Since 2013, the wars in Iraq and Syria have revived the persecution of Christians of the Middle East as an issue in Europe, in the context of Islamist targeting of religious minorities. The French case is especially revealing of the NGO, ideological and political movements that have emerged to advocate for the "Christians of the Middle East", or "Oriental Christians" (in French, "Chrétiens d'Orient", "Christians of the Orient"). The use of this term is far from reflecting the diversity of Christian standpoints with respect to these wars. Rather, both in France and in Syria, it contributes to processes in which identities become redefined.

The Virgin Mary of the Syrian village of Maaloula was destroyed when the village was taken over in 2013. A new statue was erected at the same place on June 13, 2015. On that day, families from the village, State officials and religious leaders gathered in the main square, adorned on that occasion with the national flag and a giant portrait of Bashar El Assad. © Smallcreative/Shutterstock.com "Oriental Christians" in France and in Syria :

Political Stakes and Redefined Identities by Claire Lefort, January 2019

The translation is by Thomas Hill

INTRODUCTION

The term "Chrétiens d'Orient" ("Christians of the Middle East", hereafter: "Oriental Christians") is widely used in France to designate Christians in the Middle East. It refers to a historical tradition according to which, since Saint Louis and Francis I, France is purported to be their protector. The term first appeared in the context of the massacres of Christians in Mount-Lebanon and Damascus in 1860. These massacres had a strong impact on French public opinion, prompting military intervention and a surge of humanitarianism. Organizations to support these "Oriental Christians" were therefore set up, starting with the Catholic charitable organization L'OEuvre d'Orient. The use of this term, however, strengthened by the use of capital letters, also implies projecting upon these Christians a fixed identity, reducing their identities to religious and doctrinal variables only. It encompasses Christian populations that are in fact defined by the variety of their ecclesiastical

CLAIRE LEFORT

Claire Lefort is a student in anthropology at the ENS (Ecole normale supérieure) and the EHESS (School of Advanced Studies in the Social Sciences) in Paris. Her works deals with the specificities of faith-based humanitarian actors. She also published a book about Sabean-Mandaeans (Les Sabéens-Mandéens: premiers baptistes, derniers gnostiques, éditions du Cygne, 2017) affiliations and socio-political conditions 1 , relating to them as if they were homogenous. Far from designating a homogenous group, or one concentrated in a given territory, the term seeks to paper over a shifting social and political reality-even while many of the members of these Christian communities have often formed diasporas the world over 2 . The term "Oriental Christians" therefore channels a historical and Orientalist inheritance. Since 2013, use of the term has witnessed a massive resurgence. Since then, events in Iraq and Syria have revived the issue of the persecution of Christians in the Middle East, in the context of Islamist targeting of religious minorities. In this context, new organizations have emerged to "rescue these persecuted Christians" 3 . A variety of political players have become involved: MPs gather in working groups on the issue 4 ; town halls have been emblazoned with banners proclaiming their solidarity with these Christian communities 5 ; politicians have taken their defence at the European Parliament 6 and the United Nations 7 . My aim here is to show that the proliferation of these initiatives tends towards stripping complexity from a plural reality-and towards accelerating processes that redefine the identities of those Christian communities still present in the Middle East.

The following analysis is based on a twofold ethnographic approach, conducted both among the purveyors of this solidarity movement in favour of "Oriental Christians" and among its beneficiaries 8 . During the six months of my fieldwork, from February to August 2016, I was a member of a French NGO working on these issues (henceforth "the NGO",) enabling me to study its operations and their dynamics. In the course of 2017, I then added observations conducted during events organized by this same NGO in France, as well as interviews with former volunteers who had travelled to Syria.

FRENCH YOUTH MOBILIZED FOR THE DEFENCE OF "ORIENTAL CHRISTIANS"

In 2013, after the village of Maaloula in Syria was captured from the Syrian regime, a new NGO for the defence of "Oriental Christians" was founded. At first, it was close to farright political circles, and focused on French Catholic and conservative audiences, targeting a youth audience especially. The core of its activities involved sending "volunteers", aged on average between 18 and 30, to Syria, Iraq, Lebanon, Jordan and Egypt. This movement was embedded within a broader rationale: acting to defend Christian values and heritage deemed "under threat, both abroad 8. Laetitia Atlani-Duault, Au bonheur des autres: anthropologie de l'aide humanitaire, Paris, Société d'ethnologie, 2005. and at home" 9 . This NGO was set within the context of the conservative "Manif Pour Tous" ("Demonstration For Everyone") 10 and "French Spring" movements, that draw French youth from primarily Catholic and conservative backgrounds 11 .

These moves enter into the pre-existing construction of the Catholic imaginary; they aim to respond to the need to "organize the Catholic struggle for their own survival in a secularizing society" 12 . As Father William Marie Merchat, manager in charge of youth pastoral activities for L'Oeuvre d'Orient in Alès, put it: "More and more, among young Catholics, we witness a feeling of being in the minority. This nurtures a feeling of insecurity, and with it a stronger assertion of identity. Certain parties and movements draw on these feelings, including within the Church: traditionalist movements that in some sense use the persecution of Oriental Christians to call for a new Crusade." 13 As such, these trips to assist "Oriental Christians" enter into a policy to develop a strong identity for such youth. A former volunteer in Syria emphasizes that: "Travelling to the field on behalf of this NGO enables building up a real network. For several weeks, we gather among others of the same age cohort, from the same social environment, who share the same 9. Extract from materials provided to new volunteers (February 2016). 10. In reference to the French "Mariage Pour Tous" ("Marriage for Everyone") law passed in 2013 that legalized gay marriage. 11. One indication of this is that the"Manif Pour Tous" Twitter account added to its handle the noun, the Arabic letter "N", that was used by Daesh (Islamic State, ISIS) to label Christian houses when it took over Mosul in the summer of 2014. Adding the noun to Twitter handles has since been taken up as a symbol of support for "Oriental Christians". 12. Denis Pelletier, Jean-Louis Schlegel (eds.), À la Gauche du Christ: les chrétiens de gauche en France de 1945 à nos jours, Paris, Éditions du Seuil, 2012. 13. Camille Lons, « Une compassion très politique pour les chrétiens d'Orient sur fond de crise identitaire des Français catholiques », Orient XXI, 26 April 2016, online, consulted 23/03/18.

opinions. […]

When we get back to France, one has to be careful to take some distance from that experience in order to avoid the "syndrome of the volunteer just back from the field," who has been moulded into the perfect little pro-Assad soldier. When you're 20, you get worked up fast, outraged by the terrorists' abuses-regardless of whether they're called ISIS, Al-Nusra, or the Free Syrian Army." 14 No substantive training is provided to these youth before they leave for the field, making them especially impressionable once they arrive there. Once they are confronted with complex social and political realities that they struggle to grasp, their only reference-point is their "team leader" 15 , who is often close to traditionalist or far-right circles. In Syria, volunteers are therefore confronted with pro-Assad and pro-Russia discourse, leavened with conspiracy theories close to those promoted by the notorious 9/11 "truther" Thierry Meyssan 16 Beyond the humanitarian question, this movement advocating for "Oriental Christians" also serves conservative political goals-and a kind of moralizing proselytism. On the one hand, the stated goal is to provide "material and psychological help to Oriental Christians." On the other, it is to "sensitize French public opinion to the dangers that confront Oriental Christians today, in order to rebuild the weakened link between France and the Oriental Churches" 18 . This discourse reactivates the tradition according to which France is the protector of "Oriental Christians". But, under cover of spreading the point of view of "Oriental Christians" and offering "a different perspective" on unfolding events in the Middle East, the goal is to defend Bashar Al-Assad's authoritarian response to the 2011 uprisings. Their description of how the regime's recapture of East Aleppo, that had been in opposition hands since the summer of 2012, goes as follows: "12 December 2016 will go down in Syria's history as the first day of the liberation of Aleppo. After five years of war, five years of fighting, five years in which death and destruction were omnipresent, the armed groups that occupied the Eastern part of 18. From the NGO's official presentation on its website, consulted 20/02/2017. the city finally agreed to hand in their weapons. The streets are filling up, the locals are celebrating what they had stopped daring to hope for, and civilians of East and West mix together, one people once again, the Syrian people, that kept their dignity despite the horrors visited upon them, and that mounted incredible resistance against barbarism." 19This communications strategy thus took part in rehabilitating Bashar Al-Assad's regime. In this narrative, Assad remains a dictator, but his abuses are often glossed over or justified. Portrayed as the "Christians' protector", he is above all a "lesser evil" in the current situation, as much as he is the only figure deemed capable of "protecting Christ's disciples against the barbarism of Islam" 20 . These narratives were echoed in certain French political circles. In January 2015, for instance, the NGO organized a meeting between the Cross-Parliamentary France-Syria Friendship Group (most of whose members are from the conservative "The Republicans" party (LR) and His Eminence Jean-Abdo Arbach, the Archbishop of Homs, Hama and Yabroud, in order to "gather information on the concrete situation of the Syrian population, and especially of the Christian minorities." 21

Beyond the humanitarian question, this movement advocating for "Oriental Christians" also serves conservative political goals-and a kind of moralizing proselytism.

tians at the National Assembly, directed by Valérie Pécresse (LR) and made up of many right-wing MPs from "The Republicans", "Republic, Arise" (Nicolas Dupont-Aignan) and others affiliated with the "National Rally" (RN, ex-National Front) (Gilbert Collard). This lobbying of French parliamentarians also takes the shape of meetings organized between Bashar Al-Assad and certain MPs, such as MPs including Thierry Mariani (then-LR) and Nicolas Dhuicq (LR) in January 2017. At the end of 2014, after Robert Ménard (RN) was elected as Mayor of Béziers, the NGO's members enabled Béziers to be twinned with Maaloula, the Syrian Christian village that was turned into a symbol of the persecution of Oriental Christianity 22 . Finally, the political movement for "Oriental Christians" was also expressed through cultural events. On 23 June 2016, an "Evening at the Invalides for Oriental Christians" drew more than 2,000 participants, among them such figures as Jean d'Ormesson, Michael Lonsdale, Rachida Dati, Philippe de Villiers and François Fillon. At the time, Fillon was a candidate for the Presidency. The stance he took on this issue, expressing his willingness to collaborate with the Assad regime to enable the protection of "Oriental Christians", illustrated the impact of such advocacy in spreading a certain kind of narrative concerning Syria.

22. The majority-Christian village of Maaloula is north-east of Damascus. Its inhabitants still speak Aramaic ("the language of Christ"). Violent clashes with jihadis took place there between September 2013 and April 2014. The NGO's website (consulted 22/11/18) stated that: "In September [2013], the village of Maaloula was attacked by terrorists from the Nusra Front. Its churches have been burned, its people massacred or forced into exile, absolutely destitute".

