Johannes Wirtz
email: jwirtz@lirmm.fr

Stéphane Guindon
email: sguindon@lirmm.fr

Rate of coalescence of pairs of lineages in the spatial λ-Fleming-Viot process

Keywords: Spatial λ-Fleming-Viot process, partial differential equations, spatial population dynamics, genealogies

.

Introduction

The spatial λ-Fleming-Viot process, first described in [START_REF] Barton | A new model for extinction and recolonization in two dimensions: quantifying phylogeography[END_REF], is a tool in population genetics to model evolution on a spatial continuum. This stochastic process arises as a natural extension of the processes of the Fleming-Viot type that have become part of population genetics since the 1980s (see [START_REF] Fleming | Some measure-valued markov processes in population genetics theory[END_REF][START_REF] Ethier | Fleming-viot processes in population genetics[END_REF]). A Fleming-Viot process is a measure-valued process (µ t ) t>0 in continuous time, where µ t is a probability measure on a locally compact probability space E (one writes µ t ∈ M 1 (E), with M 1 (E) denoting the set of probability measures on E). The interesting feature of this is that the elements of E, the allelic "type space" of the model, can be uncountably infinite. For instance, E could be defined as an interval of the real numbers, and the type of an individual would be given by a real number from that interval, which could be taken to represent quantitative traits such as height or weight [START_REF] Fleming | Some measure-valued markov processes in population genetics theory[END_REF][START_REF] Ethier | Fleming-viot processes in population genetics[END_REF][START_REF] Ohta | A model of mutation appropriate to estimate the number of electrophoretically detectable alleles in a finite population[END_REF]. Classical population models such as the Wright-Fisher model typically incorporate only finitely or at most countably many different allelic types, and so do their diffusion limits; but it can be shown that under suitable conditions, for instance, a sequence of Wright-Fisher models of size N whose initial distibutions converge to some probability measure µ 0 on E, has a (weak) limit in the form of a Fleming-Viot process with values in M 1 (E) [START_REF] Ethier | Fleming-viot processes in population genetics[END_REF]. It has been shown [START_REF] Ethier | Fleming-viot processes in population genetics[END_REF][START_REF] Donnelly | Genealogical processes for Fleming-Viot models with selection and recombination[END_REF] that the approach of modeling populations via Fleming-Viot processes was also robust to incorporating not only classical evolutionary mechanisms like selection, recombination and mutation , but also generalisations of the standard reproduction mechanisms that are present in the Wright-Fisher model. In particular, it is possible to incorporate "extreme" reproduction events, in which large portions of the population are replaced by the offspring of one single individual. More precisely, "extreme" means that the variance of the number of individuals affected by such a reproduction event tends to infinity as the population size increases. Under suitable conditions, there exist Fleming-Viot-type processes representing the diffusion limits of such population models [START_REF] Griffiths | The λ-fleming-viot process and a connection with wrightfisher diffusion[END_REF][START_REF] Birkner | Measure-valued diffusions, general coalescents and population genetic inference[END_REF][START_REF] Eldon | Coalescent processes when the distribution of offspring number among individuals is highly skewed[END_REF]. These are the so-called λ-Fleming-Viot processes; their dual processes are represented by the λ-coalescents [START_REF] Pitman | Coalescents With Multiple Collisions[END_REF][START_REF] Sagitov | The general coalescent with asynchronous mergers of ancestral lines[END_REF][START_REF] Berestycki | Recent progress in coalescent theory[END_REF], a class of processes generalizing Kingman's coalescent and enabling more than two lineages to collide at the same time ("k-mergers"). In this context, λ denotes a probability distribution on [0, 1], by which the rate of occurrence of k-mergers in a sample genealogy, k ≥ 2 is determined. The spatial λ-Fleming-Viot process (from here on, ΛV) is a population model in which the type of an individual is determined on an allelic level as well as by its physical location. Suppose the set of allelic types in the population is given by some set K, and the population inhabits some metrizable two-dimensional object H called habitat. The most well-studied cases in the existing literature are H = R 2 (e.g. the seminal [START_REF] Barton | A new model for extinction and recolonization in two dimensions: quantifying phylogeography[END_REF]), H = T (where T is a torus, [START_REF] Etheridge | The spatial Λ-Fleming-Viot process on a large torus: Genealogies in the presence of recombination[END_REF]) and H = A, with A denoting a rectangular object [START_REF] Guindon | Demographic inference under the coalescent in a spatial continuum[END_REF]. The state space space is given by E = H × K (hence, an individual is characterized by allelic state and physical location; [START_REF] Véber | The spatial lambda-fleming-viot process: An event-based construction and a lookdown representation[END_REF]). An infinite number of individuals are located at each location z ∈ H and ρ(z, t) ∈ M 1 (K) denotes the distribution of types at location z and time t (This is referred to as the hight-population density limit in [START_REF] Barton | A new model for extinction and recolonization in two dimensions: quantifying phylogeography[END_REF][START_REF] Barton | Modelling evolution in a spatial continuum[END_REF]). The proportions of types at each location is modified over time by randomly occurring events facilitating reproduction and death ("REX events" [START_REF] Guindon | Demographic inference under the coalescent in a spatial continuum[END_REF]). These events occur at points z ∈ H according to a Poisson process of intensity λ > 0. There are several ways for an event to affect the population; the two most common ones are the following: Definition 1.

1. Let an event occur at time t and position z. A number r is drawn from a distribution with variance θ 2 < ∞ and a parental location w is sampled uniformly from the "disc" B r (z) := {y : y -z < r}. Then, a type k ∈ K is chosen from ρ(w, t -). At each position y ∈ B r (z), the type distribution after the event is given by

ρ(y, t + ) = δ k u 0 + ρ(y, t -)(1 -u 0 ), (1)
where δ k is the Dirac measure on the type k. This is called the discbased version of the process [START_REF] Barton | A new model for extinction and recolonization in two dimensions: quantifying phylogeography[END_REF]. 2. Let an event occur at time t, position z, and choose the parental location w according to the gaussian density

1 2πθ 2 exp - z -w 2θ 2 dw
Choose a type from ρ(w, t -) as before. At each position z ∈ H, the composition of the population after the event becomes

ρ(y, t + ) = δ k p(z , z) + ρ(z , t -)(1 -p(z , z)), (2)
with p(z , z)

:= u 0 exp(-z -z 2 /(2θ 2 )), (3)
This is called the gaussian version of the process [START_REF] Barton | A new model for extinction and recolonization in two dimensions: quantifying phylogeography[END_REF].

For the remainder of this work, we will consider the gaussian model. The parameter θ 2 , called dispersal variance (or spatial variance), is obviously a major determinant of the model. The mortality u 0 controls the impact of a REX event locally. REX events are "extreme" in the sense of Fleming-Viot processes, as the amount of the population that is replaced in a single event is of positive mass. Genealogies sampled from a population evolving in this way are not necessarily reproducible by Kingman's coalescent (for instance, due to multiple mergers). On the other hand, considering the limits of the parameters λ and θ, it can be shown that the ΛVconverges to a λ-coalescent under suitable conditions, and in some cases even to Kingman's coalescent [START_REF] Barton | Modelling evolution in a spatial continuum[END_REF]. It should be noted, though, that λ has a different meaning in the context of the ΛV. Also, note that in [START_REF] Limic | The spatial Λ-coalescent[END_REF], a slightly different model is considered despite the similarity in terminology. The ΛV possesses several favourable properties to model evolution in space; duality results and backwards-in-time formulations have been described previously [START_REF] Barton | A new model for extinction and recolonization in two dimensions: quantifying phylogeography[END_REF][START_REF] Véber | The spatial lambda-fleming-viot process: An event-based construction and a lookdown representation[END_REF], and it is not subject to the "clumping" issues [START_REF] Felsenstein | A pain in the torus: Some difficulties with models of isolation by distance[END_REF] observed in the classical Wright-Malécot model that also describes the evolution of organisms spatially distributed along a continuum [START_REF] Wright | Isolation by distance[END_REF][START_REF] Malécot | Les mathématiques de l'hérédité[END_REF]. One application of this model is the inference of the parameters λ, θ and u 0 from geo-referenced genetic data, because these parameters allow an assessment of the speed at which genetic variation disseminates across a given habitat, or also how fast newly reached one can be conquered. For example, in [START_REF] Guindon | Demographic inference under the coalescent in a spatial continuum[END_REF], estimation is conducted using a Markov Chain Monte Carlo approach. This approach relies on an extensive parameter augmentation approach in order to calculate the likelihood of spatial coordinates along a genealogy. Although standard Metropolis-Hastings operators apply here, full Bayesian inference is computationally intensive. Alternative inference approaches, based on pairwise coalescence for instance, are therefore required and motivated the work presented here. More specifically, the purpose of our work is to gain insight on the genealogical process within the ΛV, i.e., the genealogical structure of a sample from a population that evolves according to ΛV-mechanisms, in order to improve the efficiency and precision of Bayesian methods like the one mentioned above. The most basic case of a sample of size n = 2 has been discussed to some extent in [START_REF] Barton | A new model for extinction and recolonization in two dimensions: quantifying phylogeography[END_REF]. The genealogical space for n = 2 is entirely described by the time to coalescence T 0 of the two lineages given their initial distance d 0 and the location of their most recent common ancestor X T 0 . Arguably, T 0 is the more significant quantity of the two, since it relates to the amount of variation, as well as to the speed of the reproductive mechanism. In [START_REF] Barton | A new model for extinction and recolonization in two dimensions: quantifying phylogeography[END_REF], a link between T 0 and the probability of identity by descent is utilized to obtain a formula for T 0 ; however, its evaluation requires costly numerical integration on C and the use of nontrivial functions. In this work, we will take a look at T 0 from a different angle and describe computationally feasible ways of approximating it. While this certainly will have to be extended to larger sample sizes in the future (with which the presented methodology may be helpful), even analyses based upon pairs of samples have proven to be effective in practice (e.g. the whole literature on Tajima's D [START_REF] Tajima | Statistical method for testing the neutral mutation hypothesis by dna polymorphism[END_REF], or, more recently, [START_REF] Barroso | Inference of recombination maps from a single pair of genomes and its application to archaic samples[END_REF]). Indeed, being able to describe the situation for n = 2 already enables a statistical assessment of geo-referenced genetic data. Additionally, our approach to this problem sheds some light on other features of the process; for instance, we can answer the question whether the ΛV "comes down from infinity" negatively (which is in line with a similar result obtained in [START_REF] Véber | The spatial lambda-fleming-viot process: An event-based construction and a lookdown representation[END_REF]. In sections 2 we will review the dynamics of the ΛV. We will consider the case H = R 2 as well as H = A, since the latter relevant from a practical point of view. Afterwards, we will describe the distance process (denoted by (Z t ) t≥0 ) between two lineages backward in time under the ΛV dynamics. Importantly, we will see that Z t is linked to the distribution of T 0 via its moments. Section 5 is devoted to describing numerical approximations.

