R sinc 2 (πω)γ ij (ω/T )dω ----→ T →∞ γ ij (0) R sinc 2 (πω)dω = γ ij (0).
Asymptotically periodic functions.

Assume each f i is of the form f i (•) = h i (•) + g i (•), with h i τ -periodic functions such that d i=1 τ 0 h i (t)dt ̸ = 0 and g i bounded and integrable functions. Then

σ 2 T = T τ 2 d i=1 d j=1 k∈Z Fh i [0,τ ] k τ Fh j * [0,τ ] k τ γ ij k τ + o(T ),
so that the process σ -1/2 T S T u , u ∈ (0, 1) has the desired convergence to the standard Brownian motion.

Proof. Choose T = nτ with n ∈ N + . Then,

Fh i [0,T ] (ξ) = n k=1 kτ (k-1)τ e -2iπξx h i (x)dx = n k=1 e -2iπξ(k-1)τ Fh i [0,τ ] (ξ),
so that the leading term in the variance ( 5) is given by

d i=1 d j=1 R Fh i [0,T ] (ξ)Fh j * [0,T ] (ξ)γ ij (ξ)dξ = d i=1 d j=1 R n k=1 n l=1 e -2iπξ(k-l)τ Fh i [0,τ ] (ξ)Fh j * [0,τ ] (ξ)γ ij (ξ)dξ.
A bit of calculus gives that

n k=1 n l=1 e -2iπξ(k-l)τ = n + 2 Re e 2iπξnτ n-1 k=1 ke -2iπξkτ = 1 -cos(2πξnτ ) 1 -cos(2πξτ )
so that, for any ϕ absolutely integrable,

1 n R 1 -cos(2πξnτ ) 1 -cos(2πξτ ) ϕ(ξ)dξ = 1 n k∈Z 1/2τ -1/2τ 1 -cos(2πξnτ ) 1 -cos(2πξτ ) ϕ ξ + k τ dξ = 1 2πn 2 τ k∈Z πn -πn 1 -cos(ξ) 1 -cos(ξ/n) ϕ ξ 2πnτ + k τ dξ = 1 2πτ k∈Z ∞ -∞ 1 [-πn,πn] (ξ) n 2 1 -cos(ξ) 1 -cos(ξ/n) ϕ ξ 2πnτ + k τ dξ.
Since the integrand is bounded above by ξ → |ϕ(ξ/(2πnτ ) + k/τ )| which is integrable, and

lim n→∞ 1 [-πn,πn] (ξ) n 2 1 -cos(ξ) 1 -cos(ξ/n) ϕ ξ 2πnτ + k τ = 2 1 -cos(ξ) ξ 2 ϕ k τ ,
the dominated convergence theorem yields that

lim n→∞ ∞ -∞ 1 [-πn,πn] (ξ) n 2 1 -cos(ξ) 1 -cos(ξ/n) ϕ ξ 2πnτ + k τ dξ = 2 ϕ k τ ∞ -∞ 1 -cos(ξ) ξ 2 dξ = 2πϕ k τ .
Hence,

ξ → 1 n 1 -cos(2πξnτ ) 1 -cos(2πξτ ) converges, in the sense of distributions, to ξ → 1 τ k∈Z δ k/τ (ξ),
yielding the desired variance.

Finally, when T ̸ = nτ , write T = nτ + ε with ε < τ so that the leading term in the variance remains the same.

4 Proof of Theorem 3.1 Most of the proofs follow those of [START_REF] Cheysson | Spectral estimation of Hawkes processes from count data[END_REF], extended here to the multivariate setup. We rst present the layout of the proof, introducing auxiliary results when needed, then in subsequent subsections prove these results.

Layout of the proof

Recall that the strong mixing coecient of a point process N is dened by

α N (τ ) = α(E t -∞ , E +∞ t+τ ) = sup t∈R sup A∈E t -∞ B∈E ∞ t+τ Cov 1 A (N), 1 B (N) .
The rst step of the proof is to bound the covariance of the indicator functions with that of the point process itself, using the positive association of Hawkes processes (Gao and Zhu, 2018, Section 2.1, key property (e)). Recall that a X -valued random variable X is positively associated if for each pair of bounded, Borel measurable, non-decreasing functions f, g : X → R, we have Cov(f (X), g(X)) ≥ 0. This property can be used to bound the covariance of an associated point process, using Theorem 2.5 from Poinas et al. ( 2019), which we extend to the multivariate setup.