FROM CHRISTIANS IN SYRIA TO "ORIENTAL CHRISTIANS": THE SELECTIVE REPRESENTATION OF IDENTITY

In the humanitarian field, this advocacy movement for "Oriental Christians" works according to a selective principle of solidarity, prioritizing members of the same faithbased community. The movement has little involvement in humanitarian coordination networks; it conducts its activities through direct communication with representatives of local churches that support the regime 23 , and only takes into account lists of beneficiaries provided by these churches. Save for a few projects directed at groups of Sunnis, Yezidis and Sabean-Mandaeans, this approach rests on targeting beneficiaries according to religion, at the expense of other kinds of belonging or other social links. It therefore makes the choice of a sect-based analysis of the conflict, widening divisions between communities and setting up Christians as the privileged of the humanitarian field.

23. Christians make up around 6% of Syria's population. Their religious representatives and the regime are strongly linked together: since the 1970s, the election of these representatives by a synod is only formalized after ratification by the regime. See Anna Poujeau, Églises, Monachisme et Sainteté. Construction de la communauté chrétienne en Syrie, PhD thesis in Social Anthropology and Ethnology, Université de Nanterre -Paris X, 2008.

The very use of "Oriental Christians" as a category elides the varied stances of Christian Syrians.

Beyond this, the very use of "Oriental Christians" as a category elides the varied stances of Christian Syrians. Indeed, the closeness of many religious leaders to the regime is far from being universally shared. It has prompted a divide between these leaders and some faithful-a divide that only grew wider in the wake of the 2011 uprisings. At first, the uprisings portrayed themselves as being a cross-community movement. As such, while leaders of the Orthodox and Catholic churches officially endorsed Bashar Al-Assad's regime, other leading Christian figures, such as the filmmaker Bassel Shehadeh, were becoming symbols of the revolution.

Yet the fact that some leading figures in the Christian community opposed the regime is rarely mentioned in Western media. Many in politics and the media tend to favour a sectarian analysis of the conflict in Syria, echoing the regime's strategy of representing Syrian minorities as the victims of an Islamist threat, in order to divide the opposition. The solidarity movement for "Oriental Christians" partakes of the same approach. Through its close collaboration with Christian prelates who are often close to the regime, it reports the testimonies of only one segment of the Syrian Christian community. It therefore contributes to the portrayal of "Oriental Christians"' support for Bashar Al-Assad's regime as the choice of a "lesser evil" 24 faced with the jihadi menace-and as justified by a necessary solidarity between "minorities" against the Sunni majority's political demands.

CONCLUSION

The advocacy movement for "Oriental Christians" illustrates how activism on behalf of this specific category is constructed on an ideological basis. "Oriental Christians" have thereby become a specific category for 24. Extract from an interview, 08/02/2017 the activity of these NGOs, and for the political goals of conservatives and the Syrian regime, which grants these players distinctive status. It also echoes the tensing-up of certain Catholic conservative circles on the question of identity, worried about declining religious observance, the rise of Islam, and the negation of the purportedly Christian values that France-as the Church's "eldest daughter"-is here supposed to embody. But the use of the term "Oriental Christians" has further consequences. In giving voice to only one segment of the Syrian Christian communities, it performatively takes part in the interpretation of the war as being religious and civilizational.

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  Maaloula, the convent of Mar Taqla (Holy Thecla). Located northeast of Damascus. This predominantly Christian village, where Aramaic («the language of Christ»), is still spoken was the scene of violent fighting against jihadists between September 2013 and April 2014. © Pavel Sepi/Shutterstock.com

  . A former volunteer emphasizes that "The NGO's goal is very clear: to spread and to normalize the Assad regime's narrative, by demonstrating its logic and its coherence. It goes like this: granted, Assad has perpetrated many abuses, starting with what goes on in the regime's prisons. But given that the United States do the same thing… […] This side of the NGO's activities is quite blatant: many new volunteers first show up quite suspicious towards the regime, having absorbed Western media narratives. But after some time with us, they leave again with far more conciliatory ideas [with respect to the Assad regime]" 17 . 14. From an interview, 12/02/2017. 15. The employee tasked with managing volunteers and the NGO's activities in the field. 16. See e.g. this article from 11/09/18. 17. From an interview, 08/02/2017

	THE FRENCH POLITICAL
	ADVOCACY MOVEMENT
	FOR "ORIENTAL
	CHRISTIANS"

  Members of the NGO also took part in several meetings of the Study Group on Oriental Chris-19. From the NGO's official presentation on its website, consulted 26/02/2017 20. From an interview, 08/02/2017 21. See the account published on 21/01/2015 and available online.


CEPED Claire Lefort-Rieu

Increasing or diverting control? Refugee self-reliance, political stakes, and international aid to forcibly displaced people in Cameroon

In recent years, assistance to forcibly displaced people underwent what has been described as a neoliberal shift: instead of providing refugees and internally displaced people (IDP) with protection and assistance, it now focuses on giving them skills and equipment so they can be "self-reliant" and integrate or contribute to the local socioeconomic context. This reactivates topics mobilised in the 1970s, that considered refugees as a workforce to stimulate growth and development efforts in their host areas; it also echoes the United Nations Refugee Agency (UNHCR)'s efforts to tackle the aid 'dependency syndrome'. Valuing the economic potential of forced displaced people is also a way of reducing costs, assistance being replaced by the promotion of access to the labour market. In Cameroon, 'self-reliance' projects for refugees and IDPs are overrepresented. Echoing its global strategy for 2019-2023entitled 'Refugee Livelihoods and Economic Inclusion' -, UNHCR Cameroon signed in 2021 a partnership agreement with the Ministry of Employment and Vocational Training and with the International Labour Organisation to facilitate refugees' access to the labour market. However, such a humanitarian assistance is also apprehended as both a political stake and a power resource by state actors and refugees.

In Cameroon, employment issues are highly politicised. A strong politico-economic collusion enables the authoritarian government to maintain its hold over the business community, so that it does not become the core of a competing political group. Creating jobs and defending the existing ones are also recurrent political promises of Paul Biya's regime, which has held power for four decades. International assistance does not escape these strong political stakes. By providing refugees and IDPs with skills and equipment, self-reliance projects directly echo the priorities of the national development strategy. Furthermore, given that the material and financial means of state services are often insufficient, the resources provided by humanitarian interventions may be considered and used as a "manna": state services turn them into an additional means to enable them to achieve their development goals. The Cameroonian state apparatus thus implements recuperation strategies aimed at reintegrating into the public sphere services that were first developed by humanitarian Routed Magazine, Oxford Migration Conference 2022 Special Issue, 04/06/2022: https://www.routedmagazine.com/omc22-politics-aiddisplaced-cameroon aid. For instance, in the Eastern region, Central African refugees involved in socioeconomic integration projects are required by state services to comply with the 'procedures of the existence declaration'. Such administrative procedures (compilation of files detailing the members and activities of their micro-enterprise, obligation to pay property tax and tax on the lease, etc.), for which refugees can count on the technical and financial support of international aid actors, aim to enrol forced displaced people in the formal economy, turning their self-employment situation into small and mediumsized enterprises. State authority is exercised through a sovereign prerogative (tax) which integrates humanitarian projects into the national development strategy and in the local economic fabric. This international assistance, initially designed for refugees and their host population, serves not only the national development strategy and objectives but is also a way to strengthen the elements on which Paul Biya's regime relies to maintain itself in power.

In addition to state services, refugees may also reinvest international assistance to serve their own objectives. In a context of migration securitisation, one way to skirt the mobility restrictions they face is to obtain a document issued by the UNHCR, which serves as a pass from the police or at the checkpoints. By providing refugees with this documentso that they can move between their homes and places of training, internship, or employment during the humanitarian project -, international aid gives them opportunities to regain mobility. This travel facilitation applies to the direct 'beneficiaries' of the project but also to their relatives and community leaders, who are invited to leave the refugee camp to visit the vocational training centres in the regional capital city. In addition to the issuance of the UNHCR pass, all the costs (travel, accommodation, food) related to their stay are covered by the international aid actors: these facilitated trips are also used by the refugees to skirt mobility restrictions and to deal with various matters (family visits, negotiations with suppliers, procurement, etc.). Consequently, the humanitarian manna can also be understood in a spatial sense: the state of exception that defines refugee assistance exempts them from mobility restrictions imposed by the Cameroonian state.

The humanitarian manna can also be used by refugees to divert the norms that govern the field of international aid. In a neoliberal logic, self-reliance projects focus especially on promoting the "entrepreneur" refugee model. But instead of applying the rules of business management which they have been taught, refugees may, for instance, use their micro-enterprise funds to finance another activity, or close their shop for several weeks to attend family obligations in another part of the country. To them, socioeconomic integration projects are one resource among many others, in a landscape characterised by plural and varied strategies, embedded in networks of social and family obligations. The standardised and quantitative tools of international aid actors, designed to monitor the evolution of refugees' start-up businesses according to a

Routed Magazine, Oxford Migration Conference 2022 Special Issue, 04/06/2022: https://www.routedmagazine.com/omc22-politics-aiddisplaced-cameroon neoliberal scheme, are challenged by the diversity of practices and activities they implement. This example illustrates the plurality of daily practices implemented by refugees to live, build social relations, develop networksi.e. a multiplicity of uses and tactics, of ways of dealing with the layers of power and agency that affect forcibly displaced people. Faced with strong stigmatisation in spaces and social conditions that state and humanitarian institutions define as a state of exception, individuals implement ordinary acts of resistance enabling them to transgress everyday hegemonies. These practices combine expertise developed over the years (especially during their experience of living in exile) and limited power-to-do (granted by authorities and aid actors in the host country). Despite the restrictions they face, refugees are able to divert or skirt the rules, to bend themnotably, but not exclusively, through the opportunities offered by humanitarian interventions. Their ability to play with the policies they are subject to also enables them to bring to the fore forms of (political) mobilisations.

Bionote: Graduated with a Master of social anthropology from Ecole Normale Supérieure and Ecole des Hautes Etudes en Sciences Sociales, Claire Lefort-Rieu is a PhD candidate in critical anthropology of international aid (Ceped, Institut de Recherche pour le Développement / Université Paris Cité). She studies forced migrations governance in Cameroon, through extensive fieldwork and a methodology of double ethnography led with both aid actors and their so-called beneficiaries. Email: claire.lefort@etu-u.paris.fr


Martin M Öhle

ON BERNOULLI TRIALS WITH UNEQUAL HARMONIC SUCCESS PROBABILITIES

Keywords: . 2020 Mathematics Subject Classification. Primary: 60J10; Secondary: 60C05 Bernoulli variables, Ewens-Pitman sampling formula, Markov chains, Rényi's records, Sibuya distribution, Stirling numbers, Yule-Simon distribution

A Bernoulli scheme with unequal harmonic success probabilities is investigated, together with some of its natural extensions. The study includes the number of successes over some time window, the times to (between) successive successes and the time to the first success. Large sample asymptotics, statistical parameter estimation, and relations to Sibuya distributions and Yule-Simon distributions are discussed. This toy model is relevant in several applications including reliability, species sampling problems, record values breaking and random walks with disasters.