ΛV Dynamics

We first take a look at the model in a finite-habitat setting. We will see that in letting the habitat size tend to infinity, one naturally recovers the original model of [START_REF] Barton | A new model for extinction and recolonization in two dimensions: quantifying phylogeography[END_REF]. As per usual with coalescent processes, we consider that time is running backward, i.e. t > 0 corresponds to a point in time t units of time in the past compared to the origin (where t = 0). The habitat is defined by a rectangle A of width w and height h. |A| = w • h denotes its area. Let λ denote the intensity of a Poisson process governing the frequency at which REX events take place. In a time interval of length h, there is a probability

Pr(N h = k|α) = α k k! exp(-α), (4)
with α := λ|A|h, that the number N h of REX events on A is k.

The center of a REX event (denoted by the random variable Z) is uniformly distributed on A, i.e., the density of Z is p Z (z) = 1/|A|. We have

lim h→0 Pr(N h = 1)h -1 = λ|A| (5)
and lim h→0

Pr(N h = k)h -1 = 0 (6)
for k > 1. The quantity on the right-hand side of Eq. 5 is the rate of events of the process, i.e., the waiting times between events are exponentially distributed with parameter λ|A|. If a REX event occurs at some position z ∈ A and time t ≥ 0, the spatial composition of the population is altered in the way described in Eq. 2.

Below, we introduce new notations and fundamental quantities that will be used throughout the article.

Definition 2.

1. A lineage X = (X t ) t≥0 is the stochastic process of the location X t ∈ R 2 of the ancestor of an individual located at x 0 in the present, that lived t units of time in the past. 2. Given two lineages X, Y at an initial distance d 0 , let the random variable T 0 denote the time at which X and Y coalesce, i.e. T 0 ∈ (0, ∞].

3. For two lineages X, Y , we let

D t := X t -Y t 2
denote the random variable describing the squared euclidean distance between the lineages at time t. For T 0 ≤ t, we define D t = 0. 4. Similarly, we let

Z t := X t -Y t 2 /(4θ 2 )
denote the distance between the lineages at time t standardized with respect to the rate of dispersal. 5. Occasionally, we will denote by Z t the random variable obtained by "conditioning" Z t on T 0 > t. Formally, let f Zt (x) denote the density of the random variable Z t evaluated at x. Then Z t has density

f Zt (x) =    f Zt (x)/ Pr(T 0 > t d 0 ) x > 0 δ 0 • Pr(Z t = 0 ∩ T 0 > t d 0 )/ Pr(T 0 > t d 0 ) x = 0
where δ 0 denotes a Dirac measure at 0.

Consider a lineage X located at X t = x t ∈ A at time t > 0. A "jump" of X is its movement to an updated ("older") ancestral position when it is affected ("hit") by an event. Assume that an event takes place at time t. The probability that X is hit by the event can be obtained as follows:

Pr(X hit by the event) = 1 |A| z∈A u 0 exp -z -x t 2 /2θ 2 dz, [START_REF] Griffiths | The λ-fleming-viot process and a connection with wrightfisher diffusion[END_REF] integrating the right-hand side of Eq. 3 over all possible locations z for the event center. Since A is finite, the integral can be calculated, although it involves the error function.

Given two lineages X and Y with locations X t = x t and Y t = y t , the probability that they are hit by the same REX event, i.e., the probability that they coalesce, is obtained as follows:

Pr(X, Y hit by the event) = 1 |A| z∈A u 2 0 exp -

x t -z 2 + y t -z 2 2θ 2 dz
(8) Making use of Eq. 7, we can calculate the rate ρ X at which lineage X located at X t = x t gets hit by a REX event:

ρ X = lim h→0
Pr(x t hit by any event in an interval of length h)

• h -1 = lim h→0 Pr(N h = 1)h -1 1 |A| z∈A u 0 exp -z -x t 2 /2θ 2 dz = λ z∈A u 0 exp -z -x t 2 /2θ 2 dz (9)
because of Eq. 5 and Eq. 6. Similarly, the rate of coalescence between two lineages located at x t and y t is obtained as follows:

ρ X∧Y = λ z∈A u 2 0 exp - z -x t 2 + z -y t 2 2θ 2 dz ( 10
)
We now consider H = R 2 . Again, REX events are generated by a Poisson point process of intensity λ > 0. This means that on any Borel set U ⊂ R 2 of finite measure |U |, the number of REX events encountered on U in an time interval of length h is Poisson-distributed with parameter λ|U |h. If at time t, a lineage X is located at position x t ∈ R 2 , the rate at which a REX event appears and affects this lineage can be calculated as the limit of the same rate on a rectangle, letting its size tend to infinity (we write lim |A|→∞ assuming that both w and h become infinite):

ρ * X = lim |A|→∞ λ z∈A u 0 exp -z -x t 2 /2θ 2 dz = λ R 2 u 0 exp -z -x t 2 /2θ 2 dz = 2πθ 2 u 0 λ (11) := ∆λ, (12)
where z denotes the location of the event.

When the ancestral lineage located at x t -is hit by an event taking place at z and time t, it changes its position to X t + . This random variable is approximately distributed as a bivariate normal with mean z, and covariance matrix θ 2 I. As noted in [START_REF] Barton | Modelling evolution in a spatial continuum[END_REF], the normal approximation becomes exact in the limit of high population density, which is the case that we are considering in the present study.

The ancestral process of a single lineage can be thought of as that of a particle on the plane changing position according to a Poisson process. If two lineages X and Y are considered, then they both move through the plane with the possibility of a coalescence. The coalescence rate can again be retrieved as the coalescence rate on the rectangle in the limit of an infinite size:

ρ * X∧Y = πθ 2 u 2 0 λ exp - x t -y t 2 4θ 2 (13) = ∆λ 2 u 0 exp - d t 4θ 2 (14)
where d t is the value of the squared distance D t between X and Y at time t.

Note that we could also replace dt 4θ 2 by z t in this and the following formulae, where z t is the value taken by the random variable Z t (see Definition 2). In any case, on R 2 , the rate of coalescences between two lineages at some time t depends only on the squared euclidean distance D t rather than the locations themselves.

While each of the two lineages is hit at a rate ∆λ, the total rate of events (hitting either X or Y ) is

ρ * X∨Y = 2∆λ 1 - u 0 4 exp - d t 4θ 2 (15)
The reason for this is that if we were to simply add the rates associated with each lineage, we would put double weight on the events affecting both lineages simultaneously, i.e., the coalescences. One should also note that even though the distribution of X t + posterior to a REX event of center z is normal with variance θ 2 , this is not the case if we consider the distribution of X t + posterior to a REX event and assuming that the same event did not hit Y . The probability density of X t + in such a case can be written down as follows:

p {X t + | !Y } (w) = R 2 exp -z-xt 2 2θ 2 -exp -z-xt 2 + z-yt 2 2θ 2 ∆ 2 1 -1 2 exp -dt 4θ 2 exp - z -w 2 2θ 2 dz ( 16
)
where, with a slight abuse of notation, we signify by writing !Y t , that the event does not affect the lineage Y . Inspection of the right-hand side reveals that the exponential function involving d t is only of significant magnitude in comparison to the leading term if x t and y t are close, or if θ 2 is large. If on the other hand θ 2 is small or the lineages can be assumed to be sufficiently distant from each other, the conditional distribution above remains wellapproximated by a normal.

The ΛV on the rectangle and on the plane are very similar in the initial stages. As time progresses, boundary effects come into play on the rectangle, while lineages can expand indefinitely on R 2 . This is illustrated in Figure 1, where the average squared euclidean distance between two lineages is shown for both cases as time progresses. Also, on the rectangle, one can see that the average distance approaches some equilibrium value, while in the plane, the distance seems to grow almost linearly. This will be verified in Section 5.2. Since T 0 on R 2 depends only on D t , it suffices to simulate D t , or alternatively, the "standardized" Z t . Figure 1 depicts 10000 such trajectories of Z t on R 2 . Individual trajectories resemble Brownian motions with a drift term of strength 2∆. We may approximate the cumulative distribution of T 0 from such simulations of D t (see Figure 2). Increasing the dispersal variance θ 2 seems to accelerate the process in the long term (however, this effect is different from increasing λ, which acts as a scaling parameter). Changing the initial distance d 0 appears to affect the limiting probability of coalescence 1 -p * . For large t, the distributions seem to run in parallel. Sections 4 and 5 will be devoted to finding numerical approximations of these curves.

Derivation of the coalescence probability

In the following, we describe two ways of approaching the distribution of T 0 by symbolical methods. We first assume that the habitat is R 2 .

