Jean Bertoin
email: jean.bertoin@upmc.fr

A limit theorem for trees of alleles in branching processes with rare neutral mutations

Keywords: Weak convergence, branching process, neutral mutations, allelic partition, Lévy-Itô decomposition

We are interested in the genealogical structure of alleles for a Bienaymé-Galton-Watson branching process with neutral mutations (infinite alleles model), in the situation where the initial population is large and the mutation rate small. We shall establish that for an appropriate regime, the process of the sizes of the allelic sub-families converges in distribution to a certain continuous state branching process (i.e. a Jiřina process) in discrete time. Itô's excursion theory and the Lévy-Itô decomposition of subordinators provide fundamental insights for the results.

Introduction

Poisson point processes are a cornerstone of at least two fundamental contributions of Professor Kiyoshi Itô to Probability Theory, namely the Lévy-Itô decomposition of Lévy processes (Chapter 1 in [START_REF] Itô | Stochastic processes[END_REF]) and Itô's excursion theory [START_REF] Itô | Poisson point processes attached to Markov processes[END_REF]. The law of rare events, which stresses that Poisson variables arise as limiting distributions for the number of successes in a large number of independent trials where each trial has the same small probability of success, explains their prominent role amongst stochastic processes. For instance, Lévy processes can be viewed as weak limits of rescaled random walks and their jumps correspond to rare large steps of the latter. Informally, the law of rare events thus suggests the Poissonian structure of the jumps of Lévy processes, and this is indeed the core of the Lévy-Itô decomposition. A related but more delicate heuristic also applies to Itô's description of the excursions of Markov processes, as in discrete time, the succession of the excursions of a Markov chain away from a recurrent point forms an i.i.d. sequence of paths.

In this paper, we shall point out the relevance of this paradigm to study a question motivated by genetics. Recall that Bienaymé-Galton-Watson branching processes [START_REF] Athreya | Branching processes[END_REF][START_REF] Harris | The theory of branching process[END_REF][START_REF] Jagers | Branching processes with biological applications[END_REF] model a population in which at every generation each individual begets according to a fixed offspring distribution and independently of the other individuals, and then dies. Imagine that neutral mutations may happen, so that a child can be either a clone of its parent or a mutant, and the reproduction laws of clones and of mutants are identical. We shall further suppose that each time a mutation occurs, it produces a mutant with a genetic type (allele) which has never been observed before; this setting has been referred to as the infinite alleles model by Kimura and Crow .

The allelic partition consists in decomposing the entire population into sub-families of individuals carrying the same allele. One important issue in the study of random population models with mutations (cf. the celebrated sampling formula of Ewens [START_REF] Ewens | The sampling theory of selectively neutral alleles[END_REF] for the Wright-Fisher model) concerns statistics of this allelic partition: what is the probability of observing allelic clusters of certain sizes, how to describe the random genealogical structure connecting these clusters to each other, ... Our main concern here will be to investigate asymptotics when the size of the population is large (typically because the number of ancestors is large) and mutations rare. We shall see that, under some mild conditions and for an appropriate regime, a non-degenerate limit exists and is conveniently described in terms of a certain continuous state branching process in discrete time [START_REF] Jiřina | Stochastic branching processes with continuous state space[END_REF]. It is well-known that continuous state branching processes bear close connexions to certain infinitely divisible distributions; in particular we shall provide a representation of the limiting allelic partitions in terms of Poisson point measures appearing in the Lévy-Itô decomposition of the jumps of an underlying Lévy process.

Let us give a rough idea of the orders of magnitude of the quantities involved. We shall consider a fixed reproduction law with unit mean and finite variance, and let the Galton-Watson process start from n ancestors having all the same genetic type. It is well-known that if n generations represent one unit of time and if we rescale the population at each generation by a factor 1/n, then the rescaled Galton-Watson process converges in distribution as n tends to infinity to a Feller diffusion. We also suppose that neutral mutations affect each child with probability 1/n. The scaling between population sizes, generations and mutation rates should not come as a surprise since it is precisely the regime of interest for other standard population models, such as the Wright-Fisher model and Kingman coalescent [START_REF] Kingman | The coalescent[END_REF]. Recall that such a critical Galton-Watson process becomes extinct after roughly n generations, and that the total population is of order n 2 . So there are only a few mutations at each generation and thus about n different alleles; furthermore the largest allelic sub-families have size of order n 2 .

Our main result can be described as follows. We use the universal tree U, that is the set of finite sequences of integers (including the empty sequence ∅ that serves as the root of U) to record the genealogy of alleles, and define the tree of alleles as a random process A on U, such that the values at vertices are given by the sizes of the corresponding allelic sub-families, with the convention that sizes are ranked in the decreasing order on each sibling. We consider a fixed reproduction law which is critical and has finite variance, and for every integer n, a Galton-Watson process with this reproduction law, started from n ancestors and in which mutations occur at random with rate 1/n. We write A (n) for the process on U that describes the corresponding tree of alleles. Then as n tends to infinity, the rescaled tree of alleles n -2 A (n) converges in the sense of finite dimensional distributions towards a process A on U with values in (0, ∞). The latter describes the genealogy of a continuous state branching process in discrete time with an inverse Gaussian reproduction law. We stress that its law only depends on the variance of the offspring distribution of the Galton-Watson process, and hence may be viewed as a universal tree of alleles.

The plan of this paper is as follows. In Section 2, we first present the general setting, stressing the role of the general branching property for the study of Galton-Watson processes with neutral mutations. Then we compute explicitly reproduction laws related to allelic sub-families and point at a connexion with certain downward-skip-free random walks. Such questions have been addressed from a different point of view in [START_REF] Bertoin | The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations[END_REF] to which the present work can be viewed as a complement and a sequel. Section 3 provides some background on continuous state branching processes and convergence of rescaled Galton-Watson processes. The main asymptotic results, namely Proposition 2 and Theorem 1 are stated and then proved in Section 4.

2 Galton-Watson processes with neutral mutations

Basic definitions and branching properties

In a Galton-Watson process with neutral mutations, every individual reproduces according to the same distribution and independently of the other individuals, no matter whether it is a mutant or a clone. Of course, a clone child of a mutant bears the same allele as its parent. Recall also that we are working in the infinite alleles setting, i.e. the same genetic type cannot be recovered from a cycle of mutations. Our basic data are hence provided by a pair of nonnegative integer-valued random variables

ξ = (ξ (c) , ξ (m) )
which describes the number of clone-children and the number of mutant-children of a typical individual. In this paper, we shall mainly be interested in a special situation which appears commonly as a model in population genetics, namely where mutations affect each child according to a fixed probability and independently of the other children (in other words, the conditional distribution of ξ (m) given ξ (c) + ξ (m) = ℓ is binomial with parameter (ℓ, p)). However the first steps of the analysis can be carried on without difficulties using the general framework. We assume throughout this work that

E(ξ (c) ) ≤ 1 ,
i.e. the process of clones is critical or sub-critical1 ; and we further implicitly exclude the degenerate cases when ξ (c) ≡ 0, or ξ (m) ≡ 0. For every integer a ≥ 1, we denote by P a the law of a Galton-Watson process with neutral mutations, started from a ancestors having the same genetic type and with reproduction law given by that of ξ = (ξ (c) , ξ (m) ).

The basic branching property states that for every fixed generation, conditionally on the number of individuals at that generation, the descents of those individuals are given by independent copies of the initial process, independently of the preceding generations. It is natural to expect that this branching property should hold more generally for certain stopping rules, and that this is indeed the case will play an important role in our analysis. For the sake of simplicity, we shall now present such an extension in a rather informal way, referring to Chauvin [START_REF] Chauvin | Sur la propriété de branchement[END_REF] for technical details.

The genealogy of each ancestor is conveniently described by a planar rooted tree, with edges connecting parents to children. More precisely, this requires an additional ordering of the children of each individual, and in this direction we may decide to rank siblings uniformly at random. A line is defined as a family of edges such that every branch from the root (i.e. the ancestor) contains at most one edge in that family. For instance, the edges between parents at generation k ∈ Z + and children at generation k + 1 form a line. A stopping line should be thought of as a random line such that for every edge in the tree, the event that this edge is part of the line only depends on the marks found on the path from the root to that edge. Recall that every edge of the tree corresponds to a pair of individuals (parent,child), and denote by C τ the subset of children in the family of edges of some stopping line τ . By removing the edges of τ , we disconnect the genealogical tree into sub-trees whose roots are formed on the one hand by the ancestor, and on the other hand by the individuals in C τ . The general branching property then states that conditionally on C τ , the sub-trees rooted at the individuals in C τ are independent copies of the initial genealogical tree, and also independent of the initial tree pruned along τ .

We now take into account mutations by assigning marks to the edges between parents and their mutant children. Since we are interested by the genealogy of alleles (or equivalently, of mutants), it is convenient to say that an individual has the k-th type if its genotype has been affected by k mutations, that is if its ancestral line comprises exactly k marks. Plainly, the family τ (k) of edges connecting a parent of the (k -1)-th type to a mutant child is a stopping line, and the set C τ (k) coincides with that of the mutants of the k-th type. We denote by T k the total population of individuals of the k-th type and by M k the total number of mutants of k-th type, agreeing that mutants of the 0-th type are the ancestors (so M 0 = a, P a -a.s.). The general branching property should make the following statement obvious; we refer the reader to e.g. Chapter Ten in Taib [START_REF] Taïb | Branching processes and neutral evolution[END_REF] for a rigorous argument.