Introduction

Introduce two weights w 1 > 0 and w 2 ≥ 0, put w := w 1 + w 2 > 0, and let I 1 , I 2 , . . . be independent Bernoulli random variables with 'harmonic' success probabilities [START_REF] Arratia | Poisson process approximations for the Ewens sampling formula[END_REF] P(I m = 1) := w 1 w + m -1 , m ∈ N := {1, 2, . . .}, decreasing inversely proportional to the number m of the trial. We note the following property of Bernoulli trials with such success probabilities: the first success time K + 1 := inf{m ∈ N : I m = 1} is either a small units number or a very large one due to power-law tails of this random variable, see [START_REF] Rényi | Théorie des éléments saillants d'une suite d'observations[END_REF] below. In words indeed, if the I m 's fail to take the value 1 in the first steps, this tendency will be enhanced in the forthcoming steps resulting, for such models, in large (heavy-tailed) values of K + 1 . So K + 1 either will take small values close to 1 (the mode of K + 1 is at 1 with probability mass w 1 /w decreasing with w 2 /w 1 if w 2 > 0) or very large values (responsible of its heavy-tailedness with tail index w 1 ): small values of w 2 /w 1 favors early first success time while small values of w 1 favors late first success. So, the larger the number of steps for which no success was observed, the smaller the probability to see a success in the next step even though this probability is relatively large (harmonic decay in our case). This may be seen from the following argument:

Let J m := 1 -I m , m ∈ N, and let M n = n m=1 J m . The event M n = 1 is realized when no success was observed till time n. M n is a multiplicative random walk

M n+1 = M n J n+1 , M 0 = 1,
for which the probability of a success at step n + 1 given no success till n is P(M n+1 = 0 | M n = 1) = P(I n+1 = 1). If w 2 = 0, then the probability P(M n = 1) = n m=1 P(I m = 0) of no success by time n ∈ N is obviously equal to 0, since I 1 = 1 in this case. If w 2 > 0 then this probability is equal to

n m=1 P(I m = 0) = n m=1 w 2 + m -1 w + m -1 = Γ(w)Γ(w 2 + n) Γ(w 2 )Γ(w + n) ∼ Γ(w) Γ(w 2 ) 1 n w1 , n → ∞,
since Γ(c + n) ∼ n c Γ(n) as n → ∞ for any c > 0. Thus, the probability of no success by time n is small for large n, since w 1 > 0.

Examples of such enhancement mechanisms are

• I m = 1 if some paper is cited the day m after its publication. Oversight.

• I m = 1 if some new species is discovered the day m after a systematic daily sampling campaign. Rareness. • I m = 1 if some new word is used (or created) as the m-th word of some ongoing book. Scarcity. • I m = 1 if some individual renews its support to some political party the day (month) m after its creation. Weariness. • Time unit increases by 1 when some athlete attempts to improve some record previously established. I m = 1 if he/she succeeds at m-th trial: higher records become more and more difficult to break.

In several situations a success is actually a failure. Examples are

• I m = 1 if some device breaks down the day m after it was put into service.

Resilience. • I m = 1 if some population collapses the day m after it came to birth.

Resilience. • I m = 1 if some patient contracts some illness the day m after birth date.

Immunity. • I m = 1 if some driver has an accident the day m after obtaining its driving licence. Experience.

The number n of observations can be finite (possibly large though, depending on the time scale) or infinite. For instance, a typical driver only has finitely many driving days in his life (possibly randomly finite), but the attempts to break a record are potentially infinitely many.

For 'harmonic' Bernoulli sequences of the form (1) we study the number S n := n m=1 I m of successes among the first n ∈ N 0 trials, the time K + l := inf{m ∈ N : S m = l} of the l-th success, l ∈ N 0 , and the times L + l := K + l -K + l-1 elapsed between successive successes, l ∈ N, and analyse the associated Markov chains. It turns out that Sibuya distributions play an important role in this context. The twoparameter (w 1 , w 2 )-Sibuya distribution arises as the distribution of the waiting time till the first success. The shifted (w 1 , w 2 )-Sibuya distribution has many appealing properties, among them discrete self-decomposability and heavy-tailedness, [START_REF] Kozubowski | A generalized Sibuya distribution[END_REF]. It includes the 'bare' Sibuya distribution (w 1 + w 2 = 1, see [START_REF] Sibuya | Generalized hypergeometric, digamma and trigamma distributions[END_REF]) and the Yule-Simon distribution (w 2 = 1, see [START_REF] Yule | A mathematical theory of evolution based on the conclusions of Dr[END_REF]). The case w 2 = 0 is degenerate as far as the waiting time for the first success is concerned, but it appears to make sense from the point of view of the number of successes in the Ewens species sampling problem [START_REF] Ewens | The sampling theory of selectively neutral alleles[END_REF]. The case (w 1 , w 2 ) = (1, 0) also appears in the study of the number of record values stemming from an arbitrary independent and identically distributed (iid) sequence of observations, see [START_REF] Neuts | Waitingtimes between record observations[END_REF][START_REF] Rényi | On outstanding values of a sequence of observations[END_REF][START_REF] Rényi | Théorie des éléments saillants d'une suite d'observations[END_REF].

Number of successes

In this section we are mainly interested in the number S n := n m=1 I m of successes among the first n ∈ N 0 trials. Note that 0 ≤ S n ≤ n for n ∈ N 0 . In particular, S 0 = 0.

In the following, s(n, k), n, k ∈ N 0 := {0, 1, . . .}, denote the Stirling numbers of the first kind. Recall that the unsigned Stirling numbers of the first kind

|s(n, k)| are characterized via [z] n = k≥0 |s n,k |z k , z ∈ R, n ∈ N 0 , where [z] 0 := 1 and [z] n := z(z +1) • • • (z +n-1), n ∈ N. These numbers satisfy the recursion |s n+1,k | = n|s n,k | + |s n,k-1 | with |s n,k | = 0 for k > n, |s n,n | = 1 and |s n,0 | = δ n,0 (Kronecker symbol). 2.1. The Markov chain (S n , n ∈ N 0 ).
Clearly, (S n , n ∈ N 0 ) is a time-inhomogeneous Markov chain with state-space N 0 and transition probabilities

(2) P(S n+1 = k + 1 | S n = k) = 1 -P(S n+1 = k | S n = k) = w 1 w + n , n, k ∈ N 0 .
Note that the probability (2) that the chain moves from state k at time n to state k + 1 at time n + 1 does not depend on the current state k. The increments S n -S n-1 = I n , n ∈ N, are independent but not identically distributed. The chain (S n , n ∈ N 0 ) also coincides with the chain studied in the restaurant process with a cocktail bar [12, Section 6.1] with parameters (α, θ 1 , θ 2 ) := (0, w 1 , w), where S n counts the number of occupied tables after n customers have entered the restaurant. The probability-generating function (pgf) z → f n (z) := E(z Sn ) of S n is given by

f n (z) = n-1 m=0 w 1 z + w 2 + m w + m = [w 1 z + w 2 ] n [w] n = [w 1 (z -1) + w] n [w] n , z ∈ R,
Clearly, f n is a polynomial of degree n of the form

f n (z) = 1 [w] n n l=0 |s n,l |(w 1 z + w 2 ) l = 1 [w] n n k=0 z k w k 1 n l=k l k |s n,l |w l-k 2 .
Denoting by

[z k ]f n (z) the coefficient in front of z k of f n yields (3) P(S n = k) = [z k ]f n (z) = w k 1 [w] n n l=k l k |s n,l |w l-k 2 , k ∈ {0, . . . , n}. With (n) 0 := 1 and (n) l := n(n -1) • • • (n -l + 1) for l ∈ N, S n has the l-th descending factorial moment (4) E((S n ) l ) = l![(z -1) l ]f n (z) = w l 1 [w] n n k=l (k) l |s n,k |w k-l , l ∈ N 0 .
Note that E((S n ) l ) = 0 for l > n. The distribution π n (k) := P(S n = k) of S n can be recursively computed via π 0 (k) = δ k,0 and

(5)

π n+1 (k) = w 1 w + n π n (k -1) + w 2 + n w + n π n (k), n, k ∈ N 0 .
Note that π n (k) = 0 for k / ∈ {0, . . . , n}. Comparing (5) with the recursion [7, Theorem 1] s r (n + 1, k) = s r (n, k -1) + (n + r)s r (n, k) for the generalized Stirling numbers s r (n, k) := S(n, k; -1, 0, r), n, k ∈ N 0 , r ∈ R, in the notation of [START_REF] Hsu | A unified approach to generalized Stirling numbers[END_REF] having vertical generating functions [START_REF] Hsu | A unified approach to generalized Stirling numbers[END_REF]Theorem 2]

k! n≥0 s r (n, k)t n /n! = (1 - t) -r (-log(1 -t)) k , r ∈ R, k ∈ N 0 , |t| < 1,
it follows that (3) can be alternatively expressed in terms of these generalized Stirling numbers as [START_REF] Goncalves | Keeping random walks safe from extinction and overpopulation in the presence of life-taking disasters[END_REF] π

n (k) = w k 1 [w] n s w2 (n, k), k ∈ {0, . . . , n},
in agreement with [12, Eq. ( 14)] for (α, θ 1 , θ 2 ) := (0, w 1 , w). Similarly, (4) can be written as

(7) E((S n ) l ) = w l 1 [w] n l!s w (n, l), l ∈ N 0 .
Introducing the superdiagonal stochastic transition matrices

Π n :=      w2+n w+n w1 w+n 0 • • • 0 w2+n w+n w1 w+n 0 0 0 w2+n w+n • • • . . . . . . 0 . . .      , n ∈ N 0 , the distributions π n := (π n (k), k ∈ N 0 ) of S n , n ∈ N 0 , satisfy the recursion π n+1 = π n Π n , n ∈ N 0 . Thus, π n = π 0 n-1 m=0 Π m , n ∈ N 0 , with π 0 = (1, 0, 0, . . .). Remark 1. The pgf f n of S n has only real zeros -(w 2 + m)/w 1 , m ∈ {0, . . . , n -1}.
By [15, Proposition 1], (π n (0), . . . , π n (n)) is a Pólya frequency sequence. Thus, the infinite matrix M := (π n (k -l)) k,l∈N0 (where π n (k) = 0 for k / ∈ {0, . . . , n}) is totally positive of any arbitrary order, i.e., all principal minors of any arbitrary order of M have nonnegative determinant.

Special cases.