Cov f (N |A ), g(N |B ) ≤ ∥f ∥ A ∥g∥ B Cov ∥N(A)∥ 1 , ∥N(B)∥ 1 ,
where

∥f ∥ A := sup N∈N sup 1≤i≤d x∈A |f (N |A + δ ix ) -f (N A )|,
and δ ix denotes the multivariate point process with a single atom at component i and position x.

For the Hawkes process, this leads to the following lemma, whose proof uses the Poisson cluster structure of the process. Lemma 4.1. Let N be a multivariate Hawkes process satisfying Assumption 2.1. Let s, t, v ∈ R and τ > 0 such that s < t < t + τ < v, A ∈ E t s , B ∈ E v t+τ , and A = (s, t] and B = (t + τ, v]. Then

Cov 1 A (N), 1 B (N) ≤ d i=1 d j=1 d k=1 Cov N i (A | y), N j (B | y) M c k (dy), (6)
where M c k denotes the rst-order moment measure of the k-th component N c k of the cluster centre process.

Without loss of generality, we consider a cluster whose immigrant is located at time 0. Let Z ki denote the number of points of generation k of the i-th component of the point process, and Z ki (A) those that are located in the set A ⊂ R. By denition, we have

N i (A | 0) = ∞ k=0 Z ki (A).
Then, the covariance between two components of a single branching process can be written

Cov N i (A | 0), N j (B | 0) = ∞ k=0 ∞ l=0 Cov Z ki (A), Z lj (B) . (7)
As Z ki (A), Z lj (B) ≥ 0, the terms appearing in the sum on the right hand side can be bounded by

|Cov(Z ki (A), Z lj (B))| = E Z ki (A)Z lj (B) -E Z ki (A) E Z lj (B) ≤ max E Z ki (A)Z lj (B) , E Z ki (A) E Z lj (B) .
Let T m lj denote the m-th time of arrival of generation l and type j, and consider the rst term appearing in the maximum above. Noting that for a non-negative integer I, I = ∞ n=1 1 {I≥n} , we have that

E[Z ki (A)Z lj (B)] ≤ E[Z ki (R)Z lj (B)] = ∞ n=1 ∞ m=1 E 1 {Z ki ≥n} 1 {Z lj ≥m} 1 {T m lj ∈B} ≤ ∞ n=1 ∞ m=1 E 1 {Z ki ≥n} 1/p E 1 {Z lj ≥m} 1/q E 1 {T m lj ∈B} 1/r = ∞ n=1 ∞ m=1 P(Z ki ≥ n) 1/p P(Z lj ≥ m) 1/q P(T m lj ∈ B) 1/r . ( 8
)
where the second line follows from Hölder's inequality with any p, q, r ≥ 1,

1 p + 1 q + 1 r = 1.
Remark. Note that the use of Hölder's inequality and the need for an exponential bound on P(Z ki ≥ n) and P(Z lj ≥ m) dier from the univariate case (see [START_REF] Cheysson | Spectral estimation of Hawkes processes from count data[END_REF]), for which the times of arrival in a generation k and the number of points in a generation l are independent, for all k and l. This is not the case in the multivariate setup, since the distribution of an arrival time may depend on its genealogy, through the types of its ancestors.

We can establish a simple upper bound on P(T m lj ∈ B) using Markov's inequality.

Lemma 4.2. Let β > 0 and assume that

ν 1+β = sup 1≤i,j≤d R t 1+β h * ij (t)dt < ∞, with h * ij = h ij /∥h ij ∥ L 1 . Then, P(T m lj ∈ B) ≤ l 1+β ν 1+β (t + τ ) 1+β . ( 9
)
The following two lemmas form the heart of the proof, and make it distinct to the univariate case: in order to bound the right-hand side of ( 8) with the fastest rate of convergence to zero, the term P(T m lj ∈ B) 1/r = O τ -(1+β)/r must converge as fast as possible, so that r should be chosen as small as possible. Consequently, in Hölder's inequality, p and q may be arbitrary large. To ensure that the terms P(Z ki ≥ n) 1/p and P(Z lj ≥ m) 1/q are summable with respect to n and m, we rst prove an exponential inequality for the multitype Galton-Watson process (Z ki ), through its Laplace transform, yielding the following lemma.