Lemma 1 Under P a , (M k , k ∈ Z + )
is a standard Galton-Watson process with reproduction law P 1 (M 1 ∈ •). More generally,

((T k , M k+1 ), k ∈ Z + )
is a Markov chain with transition probabilities

P a (T k = n ′ , M k+1 = m ′ | T k-1 = n, M k = m) = P m (T 0 = n ′ , M 1 = m ′ ) .
Remark. We stress the fact that the chain (T k , k ∈ Z + ) of the sizes of sub-populations with given types is not Markov; nonetheless it can be viewed as a hidden Markov chain. In this direction, we also point out that ((T k , M k ), k ∈ Z + ) is Markovian, since the transition probabilities of the chain ((T k , M k+1 ), k ∈ Z + ) only depend on the second coordinate. Indeed, by a straightforward application of the general branching property, one gets (assuming implicitly that the events on which we condition have positive probability)

P a (T k+1 = n ′ , M k+1 = m ′ | M k = m, T k = n) = P m ′ (T 0 = n ′ ) P a (T k = n, M k+1 = m ′ | M k = m) P a (T k = n | M k = m) = P m ′ (T 0 = n ′ ) P m (T 0 = n) P m (T 0 = n, M 1 = m ′ ) .
Next, observe that in an infinite alleles model, the genealogy of individuals naturally induces a genealogy for the alleles in that population. Indeed, we may identify alleles and mutants, which enables us to use the set of new mutants plus a root corresponding to the ancestors of the population (recall that we assume that all the ancestors have the same genetic type) as the set of vertices. We draw an edge between the root and mutants of the 1st type, and for every k ≥ 1 we also draw an edge between a mutant of the k-th type and a mutant of the (k + 1)-type if and only if the path connecting these individuals in the genealogical tree does not contains other mutants. Hence the set of alleles has a natural structure of rooted tree. Note that for k ≥ 1, M k corresponds to the number of vertices at the k-th level2 in the tree of alleles.

Our main goal in this paper is to establish asymptotic features on the genealogy of allelic sub-families, and in this direction, it will be convenient to view the latter as random processes indexed by the universal tree. More precisely, introduce the set of finite sequences of positive integers

U := k∈Z + N k ,
where N = {1, 2, . . .} and N 0 = {∅}. Let us briefly recall some standard notation in this setting: if u = (u 1 , . . . , u k ) is vertex at level k ≥ 0 in U, then the children of u are uj := (u 1 , . . . , u k , j) for j ∈ N. We also denote by |u| the level of the vertex u, with the convention that the root has level 0, i.e. |∅| = 0. We now take advantage of the natural tree structure of U to record the genealogy of allelic sub-families together with their sizes.

Given a Galton-Watson process with neutral mutations, we construct recursively a process A = (A u : u ∈ U) as follows. First, A ∅ = T 0 is the size of the sub-population without mutation. Next, recall that M 1 denotes the number of mutants of the first type. We enumerate the M 1 allelic sub-populations of the first type in the decreasing order of their sizes, with the convention that in the case of ties, sub-populations of the same size are ranked uniformly at random. We denote by A j the size of the j-th allelic sub-populations of the first type, agreeing that A j = 0 if j > M 1 . We then complete the construction at all levels by iteration in an obvious way. Specifically, if A u = 0 for some u ∈ U, then A uj = 0 for all j ∈ N. Otherwise, we enumerate in the decreasing order of their sizes the allelic sub-populations of type |u| + 1 which descend from the allelic sub-family indexed by the vertex u, and then A uj is the size of this j-th subfamily (as before, in the case of ties, sub-families are ordered uniformly at random, and empty sub-families have size 0). See Figure 1 for an example. We call the process A = (A u : u ∈ U) the tree of alleles.

• • • ♣ d d d d d d d d d d d d d d d d d d d d ♥ • ♠ d d d d d d ♥ ♠ ♠ ♦ 4 1 3 2 1 1 ¡ ¡ ¡ ¡ ¡ ¡ ¡
Figure 1: Genealogical tree with mutations (left) and tree of alleles (right). The symbols •, ♠, ♥, ♦, ♣, represent the different alleles. The labels on the tree of alleles are the sizes of the corresponding allelic sub-families; sub-families with zero size (i.e. which are empty) are omitted.

It is important to observe that the transition probabilities of the chain ((T k , M k+1 ), k ∈ Z + ) in Lemma 1 depend only on the second coordinate, and that the latter alone is a Galton-Watson process. This suggests that the tree of alleles should enjoy some kind of branching property. In order to give a formal statement, is convenient to define first the (outer) degree of the tree of alleles A at some vertex u ∈ U as

d u := max{j ≥ 1 : A uj > 0} ,
where we agree that max ∅ = 0. In words, d u is the number of allelic sub-populations of type |u|+1 which descend from the allelic sub-family indexed by the vertex u; in particular d ∅ = M 1 . We shall also need the following notation. Let γ be a random variable in N 2 , d ≥ 1 an integer, and γ (d) = (γ 1 , . . . , γ d ) where the γ i are independent copies of γ. We then denote by γ (d↓) the rearrangement of γ (d) in the decreasing order of the first coordinate, with the convention that in the case of ties, the variables γ i with the same first coordinate are ranked uniformly at random. The characterization of the probabilistic structure of the tree of alleles that we are now ready to present stems again easily from the general branching property by iteration.

Lemma 2 For every integers a ≥ 1 and k ≥ 0, the tree of alleles fulfills the following properties under P a conditionally on

((A v , d v ) : |v| ≤ k): (i) the families of variables ((A uj , d uj ) : 1 ≤ j ≤ d u ) , u vertex at level k such that A u > 0 , are independent, (ii) for each vertex u at level k with A u > 0, the d u -tuple ((A uj , d uj ) : 1 ≤ j ≤ d u ) is distributed as (T 0 , M 1 ) (du↓) under P 1 .
Of course Lemma 2 of much more informative than the sole Markovian description of the chain ((T k , M k+1 ), k ∈ Z + ) in Lemma 1 as it retains the information about the genealogy of the allelic sub-families and not merely the sizes of populations of a given type. In this direction, observe that

T k = |u|=k A u and M k+1 = |u|=k d u .

Calculation of reproduction laws

We shall now determine the transition probabilities that appear in Lemma 1. Essentially, this has been achieved recently in [START_REF] Bertoin | The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations[END_REF] using an approach that largely relies on Harris connexion between downward-skip-free random walks and standard Galton-Watson processes, extended to encompass the situation where neutral mutations occur. Here, we shall use a different route, developing calculations that involve generating functions in the case when mutants are supposed to be sterile.

We denote the law of ξ = (ξ (c) , ξ (m) ) by π = (π k,ℓ : k, ℓ ∈ Z + ), that is

π k,ℓ := P(ξ (c) = k, ξ (m) = ℓ) .
We also introduce the generating function

g(x, y) := ∞ k,ℓ=0 x k y ℓ π k,ℓ = E(x ξ (c) y ξ (m) ) , x, y ∈ [0, 1] .
As we are interested in the joint distribution of the total number of individuals of the 0-th type and the number of mutants of the 1-st type, we may imagine a two-type branching process such that clones reproduce independently of each other according to the same distribution π, while mutants are sterile, i.e. have no progeny a.s. We write ϕ for the generating function of the total population of 0-th type and the number of mutants when there is a single ancestor, i.e.

ϕ(x, y)

:= E 1 (x T 0 y M 1 ) , x, y ∈ [0, 1] ,
so that by the branching property, the generating function of (T 0 , M 1 ) under P a is ϕ a . The following result is a slight extension of Theorem 1(ii) of [START_REF] Bertoin | The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations[END_REF] (recall that here we only assume that E(ξ (c) ) ≤ 1 and have an arbitrary number of ancestors, while in [START_REF] Bertoin | The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations[END_REF] we worked with a single ancestor and assumed that E(ξ (c) + ξ (m) ) ≤ 1). It can be viewed as a generalization of the well-known Dwass formula [START_REF] Dwass | The total progeny in a branching process[END_REF] for the distribution of the total population in standard Galton-Watson processes. (ii) The distribution of (T 0 , M 1 ) is given by

P a (T 0 = n, M 1 = ℓ) = a n π * n n-a,ℓ , n ≥ a ≥ 1 and ℓ ≥ 0 ,
where π * n denotes the n-th convolution power of π (i.e. π * n is the distribution of the sum of n i.i.d. copies of ξ).

Proof: (i) A standard application of the branching property at the first generation gives

ϕ(x, y) = E 1 (x T 0 y M 1 ) = x ∞ i,j=0 P(ξ (c) = i, ξ (m) = j)E 1 (x T 0 y M 1 ) i y j = xg(ϕ(x, y), y) .
This invites us to consider the equation in the variable z ∈ [0, 1]

g(z, y) z = 1 x , (1)
where x, y ∈ (0, 1] are fixed. Our assumptions E(ξ (c) ) ≤ 1 and ξ (c) ≡ 1 imply that g(0, y) > 0, and hence lim z→0+ g(z, y)/z = ∞. On the other hand, the derivative of z → z -1 g(z, y) is z → z -2 (z∂ z g(z, y) -g(z, y)), and this derivative is strictly negative when z > 0 is sufficiently small. This ensures that for each fixed y ∈ [0, 1] and x > 0 small enough, the equation ( 1) has a unique solution z = ϕ(x, y), and this suffices to determine the law of (T 0 , M 1 ).

(ii) We shall now derive explicitly the law of (T 0 , M 1 ) under P a from its generating function ϕ a using the classical Lagrange inversion formula. For each fixed y ∈ [0, 1], the function x → g(x, y) is analytic with g(0, y) = 0. More precisely, we have

g(x, y) = ∞ k=0 a k (y)x k with a k (y) := ∞ ℓ=0 y ℓ π k,ℓ .
According to Lagrange inversion formula (see for instance Section 5.1 in [START_REF] Wilf | Generatingfunctionology[END_REF]), the a-th power of the solution to the equation ( 1) with y ∈ [0, 1] fixed and x > 0 sufficiently small, can be expressed in the form

ϕ a (x, y) = ∞ n=1 a n α * n n-a x n ,
where α * n stands for the n-th convolution power of the finite measure α = (a k (y) : k ∈ Z + ).

Observe that the generating function of α is x → g(x, y), so that of α * n is

x → g(x, y) n = ∞ k=0 x k ∞ ℓ=0 y ℓ π * n k,ℓ
.

Hence we have

α * n k = ∞ ℓ=0 y ℓ π * n k,ℓ ,
and we conclude that

ϕ a (x, y) = ∞ n=1 ∞ ℓ=0 a n π * n n-a,ℓ x n y ℓ ,
which completes the proof of (ii).

As generating functions easily yield moments of variables, one immediately deduces from Proposition 1 simple criteria to decide whether the number of mutant children M is critical, sub-critical, or super-critical, or has a finite second moment.