-w = 1:

f n (z) = E(z Sn ) = [w1(z-1)+1]n n! = 1 n! n k=0 |s n+1,k+1 |w k 1 (z -1) k showing that S n has k-th descending factorial moment E((S n ) k ) = k![(z -1) k ]f n (z) = k! n! |s n+1,k+1 |w k 1 , k ∈ N 0 .
Note that in that case, necessarily w 1 ∈ (0, 1).

-

w 2 = 1: f n (z) = E(z Sn ) = [w1z+1]n [w]n = [w1z]n+1 z[w1]n+1 = 1 [w1]n+1 n+1 k=0 |s n+1,k |w k 1 z k-1 showing that (|s n,0 | = δ n,0 ) π n (k) = P(S n = k) = |s n+1,k+1 |w k+1 1 [w 1 ] n+1 , k ∈ {0, . . . , n}.
-

w 2 = 0: f n (z) = E(z Sn ) = [w1z]n [w]n = 1 [w1]n+1 n k=0 |s n,k |w k 1 z k showing that π n (k) = P(S n = k) = |s n,k |w k 1 [w 1 ] n , k ∈ {0, . . . , n}.
If in addition w 1 = 1, then S n is the number of record values of an arbitrary iid sequence of observations appearing before n; [START_REF] Neuts | Waitingtimes between record observations[END_REF][START_REF] Rényi | On outstanding values of a sequence of observations[END_REF][START_REF] Rényi | Théorie des éléments saillants d'une suite d'observations[END_REF]. In this case the law

π n (k) = |s n,k |/n!, k ∈ {0, .
. . , n}, of S n coincides with the distribution of the number of cycles of a permutation of size n chosen uniformly at random.

Poisson approximation.

Clearly, [START_REF] Pitman | Probabilistic bounds on the coefficients of polynomials with only real zeros[END_REF] and [START_REF] Yamato | Poisson approximations for sum of Bernoulli random variables and its application to Ewens sampling formula[END_REF]), because

µ n := E(S n ) = n m=1 P(I m = 1) = w 1 n-1 m=0 1/(w + m) = w 1 log n + O(1) and
σ 2 n := Var(S n ) = n m=1 P(I m = 0)P(I m = 1) = w 1 n-1 m=0 w 2 + m (w + m) 2 ∼ w 1 log n as n → ∞. The law of S n is in total variation distance close to the law of N n d ∼ Poi(µ n ), (see
µ n -σ 2 n = w 2 1 n-1 m=0 1/(w + m) 2 µ n (see [2, Theorems 1 and 2]) with LeCam Poisson approximation of the total variation distance d T V (S n , N n ) := 1 2 k≥0 |π n (k) -µ k n e -µn /k!|
given by (see [START_REF] Sevast'yanov | Poisson limit law for a scheme of sums of dependent random variables[END_REF])

(8) 1 32 min(1, µ -1 n )(µ n -σ 2 n ) ≤ d T V (S n , N n ) ≤ (1 -e -µn ) µ n -σ 2 n µ n .
Therefore (and also by the Lindeberg-Feller central limit theorem), (

S n -µ n )/σ n → N (0, 1) in distribution as n → ∞, consistently with the fact that (N n -µ n )/σ n → N (0, 1) in distribution as n → ∞. Since n≥2 P(I n = 0)P(I n = 1)/(log n) 2 < ∞,
it follows from well-known law of large numbers results for sums of independent, but not identically distributed random variables, that (log n) -1 n m=1 (I m -E(I m )) → 0 almost surely or, equivalently, that S n /µ n → 1 almost surely as n → ∞.

Maximum likelihood estimation.

With i := (i 1 , . . . , i n ) ∈ {0, 1} n an observed sequence of I := (I 1 , . . . , I n ) and k := n m=1 i m , (9) P(I = i) = n m=1 w im 1 (w 2 + m -1) 1-im w + m -1 = w k 1 n m=1 (w 2 + m -1) 1-im [w] n .
This probability is not symmetric in i 1 , . . . , i n since the random variables I 1 , . . . , I n are not exchangeable. Using

∂ w log[w] n = n-1 m=0 1/(w + m) = Ψ(w + n) -Ψ(w), where Ψ denotes the digamma function obeying Ψ(z) = log z -1/(2z) + O(1/z 2 ) as z → ∞, the two equations ∂ wj log P(I = i) = 0, j ∈ {1, 2}, yield (10) k w 1 = n-1 m=0 1 w + m = Ψ( w + n) -Ψ( w)
and ( 11)

n-1 m=0 1 -i m+1 w 2 + m = Ψ( w + n) -Ψ( w),
where ( w 1 , w 2 ) is the maximum likelihood estimator (MLE) of (w 1 , w 2 ) based on the observed sequence i = (i 1 , . . . , i n ) and w := w 1 + w 2 . It is easily checked that the 2 × 2 Hesse matrix J of the map (w 1 , w 2 ) → P(I = i) is negative semidefinite at ( w 1 , w 2 ). As n → ∞, asymptotic normality of ( w 1 , w 2 ) is expected at rate n -1/2 , the limiting normal law having mean (w 1 , w 2 ) and covariance matrix either the inverse of the expected Fisher information matrix or the inverse J -1 of the observed information matrix J evaluated at ( w 1 , w 2 ).

If the model has two independent parameters (w 1 , w 2 ) that have to be estimated, the first equation [START_REF] Karlin | A First Course in Stochastic Processes[END_REF] gives w 1 = k/(Ψ( w + n) -Ψ( w)) as a function of w (and k) and so w 2 = w -w 1 as a function of w. Plugging this expression of w 2 into the second equation ( 11) yields an equation in the single variable w that can be solved from the data i = (i 1 , . . . , i n ). An expression of both w 1 and w 2 then follows.

When w = w 1 + w 2 = 1, there is only one parameter to estimate, say w 1 , and ( 10) and ( 11) yield

k w 1 = n m=1 1 -i m m -w 1 (entailing w 1 ∈ (0, 1)).
When w 2 = 1 or 0, only this first equation ( 10) is needed and the searched w

1 solves k w 1 = n-1 m=0 1 w 1 + m + 1 or k w 1 = n-1 m=0 1 w 1 + m .
Note that Ψ( w + n) -Ψ( w) ∼ log n as n → ∞. Thus, by [START_REF] Karlin | A First Course in Stochastic Processes[END_REF], w 1 ∼ k/ log n as n → ∞ and, by [START_REF] Kozubowski | A generalized Sibuya distribution[END_REF], a large n approximation for w 2 is the solution w of the equation

n-1 m=0 (1 -i m+1 )/(w + m) = log n. Remark 2.
(Random number of observations) It can be natural to assume that the number of observations is finite but random (and independent of I 1 , I 2 , . . .). In this case one has to replace n by a random variable N taking values in N, and E(z S N ) = n≥1 E(z Sn )P(N = n) yields the law of the number of successes over the time window N with supposedly (or not) known mean E(N ). For example, N could be geometrically distributed P(N = n) = p(1 -p) n-1 , n ∈ N, with parameter p ∈ (0, 1). For instance, it can be a good modeling issue to infer that there are only finitely many days in a species sampling campaign, geometrically distributed (without any further information but its mean number). The random variable S N then counts the total number of sampled species over the observation window N .

Times to successive successes

For l ∈ N let K + l := inf{n ∈ N : S n = l} be the time elapsed till the l-th success. Furthermore, put K + 0 := 0. The process (K + l , l ∈ N 0 ) is called the first-passage time process of the random walk (S n , n ∈ N 0 ). Such processes have been studied extensively in the literature. We refer the reader to [START_REF] Denisov | First-passage times for random walks with nonidentically distributed increments[END_REF] and the references therein. We have P(K + l > n) = P(S n < l) as the laws of (K + l , S n ) are mutual inverse in the sense of inverse sampling ([9, p. 192-194]. It follows from this, [START_REF] Huillet | On a Markov chain model for population growth subject to rare catastrophic events[END_REF], and the works [START_REF] Rényi | On outstanding values of a sequence of observations[END_REF][START_REF] Rényi | Théorie des éléments saillants d'une suite d'observations[END_REF] (see also [START_REF] Möhle | A restaurant process with cocktail bar and relations to the threeparameter Mittag-Leffler distribution[END_REF]Proposition 1]), that [START_REF] Möhle | A restaurant process with cocktail bar and relations to the threeparameter Mittag-Leffler distribution[END_REF] w

1 log K + l l a.s. → 1 as l → ∞ and w 1 log K + l -l √ l d → N (0, 1) as l → ∞,
and the law of iterated logarithm for the log K + l 's. And similarly for the time elapsed between contiguous successes, while replacing

K + l by L + l := K + l -K + l-1
in [START_REF] Möhle | A restaurant process with cocktail bar and relations to the threeparameter Mittag-Leffler distribution[END_REF] with the notable exception that the first almost sure convergence is now a convergence in probability [START_REF] Neuts | Waitingtimes between record observations[END_REF].

3.1. The laws of the times to successive successes and times elapsed between contiguous successes.

The law of K + l is easily obtained as follows. Clearly, {K + l = n} = {S n-1 = l -1, I n = 1}. The independence of S n-1 and I n thus yields ( 13)

P(K + l = n) = P(I n = 1)P(S n-1 = l -1) = w 1 w + n -1 π n-1 (l -1).
Using (3) the law of K + l is therefore given by ( 14)

P(K + l = n) = w l 1 [w] n n-1 k=l-1 k l -1 |s n-1,k |w k-l+1 2 , n ≥ l.
We also conclude that L + l+1 := K + l+1 -K + l = i is realized if and only, for some n ≥ l: S n-1 = l -1 and I n is a success and S n+i-1 = l and I n+i is a success. Hence, with i ≥ 1, ( 15)

P(L + l = i) = n≥l w 1 w + n -1 w 1 w + n + i -1 π n-1 (l -1)π n+i-1 (l),
where π n (l) is given by (3). When (w 1 , w 2 ) = (1, 0), it follows from (3) in [START_REF] Neuts | Waitingtimes between record observations[END_REF], developing problem 32 on p. 268 in [START_REF] Karlin | A First Course in Stochastic Processes[END_REF], that

P(L + l > i) = i k=0 (-1) k i k (1 + k) -l .
The law of K + l can be obtained on a computer by launching a three-term recursion. Indeed, from (13), the recursion (5) on π n (l) yields a recursion for

P(K + l = n) with P(K + l = n) = 0 if n < l. With n ≥ l, this is (16)
P(K + l+1 = n + 1) = w 1 w + n P(K + l = n) + w 2 + n -1 w + n P(K + l+1 = n).
Introducing the lower-triangular matrix P = (P n,l ), where P n,l := P(K + l = n), l ≤ n, we see that P n+1,l+1 can be obtained from its north-west and north neighbors. With the knowledge of the first column of P and its diagonal, this recursion becomes effective, starting from P 3,2 obtained from P 2,2 and P 2,1 . For n = l, Eq. ( 16) reduces to P(K + l+1 = l + 1) = (w 1 /(w + l))P(K + l = l), which yields the diagonal terms

P(K + l = l) = l-1 m=0 w 1 /(w + m).
The entries P(K + 1 = n) of the first column of P are given in (20) below.