The distribution as a solution to an ODE

We consider the probability that coalescence between the two ancestral lineages that started at distance d 0 takes place before or at time t + h. We may write Pr(T 0 ≤ t + h|d 0 ) = Pr(T 0 ≤ t|d 0 ) + Pr(t < T 0 ≤ t + h|d 0 ) = Pr(T 0 ≤ t|d 0 ) + Pr(T 0 ≤ t + h|d 0 , T 0 > t)(1 -Pr(T 0 ≤ t|d 0 )) which, by rearranging and considering the limit of h → 0, can be transformed into

∂ ∂t Pr(T 0 ≤ t|d 0 ) 1 -Pr(t ≤ T 0 |d 0 ) = lim h→0 1 h Pr(T 0 ≤ t + h|d 0 , T 0 > t) ( 17
)
Integrating and taking exponentials, we arrive at

Pr(T 0 ≤ t|d 0 ) = 1 -exp -lim h→0 1 h t 0 Pr(T 0 ≤ u + h|d 0 , T 0 > u))du ( 18
)
Let f u (•|T 0 > u, d 0 ) denote the density of D u , conditioned on T 0 > u and the distance at t = 0 being equal to d 0 . Then we have:

Pr(u < T 0 ≤ u+h|d 0 , T 0 > u) = ∞ 0 Pr(T u ≤ u+h|d u = x)f u (x|T 0 > u, d 0 )dx (19)
The limit in Eq. 18 may be moved inside the integral, and lim h→0 1 h Pr(T 0 ≤ h|d 0 = x) is given by the right-hand side of equation 13. Therefore, we have

Pr (T 0 ≤ t|d 0 ) = 1 -exp - 1 2 ∆λu 0 t 0 E exp (-Z u ) T 0 > u, d 0 du (20) so that log Pr(T 0 > t|d 0 ) = - 1 2 ∆λu 0 t 0 E exp (-Z u ) T 0 > u, d 0 du (21)
It is worth pointing out that one may exchange the integrals on the righthand side, allowing us to take the integral with respect to t over the density of Z t alone:

t 0 E exp (-Z u ) T 0 > u, d 0 du = t 0 ∞ 0 exp(-x)p Zu (x|T 0 > u, d 0 )dxdu = ∞ 0 t 0 exp(-x)p Zu (x|T 0 > u, d 0 )dudx = ∞ 0 exp(-x) t 0 p Zu (x|T 0 > u, d 0 )dudx
Hence, the probability that coalescence takes place before a given point in time given the initial distance between the two lineages can be understood as the integral, taken over the time period considered, of the moment-generating function of the standardized distance (term to the left of the equality sign in the equation above). It can also be understood as the moment-generating function of the random variable Z t (see Definition 2).

T 0 as a Cox Process

In this section, we provide a different take on the same problem by decomposing the ΛV into two stages:

1. In the first stage, X and Y (with fixed x 0 , y 0 ) move across R 2 , with the rate of events given by ρ * X∨Y -ρ * X∧Y ; every event affects either X or Y and Eq. 16 is used to update the positions of lineages (rather than a normal density). This induces a random path of the squared euclidean distance between them in R + 0 , i.e. a random piecewise-constant function

D t := X t -Y t 2 , t ∈ R + 0
2. Along R + 0 , "potential coalescent events" are distributed according to a non-homogenous Poisson process (i.e., a Cox Process [START_REF] Cox | Some statistical methods connected with series of events[END_REF]) with rate function

ρ(t) := 1 2 ∆λu 0 exp - D t ) 4θ 2 (22)
The first potential coalescent event encountered along R + 0 finally represents the actual coalescent event of the lineages.

The rates at which lineages change locations or coalesce are equal to the rates under the ΛV; therefore the above is an equivalent description of the process for two lineages. Obviously, we may also simulate the process in this way. Given a trajectory δ(t) of distances between lineages over time, the probability distribution of the time T 0 until we encounter a coalescence is given by

Pr(T 0 ≤ t|{δ(u), 0 ≤ u ≤ t}) = 1 -exp (-m(t)) , ( 23
)
where m(t) := t 0 ρ(u)du. The probability distribution of T 0 under the ΛV thus equals equation 23 averaged over all possible paths δ| [0,t] between 0 and t, i.e.

Pr(

T 0 ≤ t|d 0 ) = 1 -E (exp (-m(t))) (24)
Note that the random variable of this expression ist 0 1 2 ∆λu 0 expδ(u) 4θ 2 du, so the expectation here is taken over time as well as over space.

Remark 1. The similarity between equations 20 and 24 suggests that one may interchange expectation and exponential. Note however that the expectation in Eq. 24 is taken over all paths generated in the first stage of the ΛV, where coalescence events are not taken into account, whereas in Eq. 20 the expectation is conditioned on coalescence events not taking place up to u in the original process.

The coalescence process on a rectangle

Now, we consider again the case where the habitat is given by a rectangle A. We can, in fact, derive a slightly modified version of Eq. 20. Here, the coalescence probability depends on the lineage position relative to the border of A, which is why one needs to condition on X t and Y t (rather than D t or Z t ). More precisely, Eq. 19 becomes

Pr(u < T 0 ≤ u + h|x 0 , y 0 , T 0 > u) (25) = ∞ 0 Pr(T u ≤ u + h|X u = x, Y u = y, T 0 > u, x 0 , y 0 ) • f u (x, y|T 0 > u, x 0 , y 0 )d(x, y)
where f u (X u , Y u |T 0 > u, x 0 , y 0 ) denotes the joint density of X u , Y u conditioned on T 0 > u and x 0 , y 0 , and we can again move the limit into the integral and evaluate it, with Eq. 10 substituted for Eq. 13. The result is log (Pr(T 0 > t|x 0 , y 0 ))

= - t 0 E λu 2 0 exp - X u -Z 2 + Y u -Z 2 2θ 2 T 0 > u, x 0 , y 0 du ( 26
)
where Z is an event location uniformly distributed on A, and the expectation is taken over X u , Y u and Z.

Since the conditional expectation

E exp -Xt-Z 2 + Yt-Z 2 2θ 2
T 0 > t, x 0 , y 0 is bounded, it has to approach a limiting value c, with 1 ≥ c > 0 as t → ∞. If t is large, we have the following approximation for the density of coalescence times:

∂ ∂t Pr(T 0 ≤ t|x 0 , y 0 ) ∝ exp -λu 2 0 ct (27)
For large t, the density is thus proportional to that of an exponentially distributed random variable with parameter λu 2 0 c. More generally, the joint distribution of X t and Y t conditioned on T 0 > t approaches a quasi-stationary distribution [START_REF] Bartlett | Stochastic Population Models in Ecology and Epidemiology[END_REF]. Simulations suggest that it resembles the uniform distribution on (A × A). While one can evaluate the value of c numerically, we point out that it is also approximated by the equivalent term of Eq. 20, i.e.

c ≈

∆ 2 E exp (-Z t ) T 0 > t, x 0 , y 0 which relates c it to the adjusted distance Z t . Assuming X t , Y t are independent and uniformly distributed on A, and making use of a result presented in [START_REF] Philip | The probability distribution of the distance between two random points in a box[END_REF], the distribution of Z t is given by

p * Zt (x|T 0 > t, d 0 ) = 4θ 2 ×                                                -2 √ d w 2 h -2 √ d wh 2 + π wh + d w 2 h 2 , if 0 < d ≤ w 2 -2 √ d w 2 h -1 h 2 + 2 wh arcsin w √ d + 2 w 2 h √ d -w 2 , if w 2 < d ≤ h 2 -1 h 2 + 2 wh arcsin w √ d + 2 w 2 h √ d -w 2 -1 w 2 + 2 wh arcsin h √ d + 2 wh 2 √ d -h 2 -π wh -d w 2 h 2 , if h 2 < d ≤ w 2 + h 2 13
with d := 4θ 2 x and without restriction w ≤ h. c is then approximated by

E exp (-Z t ) T 0 > t, x 0 , y 0 = 0 -∞ exp(x) t 0 p Zt (x|T 0 > t, d 0 )dudx
and density of Z t is approximated by

p Zt (x|T 0 > t, d 0 ) ≈ δ(z 0 -x) exp(αt) + p * Zt (x|T 0 > t, d 0 ) (1 -exp(αt)) ( 28
)
where α is the probability that an event with uniformly chosen location on A neither affects lineage X located at x 0 nor Y at y 0 . In other words, exp(-αt) is the probability that at time t the lineages X, Y are both located at their initial positions x 0 , y 0 . It can be calculated using the formulae in Section 2. If X 0 and Y 0 are themselves uniformly sampled from A, the natural approximation for Pr(T 0 ≤ t) is 1 -exp(-λu 2 0 ct). If x 0 and y 0 are provided, we propose

Pr(T 0 ≤ t|x 0 , y 0 ) ≈ 1 -exp -λu 2 0 t exp(-αt) ∆z 0 2 + (1 -exp(-αt)) c
(29) with α defined as above. Generally, the results we obtain suggest that the approximation proposed here works best if the rectangle is not too large in relation to θ 2 . Otherwise, the fact that the quasi-stationary distribution is not exactly uniform seems to negatively affect the accuracy of determining c. The quasi-stationary distribution can also be found as the solution to a functional equation, but seems difficult to approach numerically.