Corollary 1 (i) Suppose that the mean number of clone children is sub-critical, i.e. E(ξ (c) ) < 1. Then

E a (M 1 ) = a E(ξ (m) ) 1 -E(ξ (c) ) = E(ξ (m) )E a (T 0 ) ,
and in particular

E 1 (M 1 )    < 1 = 1 > 1 ⇐⇒ E(ξ (c) + ξ (m) )    < 1 = 1 > 1 Further E 1 (M 2 1 ) < ∞ ⇐⇒ E((ξ (c) + ξ (m) ) 2 ) < ∞ . (ii) If E(ξ (c) ) = 1, then E 1 (M 1 ) = ∞.
Proof: Recall that the first moment of an integer-valued variable is given by the left-derivative at 1 of its generating function. We get from Proposition 1(i) Since ϕ(1, 1) = 1, this identity forces ∂ϕ ∂y

(1, 1) = E 1 (M 1 ) = ∞ when ∂g ∂x (1, 1) = E(ξ (c) ) = 1
(recall that E(ξ (m) ) > 0 by assumption), whereas it entails

E 1 (M 1 ) = E(ξ (m) ) 1 -E(ξ (c) ) whenever E(ξ (c) ) < 1.
Observe further that the process of the number of clone children is a branching process with offspring distribution given by the law of ξ (c) . In particular, in the sub-critical case E(ξ (c) ) < 1, the total population of clones has a finite expectation given by E a (T 0 ) = a/(1 -E(ξ (c) )). The first equivalence in (i) follows readily. Similar calculations involving the second derivative of generating functions yield the second equivalence in (i).

Construction from a random walk

The starting point of this section is the observation that the transition probabilities of the Markov chain ((T k , M k+1 ) : k ∈ Z + ) have a simple interpretation in terms of random walks. In this direction, let us first introduce some notation. We consider a sequence (ξ n = (ξ

(c) n , ξ (m) n ) : n ∈ N) of i.i.d
. variables with law π, and then the random walk started from a ≥ 1 and with steps ξ (c) -1,

S (c) n := a + ξ (c) 1 + • • • + ξ (c) n -n , n ∈ Z + .
It is convenient to use the (slightly abusive) notation P a for the law of (S (c)

n : n ∈ Z + ).
We also define the first hitting times

ς(j) := inf{n ∈ Z + : S (c) n = -j} , j ∈ Z + , and
Σ(j) := ς(j) i=1 ξ (m) i .
We stress that our basic assumption E(ξ (c) ) ≤ 1 ensures that the random walk S (c) does not drift to +∞, and hence the passage times ς(j) are finite a.s. The first identity in next lemma can be viewed as a two-dimensional extension of the well-known result of Otter and Dwass (see e.g. Section 6.2 in [START_REF] Pitman | Combinatorial stochastic processes[END_REF]) which relates the distribution of the total population in a Galton-Watson process to that of the first hitting time of 0 of a random walk.

Lemma 3 The pairs of random variables

(ς(0), Σ(0)) and (T 0 , M 1 )

have the same distribution under P a . Further, the shifted sequence (ξ ς(0)+j : j ∈ N) consists of i.i.d. variables with law π and is independent of (ς(0), Σ(0)).

Proof: Introduce for a = 1 the generating function φ(x, y)

:= E 1 (x ς(0) y Σ(0) ) , x, y ∈ [0, 1] .
Because (S n : n ∈ Z + ) is a downwards skip free random walk, an application of the strong Markov property at its first downward passage times shows readily that for an arbitrary integer

a ≥ 1 E a (x ς(0) y Σ(0) ) = φ(x, y) a , x, y ∈ [0, 1] .
Now we return to the case a = 1; by conditioning on the first step of the random walk, we get the obvious identity

φ(x, y) = E 1 (x ς(0) y Σ(0) ) = x ∞ k,ℓ=0 φ(x, y) k y ℓ π k,ℓ = xg( φ(x, y), y) ,
where g denotes the generating function of ξ = (ξ (c) , ξ (m) ). Thus φ solves the equation of Proposition 1(i), which establishes our first claim. As the hitting time ς(0) is a stopping time, an application of the strong Markov property then yields the second assertion.

Next, set T0 := ς(0), M1 := Σ(0) and define for every k ∈ N by an implicit recurrence Proof: It is immediately checked by induction that each τ k := T0 +• • •+ Tk is a stopping time in the natural filtration (G(n)) n∈N generated by the i.i.d. sequence (ξ n : n ∈ N), and that M k+1 is G(τ k )-measurable. By an application of the strong Markov property, we get that (( Tk , Mk+1 ) : k ∈ Z + ) is a homogeneous Markov chain. More precisely, the conditional distribution of ( Tk , Mk+1 ) given Tk-1 = t and Mk = m is that of (ς(0), Σ(0)) under P m . Combining these observation with Lemmas 1 and 3 completes the proof.

T0 + • • • + Tk = ς( M1 + • • • + Mk ) and M1 + • • • + Mk+1 = Σ( T0 + • • • + Tk ) = Σ(ς( M1 + • • • + Mk )) .
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • * * * * ς(0) ς (1) ς(2) ς(3) ς(4) ς(5) -2 E
More generally, we can apply Lemma 3 to construct from the i.i.d. variables ξ n a random process A ′ indexed by U with the same distribution as tree of alleles A, by making use of the characterization of the law of the latter in Lemma 2. To start with, the process A ′ fulfills the following two requirements. First, if A ′ u = 0 for some u ∈ U, then A ′ uj = 0 for all j ∈ N. Second, for every vertex u ∈ U such that A ′ u > 0, the (outer) degree of A ′ at u,

d ′ u := #{j ∈ N : A ′ uj > 0} ,
is a finite number and A ′ uj > 0 if and only if j ≤ d ′ u . We set A ′ ∅ = ς(0) and d ′ ∅ = Σ(0). Next, consider the increments λ(j) := ς(j) -ς(j -1) and δ(j) = Σ(j) -Σ(j -1)) , j ≥ 1 ,

For vertices at the first level, ((

A ′ j , d ′ j ) : 1 ≤ j ≤ d ′ ∅
) is given by the rearrangement of the sequence ((λ(j), δ(j)) : 1 ≤ j ≤ d ′ ∅ ) in the decreasing order the first coordinate λ(j) (with the usual convention in case of ties). We may then continue with vertices of the next levels by an iteration which should be obvious (but which would also be quite intricate to state explicitly). Figure 3 below may help visualizing the construction.

d d d d d d d d d d d d ς(0) λ(2) λ (1)
λ(4) λ(3) λ( 5) Figure 3: Tree of alleles constructed from the random walk S (c) and the variables ξ (m) of Figure 2. The labels on the vertices are the lengths of the excursions of S (c) above its current minimum, they correspond to the sizes of the allelic sub-families (again sub-families with size 0 are omitted).

3 Background on continuous state branching processes Before describing our main limit results for trees of alleles, we need to develop some basic material about limits of rescaled Galton-Watson processes. The Lévy-Itô decomposition of subordinators plays a crucial role for the representation of the genealogical structure of the continuous state limits.

We start with the classical convergence to Feller diffusions [START_REF] Feller | Diffusion processes in genetics[END_REF][START_REF] Jiřina | Stochastic branching processes with continuous state space[END_REF], i.e. the solutions (X(x, t), t ≥ 0) to stochastic differential equations of the type

X(x, t) = x + t 0 σ X(x, s)dB s + b t 0 X(x, s)ds , t ≥ 0 , (2)
where x ≥ 0 is the initial value, b ∈ R and σ 2 > 0 are parameters, and (B t : t ≥ 0) denotes a standard Brownian motion. For every n ∈ N, consider a Galton-Watson process (Z

(n) k : k ∈ Z + ) which starts from Z (n) 0
= a(n) ancestors and has reproduction law ρ (n) , where ρ (n) is some probability measure on Z + and a(n) a positive integer. Write

m(ρ (n) ) := ∞ i=0 iρ (n) i and var(ρ (n) ) := ∞ i=0 (i -m(ρ (n) )) 2 ρ (n) i
for the first moment and the variance of ρ (n) . In the situation where

a(n) ∼ nx , m(ρ (n) ) -1 ∼ bn -1 and var(ρ (n) ) ∼ σ 2 as n → ∞ (3)
for some x ∈ (0, ∞), b ∈ R and σ 2 > 0, it is well known that

(n -1 Z (n) ⌊nt⌋ : t ≥ 0) =⇒ (X(x, t) : t ≥ 0) ( 4
)
where the notation ⇒ refers to convergence in distribution as n → ∞ and X(x, t) is the Feller diffusion specified by [START_REF] Bertoin | The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations[END_REF].

We next turn our attention to the simpler situation where one only rescales the number of individuals and uses the generations as a discrete time parameter. For the sake of clarity, we shall deal with a framework that is slightly less general than it could be. We denote the tail distribution of ρ (n) by ρ(n) (y) := ρ (n) ((y, ∞)) for y > 0 and now assume that

lim n→∞ n -1 a(n) = x and lim n→∞ nρ (n) (ny) = ν(y) in L 1 loc ([0, ∞), dy) , ( 5
)
where ν is some locally integrable non-increasing function on [0, ∞) with ν(∞) = 0. We may thus think of ν as the tail of a Radon measure ν on (0, ∞) with (1 ∧ y)ν(dy) < ∞; ν will be often referred to as a Lévy measure. Our assumptions ensure that

n -1 Z (n) 1 =⇒ Z 1 , (6)
where Z 1 is a random variable with values in [0, ∞) which is infinitely divisible. Indeed, we have for any q > 0

E(exp(-qn -1 Z (n) 1 )) = 1 - [0,∞) (1 -e -qy/n )ρ n (dy) a(n) = 1 - q n
∞ 0 e -qy/n ρn (y)dy

a(n) = 1 -q ∞ 0
e -qy ρn (ny)dy

a(n)
, and ( 5) ensures that the latter quantity converges as n → ∞ towards the Laplace transform of an infinitely divisible variable

E(exp(-qZ 1 )) = exp(-xκ(q)) ,
where the cumulant κ is given by the Lévy-Khintchine formula

κ(q) = (0,∞)
(1e -qy )ν(dy) .