∞ n=1 P(Z ki ≥ n) 1/p ≤ C 1 L{Z k }(u) -1 1/p , ( 10
)
where

L{Z k }(u) = E[exp(Z ⊺ k u)]
denotes the Laplace transform of Z k and C 1 denotes a constant that can be chosen uniformly with respect to 1 ≤ i ≤ d.

Remark. Throughout the rest of the proof, the C k are absolute constants.

Plugging both ( 9) and ( 10) into ( 8) yields

E[Z ki (A)Z lj (B)] ≤ ∞ n=1 ∞ m=1 P(Z ki ≥ n) 1/p P(Z lj ≥ m) 1/q P(T m lj ∈ B) 1/r ≤ C 2 L{Z k }(u) -1 1/p L{Z l }(u) -1 1/q l 1+β ν 1+β (t + τ ) 1+β 1/r
.

The same reasoning can be applied to E[Z ki (A)] E[Z lj (B)], leading to an analogous Hölder inequality, so that we only detailed the proof for E[Z ki (A)Z lj (B)] which is the slightly more complicated case. Then, from ( 7),

Cov N i (A | 0), N j (B | 0) ≤ ∞ k=0 ∞ l=0 C 2 L{Z k }(u) -1 1/p L{Z l }(u) -1 1/q l 1+β ν 1+β (t + τ ) 1+β 1/r . ( 11
)
We now focus on proving the summability of the series k L{Z k }(u) -1 1/p . The diculty of this proof is algebraic in nature, as even though the spectral radius of the reproduction matrix M := (∥h ij ∥ L 1 ) 1≤i,j≤d is strictly less than 1 (by Assumption 2.1), it does not ensure that ∥Mu∥ 1 ≤ ∥u∥ 1 for all u. However, this holds asymptotically, such that for n large, ∥M n u∥ 1 ≤ ∥u∥ 1 . Then, a recurrence on the Laplace transform of Z k with a good choice for u yields the summability of the series.

Lemma 4.4. There exists u = (u 1 , . . . , u d ) with u j > 0 for all 1 ≤ j ≤ d such that k L{Z k }(u)-

1
1/p and l l (1+β)/r L{Z l }(u) -1 1/q are summable.

The inequality in (11) can then be simplied to

Cov N i (A | 0), N j (B | 0) ≤ C 3 ν 1+β (t + τ ) 1+β 1/r . (12)
Finally, choose r = (1 + β)/(1 + γ) and p = q = 2 (1 + β)/(β -γ), such that (1 + β)/r = 1 + γ > 1. Then, plugging (12) into (6) yields

Cov 1 A (N), 1 B (N) ≤ d i=1 d j=1 d k=1 Cov N i (A | y), N j (B | y) η k dy ≤ C 4 d 3 ν 1/r 1+β t -∞ 1 (t + τ -y) 1+γ sup 1≤k≤d {η k } dy ≤ C 5 τ -γ .
which is the desired bound.

4.2

Proof of Proposition 4.1

The layout of this proof is identical to that of [START_REF] Poinas | Mixing properties and central limit theorem for associated point processes[END_REF], but for clarity we detail it within the multivariate setup. Consider a positively associated multivariate point process N. We consider the functions f + , f -: N → R, E(A)-measurable, and g + , g -:

N → R, E(B)-measurable, dened by f ± (N) = f (N |A ) ± ∥f ∥ A ∥N(A)∥ 1 , g ± (N) = g(N |B ) ± ∥g∥ B ∥N(B)∥ 1 .
For all 1 ≤ i ≤ d and x ∈ A,

f + (N + δ ix ) -f + (N) = f (N |A + δ ix ) -f (N |A ) + ∥f ∥ A
which is positive by the denition of ∥f ∥ A . Therefore, f + is an increasing function respective to the partial order dened by N ≤ M if and only if for all

1 ≤ i ≤ d, ∀A ⊂ R, N i (A) ≤ M i (A).
With the same reasoning, g + is also increasing, and f -, g -are decreasing. While f + is not bounded, it is non-negative and almost surely nite so that by a limiting argument, using the sequence of functions min(f + , k) when k goes to innity, the association inequality can be applied to f + . The other functions can be treated the same way, which leads to the inequalities