The dynamics of Z t

The distribution of coalescence times on a rectangle could be approximated by relatively simple symbolical methods; to achieve the same for R 2 , analyzing the process of Z t , given the initial distance d 0 , turns out to be instructive. We assume u 0 = 1 from here on; the results may be reproduced in the same way for other values of this parameter. Consider the following function:

M Zt,d 0 (s) := E exp (-sZ t ) 1 T 0 >t d 0 (30)
Usually, we will denote this function by M Zt (s) for short, unless we want to consider multiple values of d 0 . One can think of this function as the moment-generating function of the random value Z t , multiplied by an additional indicator function that returns 0 if the pair of lineages has coalesced 14 at time t and 1 if it hasn't. This function conveys a lot of information on the ΛV in general and on coalescence times in particular. For example, we have

Pr T 0 ≤ t d 0 = 1 -M Zt (0) (31)
Furthermore, it holds that

∂ ∂s k M Zt (s) = E (-Z t ) k • exp (-sZ t ) 1 T 0 >t d 0 (32)
for all k ∈ N, and therefore

∂ ∂s k M Zt (0) = E (-Z t ) k 1 T 0 >t d 0 (33)
Using Eq. 31, we may even write

  ∂ ∂s k M Zt (0)   • (M Zt (0)) -1 = E (-Z t ) k T 0 > t, d 0 (34)
In particular, we can compute the expectation appearing in Equation 20:

E 1 2 ∆λu 0 exp(-Z t ) T 0 > t, d 0 (35) = E 1 2 ∆λu 0 exp(-Z t )1 T 0 >t d 0 Pr T 0 > t d 0 = - 1 2 ∆λu 0 M Zt (1) M Zt (0)
It is possible to obtain a closed form of the derivative of M Zt (s) with respect to t for any s. After close inspection of ∂ ∂t M Dt (s) and a number of algebraic manipulations (see Appendix), we arrive at

∂ ∂t M Zt (s) (36) = 2∆λ 1 s + 1 M Zt s 1 + s -M Zt (s) - 2 3s + 4 M Zt 1 + s 3s + 4 + ∆λ 2 M Zt (1 + s) (37)
The above is a "partial differential-functional equation" and to our knowledge not analytically solvable. For certain values of s, we still gain some insight on the process; for instance, plugging in 0 for s yields

∂ ∂t Pr(T 0 > t|d 0 ) = ∂ ∂t M Zt (0) = - 1 2 ∆λu 0 M Zt (1) (38)
Two more crucial features of M Zt (s) are:

Lemma 1. a) lim t→∞ ∂ ∂s k M Zt (s) = 0 for all s > 0, k ≥ 0. b) lim t→∞ M Zt (0) = p * > 0
We defer the proof of this lemma to the appendix. Lemma 1b) states that lineage pairs are not required to coalesce on R 2 . In some cases Z t never ceases to grow, and consequently the intensity of the coalescent process tends to 0 so quickly that coalescence never occurs. This result also suggests that the ΛV does not "come down from infinity", which is to say that if a sample of infinite size is taken from the population, then, looking at the genealogical process of this sample, one will always encounter an infinite amount of lineages ("dust") that have not coalesced up to any time t back in the past. This problem is also treated in [START_REF] Véber | The spatial lambda-fleming-viot process: An event-based construction and a lookdown representation[END_REF], where it is proven explicitly that the ΛV under its disc-based definition (see Definition 1) does not come down from infinity. Despite their differences, it seems only natural to expect a similar statement to hold for the gaussian version.

Approximation of the probability distribution

In order to approximate the numerical values of M Zt (0) = Pr(T 0 > t), we propose an approach that relies on the Taylor expansion of the function of interest, and one exploiting its representation as an ODE. Combining the two, one obtains a good approximation of M Zt (0). Further approaches (such as numerically solving Eq. 36 by a Runge-Kutta scheme) can be envisioned, but either seem less accurate or computationally unfeasible.

Calculation of the Taylor expansion

We consider the Taylor series expansion of M Zt (0) at t 0 = 0:

M Zt (0) = j∈N t j j! ∂ ∂t j M Zt (0) t=0 = j∈N g j t j (39) with g j := 1 j! ∂ ∂t j M Zt (0) t=0
. We obtain these coefficients by considering the differential equation Eq. 36 that is solved by M Zt (s). Multiple derivation with respect to t yields

∂ ∂t j M Zt (s) (40) = 2∆λ   1 s + 1 ∂ ∂t j-1 M Zt s 1 + s - ∂ ∂t j-1 M Zt (s)   + 2∆λ   - 2 3s + 4 ∂ ∂t j-1 M Zt 1 + s 3s + 4 + 1 4 ∂ ∂t j-1 M Zt (1 + s)  
Setting s = 0 in Eq. 40, we obtain, ∀j > 0

g j • j! = ∂ ∂t j M Zt (0) t=0 (41) = 2∆λ   ∂ ∂t j-1 M Zt (0) t=0 - ∂ ∂t j-1 M Zt (0) t=0   + ∆λ   - ∂ ∂t j-1 M Zt (1) t=0 + 1 2 ∂ ∂t j-1 M Zt (1) t=0   = - ∆λ 2   ∂ ∂t j-1 M Zt (1)   t=0
By virtue of Eq. 41, any term of the form

∂ ∂t k M Zt (σ) t=0
, σ > 0, may be expressed by terms of the form

∂ ∂t k-1 M Zt (τ ) t=0
, τ > 0. The repeated application of this equation results in an expression of the form

g j • j! = 4 j i=1 β i   ∂ ∂t 0 M Zt (σ i )   t=0 ( 42
)
where

∂ ∂t 0 M Zt (s) t=0 = M Z 0 (s) = exp -d 0 s 4θ 2
and

β i ∈ R, σ i > 0; so ultimately, g * 0k • k! = 4 k i=1 β i exp - d 0 σ i 4θ 2 (43)
Let γ (J) (t) := J j=0 g j t j denote the J-th order Taylor polynomial of M Zt (s). These polynomials approximate M Zt (s) very well for small values of t (Fig-ure 6). However, calculating successively higher orders quickly becomes computationally intense. We will therefore consider another strategy of approximating M Zt (s), sacrificing accuracy initially in exchange for being able to correctly display the long-term behavior of M Zt (s). An additional ingredient we will need is the asymptotic behavior of the first two moments of Z t .

Asymptotic of first and second moment

Recalling Eq. 32 and Eq. 33, we have

∂ ∂s M Zt (s) = E -Z t • exp (-sZ t ) 1 T 0 >t d 0 (44) ∂ ∂s 2 M Zt (s) = E Z 2 t • exp (-sZ t ) 1 T 0 >t d 0 ( 45
)
and setting s = 0,

∂ ∂s M Zt (s) s=0 = E -Z t 1 T 0 >t d 0 (46) ∂ ∂s 2 M Zt (s) s=0 = E Z 2 t 1 T 0 >t d 0 (47)
Performing the same derivations on the right-hand side of Eq. 36 and setting s = 0 leads to differential equations for the first and second moments of Z t :

∂ ∂t E Z t 1 T 0 >t d 0 = 2∆λ M Zt (0) - 1 8 3M Zt (1) + ∂ ∂s M Zt (s) s=1 (48) ∂ ∂t E Z 2 t 1 T 0 >t d 0 = 4∆λ M Zt (0) + 2E Z t 1 T 0 >t d 0 - 9 8 ∆λM Zt (1) + 1 8 ∆λ   6 ∂ ∂s M Zt (s) s=1 + 7 2 ∂ ∂s 2 M Zt (s) s=1   ( 49
)
The right-hand side of Eq. 48 is nonnegative, because M Zt (1) ≤ M Zt (0) as well as ∂ ∂s M Zt (s) s=1 ≤ M Zt (0), so unsurprisingly, E Z t 1 T 0 >t d 0 is monotonously increasing. Also, because M Zt and all its derivatives with respect to s evaluated at s > 0 vanish as t → ∞ (Lemma 1), we have the following approximation for large t:

∂ ∂t E Z t 1 T 0 >t d 0 ≈ 2∆λM Zt (0) (50) ∂ ∂t E Z 2 t 1 T 0 >t d 0 ≈ 4∆λ M Zt (0) + 2E Z t 1 T 0 >t d 0 (51)
Furthermore, Lemma 1b) states that M Zt (0) can be treated like a nonzero constant for large t. This allows us to solve the system exactly (substituting equalities for both "≈"):

E Z t 1 T 0 >t d 0 ≈ 2∆λM Zt (0) • t + c 1 (52) E Z 2 t 1 T 0 >t d 0 ≈ 8∆ 2 λ 2 M Zt (0) • t 2 + 4∆λ(M Zt (0) + 2c 1 ) • t + c 2 (53)
with initial values c 1 , c 2 > 0 (for which we could use

c 1 = E Z 0 1 T 0 >0 d 0 = d 0 /4θ 2 and c 2 = E Z 2 0 1 T 0 >0 d 0 = d 0 4θ 2

2

).

Let

µ 1 (t) := E Z t T 0 > t, d 0 , µ 2 (t) := E Z 2 t T 0 > t,
d 0 and σ 2 (t) denote the first and second moment under the condition of no coalescence up to time t, and the conditional variance respectively. We obtain these moments by dividing the above equations by Pr(T 0 > t d 0 ) = M Dt (0). From the above approximation, we get:

µ 1 (t) ≈ 2∆λ • t + c 1 M Zt (0) (54) µ 2 (t) ≈ 8∆ 2 λ 2 • t 2 + 4∆λ 1 + 2 c 1 M Zt (0) • t + c 2 M Zt (0) (55)
σ 2
t is obtained by applying V(X) = E(X 2 )-E(X) 2 . These approximations are not very precise and should only be taken to reflect the asymptotic behaviour. Simulations suggest that their accuracy increases considerably if the initial distance d 0 is taken to be large (see Figure 4). In any case, if t is large, M Zt (0) is close to p * (Lemma 1b), so c 1 /M Zt (0) and c 2 /M Zt (0) are almost constant, which shows that µ 1 (t) is asymptotically linear (and µ 2 (t) quadratic).

