PARETO GENEALOGIES ARISING FROM A POISSON BRANCHING EVOLUTION MODEL WITH SELECTION

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1 PARETO GEEALOGIES ARISIG FROM A POISSO BRACHIG EVOLUTIO MODEL WITH SELECTIO THIERRY E. HUILLET Abstract. We stuy a class of coalescents erive from a sampling proceure out of i.i.. Paretoα ranom variables, normalize by their sum, incluing β size-biasing on total length effects β < α. Depening on the range of α, we erive the large limit coalescents structure, leaing either to a iscrete-time Poisson-Dirichletα, β Ξ coalescent α [,, or to a family of continuous-time Beta2 α, α β Λ coalescents α [, 2, or to the Kingman coalescent α 2. We inicate that this class of coalescent processes an their scaling limits may be viewe as the genealogical processes of some forwar in time evolving branching population moels incluing selection effects. In such constant-size population moels, the reprouction step, which is base on a fitness-epenent Poisson Point Process with scaling power-lawα intensity, is couple to a selection step consisting of sorting out the fittest iniviuals issue from the reprouction step. Running title: Pareto genealogies in a Poisson evolution moel with selection. Keywors: Pareto coalescents, scaling limits, Poisson-Dirichlet, Kingman an Beta coalescents, Poisson Point Process, evolution moel incluing selection.. Introuction We first investigate iscrete-time finite coalescents erive from sampling from i.i.. Paretoα ranom variables, normalize by their sum, with α >. We inclue size-biasing on total length effects involving a parameter β < α. Depening on the range of α, we erive the large limit coalescents structure: The case α [, leas to a iscrete-time Poisson-Dirichletα, β Ξ coalescent with no time-scaling. The case α = gives rise to a continuous-time beta, β Λ coalescent, involving a logarithmic time scaling log. The case α, 2 leas to a continuous-time Beta2 α, α β Λ coalescent, involving a power-law time scaling accoring to α. The range α 2 gives rise to the stanar Kingman coalescent with time scaling if α > 2 an / log in the critical case α = 2. We give for each case the exact spees of convergence. We establish a loose link with a Generalize Central Limit Theorem for stable ranom variables an we briefly recall the main statistical features akin to general Λ coalescents. We inicate that the above special classes of coalescent processes an their scaling limits may be viewe as the genealogical processes of some forwar in time evolving

2 2 THIERRY E. HUILLET branching population moels incluing selection effects. These moels are closely relate in spirit to the aitive exponential moel iscusse in Brunet et al [6], [7]. In the moels we first consier, the size of the population is kept constant over the generations. Each alive iniviual is assigne some positive fitness x >. In each generation an for each of the offspring alive inepenently, the reprouction step is base on a fitness-epenent Poisson Point Process PPP with scaling power-lawα intensity; this proceure assigns new fitnesses to the potentially infinitely many iniviuals of the next generation, in a multiplicative way. We call f x = x α the output fitness of an iniviual with fitness x. The selection step then consists of sorting out the fittest iniviuals issue from the reprouction step. The process is iterate inepenently over the subsequent generations. To the first large approximation, the logarithm of the mean output fitness within each generation k, scale by the generation number k, is shown to shift to the right at spee v = log log as k. While aopting a sampling point of view base on the intensity of the PPP to compute the coalescence probabilities that some offspring is the one of a parental iniviual with given fitness, it is shown that the genealogy of the branching moel with selection is in the omain of attraction of the beta, β coalescent in the large limit Bolthausen-Sznitman coalescent if β =. It is also shown that the full class of the Pareto-coalescents iscusse earlier can be obtaine while consiering the PPP which is the output image of the original one, given by the output map f x = x α in fitness space. In this setup, the large limit computations of the coalescence an merging probabilities are base not on the fitnesses but on the output eforme fitnesses. 2. Coalescents erive from Pareto-Sampling 2.. Pareto sampling an coalescents. Let X,.., X be i.i.. Paretoα ranom variables rvs with P X > x = x α, α > an x. Let F X x = P X > x enote its probability istribution function pf. The ensity of X is f X x = αx α+. Let S n := X n / X n, n =,..., efine a ranom partition of the unit interval, upon normalizing the Xs by their sum. The S n s are ientically istribute but not inepenent of course as they sum up to ; by oing so, the unit interval [, ] is thus broken into ranom pieces subintervals or segments of sizes S n, n =,...,. By sampling the Ss, we mean that we raw inepenently i uniform ranom variables on the unit interval with i, looking at the subintervals which are being hit in the process. From this proceure, for instance, the probability that the i sample hits any one of the S n s only once is P i, = E Sn i = E S i. n= This particular way of introucing selection in a ranomly evolving branching population with constant poulation size seems to appear first in [5]. It has nothing to o with the way selection is classically introuce in population genetics; see [28], [4] an [8].