Cov f + (N), g + (N) ≥ 0 and Cov f -(N), g -(N) ≥ 0.
We can then expand these expressions

Cov f + (N), g + (N) = Cov f (N |A ), g(N |B ) + ∥f ∥ A ∥g∥ B Cov ∥N(A)∥ 1 , ∥N(B)∥ 1 + ∥g∥ B Cov f (N |A ), ∥N(B)∥ 1 + ∥f ∥ A Cov ∥N(A)∥ 1 , g(N |B ) and Cov f -(N), g -(N) = Cov f (N |A ), g(N |B ) + ∥f ∥ A ∥g∥ B Cov ∥N(A)∥ 1 ), ∥N(B)∥ 1 -∥g∥ B Cov f (N |A ), ∥N(B)∥ 1 -∥f ∥ A Cov ∥N(A)∥ 1 , g(N |B ) .
Adding these two expressions together yields the lower bound:

Cov f (N |A ), g(N |B ) ≥ -∥f ∥ A ∥g∥ B Cov ∥N(A)∥ 1 , ∥N(B)∥ 1 . (13)
The upper bound is obtained similarly by replacing f by -f in the previous expression.

4.3

Proof of Lemma 4.1

Using Proposition 4.1 with f = 1 A and g = 1 B , we have

Cov 1 A (N), 1 B (N) ≤ Cov ∥N(A)∥ 1 , ∥N(B)∥ 1 ≤ d i=1 d j=1 Cov N i (A), N j (B) .
Then, denoting by M c k the rst-order moment measure of the k-th component N c k of the cluster centre process, and by C c kl the covariance measure between the k-th and l-th components of the cluster centre process (1 ≤ k, l ≤ d), we have, by conditioning on the cluster centre process (see for example Daley and Vere-Jones, 2003, Exercise 6.3.4),

Cov N i (A), N j (B) = d k=1 Cov N i (A | y), N j (B | y) M c k (dy) + d k=1 d l=1 E(N i (A) | x) E(N j (B) | y)C c kl (dx × dy).
Since the centre process is Poisson, C c kl ≡ 0 and the second term vanishes.

4.4

Proof of Lemma 4.2: an upper bound for P(T m lj ∈ B)

Consider the inter-arrival times between a parent and its osprings. Since an individual T n ki of generation k and type i generates osprings of generation k + 1 and type j according to an inhomogeneous Poisson process with intensity h ij (• -T n ki ), these inter-arrival times are independent of each other, and their distributions only depend on the type i of the parent and the type j of the ospring. Consequently, any single arrival time T m lj can be written as the sum of l independent inter-arrival times,

T m lj = l ι=1 ∆ ι (T m lj ),
where the density functions of the ∆ ι (T m lj ) are amongst the (h * ij ) i,j . Hölder's inequality then gives,

T m lj = l ι=1 ∆ ι (T m lj ) ≤ l ι=1 1 β/(1+β) × l ι=1 ∆ ι (T m lj ) 1+β 1/(1+β)
, and by independence of the ∆ ι (T m lj ),

E l ι=1 ∆ ι (T m lj ) 1+β ≤ l sup 1≤ι≤l E ∆ ι (T m lj ) 1+β ≤ l ν 1+β .
Hence,

E T m lj 1+β ≤ l 1+β ν 1+β .
Finally, we can apply Markov's inequality to P(T m lj ∈ B), recalling that B = (t + τ, v]:

P(T m lj ∈ B) ≤ P(T m lj > t + τ ) ≤ l 1+β ν 1+β (t + τ ) 1+β .

4.5

Proof of Lemma 4.3: exponential inequality for P(Z ki ≥ n) and P(Z lj ≥ m)

First recall that the Laplace transform of a Poisson variable X with intensity λ is given by

L{X}(u) = E[e uX ] = e λ(e u -1) . ( 14
)
For a N-valued random variable Y and a constant α ≥ 0, dene the reproduction operator • by

α • Y = Y n=1 ξ (α) n ,
where ξ

(α) n i.i.d.
∼ P(α), independently of Y . Denoting by Z k = (Z k1 , . . . , Z kd ) ⊺ the random vector for the number of points of generation k for each component of the branching process, we have that conditionally on Z k-1 , the variables Z kj are Poisson with intensity