Using the characteristic function of the Gamma Distibution

In the following, we will consider the random variable Z t , defined as

Z t conditioned on T 0 > t. Consequently, E Z t d 0 = µ 1 (t), E Z 2 t d 0 = µ 2 (t) and V Z t d 0 = σ 2 (t).
We assume that Z t follows a Gamma distribution with shape and rate parameters α t and β t . This assumption is justified by the fact that the distribution of Z t is a convolution of χ 2 -distributions, which are of the Gamma family, and can be supported by simulations (see Figure 5). Then, we have for the characteristic function of Z t , evaluated at s = i (i representing the imaginary

χ Γ(αt,βt) (i) = E exp(-Z t ) d 0 = 1 + 1 β t -αt (56)
The parameters α t and β t can be computed from the first two moments of the distribution:

β t = µ 1 (t) σ 2 (t) -1 α t = µ 1 (t) 2 σ 2 (t) -1
Eq. 20 then becomes

Pr(T 0 ≤ t|d 0 ) ≈ 1 -exp   - t 0 1 2 ∆λ 1 + σ 2 (u) µ 1 (u) - µ 1 (u) 2 σ 2 (u) du    (57)
In this, we can use the approximations for µ 1 (t) and σ 2 (t) (Eq. 54, Eq. 55) developed in section 5.2. Equipped with this, Eq. 57 is an approximation scheme for Pr(T 0 ≤ t|d 0 ). While this usually is not particularly close to the distribution of T 0 (see Figure 7), there are several possibilities of improvement. If there are known or reasonably well approximated values of Pr(T 0 ≤ x|d 0 ) for some x > 0, we have

Pr(T 0 ≤ t|d 0 ) ≈ 1 -Pr(T 0 > x|d 0 ) exp   - t x 1 2 ∆λ 1 + σ 2 (u) µ 1 (u) - µ 1 (u) 2 σ 2 (u) du    (58)
For example, M Zt (0) can be approximated up to x > 0 by a Taylor polynomial, while for t > x one utilizes Eq. 58. Below, we show the result of this using the asymptotic approximations (Eq. 54, Eq. 55) for the moments (this procedure is dubbed the "naive" approach). It is possible to attain more precision by calculating µ 1 (x) and σ 2 (x) exactly (e.g., by another Taylor scheme), and continue µ 1 (t) linearly and σ 2 (t) quadratically for t > x, using the results of section 5.2. These approximation schemes typically results in a "knee" of the curve (i.e, a point where it visibly ceases to be smooth). One can mitigate this by calculating the (unique) values of μ1 (x) and σ2 (x) that provide a smooth continuation in Eq. 58 (they do not necessarily equal the true values µ 1 (x), σ 2 (x)). For t > x, µ 1 (t) can be extended linearly and σ 2 (t) quadratically. This "smooth" way of approximating M Zt (0) fits the distribution of T 0 rather well. Generally, a higher threshold x results in a higher accuracy. In practice, it may become necessary to approximate M Zt,d 0 (0) for several different initial values of d 0 (As a reminder to the reader, d 0 is included in the subscript in the definition of M Zt,d 0 (0), see Eq. 30, but was omitted throughout most of the previous sections for the sake of convenience). In order to do that, one can take advantage of the observation in Figure 2 that in the long run, the derivatives ∂ ∂t M Zt,d 0 (0) for different initial conditions start to closely resemble each other; i.e., their slopes become similar as t gets large. One possible explanation of this is that lineage pairs are expected to grow apart almost linearly, regardless of the value of d 0 , unless they coalesce early on. To support this claim, we refer to our simulations. Assuming we have a precomputed approximation M 0 (t) of M Zt,0 (0), a fast way of approximating M Zt,d (0) for some d > 0 is therefore the utilization of the Taylor expansion for small values of t ≤ x, and continuing the curve for t > x by the slope of M Zt,0 (0) that is obtained according to the scheme we discussed (Section 5.3). Again, this can be done in such a way that the resulting curve is smooth, e.g., by "shifting" to the point x that guarantees a smooth transition. To be precise, let γ

M d (t) =    γ (k) d (t) t ≤ x M 0 (t -x + x ) t > x (59)
is smooth. Hence, M d (t) is an approximation of M Zt,d (0) whose long-term behaviour agrees with that of M 0 (t).

Visualisations of the approximations

We show some approximations of the coalescent probability for θ 2 , λ = 1, u 0 = 1 and Z 0 = 0. The density itself is obtained from a set of 10000 simulation runs of Z t (see also Figures 1 and2).

Discussion

In this study, we describe ways of approximating the distribution of the time to coalescence Pr(T 0 ≤ t|d 0 ) for pairs of lineages under the ΛV on R 2 7: Comparison between the "naive" approximation using Eq. 57 as well as continuing with Eq. 58 after the threshold value x (left), and the "smooth" approach. Not only does the naive way introduce an unrealistic point of non-differentiability, but also deviates from the probability distribution much faster. as well as on a finite rectangle. The major difference between the two cases is the fact that T 0 is almost surely finite on a rectangle, whereas our analysis showed that on the plane, there is a chance p * , dependent on the parameter choice of the model, that lineage pairs escape the coalescent mechanism entirely (as has been suspected and investigated before in different contexts, e.g. [START_REF] Véber | The spatial lambda-fleming-viot process: An event-based construction and a lookdown representation[END_REF]). It is noteworthy that this does not necessarily pose a problem in applications, since T 0 can be conditioned on coalescence taking place by simply dividing its distribution by 1 -p * . For the ΛV on a rectangle A, Pr(T 0 ≤ t|x 0 , y 0 ) becomes proportional to an exponential function as t gets large. It is reasonable to expect this to be true not just on a rectangle, but on any compact habitat. Especially interesting cases for similar analyses appear to be spherical and toric habitats, since those are compact objects, but the reproduction mechanism can be defined such that border effects are avoided. Regarding the approximation of Pr(T 0 ≤ t|x 0 , y 0 ), there are two cases to be distinguished: First, if A is of "moderate" size in relation to θ 2 , the determining parameters of the distribution can be well-approximated, as the equilibrium distribution of the location of the two lineages is nearly uniform and the corresponding density for the distribution of pairwise distance has a closed-form formula [START_REF] Philip | The probability distribution of the distance between two random points in a box[END_REF]. If A is large, simulations show that the deviation of the equilibrium distribution, while small in absolute terms, negatively impacts the approximation scheme. However, one natural solution to this problem might be to simply rely on the results for the process on R 2 in such a case. On R 2 , we related Pr(T 0 ≤ t|d 0 ) to the process of Z t , which was defined as the squared euclidean distance D t between a pair of lineages, divided by 4θ 2 . Our analysis revealed several properties of Z t as well, for instance the asymptotically linear growth of its expectation. The function M Zt (s), defined similarly to a moment generating function, with the addition of an indicator variable, is linked to Pr(T 0 ≤ t|d 0 ) via a series of algebraic and differential equations. Perhaps similar techniques can be used to analyze survival times in other stochastic processes with spontaneous entries into absorbing states (for the ΛV, these are the coalescences). The approximation scheme we suggest relies on evaluating Pr(T 0 ≤ t|d 0 ) = 1 -M Zt (0) for small t using a Taylor polynomial, and continuing the approximation for larger t by assuming that Z t is gamma-distributed. We have ignored u 0 , the "mortality" parameter, by systematically considering its value as equal to 1 in our derivations. Intuitively, the effect of changing u 0 should be similar to changing the value of λ. Calculations presented in Section 4 can be repeated by making the parameter u 0 explicit, resulting in an additional parameter in the approximation. Yet, more work would be required in order to verify that coalescence probabilities that involve u 0 could eventually be approximated by using the very same approaches as the ones presented here. The presented results describe the ancestral process for a sample of size n = 2 under the ΛV. For bio-statistical purposes, it would certainly be helpful to extend these to bigger sample sizes. Most of the formulae in Section 2 can, at first glance, be modified to incorporate more than two lineages, so it seems possible to approach waiting times for multiple mergers at least approximately, which would allow for more sound statistical analyses in phylogeography. However, knowledge about the process for n = 2 enables statistical assessment at least for pairs of samples, with all pairs in a sample considered as independent from one another. We maintain that the presented methodology may be used to obtain estimates of the parameters θ 2 , λ from geo-referenced genomic data. At the very least, estimates of θ 2 and λ obtained in this way may serve as valuable indicators of the speed of evolution and dispersing potential of biological organisms.

Appendix

Derivation of Eq. 36. We have

∂ ∂t M Zt (s) (60) = lim h→0 1 h E exp (-sZ t+h ) 1 T 0 >t+h d 0 -E exp (-sZ t ) 1 T 0 >t d 0 = E lim h→0 1 h E exp (-sZ t+h ) 1 T 0 >t+h -exp (-sx) Z t = x, d 0 1 T 0 >t d 0
We may decompose the interior expectation by conditioning the number N h of events affecting the lineages that are encountered in the interval [t, t+h]. Any event that hits either X or Y or both contributes to the number N h . We will show that in the limit of h → 0, only the case N h = 1 is relevant.

E exp (-sZ t+h ) 1 T 0 >t+h -exp (-sx) Z t = x, d 0 = i∈N Pr(N h = i Z t = x, d 0 )E e -sZ t+h 1 T 0 >t+h -e -sx N h = i, Z t = x, d 0 (61)
If N h = 0, we have Z t+h = x and the corresponding term vanishes. For N h = 1, we can express the probability by multiplying the density with which an event occurs at t + u ∈ [t, t + h] (exponential with parameter given by Eq. 15) with the probability of no further event during the remainder of this interval and integrating over u ∈ [0, h]. Z + t+u denotes the distance of the lineages immediately after the event that occurs at time t + u; note that X and Y may coalesce due to this event, in which case Z + t+u = 0. Then, it holds that lim where the expression (64) accounts for all cases in which only one lineage is affected by the event, and the one below for those in which the lineages coalesce. Let w denote the value Z + t immediately after the event. We can assume without restriction that lineage X t is hit by the event and Y t remains at its position. Then, the density of w is given by Eq. 16. We find (66)

The evaluation of the integral is extensive, but ultimately trivial. Assembling everything, we arrive at Eq. 36: Having established that M Zt (s) converges to 0, we know that for any δ ∈ R + , the probability Pr(Z t ≤ δ, T 0 > t d 0 ) converges to zero as well (otherwise, we end up with a contradiction). Consequentially, since Therefore, there exists an α > 0 such that E Dt 4θ 2 1 T 0 >t d 0 > αt for all t larger than some t 0 . In turn, looking back at Eq. 48, this implies that Pr(T 0 > t d 0 ) = E(1 T 0 >t d 0 ) = M Zt (0) does not converge to 0.