3.2. Markov structure of (K + l , l ∈ N). The homogeneous Markov structure of the sequence (K + l , l ∈ N) follows from

P(K + l+1 -m > n | K + l = m) = n-1 k=0 w 2 + m + k w + m + k = [w 2 + m] n [w + m] n = m+n-1 k=m w 2 + k w + k ,
where m ≥ l and n > 0. The random variable L + l+1 := K + l+1 -K + l ≥ 1 is the 'time-lag' elapsed between the l-th and the (l + 1)-th success. Its law depends on K + l . It is thus expected that, for each l ≥ m, the larger m is, the larger is

P(K + l+1 -m > n | K + l = m), because P(K + l+1 -(m + 1) > n | K + l = m + 1) P(K + l+1 -m > n | K + l = m) = w 2 + m + n w + m + n w + m w 2 + m > 1.
The chain (K + l , l ∈ N) therefore obeys a sort of reinforcement property. For general information on random processes with reinforcement we refer the reader to [START_REF] Pemantle | A survey of random processes with reinforcement[END_REF]. From Stirling's formula, Γ(z + b)/Γ(z + a) ∼ z b-a as z → ∞. For fixed m n, for each m ≥ l, we indeed get

P(K + l+1 -m > n | K + l = m) = [w 2 + m] n [w + m] n = Γ(w + m) Γ(w 2 + m) (n -w1 + O(n -(w1+1) )),
translating that, given K + l = m, the tails of L + l+1 have a tail index w 1 . Given the l-th record occurred at m n, the waiting time till the (l + 1)-th has power-law tails with exponent w 1 . Note however that the probability that

K + l+1 -m = 1 is w 1 /(w + m) which is small only if m 1. Introducing c m := Γ(w + m)/Γ(w 2 + m), for each l ≤ m, c m+1 /c m = (w + m)/(w 2 + m) > 1 translating that the tails of L +
l+1 get heavier as m increases, but without affecting the tail index itself, only the prefactor.

With m > m ≥ l ≥ 1, we similarly get

P(K + l+1 = m | K + l = m) = w 1 w + m -1 m -2 n=m w 2 + n w + n , P(K + l+1 = m ) = m≥l P(K + l+1 = m | K + l = m)P(K + l = m),
with initial condition P(K + 1 = m ) given below in (20) if w 2 = 0. The homogeneous Markov structure of (K + l , l ∈ N) appears more clearly, recalling P(K + l = m) is given by [START_REF] Pemantle | A survey of random processes with reinforcement[END_REF]. For w 2 = 0 the initial condition should start with P(K + 2 = m ) given in [START_REF] Yamato | On the Donnelly-Tavaré-Griffiths formula associated with the coalescent[END_REF].

Introducing the excess time to l-th failure K l := K + l -l ≥ 0, now a shifted random variable taking values in N 0 , ( 17)

P(K l+1 = n) = w 1 w + n P(K l = n) + w 2 + n w + n P(K l+1 = n -1).
Replacing n by n + l in [START_REF] Pemantle | A survey of random processes with reinforcement[END_REF] shows that the excess time K l has distribution

P(K l = n) = w l 1 [w] n+l n+l-1 k=l-1 k l -1 |s n+l-1,k |w k-l+1 2 , l ∈ N, n ∈ N 0 .
Alternatively, the three-terms recursion (17) can be solved numerically using initially P(K 1 = n) and observing P(K l = 0) = w l 1 /[w] l . Consequently, for all n, n ≥ 0, ( 18)

P(K l+1 > n | K l = n) = 2n+n +1 m=n+1 w 2 + m w + m
and, with

P(K l+1 = n | K l = n) = w 1 w + 2n + n + 1 2n+n m=n+1 w 2 + m w + m , P(K l+1 = k ) = k≥0 P(K l+1 = k | K l = k)P(K l = k),
emphasizing the inhomogeneous Markov structure of (K l , l ∈ N) as well. Setting n, n = 0 in (18), we get in particular P(K l+1 = 0 | K l = 0) = w 1 /(w + 1). Note that K l also represents the number of failures till the observation of the l-th success, a generalized version of the negative binomial distribution.

Time to first success

If l = 1, with K + 0 := 0, the distribution of the time to the first success reads (L

+ 1 = K + 1 -K + 0 = K + 1 ) (19)
P(K + 1 > n) = [z 0 ]E(z Sn ) = [w 2 ] n [w] n ∼ Γ(w) Γ(w 2 ) n -w1 , n → ∞.
and ( 20)

P(K + 1 = n) = w 1 w + n -1 [w 2 ] n-1 [w] n-1 = w 1 w [w 2 ] n-1 [w + 1] n-1 , n ∈ N.
It is easily seen that the law of K + 1 is unimodal with mode at n = 1 having mass w 1 /w. Upon shifting, K 1 = K + 1 -1 ≥ 0 has a generalized (heavy tailed with index w 1 ) Sibuya distribution [START_REF] Kozubowski | A generalized Sibuya distribution[END_REF] with probability generating function (pgf) ( 21)

E(z K1 ) = w 1 w F (1, w 2 ; w + 1; z) observing (w + n)[w] n = w[w + 1] n , where F := 2 F 1 is the Gauss hypergeometric function F (a, b; c; z) := n≥0 ([a] n [b] n /[c] n )(z n /n!).
The initial condition to the recursion [START_REF] Pitman | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator[END_REF] giving

P(K l = k) is P(K 1 = k) = w 1 w + k [w 2 ] k [w] k = w 1 w [w 2 ] k [w + 1] k , k ∈ N 0 .
Remark 3. (Time to first success in a N -Bernoulli trial with N finite and Geometric (p)).

In that case, K + l = inf{n ∈ {1, . . . , N } : S n = l} and K + 1 = inf{m ∈ {1, . . . , N } : I m = 1}. Therefore, K + 1 = ∞ with probability P(S N = 0) = E( N m=1 P(I m = 0)) and P(K + 1 > n) = P(S n = 0 | N ≥ n) with probability P(S N > 0), where

P(S n = 0 | N ≥ n) = P(S n = 0) = [w 2 ] n [w] n .
So, if w 2 > 0, the new K + 1 has an atom at ∞ with mass

P(S N = 0) = q n≥1 [w 2 ] n [w] n p n-1 = q p [F (1, w 2 ; w; p) -1]
translating that no success was registered before N .

4.1. Special cases.

-Sibuya:

w = 1 ⇒ w 1 , w 2 = 1 -w 1 ∈ (0, 1) with E(z K1 ) = w 1 F (1, 1 -w 1 ; 2; z) = z -1 (1 -(1 -z) w1 ), equivalently, P(K 1 = k) = w 1 [1 -w 1 ] k /(k + 1)!, k ≥ 0.
-Yule-Simon: w 2 = 1, w 1 > 0 with E(z K1 ) = w1 w1+1 F (1, 1; w 1 + 2; z), equivalently,

P(K 1 = k) = w 1 k! [w1+1] k+1 .
-Ewens: w 2 = 0, w 1 > 0: this is a singular case for which

P(K + 1 = n) = δ n,1 .
In view of F (a, b; c; z) = F (b, a; c; z), the Yule-Simon distribution with a = b = 1 and c = w 1 +2 is the only one in the class [START_REF] Sibuya | Generalized hypergeometric, digamma and trigamma distributions[END_REF] to be identifiable (different parameters yield different distributions).

4.2.

Falling factorial moments of K 1 .

K + 1 = K 1 + 1 is an important random variable if one considers that the first occurrence of a success may lead to a stop of some ongoing process.

With a = 1, b = w 2 , c = w + 1, i integer, using the special values and differential identities

F (a, b; c; 1) = [c -a] a [c -a -b] a d i dz i F (a, b; c; z) = [a] i [b] i [c] i F (a + i, b + i; c + i; z), evaluated at z = 1, with (K 1 ) i = K 1 (K 1 -1) • • • (K 1 -i + 1
), when i < w 1 , we get the descending i-th factorial moments of K 1 as

E[(K 1 ) i ] = ϕ (i) (1) = i![w 2 ] i [w 1 -i] , i < w 1 ,
where ϕ(z) := E(z K1 ) = w1 w F (1, w 2 ; w + 1; z). In particular, if

w 1 > 1, E(K 1 ) = w 2 /(w 1 -1) < ∞ and, if w 1 > 2, Var(K 1 ) = ϕ (1) + ϕ (1) -(ϕ (1)) 2 = w 1 (w -1)E(K 1 ) (w 1 -1)(w 1 -2) < ∞.
Overdispersion holds. The mean E(K + 1 ) = w-1 w1-1 > 1 and the variance Var(K + 1 ) = Var(K 1 ) of K + 1 (if they exist) may be used to estimate (w 1 , w 2 ) by the method of moments provided empirical values of these quantities are available. 4.3. MLE estimator of (w 1 , w 2 ) from K + 1 . If we have an L-sample (n 1 , . . . , n L ) for the time K + 1 to first success,

P(K + 1 (1) = n 1 , . . . , K + 1 (L) = n L ) = w L 1 L l=1 1 w + n l -1 [w 2 ] n l -1 [w] n l -1 . Considering ∂ w k log P(K + 1 (1) = n 1 , . . . , K + 1 (L) = n L ) = 0 for k ∈ {1, 2} yields a MLE ( w 1 , w 2 ) for (w 1 , w 2 ) based on the histogram of the observed time to first failure sample (n 1 , . . . , n L ). With w = w 1 + w 2 , we get L w 1 - L l=1 1 w + n l -1 - L l=1 (Ψ( w + n l -1) -Ψ( w)) = 0 and - L l=1 1 w + n l -1 - L l=1 Ψ( w + n l -1) -Ψ( w) -Ψ( w 2 + n l -1) + Ψ( w 2 ) = 0.
The first equation gives w 1 as a function of w (and the data) and so w 2 = w -w 1 as a function of w. Plugging this expression of w 2 into the second equation yields an equation in the single variable w that can be solved from the data. A separate expression of both w 1 and w 2 then follows. Asymptotic normality of this estimator is proved in [START_REF] Kozubowski | A generalized Sibuya distribution[END_REF], together with an expression of the Fisher information matrix.