We underline the fact that the drift coefficient is 0; this will play an important role in the sequel. An application of the Markov property now shows that more generally

(n -1 Z (n) k : k ∈ Z + ) =⇒ (Z k : k ∈ Z + ) (8)
where (Z k : k ∈ Z + ) is a Markov chain with values in R + , started from Z 0 = x and whose transition probabilities are characterized as follows: for every k ∈ Z + and q, y ≥ 0, E(e -qZ k+1 | Z k = y) = exp(-yκ(q)) .

One refers to the limiting chain Z as a (discrete time) continuous state branching process, in short CSBP, with reproduction measure ν and started from x.

It is interesting to recast the preceding convergence in the framework of the law of rare events. In this direction, recall that the Lévy-Itô decomposition of the infinitely divisible variable Z 1 reads

Z 1 = ∞ i=1 a i , (9)
where a 1 ≥ a 2 ≥ . . . are the atoms ranked in the decreasing order of a Poisson random measure on (0, ∞) with intensity xν, with the convention that atoms are repeated according to their multiplicity and that when the Poisson random measure is finite (which occurs if and only if ν((0, ∞)) < ∞), then a i = 0 whenever the index i exceeds the total mass of the Poisson measure. Consider for every n ∈ N a family (ξ

(n) i : 1 ≤ i ≤ a(n)) of i.i.d. variables with law ρ (n) ; we should think of ξ (n) i
as the number of children of the i-th ancestor in the Galton-Watson process Z (n) . Denote by a

(n) 1 ≥ a (n) 2 ≥ . . . ≥ a (n) a(n) the decreasing reordering of the rescaled variables (n -1 ξ (n) i : 1 ≤ i ≤ a(n)).
In the regime [START_REF] Ewens | The sampling theory of selectively neutral alleles[END_REF], the law of rare events for null arrays (e.g. Theorem 14.18 in [START_REF] Kallenberg | Foundations of modern probability[END_REF]) ensures that (a

(n) 1 , a (n) 2 , . . . , a (n) a(n) ) =⇒ (a 1 , a 2 , . . .) , (10)
in the sense of finite dimensional distributions. Note also that ( 6) can be re-written in this setting as

a(n) i=1 a (n) i =⇒ i∈N a i ;
however the latter does not follow from [START_REF] Jagers | Branching processes with biological applications[END_REF].

This invites us to describe the convergence of rescaled Galton-Watson processes to (discrete time) CSBP from another point of view that takes into account the genealogy, and not merely the total sizes of populations at given generations. In this direction, we use a representation of the latter as random processes indexed by the universal tree U. For simplicity, suppose for a while that the Lévy measure ν is infinite, so a Poisson random measure with intensity cν with c > 0 has infinitely many atoms a.s. Recall from the Lévy-Itô decomposition ( 9) that almost all the individuals at the first generation in a CSBP descend from only countably many ancestors (we stress again that we are dealing with cumulants κ with no drift component), and plainly the same feature holds for the subsequent generations. Roughly speaking, vertices u ∈ U at level |u| = k ≥ 1 represent the sizes of the sub-populations at generation k in the CSBP which descent from the same parent at generation k -1. We construct a random process (Z u : u ∈ U) related to the CSBP Z, where Z uj is the size of the j-th largest sub-population at generation |u| + 1 which descents from a parent in the sub-population represented by u. We stress the process Z is by definition non-increasing on each sibling, i.e. the map j → Z uj is non-increasing on N for every u ∈ U. More precisely, conditionally on Z u = z, the Lévy-Itô decomposition [START_REF] Itô | Poisson point processes attached to Markov processes[END_REF] suggests that (Z uj : j ∈ N) should be given by the sequence of the atoms of a Poisson random measure on (0, ∞) with intensity zν, where atoms are repeated according to their multiplicity and ranked in the decreasing order. We make the construction formal in the following definition.

Definition 1 Fix x > 0 and ν a measure on (0, ∞) with (1 ∧ y)ν(dy) < ∞. A tree-indexed CSBP with reproduction measure ν and initial population of size x is a process (Z u : u ∈ U) with values in R + and indexed by the universal tree, whose distribution is characterized by induction on the levels as follows:

(i) Z ∅ = x a.s.; (ii) for every k ∈ Z + , conditionally on (Z v : v ∈ U, |v| ≤ k)
, the sequences (Z uj ) j∈N for the vertices u ∈ U at generation |u| = k are independent, and each sequence (Z uj ) j∈N is distributed as the family of the atoms of a Poisson random measure on (0, ∞) with intensity Z u ν, where atoms are repeated according to their multiplicity, ranked in the decreasing order, and completed by an infinite sequence of 0 if the Poisson measure is finite.

It should be plain that if Z is a tree-indexed CSBP with reproduction measure ν and initial population of size x, then |u|=k Z u : k ∈ Z + is a CSBP with reproduction measure ν started from x. We also point out that for every integer n, we can represent similarly the genealogy for the Galton-Watson process Z (n) as a process Z (n) indexed by the universal tree U, and one can check that under the regime (5), the following extension of (8) holds:

n -1 Z (n) =⇒ Z
in the sense of finite dimensional distributions. This should be viewed as a variation of the law of rare events [START_REF] Jagers | Branching processes with biological applications[END_REF]; the easy proof is left to the interested reader.

We now conclude this section by underlying the connexion between discrete time CSBP and subordinators (i.e. Lévy processes with values in R + ). Consider a subordinator τ = (τ t : t ≥ 0) with no drift and Lévy measure ν. Its cumulant κ is given by the Lévy-Khintchine formula [START_REF] Harris | The theory of branching process[END_REF] and we have E(e -qτt ) = exp(-tκ(q)) , for all q, t ≥ 0 .

Fix x > 0 and define a sequence (ζ k : k ∈ Z + ) by implicit iteration as follows:

ζ 0 = x , ζ 1 = τ x , ζ 1 + ζ 2 = τ x+ζ 1 , . . . , ζ 1 + • • • + ζ k+1 = τ x+ζ 1 +•••+ζ k .
Observe by an easy induction that the random times x + ζ 1 + • • • + ζ k are stopping times in the natural filtration of τ , so that the strong Markov property can be applied. It is then immediate to check that (ζ k : k ∈ Z + ) is a CSBP with reproduction measure ν and initial population of size x.

More generally, the tree-indexed CSBP Z can be constructed from the subordinator τ by making full use of the Lévy-Itô decomposition. Specifically, we know from the latter that the Stieltjes measure dτ on the random interval

I k := (x + ζ 1 + • • • + ζ k-1 , x + ζ 1 + • • • + ζ k ]
is purely atomic, and conditionally on |I k |, the sequence of the atomic masses has the same distribution as the family of the atoms in a Poisson point measure on (0, ∞) with intensity |I k |ν. These atoms should be viewed as the sizes of sub-families at level k, so it remains to identify the siblings and rank atoms corresponding to a same sibling in the decreasing order. This is straightforward for the first levels but becomes increasingly intricate for larger levels. Specifically, we let Z ∅ = x and declare that (Z j : j ∈ N) is given by the sequence of the jumps of τ on (0, x] ranked in the decreasing order. Next (Z 1j : j ∈ N) corresponds to the ranked sequence of the jumps of τ on the interval (x, x + τ Z 1 ], (Z 2j : j ∈ N) to those on the interval (x + τ Z 1 , x + τ Z 1 +Z 2 ] and so on. The algorithm may be thought of as a variant of the breadth first search in which each sibling is ordered according to the size of its progeny.

Asymptotic for rare mutations

This section contains our main results on limits of trees of alleles; we shall first present and discuss the general framework, then state the results, and finally prove the latter.

Framework and main results

We consider a fixed probability measure π (+) on Z + which serves as reproduction law for a standard Galton-Watson process denoted by Z (+) . We assume that Z (+) is critical, i.e.

∞ i=0 iπ (+) i = 1 ,
and has a finite variance

∞ i=0 (i -1) 2 π (+) i = σ 2 < ∞ .
Further, we suppose that mutations affect each child according to a fixed probability p ∈ (0, 1) and independently of the other children. That is to say that the probability measure π on Z + × Z + which gives the law of the number of clone children and the number of mutant children of a typical individual is given by

π k,ℓ = π (+) k+ℓ k + ℓ k (1 -p) k p ℓ .
We will use the notation P p a for the probability measure under which the Galton-Watson process Z (+) has a ancestors and the mutation rate is p, and L (•, P p a ) will then refer to the distribution of a random variable or a process under P p a We are interested in the situation where the mutation rate p = p(n) is small and the number of ancestors a = a(n) large when the parameter n goes to infinity. Specifically, we consider the regime a(n) ∼ nx and p(n

) ∼ cn -1 , (11)
where c, x are some positive constants. Let us start by mentioning some results of convergence in distribution for Galton-Watson processes in this setting.

First, we know from (4) that the Galton-Watson process Z (+) properly rescaled converges to a Feller diffusion on R + ; specifically

L (n -1 Z (+) ⌊nt⌋ : t ≥ 0), P p(n) a(n) =⇒ (X (+) t : t ≥ 0) , (12)
where (X (+) t

: t ≥ 0) solves the SDE (2) for the parameter b = 0. In the same direction, the marginal law of ξ (c) under P p(n) has first moment 1 -p(n) and variance close to σ 2 when n is large. Hence, if Z (c) denotes the Galton-Watson process of clones (i.e. we only consider individuals of the 0-th type), then

L (n -1 Z (c) ⌊nt⌋ : t ≥ 0), P p(n) a(n) =⇒ (X (c) t : t ≥ 0) as n → ∞, (13)
where (X (c) t : t ≥ 0) is another Feller diffusion solution to the SDE (2) for the parameter b = -c. On the other hand, recall from Lemma 1 and Corollary 1(i) that the process of the number of mutants of given types (M k : k ∈ Z + ) is a critical Galton-Watson process with finite variance. In view of the classical limit theorem stated as [START_REF] Dwass | The total progeny in a branching process[END_REF] in Section 3, one might suspect that the rescaled process (n -1 M ⌊nt⌋ : t ≥ 0) could converge to some Feller diffusion. However this is not the case; indeed an easy calculation shows that the variance of the reproduction law of M • under P p(n) is of order n, and thus the requirement (3) fails. Nonetheless one can deduce from a few lines of calculations based on Proposition 1 that the condition ( 5) is fulfilled by the reproduction law of M under P (n) a(n) , and hence

L (n -1 M k ) : k ∈ Z + ), P p(n) a(n)
converges weakly when n → ∞ towards the law of some discrete time CSBP started from a. We do not give a formal statement as the forthcoming Proposition 2 is a stronger result.