(Z ⊺ k-1 M) j = d i=1 Z k-1,i M ij (k > 0, 1 ≤ j ≤ d).
More specically, we have that

Z kj = d i=1 M ij • Z k-1,i = d i=1 Z k-1,i n=1 ξ (M ij ) n where ξ (M ij ) n i.i.d. ∼ P(M ij ) independently of all Z k-1,i . For all u = (u 1 , • • • , u d ) ∈ R d , consider now the Laplace transform of Z k , L{Z k }(u) = E exp(Z ⊺ k u) = E   exp   d j=1 u j Z kj     = E   exp   d j=1 d i=1 u j M ij • Z k-1,i     = E   exp   d j=1 d i=1 Z k-1,i n=1 u j ξ (M ij ) n     = E   d j=1 d i=1 Z k-1,i n=1 E exp u j ξ (M ij ) n Z k-1   = E   d j=1 d i=1 Z k-1,i n=1 exp (M ij (e u j -1))   (15) = E   exp   d j=1 d i=1 Z k-1,i M ij (e u j -1)     = E exp Z ⊺ k-1 M(e u -1) ,
where (15) was obtained using ( 14) and the fact that ξ

(M ij ) n
and Z k-1 are independent, and e u denotes the component-wise exponentiation of u.

Dene the function g by g(u) = M(e u -1). By recurrence, we therefore have

L{Z k }(u) = L{Z k-1 }(g(u)) = . . . = L{Z 0 }(g k (u)), (16)
where g k stands for the k-fold composition of g with itself.

Finally, we can apply Markov's inequality to Z ki for any 1 ≤ i ≤ d,

P(Z ki ≥ n) = P(e u i Z ki -1 ≥ e u i n -1) ≤ Ee u i Z ki -1 e u i n -1 ≤ L{Z k }(u) -1 e u i n -1 ,
where u = (u 1 , . . . , u d ) was chosen with strictly positive entries. Then, as n (e u i n -1) -1/p is summable, n P(Z ki ≥ n) 1/p is also summable and

∞ n=1 P(Z ki ≥ n) 1/p ≤ C 1 (L{Z k }(u) -1) 1/p ,
where C 1 is an absolute constant, which can be chosen uniformly with respect to 1 ≤ i ≤ d. Dene δ such that 1 < δ < ρ -1 , where ρ = Sp(M) < 1 denotes the spectral radius of the reproduction matrix M. Then there is a u 0 > 0 such that, for all u ∈ [0, u 0 ], e u -1 ≤ δu. Applying this to the vector u = (u 1 , . . . , u d ) with u j ∈ [0, u 0 ] for all 1 ≤ j ≤ d, we nd that

g(u) = M(e u -1) ≤ δMu, (17)
where the inequality holds term by term.

Lemma 4.5. There exists u = (u 1 , . . . , u d ) with u j > 0 for all 1 ≤ j ≤ d such that, for all

k ∈ N, g k (u) ≤ δ k M k u.
Furthermore, there exists a constant 0 < c < 1 and an integer k 0 > 0 such that, for all k ≥ k 0 ,

∥g k (u)∥ 1 ≤ c k ∥u∥ 1 . (18)
Proof. First note that, since ρ < 1, then for any matrix norm ∥•∥,

lim n→∞ ∥M n ∥ 1/n = ρ. Dene ε = 1/2 (δ -1 -ρ) > 0 such that c ε = δ(ρ + ε) < 1. Then there exists k 0 > 0 such that, for all k ≥ k 0 , ∥M k ∥ 1 ≤ (ρ + ε) k .
Therefore, for all k ≥ k 0 ,

∥δ k M k u∥ 1 ≤ δ k (ρ + ε) k ∥u∥ 1 = c k ε ∥u∥ 1 ---→ k→∞ 0. Since ∥δ k M k u∥ ∞ ≤ ∥δ k M k u∥ 1 , the sequence (∥δ k M k u∥ ∞ )
k∈N is bounded for any u. Choose u with strictly positive entries such that ∥δ k M k u∥ ∞ ≤ u 0 for all k ∈ N (such a u exists since the function u → sup k∈N ∥δ k M k u∥ ∞ is continuous and non-decreasing). Then, a straightforward recurrence from (17) yields the result.