3 PARETO GEEALOGIES I A POISSO EVOLUTIO MODEL WITH SELECTIO 3 Let Σ := n= X n enote the partial sum of the Xs. For the values of β for which E Σ β exists, we can size-bias the latter probability by the total length Σ an consier instea 2 E Σ β P n= Si n E Σ β i, = Si =. E Σ β E Σ β The latter event uner consieration correspons to an i to merger of some Markov coalescent process where i particles are ientifie to a single one share the same ancestor whenever the i sample hits the same subinterval of the unit partition. In this setup, P i, is therefore the entry i, of its one-step transition matrix. The quantity c := P 2,, which is the probability that two iniviuals chosen at ranom out of share the same common ancestor, is calle the coalescence probability. Similarly we can efine a i to j merger j i by consiering the event that the i particles hit any size j subset of the segments S constituting the partition of unity. We get 2 P i,j = j i E S i l l = j i i...i j +...+i j=i l= j j l j E S S l i, j l l= j where the star-sum in 2 runs over the i l. The quantity E l= Si l l is the probability of a i,..., i j merger from i to j. Using the same abuse of notation, we shall also write the size-biase version of the latter probability as 3 P i E Σ β j l= Si l l i,j =. j i i...i j +...+i j=i E Σ β Unless state otherwise, whenever we speak in the sequel of P i,j, we mean 3 an not 2. Clearly, i j= P i,j = an so the matrix P with entries P i,j, i =,..., an j =,..., i, is a lower-triangular stochastic matrix corresponing to some Markov iscrete-time-k coalescent pure eath process, say x k, with finite state-space 3 an state {} absorbing. Let us first investigate the expression of the size-biase probability P i,, showing that its large estimate epens on the unerstaning of the β moments of Σ. 2 We abusively use the same notation P i, in the size-biase setup as in corresponing to β =, to avoi overburen notations. 3 A true coalescent process takes values in the set of equivalence relations or partitions on {,..., } an we rather eal here an throughout with its block-counting counterpart.

4 4 THIERRY E. HUILLET Proposition. When < β < α < 2, the size-biase probability of an i to merger reas E Σ β α 4 P i, = α E Σ β Γ i α Γ α β, i 2. Γ i β Proof: We have /S = + Σ /X where Σ := n=2 X n = n= X n =: Σ. By conitioning on X, with f Σ the ensity of Σ, we get P i, = E Σ β x f X x x β+ + x Reversing the integration an after two changes of variables P i, = = E Σ β α E Σ β E = α E Σ β α Σ β z z β i z z z β i z α β z β i f Σ x z z. x β+ f X x f Σ x z x u i 2 u α u α β u. s β α f Σ s s When β < α < 2, the latter integral is upon aequate normalization by a beta function term B 2 α, α β ientifie with the orer i 2 moment of a beta2 α, α β rv. In this parameter range, for i 2 are well-efine. u i 2 u α u α β u = Γ i α Γ α β Γ i β =: B i α, α β The Pareto rv X has power-law tails with inex α. Therefore E X β only exists when β < α, with E X β = α/ α β. Clearly also, the tails of the sum Σ obey P Σ > s P X > s = s α. Because P X > s P M > s = s s s α where M = max X,.., X, this means that for large s, the event Σ > s is essentially etermine by the event M > s. ote that, as a result, Σ has the same tail inex as X, inicating that the β moment of Σ only exists for β < α. We will show below from this, that large estimates of E Σ β can be obtaine. As a result, for example, base on 4, whenever < α < 2 an β < α, it will be checke that E Σ β α c := P 2, = α B 2 α, α β α. E Σ β