The asymptotics ( 12) and ( 13) point to the fact that in the regime [START_REF] Jiřina | Stochastic branching processes with continuous state space[END_REF], the total size of the population of the Galton-Watson process should be rescaled by a factor n -2 , and in particular the asymptotic behavior of the number

T 0 = ∞ k=0 Z (c)
k of individuals of 0-th type is given by

L n -2 T 0 , P p(n) a(n) =⇒ ∞ 0 X (c) t dt .
More generally, we have the following joint convergence in distribution for the rescaled process of the sizes of sub-populations and the number of mutants of a given type.

Proposition 2 In the regime [START_REF] Jiřina | Stochastic branching processes with continuous state space[END_REF], we have

L ((n -2 T k , n -1 M k+1 ) : k ∈ Z + ), P p(n) a(n) =⇒ ((Z k+1 , cZ k+1 ) : k ∈ Z + ) where (Z k : k ∈ Z + ) is a CSBP with reproduction measure ν(dy) = c 2πσ 2 y 3 exp - c 2 y 2σ 2 dy , y > 0 ,
and initial population of size x/c.

The Lévy-Itô decomposition now suggests that conditionally on n -2 T k ∼ y, the sequence of the sizes of the sub-populations carrying a same allele of the (k + 1)-type and normalized by a factor n -2 should converge in distribution to the sequence of the atoms of a Poisson random measure on R + with intensity specified in Proposition 2. Recall also that d u denotes the outer degree at the vertex u ∈ U in the tree of alleles, and observe from Lemma 2 that for a Galton-Watson process with neutral mutations, the process ((A u , d u ) : u ∈ U) has a simpler Markovian structure than (A u : u ∈ U) alone. This leads us to our main limit theorem for the tree of alleles.

Theorem 1 In the regime [START_REF] Jiřina | Stochastic branching processes with continuous state space[END_REF], the rescaled tree of alleles n -2 A under P p(n) a(n) converges in the sense of finite dimensional distributions to the tree indexed CSBP (Z u : u ∈ U) with reproduction measure ν given in Proposition 2 and random initial population with inverse Gaussian distribution:

P(Z ∅ ∈ dy) dy = x 2πσ 2 y 3 exp - (cy -x) 2 2σ 2 y , y > 0 .
More precisely, if we also take into account the outer degrees, then we have the joint convergence in the sense of finite dimensional distributions:

L ((n -2 A u , n -1 d u ) : u ∈ U), P p(n) a(n) =⇒ (Z u , cZ u ) : u ∈ U) .

Proofs

Let us first present informally some intuitions for the proofs, which rely on the connexion with random walks in Section 2.3. Roughly speaking, we shall observe that in the regime [START_REF] Jiřina | Stochastic branching processes with continuous state space[END_REF], the random walk S (c) suitably rescaled converges to a Brownian motion with negative drift. As the lengths of the excursions of S (c) above its current minimum correspond to the sizes of subpopulations with the same allele, this suggests that in the limit, the lengths of the excursions of a Brownian motion with drift above its current minimum should describe the limit of rescaled sub-populations. According to Itô's excursion theory, these lengths can be described in terms of a Poisson point process. The comparison with the construction of the tree indexed CSBP presented in Section 3.2 should then make Theorem 1 more intuitive.

The proofs of Proposition 2 and Theorem 1 both rely on the following technical lemma.

Lemma 4 In the regime (11), we have:

(i) Let (τ x : x ≥ 0) be a inverse Gaussian subordinator with cumulant κ(q) = σ -2 c 2 + 2qσ 2 -c = c -1 ∞ 0
(1e -qy )ν(dy) , q ≥ 0 , i.e. with zero drift and Lévy measure c -1 ν where ν given in Proposition 2. Then

L (n -2 T 0 , n -1 M 1 ), P p(n) a(n) =⇒ (τ x , cτ x ) .
(ii) The behavior of the joint tail distribution of T 0 and M 1 under P

p(n) 1 is given by lim n→∞ nP p(n) 1 n -2 T 0 > t or n -1 M 1 > m = c -1 ν(min(t, m/c)) in L 1 loc (R + × R + , dt dm) ,
where ν denotes the tail function of the Lévy measure ν.

Proof: One could establish these limits from the explicit expressions in Proposition 1; however a probabilistic argument based on the construction in Section 2.3 circumvents the somewhat tedious calculations.

(i) Recall that the fixed reproduction law π (+) has unit mean and variance σ 2 . For each n ∈ N, consider a random walk (S

(n) k : k ∈ Z + ) started from S (n) 0
= a(n) and with step distribution that of ξ (+) -1. By Donsker's invariance principle and Skorohod's representation, we may suppose that with probability one lim n→∞ n -1 S (n)

⌊n 2 t⌋ = x + σB t ,
where (B t : t ≥ 0) is a standard Brownian motion and the convergence holds uniformly on every compact time-interval.

For every fixed n, we now decompose each variable ξ It is well-known that the latter is a subordinator with cumulant κ as given in the statement, and the first claim is established by an appeal to Lemma 3.

(ii) The branching property shows that the law of (T 0 , M 1 ) under P p(n) a(n) is that of the sum of a(n) i.i.d. variables distributed as (T 0 , M 1 ) under P p(n) 1

. This observation enables to deduce (ii) from (i) by an argument similar to that we use to establish [START_REF] Feller | Diffusion processes in genetics[END_REF]. Indeed, write μn (t, m) = P p(n) 1 (T 0 > t or M 1 > m) for the bivariate tail distribution of the pair (T 0 , M 1 ) under P p(n) 1

. By an elementary calculation, we have that for every q, r > 0 This entails our claim.

E p(n) an exp - q n 2 T 0 -
Proposition 2 immediately follows from Lemma 1 and Lemma 4(i), so we turn our attention to the proof of Theorem 1.

Proof of Theorem 1: Recall that A ∅ = T 0 and d ∅ = M 1 . On the one hand, we know from Lemma 4(i) that L (n -2 A ∅ , n -1 d ∅ ), P

=⇒ (τ x , cτ x ) .

On the other hand, Lemma 4(ii) and the law of rare events for null arrays (e.g. Theorem 14.18 in [START_REF] Kallenberg | Foundations of modern probability[END_REF]) entails that for any sequence of integers b(n) such that b(n) ∼ bn for some b > 0, L (n -2 T 0 , n -1 M 1 ) (b(n)↓) , P p(n) 1

=⇒ ((a 1 , ca 1 ), (a 2 , ca 2 ), . . .) ,

where the notation γ (d↓) has been defined just before Lemma 2 and (a 1 , a 2 , . . .) stands for the sequence ranked in the decreasing order of the atoms of a Poisson measure on (0, ∞) with intensity bc -1 ν.

Denote by Q x the law of a tree-indexed CSBP and initial population distributed as τ x and reproduction measure ν. We now see from Lemma 2 that L ((n -2 A u , n -1 d u ) : |u| ≤ 1), P p(n) a(n) =⇒ L (((A u , cA u ) : |u| ≤ 1), Q x ) in the sense of finite dimensional convergence. Lemma 2 enables us to iterate the argument to the subsequent levels of vertices, which establishes our claim.

Proposition 1

 1 (i) The generating function ϕ is determined by the equation ϕ(x, y) = xg(ϕ(x, y), y) , x, y ∈ [0, 1].

Figure 2

 2 Figure 2 below depicts these quantities.

Figure 2 :Corollary 2

 22 Figure 2: The graph of the random walk S (c) ; the * represent the non-zero values of the variables ξ (m) . Here M1 = 2, M2 = 2, M3 = 1 and M4 = 0.

1 )⌊n 2 ς

 12 sampling; that is conditionally on ξ (+) i = ℓ, ξ (mn) i has the binomial distribution with parameter (ℓ, p(n)). Of course, we use independent Bernoulli sampling for the different indices i, so that the pairs (ξ(cn) i , ξ (mn) i) are i.i.d. and have the law of ξ under P p(n) . If we define S = p(n) ∼ c/n and var(ξ(mn) 1 ) = O(1/n),and it is easy to verify that with probability one lim n→∞ n -1 S (mn) ⌊n 2 t⌋ = ct , uniformly on every compact time-interval. Hence the random walk S (cn) k:= a(n) + ξ t⌋ = x + σB t -ct ,where again the convergence holds a.s., uniformly on every compact time-interval. Now recall the framework of Section 2.3 and introduceς (n) (0) := inf{k ∈ Z + : S (cn) k = 0} and Σ (n) (0) := ς (n) (n) (0) .It follows readily from the preceding observations that with probability onelim n→∞ n -2 ς (n) (0) = τ x and lim n→∞ n -1 Σ (n) (0) = cτ xwhere τ denotes the process of first passage times for a Brownian motion with drift, τ y := inf{t ≥ 0 : ct -σB t > y} , y ≥ 0.

e 0 ( 1 -e 0 ( 1 -e

 0101 -qt e -rm μn (n 2 t, nm)dt dm an .We know from (i) that this quantity converges as n → ∞ towards E(exp(-(q + cr)τ x )) = exp -x ∞ e -(q+cr)y )c -1 ν(dy) , so that taking logarithms, we get lim -qt e -rm μn (n 2 t, nm)dt dm = x ∞ e -(q+cr)y )c -1 ν(dy) -qt e -rm c -1 ν(min(t, m/c))dt dm .

Note that this is weaker than assuming that E(ξ (c) + ξ (m) ) ≤ 1 which was required in[START_REF] Bertoin | The structure of the allelic partition of the total population for Galton-Watson processes with neutral mutations[END_REF].

For the sake of clarity we shall keep the name generation for the distance to the root of individuals in the genealogical tree, and use the name level when dealing with the structure of alleles.

Acknowledgments. The question of describing the asymptotic shape of the tree of alleles for large populations with rare mutations was raised by Matthias Winkel during a lecture based on [2] that I delivered at the University of Oxford. I would like to thank Matthias for having stimulated this work. This work has been supported by ANR-08-BLAN-0220-01.