Plugging (18) into the recurrence relation for the Laplace transform of Z k , ( 16), for a u, c and k 0 dened as in Lemma 4.5 and k ≥ k 0 , we nd that

L{Z k }(u) = E[exp(Z ⊺ 0 g k (u))] ≤ exp ∥g k (u)∥ 1 ≤ exp c k ∥u∥ 1 , (19)
noting for the rst inequality that for the branching process, Z 0 is a vector with only one non-zero component, equal to one. Therefore, when k ≥ k 0 , we have that

|L{Z k }(u) -1| ≤ |exp c k ∥u∥ 1 -1| = k→∞ c k ∥u∥ 1 + o(c k ).
Hence, k L{Z k }(u) -1 1/p is summable. Similarly, the series l l (1+β)/r L{Z l }(u) -1 1/q is summable.

5 Proof of Proposition 3.1

Let us rst write the functional central limit theorem proven in [START_REF] Merlevède | Functional CLT for nonstationary strongly mixing processes[END_REF] in the context of Rosenblatt's traditional α-mixing coecient [START_REF] Rosenblatt | A central limit theorem and a strong mixing condition[END_REF]

Theorem 5.1 [START_REF] Merlevède | Functional CLT for nonstationary strongly mixing processes[END_REF]. Assume that conditions (20) and ( 21) are satised for some δ > 0. Then, {σ -1 n W n (u), u ∈ (0, 1]} converges in distribution in D([0, 1]) (equipped with the uniform topology) to the standard Brownian motion.

We look to apply this result to functions of Hawkes processes. Dene the sequence {X k } k≥1 as

X k = d i=1 k k-1 f i (u) N i (du) -m i du ,
so that its partial sums S n coincide with the statistic dened in (2) when T = n is integer.

First note that

n k=1 E[X 2 k ] = n k=1 d i=1 d j=1 [k-1,k] 2 f i (u)f j (v)C ij (du × dv) ≤ n k=1 d i=1 d j=1 ∥f i ∥ ∞ ∥f j ∥ ∞ [0,1] 2 C ij (du × dv) =: nκ 1 ,
where the inequality comes from the boundedness of f i and the stationarity of N. Then,

sup n≥1 1 σ 2 n n k=1 E[X 2 k ] ≤ sup n≥1 nκ 1 nσ 2 + o(n) < ∞.
and the left part of condition ( 20) is fullled. Next, we have that where the rst two inequalities follow from Jensen's inequality and the third from the boundedness of f i and the stationarity of N i . Then, as N i admits nite exponential moments as a Poisson cluster of subcritical multitype Poisson branching processes, we nd that κ 2 < ∞, and since σ 2 n = nσ 2 + o(n) as n → ∞, the right part of condition (20) is veried. For the left hand side condition in (21), we write

E[|X k | 2+δ ] ≤ d 1+δ d i=1 E n n-1 f i (u) N i (du) -m i du 2+δ ≤ (
sup n≥1 1 σ 2 n n k=1 ∥X k ∥ 2 2+δ ≤ sup n≥1 nκ 2/(2+δ) 2 nσ 2 + o(n) < ∞,
and the right hand side is satised when (β -1)δ > 2 according to Theorem 3.1. Finally, Proposition 3.1 follows by applying the Theorem 5.1 to {X k } k≥1 , and whenever T ̸ ∈ N using Slutsky's lemma for S T = S ⌊T ⌋ + R(T ) with R(T ) = o P (σ T ).

  Proposition 4.1. Let N = (N 1 , • • • , N d ) be an associated multivariate point process and A, B ⊂ R two disjoint bounded subsets of R. Let f : N → R and g : N → R be two functions such that f (N |A ) and g(N |B ) are bounded, then

  Lemma 4.3. Let u = (u 1 , . . . , u d ) ∈ R d with strictly positive entries. Then,

  , for non-triangular sequences. Consider a sequence{X k } k≥1 of centered (E[X k ] = 0) variables, with partial sums S n = n k=1 X k . Let σ 2 n = Var S n . For 0 ≤ t ≤ 1, dene v n (t) = inf m : 1 ≤ m ≤ n, σ 2 m k ∥ 2 2+δ < ∞ and τ ≥1τ 2/δ α(τ ) < ∞.