4.4. The Ewens case w 2 = 0.

In a sampling problem from a Poisson-Dirichlet partition PD(θ) of the unit interval modeling species abundances, the law of the number S n = n m=1 I m of distinct sampled species for a size n uniform sample obeys (5), [START_REF] Ewens | The sampling theory of selectively neutral alleles[END_REF], [START_REF] Arratia | Poisson process approximations for the Ewens sampling formula[END_REF] and [START_REF] Yamato | Poisson approximations for sum of Bernoulli random variables and its application to Ewens sampling formula[END_REF], with w 1 = θ, w 2 = 0 and S 1 = 1, corresponding to K + 1 = 1. Because sampling is modeled as uniform throws on a partition of the unit interval, necessarily on day n = 1, a new species is sampled but new species with smaller abundance become increasingly unlikely to be subsequently sampled. The PD(θ) partition of the unit interval has countably many pieces, so the sampling process potentially never stops. Here K + l (l ≥ 2) is the sample size till l new species have been sampled with, from (14) ( 22)

P(K + l = n) = w 1 [z l-1 ] [w 1 z] n-1 [w 1 ] n = w l-1 1 |s n-1,l-1 | [w 1 + 1] n-1 , n ≥ l.
This distribution seems to be new. Note the resulting 'vertical' identity for the |s n,l |'s:

n≥l |s n,l |
[w1+1]n = w -l 1 for all w 1 > 0. The random variable K + 2 is the time to second non-trivial discovery of a new species (after

K + 1 = 1), with, recalling |s n-1,1 | = (n -2)!, (23)
P(K + 2 = n) = w 1 (n -2)! [w 1 + 1] n-1 , n ≥ 2,
reducing to P(K + 2 = n) = 1/(n(n -1)) when w 1 = 1. With K + 2 -1 ≥ 0 the time elapsed since K + 1 = 1, we thus have

E(z K + 2 -1 ) = w 1 z 0 F (1, 1; w 1 + 1; t) -1 t dt.
The above theory applies to this fundamental Ewens model. Given S n = k, the probability to discover a new species at time n + 1 is w 1 /(w 1 + n), decreasing inversely proportional to n and independently of k. Recall from (10) that the MLE w 1 for w 1 is characterized by k/ w 1 = Ψ( w 1 + n) -Ψ( w 1 ) and hence only depends [22, p. 41, Eq. (3.7.7)].

on k = i 1 + • • • + i n . See

An extension of the harmonic Bernoulli trial

With α > 0, consider the inhomogeneous Bernoulli trial with

P(I m = 1) = w 1 /(w+ m α -1), m ∈ N.
For α ∈ (0, 1) the successful events are more frequent than for α = 1. Then,

µ n := E(S n ) = n m=1 P(I m = 1) = w 1 n-1 m=0 1/(w + m α ) ∼ w1 1-α n 1-α as n → ∞ and σ 2 n := Var(S n ) = n m=1 P(I m = 1)P(I m = 0) = w 1 n-1 m=0 w 2 + m α (w + m α ) 2 ∼ w 1 1 -α n 1-α
and the law of S n is close in the sense of total variation distance to P n d ∼ Poi(µ n ) for this new µ n now growing algebraically with n.

Clearly also, [START_REF] Yamato | Poisson approximations for sum of Bernoulli random variables and its application to Ewens sampling formula[END_REF] w

1 (K + l ) 1-α l(1 -α) a.s. → 1 as l → ∞ and w 1 (K + l ) 1-α /(1 -α) -l √ l d → N (0, 1) as l → ∞.
The time to the l-th success occurs much sooner than when α = 1.

If α > 1, then S n converges in distribution to a Poisson random variable with finite mean µ ∞ := lim n→∞ µ n = w 1 ∞ m=0 1/(w + m α ).

A related random walk with disasters

Bernoulli trials with unequal harmonic success probabilities are also relevant in the context of growth-collapse random walks with disasters. Discrete-time integralvalued growth-collapse processes where long periods of linear growth alternate with rare catastrophic events occur in a large variety of systems. A collapse or catastrophic event is when the size of some population shrinks by a random number of units, not exceeding the current system's size. A total disaster is when the size of the system shrinks instantaneously to zero (a massive extinction event). Disastrous growth-collapse models occur as models for population growth subject to rare catastrophic extinction events.

A one-parameter version of such discrete-time models was investigated in [START_REF] Huillet | On a Markov chain model for population growth subject to rare catastrophic events[END_REF]. Here, holding probabilities were allowed (with some probability the system's size can be left unchanged) and pure reflection at the origin was assumed (once in state zero, the system's size grows by one unit with probability 1). Whenever zero is a reflection/absorption barrier, pomp periods will alternate with periods of scarcity. We herewith focus on discrete-time disastrous growth-collapse models with no holding probability and with zero either standing for a reflection or an absorption barrier. The probabilities of either growth or disastrous events will be chosen to be dependent on the current state as in the Bernoulli model with harmonic success probabilities, and this will favor large populations in the long run.

With α > 0, define q n := w 1 /(w + n α ) and p n := 1 -q n , n ∈ N 0 . With (U m , m ∈ N) an iid sequence of uniforms,

N m+1 := (N m + 1)1(U m+1 ≤ p Nm ), N 0 ≥ 0, defines a time-homogeneous Markov chain (N m , m ∈ N 0 ) that moves from state n to state n + 1 with probability p n or is sent from state n to state 0 with probability q n (a disaster event).

The transition matrix of this Markov chain with state-space N 0 is

P =         q 0 p 0 • • • q 1 0 p 1 • • • . . . . . . . . . . . . • • • q n 0 • • • 0 p n • • • . . . . . . . . . . . .         .
Let us distinguish two cases.

Case 1. Assume that w 2 = 0. In this case state 0 is absorbing. Let n ∈ N.

The probability P(N m → ∞ | N 0 = n) = m≥n p m is equal to 0 if and only if m≥n q m = ∞, which in turn holds if and only if α ≤ 1. Thus, for α ≤ 1 the chain (N m , m ∈ N 0 ), started from state N 0 ≡ n, will eventually go extinct. For α > 1 the chain, started from state n, will tend to infinity with probability m≥n p m > 0 and go extinct with complementary probability 1 -m≥n p m . The extinction time τ n,0 := inf{m ∈ N 0 : N m = 0, N 0 = n} has pgf E(z τn,0 ) = m≥n q m z m m-1 k=n p k , |z| < 1, and τ n,0 takes the value ∞ with probability m≥n p m being strictly positive if and only if α > 1.

Case 2. Assume that w 2 > 0. Then state 0 is reflecting and all states are communicating since w 1 > 0 by assumption. The chain (N m , m ∈ N 0 ) is hence irreducible and obviously aperiodic. This is a small variation of a Markov chain whose salient statistical features were studied in [START_REF] Goncalves | Scaling features of two special Markov chains involving total disasters[END_REF]. From the study in [START_REF] Goncalves | Scaling features of two special Markov chains involving total disasters[END_REF] we conclude that:

• For α > 1 the chain is transient. After a finite number of returns to 0 (excursions) the chain drifts to infinity. • For α < 1 the chain is positive recurrent with invariant probability measure π n = π 0 n-1 k=0 p k , n ∈ N 0 , where the normalizing constant π 0 is determined by

∞ n=0 π n = 1. • For α = 1 (critical case) the chain is null-recurrent if 0 < w 1 ≤ 1 and posi- tive recurrent if w 1 > 1.
For the latter case w 1 > 1 the invariant probability measure is given by π n = π 0 [w 2 ] n /[w] n , n ∈ N 0 , with normalizing constant π 0 := (w 1 -1)/(w -1), having heavy tails with index w 1 > 1.

In the recurrent case (α ≤ 1) the sample paths of (N m , m ∈ N 0 ), started at N 0 = 0, are made of iid excursions through state 0. The first excursion has length L + 1 and height L + 1 -1, where L + 1 := inf{m ∈ N : N m = 0, N 0 = 0} is the time elapsed till the first disaster. Clearly, in the positive recurrent case (α < 1 or α = 1 and w 1 ≤ 1) the invariant probability measure has the general form π n = P(L

+ 1 > n)/E(L + 1 ), n ∈ N 0 . With (L + i -1, i ∈ N)
iid copies of the first excursion height L + 1 -1, of interest for the control of overcrowding are the random variables

T 1 (n) := inf{m ∈ N : N m > n | N 0 = n 0 } and inf{i ∈ N : max j∈{1,...,i} (L + j -1) > n},
corresponding to the first (overcrossing) time the chain N m exceeds n given N 0 = n 0 < n and the number of the corresponding excursion.

Let P (n) be the truncated upper-left corner with size (n + 1, n + 1) of the full irreducible transition matrix P of N m (its north-west part). With 1 = (1, . . . , 1) and e n0 = (0, . . . , 0, 1, 0, . . . , 0) transpose row vectors with size n + 1 (with 1 in position n 0 + 1 for e n0 ), it follows from Propositions 11 and 12 of [6] that (26)

P n0 (T 1 (n) > l) = e n0 P l (n) 1,
where P l (n) is the l-th power of P (n) . P(T 1 (n) > l) = 1 for l ∈ {1, . . . , n -n 0 }. At this time T 1 (n), the state of the chain N m is n + 1 because the overshoot can only be 1. So T 1 (n) has geometric tails with decay-rate parameter the spectral radius of P (n) and

E n0 (T 1 (n)) = e n0 (I -P (n) ) -1 1.
Clearly, given N 0 = n 0 < n, with N * l = max m≤l N m the extremal process of N m , the events N * l ≤ n and T 1 (n) > l coincide, so (26) also gives the marginal law P n0 (N * l ≤ n) of N * l . The extremal chain N * l only grows (by one unit) at the record times

R k := inf{r ∈ N : r > R k-1 , N r > N R k-1 } of N m .

A more general Markov model for the number of successes

As before, let w 1 > 0 and w 2 ≥ 0 and define w := w 1 + w 2 . A more general model can be introduced by taking an additional third parameter α ∈ [0, 1] and assuming that the number S n of successes forms a Markov chain (S n , n ∈ N 0 ) satisfying S 0 = 0 and

P(S n+1 = k + 1 | S n = k) = 1 -P(S n+1 = k | S n = k) := w 1 + kα w + n , n ∈ N 0 .
In this case S n coincides with the number of occupied tables in the restaurant process with a cocktail bar [START_REF] Möhle | A restaurant process with cocktail bar and relations to the threeparameter Mittag-Leffler distribution[END_REF] after n customers have entered the restaurant.

For α = 0 we are back to the model studied before. For α > 0 the transition probabilities of the random walk (S n , n ∈ N) now depend not only on the time n but also on the current state S n = k. The distribution of S n can be expressed as (see [12, Eq. ( 14)]) The behavior of S n for α > 0 differs substantially from the case α = 0. For α > 0, as n → ∞, S n /n α converges almost surely and in L p for any p > 0 to a limiting random variable being three-parameter (α, β, γ)-Mittag-Leffler distributed, where β := w and γ := w 1 /α, see [START_REF] Möhle | A restaurant process with cocktail bar and relations to the threeparameter Mittag-Leffler distribution[END_REF]Theorem 3]. We refer the reader to [START_REF] Möhle | A restaurant process with cocktail bar and relations to the threeparameter Mittag-Leffler distribution[END_REF]Section 7] for further details on the three-parameter Mittag-Leffler distribution ML(α, β, γ). For α = 1 the limiting distribution ML(1, w, w 1 ) = β(w 1 , w 2 ) is the beta distribution with parameters w 1 and w 2 , in agreement with well-known results for standard Pólya urns.