Lionel Bombrun
email: lionel.bombrun@gipsa-lab.grenoble-inp.fr

Gabriel Vasile
email: gabriel.vasile@gipsa-lab.grenoble-inp.fr

Michel Gay
email: michel.gay@gipsa-lab.grenoble-inp.fr

Segmentation of Polarimetric SAR images based on the SIRV model and the Kummeru PDF

Dans le contexe des modèles de fouillis polarimétriques non gaussien, ce papier présente une application des récentes avancées dans le domaine des processus sphériquement invariants (SIRV) pour la l'estimation de la matrice de cohérence dans des zones de fouillis hétérogène. Les données polarimétriques radar sont décrites complètement à travers deux paramètres indépendant : le span et la matrice de cohérence normalisée. La matrice de cohérence normalisée décrit l'information polarimétrique tandis que le span montre la puissance rétrodiffusée. À l'aide du modèle SIRV, une nouvelle distance au sens du maximum de vraisemblance est introduite pour segmenter les images polarimétriques. La méthode proposée est ensuite testée sur une image acquise par le système aéroporté RAMSES.

Introduction

In a particular frequency band, the wave-media interactions over distributed areas are generally studied using the polarimetric covariance matrix. In general, POLSAR data are locally modelled by the multivariate, zero mean, circular Gaussian probability density function, which is completely determined by the covariance matrix.

The recently launched POLSAR systems are now capable of producing high quality images of the Earth's surface with meter resolution. The decrease of the resolution cell offers the opportunity to observe much thinner spatial features than the decametric resolution of the up-to-now available SAR images. Recent studies [START_REF] Greco | Statistical analysis of high-resolution SAR ground clutter data[END_REF] show that the higher scene heterogeneity leads to non-Gaussian polarimetric clutter modelling, especially for urban areas.

This paper presented a new estimation scheme for deriving normalized coherency matrices and the resulting estimated span with high resolution POLSAR images. The proposed Fixed Point estimator is independent on the span Probability Density Function (PDF) and represents an approximate ML estimator for a large class of stochastic processes obeying the SIRV model. Here, we propose to use the Fisher PDF to model the estimated texture parameter. For a Fisher distributed texture, authors prove that the target scattering vector k follows a Kum-merU PDF. For SIRV clutter, a new approximate ML distance measure is introduced for unsupervised POLSAR classification. The effectiveness of the proposed estimation/segmentation scheme is illustrated by very high resolution X-band POLSAR data.

SIRV model

Definition

With the new generation of airborne and spaceborne SAR sensors, the number of scatterers present in each resolution cell decreases considerably, homogeneous hypothesis of the Pol-SAR clutter can be reconsidered. Heterogeneous clutter models have therefore recently been studied.

In 1973, Kung Yao has first introduced the use of SIRV and their applications to estimation and detection in communication [START_REF] Yao | A representation theorem and its applications to sphericallyinvariant random processes[END_REF]. From a PolSAR point of view, the target vector k is defined as the product of a square root of a positive random variable τ (representing the texture) with an independent complex Gaussian vector z with zero mean and covariance matrix [M ] = E{zz H } (representing the speckle) :

k = √ τ z (1)
where the superscript H denotes the complex conjugate transposition and E{•} the mathematical expectation.

For a given covariance matrix [M ], the ML estimator of the texture parameter τ is given by :

τi = k H i [M ] -1 k i p . ( 2
)
where p is the dimension of the target scattering vector k (p = 3 for the reciprocal case).

The ML estimator of the normalized covariance matrix under the deterministic texture case is the solution of the following recursive equation :

[ M ] F P = f ([ M ] F P ) = p N N i=1 k i k H i k H i [ M ] -1 F P k i . (3)
Pascal et al. have established the existence and the uniqueness, up to a scalar factor, of the Fixed Point estimator of the normalized covariance matrix, as well as the convergence of the recursive algorithm whatever the initialization [7] [8]. In this paper, the trace of the covariance matrix is normalized to p the dimension of target scattering vector.

It is important to notice that in the SIRV definition, the PDF of the texture random variable is not explicitly specified. As a consequence, SIRVs describe a whole class of stochastic processes. This class includes the conventional clutter models having Gaussian, K-distributed, Rayleigh or Weibull PDFs.

Texture modeling 2.2.1 Definition

The texture parameter τ is the random power of the clutter, it characterizes the randomness induced by variations in the radar backscattering over different polarization channels. This texture parameter can be rewritten as the product of a normalized texture parameter ξ with the mean backscattered power µ by :

τ = µ ξ (4)
For a segment S containing N pixels, the normalized texture parameter for pixel i is defined by :

ξ i = τ i µ = τ i 1 N N j=1 τ j = k H i [M ] -1 k i 1 N N j=1 k H j [M ] -1 k j (5)
A common assumption is to considered τ and µ as two chisquared random variable with respectively ν 1 and ν 2 degrees of freedom. Hence, if τ and µ are independent, ξ follows a Beta Prime distribution with ν 1 and ν 2 parameters. In practice, τ and µ are not independent, but it is reasonable to assume that ξ follows a Beta Prime distribution with other parameters (L and M).

Beta Prime PDF

Let ξ be a positive random variable distributed according to a Beta Prime distribution. Its PDF is defined by two parameters L and M as :

BP [ξ|L, M] = Γ(L + M) Γ(L)Γ(M) ξ L-1 (1 + ξ) L+M (6)

Fisher PDF

The Fisher distribution is the Pearson type VI distribution. Its PDF is defined by three parameters m, L > 0 and M > 0 as the Mellin convolution of a Gamma PDF by an Inverse Gamma PDF by [START_REF] Nicolas | Introduction aux Statistiques de Deuxième Espèce : Applications des Logs-moments et des Logs-cumulants à l'Analyse des Lois d'Images Radar[END_REF] :

F [τ |m, L, M] = G [m, L] ⋆ GI [1, M] = Γ(L + M) Γ(L)Γ(M) L Mm Lτ Mm L-1 1 + Lτ Mm L+M (7)

Benefit of Fisher PDF

An urban area (80 × 35 pixels) from the L-band ESAR data over the Oberpfaffenhofen test-site has been extracted. Then, the covariance matrix [M ] F P and the texture parameter τ are estimated according to Eq.2 and Eq. 3. To see the benefit of Fisher PDF to model PolSAR clutter, the κ 2 /κ 3 plan has been plotted on Fig. 1. It shows the evolution of the second logcumulant versus the third log-cumulant. In this plan, Gamma and Inverse Gamma PDF are respectively represented by the blue and red line. Fisher PDF cover all the space between the blue and red line.

This example shows that Fisher PDF are well adapted to model high resolution PolSAR clutter.

Target scattering PDF for a Fisher distributed clutter

The PDF of the target scattering vector can be expressed with the density generator function

h p k H [M ]
-1 k by [START_REF] Gini | Covariance matrix estimation for CFAR detection in correlated heavy tailed clutter[END_REF] :

p k (k|[M ], L, M, m) = 1 π p |[M ]| h p k H [M ] -1 k (8)
where, for a Fisher distributed clutter, the density generator function h p (•) is defined by [1] :

h p k H [M ] -1 k = Γ(L + M) Γ(L)Γ(M) L Mm p Γ(p+M)U (a; b; z) (9)
with a = p + M, b = 1 + p -L and z = L Mm k H [M ] -1 k.
| • | and U (•; •; •) denotes respectively the determinant operator and the confluent hypergeometric function of the second kind (KummerU). This PDF has been named the KummerU PDF.

Maximum Likelihood (ML) estimator

If the density generator function can be mathematically established, the exact ML estimator of the normalized covariance matrix is the solution of the following recursive equation :

[ MML ] = 1 N N i=1 h p+1 k H i [ MML ] -1 k i h p k H i [ MML ] -1 k i k i k H i ( 10
)
Chitour and Pascal have proved that Eq. 10 admits a unique solution and that its corresponding iterative algorithm converges to the Fixed Point solution for every admissible initial condition [START_REF] Chitour | Exact maximum likelihood estimates for SIRV covariance matrix : Existence and algorithm analysis[END_REF]. In practice, the algorithm is initialized with the identity matrix [I p ].

For a Fisher distributed clutter, the density generator function has been obtained in Eq. 9. The exact ML estimator of the normalized covariance matrix exists. One can replace the density generator function by its expression in Eq. 10. It yields :

[ MML] = p + M N " L Mm « × N X i=1 U " p + 1 + M; 2 + p -L; L Mm k H i [ MML] -1 ki « U " p + M; 1 + p -L; L Mm k H i [ MML] -1 ki « kik H i (11)
3 Optimal GLRT segmentation

In this section, we propose to implement the SIRV estimation scheme and the KummerU PDF presented in section 2 to segment PolSAR data. This segmentation is based on the Generalized Likelihood Ratio Test formalism.

Multiple composite hypothesis test

We consider the POLSAR segmentation as the following multiple composite hypothesis test :

8 > > > < > > > : H0 : k = √ τ z, with τ ∼ F [m, L, M] and z ∼ N [0, [M ]] H1 : k = √ τ z, with τ ∼ F [m1, L1, M1] and z ∼ N [0, [M1]] . . . HC : k = √ τ z, with τ ∼ F [mC, LC, MC] and z ∼ N [0, [MC]] ( 12
)
where C is the number of segments. m, L and M are the Fisher parameters and [M ] is the normalized covariance matrix of the observed independent identically distributed N secondary data. Notice that the segmentation in C segments is equivalent to testing C + 1 hypotheses.

In this paper we suppose that the multiple hypotheses from Eq. 12 are not nested, i.e. there is a unique set of parameters (m, L, M and [M ]) characterizing each class. This implies that each class is described by a different SIRV, and hence the extended GLRT can be applied.

GLRT similarity measure

In multiple composite hypothesis test, the likelihood ratios are sufficient for optimal segmentation problem. The use of likelihood ratios referenced to a "dummy" hypothesis has been introduced in [START_REF] Vantrees | Detection, estimation, and modulation theory[END_REF]. In consequence, the secondary data observed at instance t + 1 are assigned to the segment ω, which maximizes the extended GLRT over all classes characterized by their normalized covariance matrices and Fisher parameters estimated at instance t :

ω = Argmax 1≤i≤C Λ i = Argmax 1≤i≤C p k (k 1 , . . . , k N |H i ) p k (k 1 , . . . , k N |H 0 ) = Argmax 1≤i≤C N a=1 p k (k a |H i ) N b=1 p k (k b |H 0 ) (13)
where p k (k 1 , . . . , k N |H i ) is the PDF under the H i hypothesis.