Figure 1 :

 1 Figure1: Left: Trajectories of Z t on R 2 , using λ = 1, u 0 = 1, θ 2 = 1/4, Z 0 = 0, over 1000 units of simulated time. Right: the average Z t over time on R 2 displays a linear trend, on the rectangle it approaches an equilibrium value. In both cases, trajectories undergoing a coalescence were ruled out when generating the plots.

Figure 2 :

 2 Figure 2: Left: Coalescence probability for varying values of θ 2 . Increasing θ 2 has a similar effect as speeding up time. According to these simulations, p * ≈ 0.2 for d 0 = 0. Right: Coalescent probability for θ 2 = 1/4 and d 0 = 0, 1, 2, 10. Again, λ = 1 and u 0 = 1 were used.

Figure 3 :

 3 Figure 3: Left: Probability of coalescence times on a 10 × 10 rectangle, θ 2 = 1/4, λ = 1, for fixed initial distances 0. The value of c is around 0.012. Shown in dots is the function 1 -exp(-c(t + t 0 )) (an exponential distribution with parameter c, shifted to the left by t 0 ≈ 0.8). The bigger t gets, the more similar the two curves become. Right: The quasistationary distribution of Z t on the rectangle (black line) resembles the distribution of Z t when both positions are uniform (dots), with a slight preference for higher values.

Figure 4 :

 4 Figure 4: Top left: Average of Z t (i.e., µ 1 (t)) with respect to time and d 0 = 0 (grey) and d 0 = 10 (black). Shown in dotted lines is the respective approximation obtained by the results of section 5.2. Top right: One observes that the relative error is smaller for d 0 = 10 than for d 0 = 0. Bottom: Here, we compare the average squared Z t (i.e., µ 2 (t)) with its approximation. Again, d 0 = 10 (as t increases) yields the smallest relative error.

Figure 5 :

 5 Figure 5: Gamma distributions (red) with parameters derived from the asymptotic moments are good approximates for the observed distributions (black) of Z t . Left: t = 100, right: t = 1000.

5. 4 .

 4 Utilizing the similarity of the slope of M Zt (0) for differing d 0

d

  (t) denote a Taylor approximation of M Zt,d (0). Then, for given x > 0, we may choose x such that the function

Figure 6 :

 6 Figure 6: Taylor approximations of the probability density of coalescences. Shown in color from yellow to blue are the approximations of increasing order up to 16.

Figure

  Figure7: Comparison between the "naive" approximation using Eq. 57 as well as continuing with Eq. 58 after the threshold value x (left), and the "smooth" approach. Not only does the naive way introduce an unrealistic point of non-differentiability, but also deviates from the probability distribution much faster.

Figure 8 :

 8 Figure 8: Long-term behaviour of the probability distribution and the smooth approximation for x = 2.

h→0 1 h 1 h

 11 Pr(N h = 1 Z t = x, d 0 ) for N h ≥ 2, it clearly holds that Pr(N h = k ≥ 2 Z t = x, d 0 ) ≤ (2∆λh) k /k!e -2∆λ(63)because 2∆λ is an upper bound to the total rate of events (Eq. 15). Because of that, we havelim h→0 Pr(N h = k ≥ 2 Z t = x) ≤ lim

  1+s Zt -e -sZt + 2 3s + 4 e -4(s+1) 3s+4 Zt 1 T 0 >t d 0+ E ∆λ 2 e -(1+s)Zt 1 T 0 >t d 0(67)from which the claimed identity follows by linearity of the expectation.Proof of Lemma 1. a) Since M Zt (0) ∈ [0, 1] and monotonously falling, there exists a limit c ∈ [0, 1]. Consequently, lim t→∞ M Zt (1) = 0, because of the established relationship between the two (Eq. 38).For s > 1, M Zt (1) > M Zt (s), so lim t→∞ M Zt (s) = 0 holds here as well.Concerning s < 1, we haveM Zt (s) = E exp (-sZ t ) 1 T 0 >t d 0 ≤ E exp (-Z t ) 1 T 0 >t d 0 s = (M Zt (1)) s by Markov's inequality. Therefore, M Zt (s) converges to 0 uniformly on any interval [σ, ∞), σ > 0.

  4 e -sZ t+h 4θ 2 1 T 0 >t+h -e -sx

						4θ 2	N h = 1, Z t = x, d 0
	= 2∆λE 1 -	e -x 4	e -sZ + t 4θ 2 1 T 0 >t -e -sx 4θ 2	Z t = x, d 0 , Event at t
	= 2∆λE	1 -	e -x 2	e -sw 4θ 2 -e -sx 4θ 2	p {X + t |!Yt} (w)dw Z t = x, d 0 (64)
	-∆λ	e -x/4θ 2	e -sZ t 4θ 2

  converges to 0 for any k > 0 as x → ∞, all the derivatives, given by Zt (s) = E (-Z t ) k exp (-sZ t ) 1 T 0 >t d 0 necessarily converge to 0 as well. b) Consider the function y(t) := E Zt t 1 T 0 >t d 0 . It solves the differential equation Zt (0)), the following must hold for any solution y(t) to this differential equation: := ct 0 • t -1 with limit 0 as t → ∞. Thus,

									x 4θ	k	exp -sx 4θ 2
	∂ ∂s	k M ∂ ∂t y(t) = 2∆λ M Zt (0) -	1 8	3M Zt (1) +	∂ ∂s	M Zt (s)	s=1	• t -1
		-E	Z t t	1 T 0 >t d 0 • t -1		(68)
		= 2∆λ M Zt (0) -	1 8	3M Zt (1) +	∂ ∂s	M Zt (s)
		∀t > t 0 : y(t) > z(t) if	∂ ∂t	z(t) = z(t) • t -1	(70)
			lim t→∞	y(t) = lim t→∞	E	Z t t	1 T 0 >t d 0 > 0	(71)

s=1 • t -1 -y(t) • t -1

(69)

Since the first term on the right-hand side is strictly positive (we have

M Zt (1) ≤ M Zt (0), ∂ ∂s M Zt (s) s=1 ≤ M provided some initial value c = y(t 0 ) = z(t 0 ), t 0 > 0. Moreover, if lim t→∞ z(t) = δ ∈ R, then lim t→∞ y(t) > δ. The solution to ∂ ∂t z(t) = z(t) • t -1 , z(t 0 ) = c is z(t)

Acknowledgements

This work was funded by the Agence Nationale pour la Recherche [https: //anr.fr/] through the grant GENOSPACE, and the Walter-Benjamin Program (WI 5589/1-1) of the DFG [https://dfg.de/].


Juan Barrios Rudloff

Simon Gabay
email: simon.gabay@unige.ch

Florian Cafiero
email: florian.cafiero@chartes.psl.eu

Martin Debbané
email: martin.debbane@unige.ch

Juan Barrios
email: author.juan.barrios@unige.ch

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The assumptions that the "words we use in natural language [. . . ] reveal a tremendous amount of information about our social interactions and personality" and that Natural Language Processing

Attention-deficit/hyperactivity disorder (ADHD)

ADHD is a neurodevelopmental disorder that affects 5.6% in teenagers aged 12 to 18 years [START_REF] Salari | The global prevalence of ADHD in children and adolescents: a systematic review and meta-analysis[END_REF]. Its causes are multifactorial: several genetic and environmental risk factors act together to increase both susceptibility to the disorder and the extent of psychiatric comorbidities [START_REF] Faraone | Attentiondeficit/hyperactivity disorder, Nature Reviews[END_REF]. Regarding its diagnosis, it is reliable and valid when evaluated with standard criteria for psychiatric disorders, but the expression of symptoms varies as a function of patient developmental stage in both the social and the academic contexts [START_REF] Faraone | Attentiondeficit/hyperactivity disorder, Nature Reviews[END_REF]. All this makes ADHD diagnoses a time consuming task (up to one working day) requiring a high level of expertise.

Furthermore, during adolescence, young people with ADHD are especially prone to experience difficulties in interpersonal relationships, characterised by conflicts within the family or at school. In the latter case, conflicts with classmates may in turn lead to peer rejection, social isolation or school failure [START_REF] Bagwell | Attention-deficit hyperactivity disorder and problems in peer relations: Predictions from childhood to adolescence[END_REF][START_REF] Barkley | Adolescents with ADHD: Patterns of behavioral adjustment, academic functioning, and treatment utilization[END_REF][START_REF] Simmons | Bullying and depression in youth with ADHD: A systematic review[END_REF]. Besides these relational problems, adolescents with ADHD are also exposed to higher risks of neurocognitive dysfunction, substance abuse, low self-esteem and social disability [START_REF] Faraone | Attentiondeficit/hyperactivity disorder, Nature Reviews[END_REF]. In summary, this leads to a situation in which a complex diagnosis with extremely serious health and social implications for patients must be made, involving time and resources that are not always available and in a context of high prevalence.

Psychology and Natural Language Processing (NLP)

For more than ten years now, the use of NLP has established itself as a valid method for different types of clinical applications in neuroscience and psychiatry [START_REF] Crema | Natural language processing in clinical neuroscience and psychiatry: A review[END_REF], which explains an upward trend for this approach and a growing need to develop new detection methods [START_REF] Zhang | Natural language processing applied to mental illness detection: a narrative review[END_REF]. Clinical applications include psychological profiling [START_REF] Noecker | Psychological profiling through textual analysis[END_REF][START_REF] Kim | College students with ADHD traits and their language styles[END_REF], or the detection of depression [START_REF] Eichstaedt | Facebook language predicts depression in medical records[END_REF][START_REF] Pérez | Automatic depression score estimation with word embedding models[END_REF] and psychological distress [START_REF] Manabe | Estimation of psychological distress in japanese youth through narrative writing: Text-based stylometric and sentiment analyses[END_REF] with different methods: word embeddings, sentiment analysis or stylometry. The latest research makes it possible not only to identify mental health risks (like patients who exhibit suicidal risk behaviour [START_REF] Tchounwou | Environmental research and public health[END_REF]), but also to predict depression [START_REF] Eichstaedt | Facebook language predicts depression in medical records[END_REF] or schizophrenia in first-episode of psychosis [START_REF] Figueroa-Barra | Automatic language analysis identifies and predicts schizophrenia in first-episode of psychosis[END_REF][START_REF] Tang | Natural language processing methods are sensitive to subclinical linguistic differences in schizophrenia spectrum disorders[END_REF][START_REF] Rezaii | A machine learning approach to predicting psychosis using semantic density and latent content analysis[END_REF] on a linguistic basis.