P(S n = k) = [w 1 |α] k [w] n S(
If w 2 = 0 then S n counts the number of distinct species in a sample of size n taken from Pitman and Yor's [START_REF] Pitman | The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator[END_REF] two-parameter stick-breaking PD(α, w 1 )-partition of the unit interval, extending the Ewens case. We refer the reader to Chapter 3 of Pitman's lecture notes [START_REF] Pitman | Combinatorial Stochastic Processes[END_REF] for further information on the two-parameter model and Observing that n-n0 k=0 S(n -n 0 , k; -1, -α, 0)[w 1 |α] k = [w 1 ] n-n0 , the probability that, in a sample of size n, there are n 0 ≤ n visits to the 'fictitious species' is thus the beta-binomial probability P(N n (0) = n 0 ) = n n0 [w 2 ] n0 [w 1 ] n-n0 /[w] n , in agreement with the explanations above.

1 i=0 (w 1 +

 11 n, k; -1, -α, w 2 ), k ∈ {0, . . . , n}, where [w 1 |α] 0 := 1, [w 1 |α] k := niα) for k ∈ N and S(n, k; -1, -α, w 2 ) denote the generalized Stirling numbers in the notation of Hsu and Shiue [7], which can be calculated as follows. For α = 0 it follows from (3) that S(n, k; -1, 0, w 2 ) = n l=k l k w l-k 2 |s n,l |, k ∈ {0, . . . , n}. For α = 0, the Dobiński-type formula [7, Theorem 4] yieldsS(n, k; -1, -α, w 2 ) = α -k k! k l=0 (-1) l k l [w 2 -lα] n , k∈{0, . . . , n}. Note that P(S n = 0) = [w 2 ] n /[w] n does not depend on α ∈ [0, 1]. In particular, for any n ∈ N, P(S n = 0) = 0 if and only if w 2 = 0. Formulas for the moments of S n are provided in [12, Section 6.1] for α = 0 and in [12, Corollary 1] for α > 0.

acknowledgmentT.

  Huillet acknowledges partial support from the Chair 'Modélisation mathématique et biodiversité' of Veolia-Ecole Polytechnique-MNHN-Fondation X and from the labex MME-DII Center of Excellence (Modèles mathématiques et économiques de la dynamique, de l'incertitude et des interactions, ANR-11-LABX-0023-01 project). This work was also funded by CY Initiative of Excellence (grant 'Investissements d'Avenir ' ANR-16-IDEX-0008), Project 'EcoDep' PSI-AAP2020-0000000013. Thierry Huillet Laboratoire de Physique Théorique et Modélisation CY Cergy Paris University, CNRS UMR-8089 Site de Saint Martin, 2 avenue Adolphe-Chauvin 95302 Cergy-Pontoise, France E-mail address: thierry.huillet@cyu.fr Martin Möhle Fachbereich Mathematik Eberhard Karls Universität Tübingen Auf der Morgenstelle 10 72076 Tübingen, Germany E-mail address: martin.moehle@uni-tuebingen.de

to Yamato and Sibuya [START_REF] Yamato | Moments of some statistics of Pitman sampling formula[END_REF] and Yamato, Sibuya and Nomachi [START_REF] Yamato | Ordered sample from two-parameter GEM distribution[END_REF] for some further related works.

Assume now that w 2 > 0. In this case S n may no longer be seen, stricto sensu, as the number of new species in a sample of size n taken from a partition of the unit interval. However (see [START_REF] Möhle | A restaurant process with cocktail bar and relations to the threeparameter Mittag-Leffler distribution[END_REF]Theorem 2]), S n is the number of new species (excluding a 'fictitious species' 0 with beta distributed 'abundance' B 0 d = β(w 2 , w 1 )) in a sample of size n drawn from a kind of three-parameter Poisson-Dirichlet partition PD(α, w 1 , w 2 ) := (B 0 , (1 -B 0 )PD(α, w 1 )), where B 0 is independent of PD(α, w 1 ).

Note that K + 1 := inf{n ∈ N : S n = 1} has distribution

) with the number of new species from a PD(α, w 1 )-partition of the unit interval. Whenever a sample hits the 'fictitious species' 0, sampling simply fails to draw any new species: this event thus represents the possibility of a failure of the sampling process from scratch. The probability that in a sample of size n there are n 0 failure events clearly is the beta binomial probability mass function n

is the number of new species (excluding the 'fictitious species' 0 with 'abundance' B 0 ) in a sample of size n drawn from the partition PD(0, w 1 , w 2 ) = (B 0 , (1 -B 0 )PD(0, w 1 )), extending the Ewens case.

Let n 0 ∈ N 0 and n 1 , . . . , n k ∈ N and put n :

is the joint distribution that there are n 0 visits to the reservoir set with size B 0 (accounting for early failure events of the sampling process, or missed samples) and S n = k distinct visited species in order of appearance with positive sample sizes n 1 , . . . , n k not in the reservoir. For w 2 = 0, [START_REF] Yule | A mathematical theory of evolution based on the conclusions of Dr[END_REF] reduces to the two-parameter Donnelly-Tavaré-Griffiths distribution DTG(w 1 , α) (see [START_REF] Yamato | Ordered sample from two-parameter GEM distribution[END_REF]Theorem 1])

For α = 0, (28) reduces to


Douglas A George

Ryan P Mulligan

Crossing the boundaries: how key advancements in understanding of headland sediment bypassing improves definition of littoral cells Authors

Headland sediment bypassing connects short-term physical forcing and long-term morphological response. The contributions from King et al., 2021 are a substantial step in understanding the complexities by combining field observations, process-based numerical modeling, and empirical generalization with the development of an improved parametrization of headland sediment bypassing. This study adds to the growing body of knowledge about littoral cell boundaries, or as proposed in this commentary, littoral cells that respond to varying wave energy and water levels that may influence the results from shoreline position models. Evolution of the shoreline in the context of climate change can be better understood by considering variability in littoral cell boundaries.

Plain Language Summary

To better understand how shorelines will change with sea level rise and more storms, we need to widen how we view sediment moving along the coast. An important step involves using field research and computer models to improve our knowledge about how rocky headlands affect sand and mud moving in the coastal areas. The paper we write about provides a roadmap to do those types of studies about sediment movement around headlands. The next step is to consider changes to the size and shape of regions of the coast that sediment pulses through instead of boxes that trap sediment.

Key Points

1. Headland sediment bypassing connects short-term physical forcing and long-term morphological response, which is vital for predicting shoreline position. 2. The paper we comment on creates a useful guide to examine headland sediment bypassing and littoral cell boundaries. 3. The concept of littoral cells integrates temporal and spatial variability and anticipated climate change-induced evolution of the shoreline.

Body

1. Background Along rocky shorelines with embayed beaches, physical and geological processes together play key roles in shaping and responding to coastal geomorphology. One example is headland sediment bypassing, defined as the sediment connectivity between adjacent embayments driven by sediment migration around the headland from the updrift to the downdrift beach compartments [START_REF] Klein | Headland bypassing and overpassing: Form, processes and applications[END_REF]. Therefore, headland sediment bypassing is a critical component to understand coastal processes across the full beach system (i.e., backbeach to nearshore beyond the surf zone). The primary parameters that affect bypassing include geomorphology (headland size, shape, orientation, offshore bathymetry) [START_REF] George | Classification of rocky headlands in California with relevance to littoral cell boundary delineation[END_REF], physical forces (waves, tides, currents) (Veira da [START_REF] Vieira Da Silva | The influence of wave-, wind-and tide-forced currents on headland sand bypassing -study case: Santa Catarina Island North Shore, Brazil[END_REF], and sediment (availability, grain characteristics) [START_REF] Goodwin | An insight into headland sand bypassing and wave climate variability from shoreface bathymetric change at Byron Bay, New South Wales, Australia[END_REF][START_REF] George | Classification of rocky headlands in California with relevance to littoral cell boundary delineation[END_REF][START_REF] George | Modelling sediment bypassing around idealized rocky headlands[END_REF]. Bypassing is also an important process in sandy environments where, for example, sand can bypass coastal structures or inlets and influence beach and barrier island morphology [START_REF] Fitzgerald | Control of barrier island shape by inlet sediment bypassing: East Frisian Islands, West Germany[END_REF] since bypassing influences sediment availability [START_REF] Ciarletta | Quantifying thresholds of barrier geomorphic change in a cross-shore sediment-partitioning model[END_REF]. Temporal and spatial variability across all of these adds a transient element into bypassing driving the development of a "leaky" boundary [START_REF] Davies | The coastal sediment compartment[END_REF][START_REF] Van Rijn | Coastal erosion control based on the concept of sediment cells[END_REF]. The bypassing can be episodic as when sediment accumulates to a critical volume to trigger a bypassing event [START_REF] Goodwin | An insight into headland sand bypassing and wave climate variability from shoreface bathymetric change at Byron Bay, New South Wales, Australia[END_REF][START_REF] Battalio | Littoral Processes along the Pacific and bay shores of San Francisco, California, USA[END_REF], on a seasonal basis when dominant wave direction shifts [START_REF] Short | Handbook of beach and shoreface morphodynamics[END_REF], or even on multi-year to multi-decadal timescales in relation with climate patterns of atmospheric variability [START_REF] Da Silva | Headland bypassing timescales: Processes and driving forces[END_REF]. Quantifying the net transport rate and the leakiness associated with a rocky headland is critical when that headland is used as a littoral cell boundary, which will be discussed in this commentary.

The complexity of headland sediment bypassing needs deeper understanding, especially how temporal variability relates short-term physical forcing and long-term morphological response. With improved understanding, more sustainable coastal engineering and climate change adaptation techniques can be developed using headland sediment bypassing knowledge for longevity of investments in regional sediment management (e.g., beach nourishment). Interest in this topic has grown rapidly since the mid-2010s. [START_REF] Klein | Headland bypassing and overpassing: Form, processes and applications[END_REF] identified 100 studies published since 2014 that investigated either subaqueous or subaerial transport pathways, or both, through (1) field (direct and remote sensing), (2) modeling, or (3) theoretical approaches resulting in the development of conceptual models of headland sand bypassing (for comparison there were only 59 studies between 1943 and 2014). Each of these approaches present benefits and drawbacks and for a more comprehensive discussion, see [START_REF] Klein | Headland bypassing and overpassing: Form, processes and applications[END_REF] In summary, the dynamic nature of the ocean around headlands creates extremely challenging conditions for observations, especially at the apex where rapidly changing bathymetry and exposed hard substrates become dangerous for instrument deployment and recovery. Consequently, processbased modeling approaches have tried to address field observational study gaps, ultimately to develop parametric expressions of headland sand bypassing quantities (e.g. [START_REF] Mccarroll | An XBeach derived parametric expression for headland bypassing[END_REF]. Such empirical or semi-empirical expressions provide the opportunity to unify and generalize concepts, if desired. However, their calibration and validation require reliable field measurements while concurrently, simplifying essential longshore transport processes. Hence, all three approaches are needed to develop a more comprehensive and nuanced understanding of headland sediment bypassing.