By taking the natural logarithm of Eq. 13, one can prove that, for a Fisher distributed clutter, Eq. 13 is equivalent to :

ω = Argmin 1≤i≤C ln |[ Mi]ML| |[ M ]ML| ! + ln Γ( Li)Γ( Mi) Γ( Li + Mi) ! + ln Γ( L + M) Γ( L)Γ( M) ! + p ln L M m Mi mi Li ! + ln Γ(p + M) Γ(p + Mi) ! - 1 N N X a=1 ln ( U p + Mi; 1 + p -Li; Li Mi mi k H a [Mi] -1 M L ka !) + 1 N N X b=1 ln ( U p + M; 1 + p -L; L M m k H b [ M ] -1 M L k b !) (14)
where Li , Mi , mi and [ Mi ] M L at instance t are respectively the ML estimates of L i , M i , m i and [M i ] M L for the segment i at instance t -1 (Eq. 10).

4 Segmentation results

On synthetic data

To construct the data-set, four 100 × 100 Gaussian realizations (speckle) are generated. Then, the speckle is multiplied with a texture image containing Fisher realizations. The same Fisher PDF is used in the whole image.

On very high resolution PolSAR data

Here, the proposed classifier is applied on a very high resolution PolSAR image. This data-set has been acquired by the X-band RAMSES airborne sensor with a resolution of 0.57m.

Conclusion

In this paper, authors have proposed to implement the SIRV estimation scheme to derive the normalized covariance matrix and the texture parameter. Based on some statistical consideration, the Beta Prime PDF has been introduced to describe the normalized texture component, which implies a Fisher distributed texture. Then, this statistical scalar model has been successfully validated to characterize high resolution urban clutter. Based on this model, the target scattering vector follows a KummerU PDF. Next, this PDF has been employed to derive an optimal segmentation algorithm based on the GLRT formalism. Results on synthetic and real data have shown that the proposed algorithm gives the best performances for the segmentation of high resolution data.

FIG. 1

 1 FIG. 1 -κ 2 /κ 3 plan for an urban area over the Oberpfaffenhofen test-site (ESAR, L-band). If ξ follows a Beta Prime PDF with L and M parameters, the texture parameter τ is Fisher distributed with m = µL M , L and M parameters.

  Fig. 2(a) shows a colored composition in the Pauli basis of the target scattering

FIG. 2 -

 2 FIG. 2 -Classification results of a simulated data-set (200×200 pixels) : (a) Colored composition of the target vector [k] 1 -[k] 3 -[k] 2 , (b) SIRV criterion, (c) KummerU criterion

Fig. 3 (

 3 a) shows a colored composition in the Pauli basis of target scattering vector. Classification results shown on Fig. 3(b), Fig. 3(c) and 3(d) are respectively done with the Wishart, SIRV and KummerU criterion. Each algorithms are first initialized with a partition obtained by the H/α plan. H and α represent respectively the entropy and the mean scattering angle [5].


Frédérique Krupa
email: f.krupa@lecolededesign.com

Karl Pineau
email: k.pineau@lecolededesign.com

Clémence Montagne
email: c.montagne@lecolededesign.com

Collective Interest Matrix: Can design be sustainable within capitalism?

Keywords: Capitalism, Sustainable Design, Green Design, Ecological Design, Collective Interest Matrix, Stakeholder Impact Matrix, Design for Good, Design for Emergency

 & 2019), "design for emergency" will face the same hurdles that "design for good" currently faces. When designing for adaptation, are we blind to the economic interests of our interventions? How do we reconcile conflicts between the human and environmental impact of design interventions? Should discussions of collective versus individual interests be positioned against public and private profits, against beneficial and adverse human and environmental impacts ? How do we resolve conflicts and identify our assumptions, bias, and blind spots that may limit the full exploration of possibilities? Should these discussions become the norm in design education, and if so, what methods should we employ? This paper proposes a Collective Interest Matrix for positioning design interventions.

Introduction

"…The economic advancement of the past two centuries is clearly remarkable, yet it is difficult to sustain, because by definition it requires constant innovation. The enduring bedrock principle of modern capitalism is this ingenuity. It produces new things, which, if successful, become durable, stable, and permanent features in the lives of modern people. Often, when such things are first experienced, they feel like "the shock of the new," but what at first is astonishing eventually becomes routine." [START_REF] Murphy | Design capitalism: Design, economics and innovation in the auto-industrial age[END_REF] Capitalism presents some unresolvable contradictions. Capitalism has growth imperatives for stability but cannot grow indefinitely for ecological reasons. Providing unprecedented wealth and productivity for the past 200 years to advanced economies along with extremely high standards of living, capitalism has raised many developing nations out of absolute poverty but without the ecological possibility for the global south to consume resources like the global north. [START_REF] Alexander | Post-capitalism by design not disaster[END_REF][START_REF] Hickel | The sustainable development index: Measuring the ecological efficiency of human development in the anthropocene[END_REF] While we hold on to technological hopes of "green growth" that would allow for economic expansion decoupled from resource use and green-house-gas emissions, researchers are quickly proving that there is an absolute limit to this optimistic scenario. (Hickel, 2018;Hickel & Kalis, 2019;Hickel, 2020a;[START_REF] Nardi | Design in the age of climate change[END_REF] In short, green growth has not been able to decouple progress and resource consumption anywhere near the levels required, because technology feeds the problem it is supposed to resolve through its incompressible, boundless resource and energy requirements.

This in turn has prompted a number of researchers to propose post-capitalist solutions to encourage frugal, sustainable, regenerative design. [START_REF] Mcdonough | Cradle to Cradle, Remaking the Way We Make Things[END_REF][START_REF] Manzini | Scenarios of sustainable wellbeing[END_REF], 2015. Mang & Als, 2012, 2015[START_REF] Klein | This changes everything: Capitalism vs. the climate[END_REF][START_REF] Alexander | Post-capitalism by design not disaster[END_REF][START_REF] Hickel | What does degrowth mean ? A few points of clarification Johnson[END_REF][START_REF] Fry | Defuturing: a new design philosophy[END_REF]. Monnin & Als, 2020[START_REF] Miller | Capitalism: an ethnographic approach[END_REF]. Design Council, 2020[START_REF] Miller | Capitalism: an ethnographic approach[END_REF] Lifecycle management, bio-design, degrowth by designrefocusing our ambitions and processes towards more frugal, eco-responsible, collaborative practices and solutions -have yet to become corporate norms due to the efforts that would be required to adhere to consumer demands for our currently-affordable privileges, conveniences and desires -like low-cost intercontinental tourism, video-on-demand, exotic and out-of-season foods, fast fashion, ML-enhanced home automation, etc. Willfully forgoing these modern marvels of international logistics is difficult indeed. "... degrowth captures an essential insight: it directly evokes, more clearly than any other term, the need for planned contraction of the energy and resource demands of overgrown or 'developed' economies. That is an agenda that mainstream environmental and social discourse refuses to acknowledge, because significant contraction of energy and resource demands is incompatible with ongoing growth in GDP." [START_REF] Alexander | Post-capitalism by design not disaster[END_REF] Some companies are moving away from consumption driven by individual ownership and programmed obsolescence to a subscription-based service model that incentivizes reduced consumption, repair and shared ownership, like the Porsche Drive program 1 . Software companies like Adobe are also moving into this arena, reducing their need to release software updates to maintain recurring revenues from their existing user-base. Now clients pay to access its cloud-based services, yielding record profits for Adobe in the first quarter of 2022. This extractive financial model allows market leaders to maintain their positions with less investments or requirement to innovate their products, and while convenience and security are enhanced with cloud-based computing compared to using a traditional hard drive, the global network of data centers needed to power these applications requires about a million times more energy than using a local hard drive. [START_REF] Adamson | Carbon and the Cloud[END_REF] Short of a cataclysm, capitalism will try to fill any vacuum of unmet need with most profitable cost margins, ignoring ecological imperatives when convenient.

Design for « Good »

Designers working in industry are service providers. In the best case scenario, they help define strategies and corporate visions. In worst case scenarios, they are mere executors of others specifications. Papanek and others [START_REF] Garland | First things first manifesto[END_REF] put forth perceived need versus frivolous, marketing-related design as a litmus test between good and bad design practices. However, context often blurs these distinctions, especially when dealing with inclusive design that may appear like trivial conveniences to some but are performance-enhancing options to others. A smart speaker to control home lighting is a convenience for the able-bodied but a gamechanging tool to anyone with mobility-impairment. Two hundred years ago, there were a halfdozen formulas for soap that cleaned everything in the house, people included. [START_REF] Forty | Objects of desire: design and society since 1750[END_REF]. Now it seems normal to have a different soap for hands, faces, hair, and intimate body parts, for different laundry, and all the different surfaces and rooms in the house. How much of this evolution is dictated by marketing versus improved performance? Would eczema-prone, acnesuffering individuals consider soap-free cleansers frivolous? We will not pretend to make these distinctions. We simply point out that utility value judgements in complex socio-technical systems are nuanced, with valid arguments on opposite sides of the debate. [START_REF] Norman | DesignX: complex sociotechnical systems[END_REF] In this ethically-charged, possibly guilt-inducing discourse, design students have a hard time identifying their ethical stances because like all nuanced discussions, hard and fast boundaries are not easy to identify or defend. To help students clarify their design's positioning within this complex system, we propose two successive intermediate design methods and artefacts -the Stakeholder Impact Matrix (SIM) -which is a reworking of the methods proposed by [START_REF] Olander | Stakeholder impact analysis in construction project management[END_REF] -and the Collective Interest Matrix (CIM) which we developed and explain in in this paper. These methods help identify assumptions, bias, and blind spots that may otherwise limit the full exploration of possibilities and to discuss potential conflicts between: collective versus individual interests; public and private profits; and beneficial and adverse human and environmental impacts.