5 PARETO GEEALOGIES I A POISSO EVOLUTIO MODEL WITH SELECTIO 5 We will show that, in that case, for each i 2, the limits lim c with c P Γ α β Γ i α i, = B 2 α, α β Γ i β P u i 2 Λ u >, i, exist, where Λ u has the ensity on [, ]: u α u α β /B 2 α, α β, which is a beta2 α, α β ensity. Because in the large limit, simultaneous multiple collisions will be seen to be negligible, we conclue, using similar arguments to the ones in [36] an [3], that, in the range < α < 2 an β < α, a time-scale version of the finite iscrete-timek coalescent x k arising from size-biase sampling out of Paretoα partitions converges weakly as to a continuous-time-t Λ coalescent x t where Λ is a beta2 α, α β probability measure on [, ] ; namely x t = lim x 4 [t/c ]. In other wors, if t is continuous-time, the appropriate scaling is k [t/c ] the integral part of t/c, showing that time t shoul be measure in units of e := c the effective population size. The limiting process x t is a continuous-time pure eath Markov process on = {, 2,...}, absorbe at state {} an with transition rates from i to j given by c P i,j λ i,j := The rate terms λ i,j may also be written as λ i,j := i u i j u j Λ u, j < i. j j l= l l! i j l q l+i j+, where q l := ul 2 Λ u are the l moments of u 2 Λ u an also the rates of l to mergers λ l, Generalities on Λ coalescents. Whenever one gets a family of rates λ i,j as above for some finite probability measure Λ on [, ], one speaks of continuoustime-t Λ coalescents see [34] for a precise efinition. These Λ coalescents are non-increasing pure eath Markov processes, say x t, on the state-space, an the λ i,j s are the rates at which i to j < i mergers for x t occur. The state {} is absorbing. In such processes, multiple collisions of any orer when j < i can occur, but never simultaneously. The total eath rate at which some merger occurs, starting from state i, is λ i := i j= λ i,j. One can check that when Λ = δ corresponing to the Kingman coalescent, λ i,j only when j = i with λ i = λ i,i = i 2. The general expression of λi is λ i = u 2 u i iu u i Λ u. [,] 4 Here, because two processes are involve, the symbol = means convergence of all the finiteimensional istributions of x, t to the ones of xt, t. [t/c ]

6 6 THIERRY E. HUILLET When Λ {} = excluing thereby the Kingman coalescent, the precise ynamics of x t when starte at i is given by x = i an [3] 5 x t x = B x s, u Bxs,u> s u =,t],],t],] B xs, u s u. + Here, x + = max x,, is a ranom Poisson measure on [,, ] with intensity s u 2 Λ u an B x s, u bin xs, u is a binomial rv with parameters xs, u. As a result, with 6 r i :=,] ui + u i u 2 Λ u, i >, upon taking the expectation in 5, it hols that E x t x t = r xt t. From this, the quantity r i is the rate at which size i blocks are being lost as time passes by. Clearly r is also i i r i = iλ i jλ i,j = i j λ i,j. j= Consequently, the reciprocal function /r i of the rate r i interprets as the expecte time spent by x t in a state with i lineages an therefore x =i j=2 /r j will give a large i estimate of the expecte time to the most recent common ancestor the height of the coalescent tree: j= τ i, := inf t R + : x t = x = i. There are lots of etaile stuies in the literature see a precise partial list below on other functionals of x t such as the total branch length L i of the Λ coalescent, its total external branch length L e i, the length l i of its external branch the time till first collision of a branch chosen at ranom out of i, the number of collisions C i till time to most recent common ancestor,... All these functionals obey some istributional ientities which prove useful to obtain some insight on their limit laws as i. Given x = i, all involve the number U i of singletons taking part in the first collision occurring at time T i expλi, giving x Ti = i U i singletons not participating to the first collision with T i, x Ti inepenent. The quantity U i is important in itself because, ue to P U i = j = λ i,i j+ /λ i, j = 2,.., i r i = λ i E U i where E U i = i λ i,] u u i Λ u. u u Provie Λ has no atom at point {}, the conition i=2 /r i < is the necessary an sufficient conition for x t to come own from infinity, [38].