Data and features

Corpus

The corpus is made up of a series of 198 Self-Defining Memories (SDMs, cf. section 2.3) collected in two samples, one consisting of adolescents with a diagnose of ADHD and the other with control participants (cf. section 2.2), who all had to write a total of three SDMs each. All texts are written in French.

SDMs were handwritten by the participants and then transcribed by the person in charge of the experiment. If spelling errors can provide valuable linguistic information, we noticed that some errors were introduced or removed during computer entry of the text. We have therefore decided to correct all of them: a first time with a professional spell checker, a second time with the Hunspell package [START_REF] Ooms | hunspell: High-Performance Stemmer, Tokenizer, and Spell Checker[END_REF].

Participants

ADHD group Adolescents with ADHD were recruited as part of a research project conducted at the Unit of Developmental Clinical Psychology at the University of Geneva. The project was advertised in local parents' associations for children with ADHD and through collaborations established with local child psychiatrists. Diagnostic criteria were investigated by detailed anamnestic interviews and confirmed by a semi-structured parents interview using the ADHD Child Evaluation cf. [START_REF] Young | ADHD Child Evaluation (ACE), A diagnostic interview of ADHD in children[END_REF]. All diagnostic assessments were conducted by experienced clinical psychologists specialised in ADHD.

Control group

The control sample was recruited from the general population in Geneva by means of advertisements and personal contact. Participants were volunteers, native French or fluent French speakers and received a compensation for their participation.

Inclusion/Exclusion criteria Inclusion criteria were age [START_REF] Zhang | Natural language processing applied to mental illness detection: a narrative review[END_REF][START_REF] Noecker | Psychological profiling through textual analysis[END_REF][START_REF] Eichstaedt | Facebook language predicts depression in medical records[END_REF][START_REF] Pérez | Automatic depression score estimation with word embedding models[END_REF][START_REF] Manabe | Estimation of psychological distress in japanese youth through narrative writing: Text-based stylometric and sentiment analyses[END_REF][START_REF] Tchounwou | Environmental research and public health[END_REF], fluency in French and, for the ADHD group, meeting current diagnostic criteria for ADHD [START_REF] Association | Diagnostic And Statistical Manual Of Mental Disorders, Fifth Edition[END_REF]. Exclusion criteria were history of psychotic disorders, borderline personality disorder, autism spectrum disorder or neurological disorders.

Self-defining memories

SDMs are written texts that represent a specific type of autobiographical memories associated with the self-concept and a sense of coherence and continuity in one's ongoing individual history [START_REF] Blagov | Self-defining memories-narrative features in relation to adaptive and maladaptive personality traits (replication and extension of blagov & singer[END_REF]. They were collected with the SDM Task [START_REF] Thorne | Manual for Coding Events in Self-Defining Memories[END_REF][START_REF] Singer | Classification System & Scoring Manual for Self-Defining Memories[END_REF], during which participants are asked to evoke personal memories of events with specific attributes. These events 1. occurred at least one year ago; 2. are important and generally vividly represented; 3. are meaningful and useful to help the participant (or a significant other) to understand who s.he is; 4. are related to an important and enduring theme; 5. are either positive or negative but must generate strong feelings; 6. were recalled many times on a voluntary basis or spontaneously.

While listening to this description, participants had a sheet of paper in front of them that summed up these principal points. Participants were then told to imagine a situation where they meet someone they like very much and with whom they agree during a walk to talk about who they really are, their "Real Me", sharing several personal past events that powerfully convey how they have become the person they currently are. Participants were then given three sheets of paper on which they had to write down a SDM on each of them.

Prediction task

As already said, previous studies have shown that persons with ADHD have a different linguistic style from non-ADHD groups [START_REF] Yoder | Predicting lexical density growth rate in young children with autism spectrum disorders[END_REF][START_REF] Kim | Distinctive linguistic styles in children with ADHD[END_REF][START_REF] Kim | College students with ADHD traits and their language styles[END_REF]. It should therefore be possible to classify texts from the ADHD and control groups with stylometric methods.

Hypothesis

Function words (FW) are words used both in stylometry [START_REF] Kestemont | Function Words in Authorship Attribution: From Black Magic to Theory?[END_REF] and in psychology [START_REF] Chung | The psychological functions of function words[END_REF]. These words have little lexical meaning and a grammatical role in the sentence (e.g. articles, prepositions, conjunctions, auxiliary verbs. . . ). As such, FW are opposed to content words, which have a semantic content (e.g. nouns, adjectives, verbs. . . ), but there is no clear delimitation between the two groups. Indeed, some words can be classified in one or the other category such as pronouns. Pronouns have indeed attracted much attention from researchers, who have emphasised their particular status first in literature [START_REF] Muller | Sur quelques scènes de Molière, essai d'un indice du style familier[END_REF][START_REF] Sjöblom | L'indice pronominal est-il encore d'actualité ?[END_REF] and then in psychology [START_REF] Pennebaker | The Secret Life of Pronouns: What Our Words Say About Us[END_REF].

According to Chung and Pennebaker [START_REF] Chung | The psychological functions of function words[END_REF], examining the use of FW in natural language samples has provided a non-reactive way to explore personality processes. For example, it has been found that use of specific FW is related with affective states [START_REF] Weintraub | Verbal Behavior in Everyday Life[END_REF][START_REF] Chung | The psychological functions of function words[END_REF], depression [START_REF] Rude | Language use of depressed and depressionvulnerable college students[END_REF][START_REF] Mehl | The lay assessment of subclinical depression in daily life[END_REF], reactions to individual life stressors [START_REF] Pennebaker | Language use and personality during crises: Analyses of mayor rudolph giuliani's press conferences[END_REF], reactions to socially-shared stressors [START_REF] Campbell | The secret life of pronouns: Flexibility in writing style and physical health[END_REF][START_REF] Stone | Trauma in real time: Talking and avoiding online conversations about the death of princess diana[END_REF][START_REF] Gortner | The archival anatomy of a disaster: Media coverage and community-wide health effects of the texas A&M bonfire tragedy[END_REF][START_REF] Cohn | Linguistic markers of psychological change surrounding september 11[END_REF], deception [START_REF] Pennebaker | Linguistic styles: Language use as an individual difference[END_REF][START_REF] Newman | Lying words: Predicting deception from linguistic styles[END_REF], status [START_REF] Pennebaker | Pronoun use and dominance[END_REF], sex [START_REF] Newman | Lying words: Predicting deception from linguistic styles[END_REF] and age [START_REF] Pennebaker | Words of wisdom: Language use over the life span[END_REF]. We think that the use of FW differs between our two working groups (ADHD vs control) and that it is therefore possible to automatically classify one and the other group on the basis of FW. For years now, stylometry has been used to classify all types of documents [START_REF]Affaires de style : du cas Molière à l'affaire Grégory, la stylométrie mène l'enquête[END_REF], and seems to be the most suitable method for this task.

In addition to FW, we also propose to use another traditional feature of stylometric research: characters 3-grams [START_REF] Koppel | Computational methods in authorship attribution[END_REF], which can capture lexical, and even grammatical preferences. Such a feature has already shown its relevance for French texts in previous studies on authorship attribution [START_REF] Cafiero | Why Molière most likely did write his plays[END_REF], and has also proven its capacity to capture more than the authorial signal [START_REF] Gabay | Beyond idiolectometry? on racine's stylometric signature[END_REF].

Set up

Support Vector Machine (SVM) Often used in stylometry, unsupervised approaches [START_REF] Camps | Setting bounds in a homogeneous corpus: a methodological study applied to medieval literature[END_REF][START_REF] Gómez-Adorno | Hierarchical clustering analysis: the best-performing approach at pan 2017 author clustering task[END_REF][START_REF] Mccarthy | Who wrote wuthering heights?[END_REF][START_REF] Cafiero | Why Molière most likely did write his plays[END_REF][START_REF] Schmidt-Petri | Who authored on liberty? stylometric evidence on harriet taylor mill's contribution[END_REF] seemed less adapted than supervised techniques for this profiling task. A recent survey [START_REF] Hacohen-Kerner | Survey on profiling age and gender of text authors[END_REF] has shown that classical machine learning methods still perform better for profiling in similar settings (short texts, boolean or few categories) than deep learning. We turned to Support Vector Machines (SVMs) rather than random forest [START_REF] Ikae | Unine at pan-clef 2022: Profiling irony and stereotype spreaders on twitter[END_REF] or logistic regression [START_REF] Modaresi | Exploring the effects of cross-genre machine learning for author profiling in pan[END_REF][START_REF] Ward | France's citizen consultation on vaccination and the challenges of participatory democracy in health[END_REF], as it allows for easy interpretation and have established themselves as a standard method in stylometry [START_REF] Eder | Rolling stylometry[END_REF][START_REF] Cafiero | Psyché' as a Rosetta stone? assessing collaborative authorship in the french 17th century theatre[END_REF].

SuperStyl In this study, all analyses were implemented with the Python SuperStyl package [START_REF] Camps | SUPERvised STYLometry (SuperStyl)[END_REF]. This package has been used to build stylistic profiles with very good results [START_REF] Cafiero | Who could be behind QAnon? authorship attribution with supervised machine-learning[END_REF]. SuperStyl use internally the SVM and pipeline facilities from scikit-learn [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF].