Stakeholder Impact Matrix

Freeman (1984) defines stakeholders as "any group or individual who can affect, or is affected by, the achievement of a corporation's purpose." By limiting stakeholders to humans -an issue we will address later -we can build upon Olander's stakeholder impact analysis tools derived from the science of construction management and economics on the decision-making power and ability to impact project outcomes. These methods and their structural underpinnings can be used to explore stakeholder networks and impacts in socio-technical systems. We build upon his approach, simplifying and reversing the order of some steps, but ultimately, we also question some underlying ground-truths. [START_REF] Mitchell | Toward a theory of stakeholder identification and salience: defining the principle of who and what really counts[END_REF] defines seven different stakeholder classes:

1. Dormant stakeholders possess power to impose their will, but do not have any legitimate relationship or urgent claim. Their power remains unused.

2. Discretionary stakeholders possess the attribute of legitimacy, but they have no power or urgent claim. There is no absolute pressure for managers to engage in an active relationship, although they may choose to do so. 3. Demanding stakeholders possess an urgent claim, but have no power or legitimate relationship. This is bothersome, but does not warrant more than passing management attention. 4. Dominant stakeholders are both powerful and legitimate. It seems clear that the expectations of any stakeholders perceived by managers to have power and legitimacy will matter. 5. Dangerous stakeholders lack legitimacy, but possess power and urgency.

They will be coercive and possibly violent, making the stakeholder 'dangerous'. 6. Dependent stakeholders have urgent and legitimate claims, but possess no power. These stakeholders depend upon others for the power necessary to carry out their will. 7. Definitive stakeholders are those that possess both power and legitimacy.

They will already be members of an organization's dominant coalition. When such a stakeholder's claim is urgent, managers have a clear and immediate mandate to attend to and give priority to that claim. (Olander, 2007, 279.) Once the full stake-holder network has been identified (including those in the realm of production, mediation, distribution, reception, oversight, funding), [START_REF] Bourne | Visualising and mapping stakeholder influence[END_REF] propose the vested interest-impact index (ViII = √(v*i/25)), where the vested interest levels (vprobability of impact) and the influence impact levels (i -level of impact) are qualitatively defined on the following scale : 5 very high, 4 high, 3 neutral, 2 low and 1 very low. [START_REF] Olander | Stakeholder impact analysis in construction project management[END_REF]). These vested interest-impact values can used to plot the various stakeholder within the stakeholder impact/probability matrix (Fig. 1.)

Figure 1. The stakeholder impact/probability-matrix [START_REF] Olander | Stakeholder impact analysis in construction project management[END_REF]Johnson & Als., 1999).

These stakeholder mapping tools help identify obstacles to development, taking for granted that development is the end-goal. This basic assumption is unsatisfactory and ecologically inadequate regarding the production of reliable, comprehensive information to help environmentally conscious policy makers, producers, and consumers in selecting sustainable products and production processes.

While human-centered stakeholder networks may be mapped and evaluated with the SIM, we argue that a more expansive vision of stakeholder -beyond people -is necessary to evaluate impacts to organic ecosystems, as well as artificial, intangible ones like a community's reputation, attractivity, diversity, credit rating, etc. Lifecycle assessments and carbon footprint measurements have developed in this void, but this terrain is still fraught with shortcomings and requires our ongoing attention. (Roy & Al., 2008)

Collective Interest Matrix (CIM)

To identify a design proposition's economic positioning, we would like to propose a second intermediate design tool, the Collective Interest Matrix (CIM), that inserts itself into the ideation phase. CIM looks at two axes: sources of capital and value creation. The first axis looks at a project's funding sources and identifies who assumes the risk during the project's development phase. The second looks at the value creation: who benefits most from this project and who owns the intellectual property (IP). Note that there may be multiple financing sources as well as multiple recipients of value creation and IP ownership in a single project. Below is a blank CIM template (Fig. 2). The CIM template tries to identify if there may be conflicts of interest between a project's capital providers and its end-goals. CIM's Capital axis is split between three types of financing sources:

1. Private financing, which includes private equity investments, as well as inhouse corporate financing and business loans.

Public Funds

Seed, Grant & Donations

Private Equity

Corporate Government /Associations Commons

Collective Interest Matrix

Where is the financing (risk )coming from?

Capital

Value Who derives the benefits and owns the intellectual property?

2. Public financing derived from taxes, bonds, investments, etc. which are controlled and distributed through public agencies. 3. Seed Money, Donations or Grants (SDG) -funding from drawing upon personal savings, grants, and donations of time or money.

On the Value axis, we also identified three categories of value recipients, those who derive the positive impact from the project. These value recipients may be independent and different from those that own the intellectual property.

1. Corporate -When the end-goal is corporate profits, possibly branding-related. IP ownership and value creation are often directly related. 2. Government/Associations -The government and association realm organizes itself to serve specific needs or goals in the community that may be necessary but not be financially lucrative enough for corporate interests. 3. The Commons -Work towards the common good, by adding to the cultural and natural resources accessible to all members of a society. Work by international nongovernment organizations (NGO) and work produced under Creative Commons licensing typically falls under this category.

Together, these two axes create a 3 by 3 matrix whose intersections could help identify some new, creative financing options or highlight potential conflicts of interest between the project's end goals and its financial backers. CIM is an internal, intermediate object in that it is meant to be used to structure one's reflection rather than as a tool to communicate with external stakeholders. Like a comprehensive SWOT analysis, some unflattering analysis could be integrated into this matrix and should be considered an internal working document.

In figure 3, we have identified the types of organizations and projects that may be located at the different matrix entries. Starting from the upper left, let us look at the dynamics between capital and value at each junction. Each entry will be listed as Capital / Value : example of a project or funding organization.

Corporate

Government /Associations Commons Where is the financing (risk )coming from?

Capital

Value Who derives the benefits and owns the intellectual property?

Collective Interest Matrix

Conclusion

In this paper, we propose addressing the difficult tensions between capitalism and ecoresponsible, sustainable design by building upon on two intermediate design tools: the Stakeholder Impact Matrix (SIM) which is derived from [START_REF] Olander | Stakeholder impact analysis in construction project management[END_REF] and Collective Interest Matrix (CIM), which we developed.

First with the SIM, we identify the seven categories of stakeholders: Dormant, Discretionary, Demanding, Dominant, Dangerous, Dependent, and Definitive. Once the stakeholder network has been identified, values can be assigned using the vested interest-impact index (ViII = (v*i/25)) and plotted on the stakeholder impact/probability matrix (Fig. 1.) Moving beyond its inherent development-driven ground-truth, non-human stakeholders who will be impacted by a design proposal should be identified to evaluate impacts to organic and socio-technical ecosystems. Refining this process will be the next step in our research.

Once the stakeholders have been identified, the Collective Interest Matrix (Fig. 2) can be plotted to identify possible sources of capital, identify who will assume the risks, who derives the benefits from the project, and who owns the intellectual property. When aiming for a socially and environmentally responsible design project, the CIM can identify potential conflicts of interest while still in the ideation stage, open up ideas towards other funding possibilities, and help guide the development towards mutually-beneficial socially and ecologically responsible cooperation between the sources of capital, design's beneficiaries, and the IP owners. However, it remains the designer's responsibility to assess and ultimately determine the social, economic and environmental end-goals, impact, and trade-offs of their propositions.

We humbly propose these intermediate methods and tools to the design community to extend ways of looking at design's impacts, and look forward to getting feedback on possible refinements and improvements.

Figure 2 .

 2 Figure 2. Blank template to be filled out in the design ideation stage.

  Public / Corporate : public financing to boost corporate innovation through research and development funding, managed through public agencies like national research organizations. The belief is that investment in corporate innovation leads to job creation and an increased tax base. 2. Public / Government-Associations : projects for and funded by public agencies. This can include government investment banks, the military, utility providers, etc. Public funding of NGOs may make them less politically independent. Government-Associations : Consortium-based organizations like the W3C that works towards shared IP and cooperation for international standards, and the open-source projects that may arise from their common goals. International tensions may arise from over-represented national in certain consortiums. 9. Private / Commons : Open-source corporate-driven projects like Apple's Swift programming languages, Google Chromium and TensorFlow, Pytorch, publication of corporate machine learning algorithms, etc. Harnessing improvement to source code by releasing it to the community in exchange for relinquishing corporate IP rights. This does produce some tension within the open source community.

	7. Private / Corporate : Traditional corporate and capitalist activities that focus on
	growth, maximizing return on investment (ROI), and protection and expansion of IP
	assets. Private funding and IP ownership brings risks for conflicts of interest when
	developing social and environmentally responsible projects. This capital/value conflict
	is most often the source greenwashing and ethics-washing charges against
	corporations.
	3. Public / Commons : projects sustaining cultural production, transformation and protection of natural resources, investments in public infrastructure, enacted through public policy and investments. United Nations sustainable development goals (UNSDGs) push for sustainability but with economic development in SDG8 that is not sustainable. (Hickel, 2019a) 4. Seed, Grant, Donations / Corporate : tax-refund-financed corporate research and development, Kickstarter and go-fund-me projects, self-financed start-ups, etc. 5. SGD / Government-Associations : Associative projects for community development, funded by grants and donations; however, special interest groups and political action committees also fall in this sector. Projects initiated by universities and students, like the Linux kernel, and other GNU general public license software development (self-Public Funds Public Innovation Financing (e.g. ANRT & CIFRE, RFI-OIC) Governement institutions Public Policy & Infrastructure (Nature reserves, United Nations Sustainable Development Goals) Seed, Grant & Donations R&D financing (tax refunds) Start-ups associations focusing on community action (may be political) NGOs focusing on ecosystems (Greenpeace) 8. Private / Figure 3. Collective Interest Matrix with various types of projects, design interventions, funding Private Equity All coprorate IP (Green & ethics washing risk) Open-source Corporate-driven Consortium based organization Projects (W3C, Mozilla, etc.) (e.g. Swift)
	funded and open source). partners and/or recipients found in each respective entry.
	6. SGD / Commons : NGO projects for ecosystem protection and cultural preservation,
	funded by grants and donations.

1.

https://www.porsche.com/usa/accessoriesandservices/porschedrive/subscription/ (retrieved 12/06/2022)