Given the rapid expansion of research, gaps are being identified and addressed with each new contribution. Assuming the primary parameters presented above (geomorphology, physical forces, and sediment) are sufficiently broad to incorporate most areas for improvement, substantial focus has been placed on hydrodynamics and their interactions with the physical shape of a headland (e.g. Mouragues et al., 2021). Lesser attention has been given to geological controls, such as erosional differences leading to geomorphic structures (e.g., exposed bedrock), or resolving dynamics at headland apexes in the field. Additionally, multi-approach studies (e.g., field-modeling or modeling-theoretical) can amplify limitations of the individual approach if not constrained appropriately, for example by incorrect calibration of model parameters. Hence, place-based studies that build from field observations to numerical modeling and then generalization are welcome additions to refining headland sediment bypassing understanding. [START_REF] King | Wave, tide and topographical controls on headland sand bypassing[END_REF]Given this context, King et al., 2021 is a substantial step forward in understanding headland sediment bypassing. The authors build on several threads described above, primarily in the physical forcings topic, while utilizing all three research approaches of field observations, processbased numerical modeling, and generalization with the development of an improved parametrization of headland sand bypassing that also includes headland toe depth and areal sand coverage for 29 headlands in the United Kingdom. In their own words, "this study tested the influence of wave, tide and morphological controls on instantaneous headland sand bypassing using a coupled wave-tide numerical model, and tested the performance of an existing parameterization when applied to realistic headland morphologies and sediment coverage." The exploration of the combined wave and tide forces refine the nuances for how tidal currents are a significant factor in some locations under particular wave conditions but not others. The variation in tidal range from micro-to macro-tidal environments affecting currents is important to properly apply sediment bypassing parameters. [START_REF] King | Wave, tide and topographical controls on headland sand bypassing[END_REF] explore this variability in the context of changing wave conditions for a macrotidal environment and delineate significant wave height thresholds that trigger non-linear wave-current interactions.

Contributions from

One of the most substantial scientific contributions from [START_REF] King | Wave, tide and topographical controls on headland sand bypassing[END_REF] is the refinement of the relationship between morphology, surf zone width, and availability of sediment. The authors develop thresholds based around parameters that can be derived from aerial imagery, wave records, bathymetry, and sediment maps. The suite of equations describing headland sediment bypassing rates establish a set of relationships that are responsive to changing conditions. For example, surf zone width varies throughout the year so the ratio between width and headland length must be dynamic to accurately characterize bypassing. [START_REF] King | Wave, tide and topographical controls on headland sand bypassing[END_REF] present that option within the suite of equations. Similarly, the amount of sediment available may fluctuate, which would be resolved by sediment maps from different periods. The suite of equations allows for spatially and temporally varying surficial sediment deposits. Taken together, these thresholds provide a new and more general approach to predict sediment bypassing rates around headlands that also incorporates the potential for leaky boundaries. [START_REF] King | Wave, tide and topographical controls on headland sand bypassing[END_REF] also provide a very practical, and important, 4-step guidance for analytically assessing headland sediment bypassing with implications for modeling shoreline change. The steps involve quantifying morphometric parameters (e.g., beach orientation, headland dimensions, headland underwater toe depth, sediment availability), transforming waves from offshore to breakpoint, estimating longshore flux, and applying the bypassing formulations. The elegance of this guide enables coastal science practitioners and managers to rely less on computationally expensive process-based models. If a numerical model is available for a location or stretch of coastline containing many headlands, the analytical model results can be crosschecked for the degree of agreement. Alternatively, if observational data from the field is available, it too could be cross-checked with the analytical model results. These cross-checks are important to constrain the analytical model results. Such future work will be important to verify the validity of the equations in a wide range of coastal environments where there are limits to sediment availability and/or absence of high-resolution observational datasets. Even without those opportunities, however, the guidance is important to expand where headland sediment bypassing rates have yet to be estimated.

Future Research Considerations for Headlands and Littoral Cells

Quantifying headland sediment bypassing is an essential component to defining littoral cells. A littoral cell is defined as an alongshore region in which sand is retained and recirculated without alongshore export [START_REF] Rosati | Concepts in sediment budgets[END_REF]. Implicit to this definition are blockages in the cross-shore direction, which have typically been associated with headlands. As the dozens of studies reviewed in [START_REF] Klein | Headland bypassing and overpassing: Form, processes and applications[END_REF] indicate, however, headlands should not be assumed to successfully block alongshore transport. Littoral cell boundaries at headlands could evolve as wave energy and incident angles fluctuate resulting in substantial changes to beaches and shoreline geomorphology [START_REF] George | Modelling sediment bypassing around idealized rocky headlands[END_REF]. [START_REF] Stul | Coastal Sediment Cells between Cape Naturaliste and the Moore River[END_REF] described littoral cells as tiered according to primary, secondary, and tertiary levels along the coast of Western Australia with sediment exchange possible among the lower levels, especially on a seasonal basis. Subsequent work expanded these delineations to the coast of Australia [START_REF] Thom | National sediment compartment framework for Australian coastal management[END_REF][START_REF] Short | Australian coastal systems: beaches, barriers and sediment compartments[END_REF] with [START_REF] Kinsela | Second-pass assessment of potential exposure to shoreline change in New South Wales, Australia, using a sediment compartments framework[END_REF] providing high-resolution tertiary level maps for the state of New South Wales. The nuanced conclusions of the studies in Australia contrast with the extensive work on littoral cells along the coast of southern England that implies static boundaries dependent on headlands (New Forest District Council, 2017). The [START_REF] King | Wave, tide and topographical controls on headland sand bypassing[END_REF] study provides additional support for revisiting the usefulness of headlands as littoral cell boundaries, and therefore is relevant to understanding morphological changes in many other coastal regions around the world.

Littoral cells should therefore be presented as more dynamic regions with variable boundaries that have allowances for sediment grain sizes, seasonality of physical forces, and exclusivity, which is a measure of the amount of connectedness to other cells [START_REF] Davies | The coastal sediment compartment[END_REF]. The littoral cells therefore should be viewed as variable in size and shape as a function of the wave conditions which affect the sediment transport pathways. Conceptually, this would be represented as Figure 1, with wave conditions driving the variable scales of the littoral cell. As conditions exceed lowlevel thresholds, the smaller cells would merge into medium-sized cells. Subsequently, larger wave conditions would push littoral cell boundaries offshore causing the amalgamation of medium-sized cells into more expansive regions that would activate transport across usually isolated embayments. The frequency of these events could be predicted using wave climate records and the magnitude of the sediment bypassing estimated using the [START_REF] King | Wave, tide and topographical controls on headland sand bypassing[END_REF] relationships, in particular the width of the surf zone to identify when a headland loses the ability to block alongshore transport. While the regional approach departs from the visually simpler boundary approach, it better reflects the current understanding of headland sediment bypassing, including development and practice of tiered zones [START_REF] Thom | National sediment compartment framework for Australian coastal management[END_REF][START_REF] Kinsela | Second-pass assessment of potential exposure to shoreline change in New South Wales, Australia, using a sediment compartments framework[END_REF]).

The regional approach still relies upon understanding headland sediment bypassing and therefore additional research is needed. Some recommended lines of inquiry to refine the conceptual model include examining the influence of islands and reef structures, considering the volume of available sediment (or thickness of sediment deposits from seafloor surface to a basement depth), parameterizing connectivity on and between compound headlands (e.g., a large peninsula consisting of a variety of headlands), studying the influence of the configuration and orientation of the adjacent embayments, improving field observations of seasonality and episodic events, investigating the criticality of understanding shoreface sediment pathways, and testing the [START_REF] King | Wave, tide and topographical controls on headland sand bypassing[END_REF] suite of equations in other locations with extensive data (e.g., California, USA, or Australia). This last recommendation would demonstrate applicability and by extension, offer insight on how to best develop littoral cell boundaries for different coastal segments. As acknowledged by [START_REF] King | Wave, tide and topographical controls on headland sand bypassing[END_REF], generic headland sand bypassing parameterizations are important to improve hybrid shoreline models. Such models often do not include headland bypassing, or at best in a too simplistic way [START_REF] Roelvink | Efficient modeling of complex sandy coastal evolution at monthly to century time scales[END_REF]. It has been shown with such models that frequency and degree of headland sand bypassing can control both time-averaged embayment planshape and beach rotation patterns [START_REF] Castelle | Modelling of embayed beach equilibrium planform and rotation signal[END_REF]. Including such parameterization in state-of-the-art hybrid shoreline models will allow simulating coastal embayments on short to long timescales, including the effect of climate change, where intensity and frequency of headland bypassing events will change in time, ultimately switching off and on new embayment connectivity. In addition, bypassing is a process that can govern large-scale coastal change during extreme events (e.g., [START_REF] Sherwood | Modeling the Morphodynamics of Coastal Responses to Extreme Events: What Shape Are We In[END_REF] and should be evaluated using process-based morphological models over a range of scales and conditions.

Overall, using littoral cells as a basis for coastal planning units better connects natural processes to coastal management. One benefit of this approach is to better incorporate natural processes for sustainable coastal engineering, regional sediment management, and ecosystem restoration. Linking littoral cells to future beach conditions and shoreline positions under climate change scenarios is essential to minimize damages to communities and habitats, at least over the next several decades. Eventually, sea level rise will alter coastal geomorphology by submergence and attendant erosion but headland sediment bypassing will continue to be a feature of coastal processes. Understanding it now may provide tools, similar to the ones developed by [START_REF] King | Wave, tide and topographical controls on headland sand bypassing[END_REF], to develop resilience for the future.

Figure 1 .

 1 Figure 1. Conceptual diagram of littoral cell boundary expansion and sediment pathways along a series of coastal embayments. Littoral cells merge and increase in size during periods with increased incident wave energy. Headland sediment bypassing progressively activates over a larger region when boundaries expand offshore past the longer headlands.

Acknowledgments

The authors thank Mike Kinsela, Carlos Loureiro, and an anonymous reviewer for comments that improved the manuscript. No new data was used in producing this manuscript.