Data All participants' SDMs are collected in a single file, with the exception of those written by two people from the control group and two from the ADHD group for a final blind test. Several word sample sizes are tested (1'000, 1'250, 1'500 and 2'000).

Parameters We tried to use two types of features: FW and character 3-grams for the reasons previously exposed. Because the length of SDMs varies a lot and we have more SDMs in the control group than in the ADHD groups (cf. section 4.3), we have tested different sampling methods (no sampling, downsampling, upsampling, Tomek links) and the use of class weights. All tests are conducted with a linear kernel and a 10-fold validation, on data normalised by using z-scores for variables and applied Euclidean vector-length normalisation (L2 normalisation).

Results

Scores Best results are achieved with 1'500 words samples, 10-fold validation, class weights and Tomek Links (for FW) / downsampling (for 3-grams). Accuracy is slightly lower with FW (0.85) than with 3-grams (0.92), which appears here as a promising indicator for research in psycholinguistics. However, the results remain surprisingly good in both cases. The recall for the ADHD group deserves special attention: for obvious reasons, a maximum number of people with ADHD must be identified and a minimum must be misclassified. With 0.75, results are not satisfactory yet, but promising.

Lexical exploration

In a medical context, predicting is not enough: it is necessary to explain the results. The study of our corpus could make it possible to give initial explanations as to the quality of the prediction, but also to confirm several hypotheses.

Hypotheses

Two standard measures have been used to explore the lexical information in the corpus:

1. Lexical Density measures the structure and complexity of a text. It provides a measure of the proportion of lexical items (types) in relation to the total number of tokens in the text [START_REF] Johansson | Lexical diversity and lexical density in speech and writing[END_REF], and thus helps evaluating the amount of information in a given document [START_REF] Halliday | Spoken and written language[END_REF]. Diversity evolves during human lifespan and it is influenced by different factors like education or the communication style of the family. Other studies in the field of psychology have found a relationship between neurodevelopmental disorders and lexical density, where children with autism spectrum disorders tend to have a lower density than the average population [START_REF] Yoder | Predicting lexical density growth rate in young children with autism spectrum disorders[END_REF]. To the best to our knowledge, no study has yet explored lexical density in the linguistic production of adolescent with ADHD. Considering the fact that ADHD is a neurodeveloppmental disorder, measures of lexical density could differentiate adolescents with ADHD from those with a typical development. 2. Lexical Diversity measures how many types are used in each adolescent's narrative.

SDMs that are lexically diverse use a wide range of vocabulary, of synonyms and a precise language to describe their memories. Previous studies have shown that measures of lexical diversity do not differentiate adolescents with ADHD from those with a typical development [START_REF] Redmond | Conversational profiles of children with ADHD, SLI and typical development[END_REF], we should therefore expect a similar result.

Set up

Pairwise matching is used to create samples balancing the ADHD and the control groups with respect to the means of participants' sex 1 and age in order to make them comparable. We matched the two samples using the cardinality matching method to find the largest matched set (in this case by age and sex) with the additional constraint that the ratio between the number of adolescents in both groups had to be equal to 1. This method allowed us to avoid differences between groups by sex or age with minimal loss of ADHD cases selecting the best-fitting control cases in function of the ADHD group.

Final samples

In both groups the final sample meeting and pairwise inclusion criteria consisted of 24 adolescents (12 females and 12 males) in both groups (cf. tab. 2). The age mean weight in ADHD group was 15.14 (𝜎 = 1.83) and 15.21 (𝜎 = 1.44) for the control group. A Student two-samples t-test showed that the difference was statistically not significant (t(43.65) = -0.16, p = 0.87). Finally, as a result of this pairwise matching, the total number of SDMs per group has been reduced to 72 SDMs (24 participants × 3 SDMs) per group. Lexical diversity (TTR = 𝑉 𝑁 ) is higher for the control group (209, 𝜎 = 63,9), indicating its members use significantly more different words to describe their memories than the ADHD group (153, 𝜎 = 54,5, p < 0.001), contradicting the results of Redmond [START_REF] Redmond | Conversational profiles of children with ADHD, SLI and typical development[END_REF]. This fact could be explained by the significant difference in size between the SDMs of the two groups (cf. fig. 1), those of the control group being generally longer (𝑥 ¯>100 token) than those of the ADHD group (𝑥 ¯≈60): as the length of the text increases, there is a greater statistical likelihood of finding a new word.

Results

On the contrary lexical density (𝐿 𝑑 = 𝑁 𝑙𝑒𝑥 𝑁 * 100) is not significantly different between the control group (0.027, 𝜎 = 0.008) and the ADHD group (0.029, 𝜎 = 0.01), which invalidates our hypothesis based on Yoder [START_REF] Yoder | Predicting lexical density growth rate in young children with autism spectrum disorders[END_REF].

Functors as markers

Analyzing which features allowed our SVM to distinguish between our two groups can help us understand further the differences between ADHD-diagnosed respondents and individuals from the control group. The size of our corpus does not allow us to comment on content words, not frequent enough to be considered as reliably interpretable. But we can gain significant insights from function words (cf. fig. 2).

For the ADHD group the markers include the neutral pronoun (on) combined with the third person auxiliaries and the abundance of words with syntactic function (donc, et, avec).

Marker

Context of meaning "donc" (...) nous avons donc été confiné ensemble (...)" "et" (...) on a pris de l'extasie et de la cocaïne et on s'est baladé toute la nuit (...) "sur" (...) On s'était couché surla plage et d'un coup, un nuage de libellules est arrivé (...) "avec" (...) C'était en 2020, en France, à la campagne avec toute ma famille (...) Table 3 Examples of markers in their context of meaning (ADHD group).

On the other hand, the control group is very marked by first person pronouns (je, me/m) and the plural (des, plus).

Marker

Context of meaning "je" (...) j'avais peur de faire des toboggans quand un ami m'a forcé à le descendre. " "des" (...) C'était compliqué de se concentrer à cause des distractions de la maison (...) "plus" (...) j'ai rencontré ma copine et ça m'a rendu plus heureuse et sûre de moi (...) "fois" (...) C'était la première fois que j'assistais à un enterrement. (...) Table 4 Examples of markers in their context of meaning (Control group).

Discussion

Our findings indicate that adolescent's with ADHD diagnosis show significant differences in their autobiographical narratives style of language from their control group (non-ADHD cohort), which is enough to have them detected by the machine. Additionally, different markers for each group were found which are meaningful from a psychological perspective. Indeed, the use of the neutral pronoun in the ADHD group is very different from the use of the first singular pronoun in the case of the Control group. In terms of agency, the latter identifies the author as the subject performing and feeling things while the narrated event occurred. In the case of the ADHD group, the agent is subsumed in an undefined neutral mass (on) that performs things in a distinctly less personal way. In their model of the relationships between self, memory, and visual perspective, Sutin and Robins [START_REF] Sutin | When the "I" looks at the "Me": Autobiographical memory, visual perspective, and the self[END_REF] argued that in the first person perspective, individuals see the event through their own eyes, while, in the third person perspective, individuals see themselves and the event from the perspective of an external observer. According to their model, a reduced use of the first person may serves a distancing function helping to reduce emotional reliving and to distance the current self from the self in the memory. In this sense, adolescents with ADHD show at least difficulties to connect with their emotional experiences, either by way of a defence mechanisms or by functional difficulties related to executive functions. This result is even more relevant to consider if one takes into account that the task asks for memories lived by the person that have marked him/her and that, therefore, easily elicit first-person experiences. With respect to lexical density, our results are not consistent with previous finding and does not confirm differences between groups based on the presence or absence of ADHD. However, this could be due to the small size of the ADHD group narratives.

Further work

Future studies are needed to increase the accuracy and fine grain detection of ADHD. First the potential differences by sex or gender in language markers should be addressed. Second, refining the precision of the detection in order to identify the different modalities of ADHD (predominantly inattentive, hyperactive or combined) is another relevant challenge to consider for future studies. Third, self-reported affect reported by participants in the Task of SDMs should be considered in order to evaluate to what extend this metadata could improve the detection of the signal of ADHD in the adolescent's autobiographical narratives. Finally, considering the potential challenge for young adolescents to write a memoir, another interesting aspect would be to modify the task to make it a verbal task and, eventually, also modify the instruction to make it simpler and eventually more stimulating for the target population.

Figure 1 :

 1 Figure 1: Number of tokens per SDM in ADHD group vs control group

Figure 2 :

 2 Figure 2: Most important FW for stylometric classification. In red the control group, in blue the ADHD group.

Table 1

 1 Results of the experiment for FW (left) and character 3-grams (right).

		Precision Recall f1-score support		Precision Recall f1-score support
	control	0.89	0.89	0.89	9	control	0.90	1.00	0.95	9
	TDAH	0.75	0.75	0.75	4	TDAH	1.00	0.75	0.86	4
	Accuracy			0.85	13	Accuracy			0.92	13
	Test The SDMs of four participants (2 ADHD and 2 control) were not used for training and
	kept for a final test. The model perfectly classify ADHD and non-ADHD SDMs.

Table 2

 2 Effect of pairwise matching on the distribution of participants (× 3 SDMs).

We do not use gender, but biological sex as a category, as many other studies do. Our objective here is not to study the difference between these two groups (male vs. female) but to obtain two samples as comparable as possible (ADHD vs. control). The question of gender could, however, be of interest in future studies, especially because ADHD is possibly because ADHD is generally more likely to be diagnosed and treated for boys.

Acknowledgements

JB was funded by the Chilean National Agency for Research and Development (ANID Chile) through the PhD Abroad Scholarship, 2018 award. The PI (Prof. Martin Debbané) was funded by the Swiss National Science foundation (Grant number 100014_179033), as well as the Marina Picasso Prize from AEMD Foundation 2018.