7 PARETO GEEALOGIES I A POISSO EVOLUTIO MODEL WITH SELECTIO 7 Clearly inee 5, τ i, = Ti +τ xti,, L i = iti +L xti, L e i = it i +L e x Ti, C i = +CxTi an l i = Ti + B i l xti where B i is a Bernoulli rv, given by: P B i = x Ti = x Ti /i with B i l xti inepenent of T i. Famous examples that we shall eal with in the sequel, inclue Λ coalescents for which: - Λ u = B a, b u a u b [,] u u, with a, b > an with B a, b the beta function: we get the betaa, b coalescents. - Λ u = B 2 α, α u α u α [,] u u, α,, 2; this is the beta2 α, α coalescent. - Lebesgue Λ u = [,] u u : this is the Bolthausen-Sznitman coalescent or beta, coalescent. - Λ u = δ : we get the Kingman coalescent where only binary mergers can occur j = i one at a time, [26]. - Λ u = δ : we get the star-shape coalescent involving a single big collision. Using the above istributional ientities, it can be shown for instance that, for the Kingman coalescent an for large i, to leaing orer of magnitue, rough estimates are: τ i, 2 /i [4], L i 2 log i [2], L e i 2 [22], C i = i an l i /i [9]. For the Bolthausen-Sznitman coalescent: τ i, log log i [7], L i i/ log i [2], C i i/ log i [2] an l i / log i [5]. For the beta2 α, α coalescent with < α <, L i i [29] an L i /L e i converges in probability to [3] an l i O [6]. For the beta2 α, α coalescent with < α < 2, L i /i α 2 [24], L e i /iα 2 [] an l i /i α []. This information is important to grasp the general shape of the coalescent trees in each case. 3. GCLT for Pareto sums In this Section, we first sketch a loose connection of the large estimate of c with the Generalize Central Limit Theorem for ranom variables in the omain of attraction of stable laws, [4]. Let Σ = n= X n be the partial sums of the i.i.. Xs with Paretoα istributions, α >, on,. Let S α be skewe α stable rvs with E e λsα = e λα if α,, λ 5 = means equality in istribution between ranom variables.

8 8 THIERRY E. HUILLET the Laplace-Stieltjes transform LST of a one-sie α stable rv, E e iλs = e λ +isignλ 2 π log λ the characteristic function c.f. of a skewe stable Cauchy rv on R E e iλsα = e λ α isgnλ tan πα 2 if α, 2, λ R the characteristic function c.f. of a skewe α stable Cauchy rv on R an the c.f. of a stanar normal rv on R. E e iλsα = e λ2 /2 if α 2, λ R The following Generalize Central Limit Theorem GCLT then hols Theorem 2. [4], [42]: Let Σ enote the partial sum sequence of i.i.. Paretoα ranom variables. Then, 7 where, with Σ a b S α C α = πα Γ α cos 2 /α if α, 2 \ {}, b is given by C = π 2, C α = α α 2 2 /2 α if α > 2, α i b = C α max/α,/2 if α 2 ii b = log /2 if α = 2 an, with γ the Euler constant an E X = µ := α/ α, a is given by i a = if α, ii a = π2 2 sin 2x π iii a = µ if α,. F X x log + γ log 2 π if α = When α, the characteristic values of Σ can be guesse to be what they are claime to be /α if α < an log if α = while estimating m xf X x x where m is the moe of M which is seen to grow like /α see [4] an similarly for the fluctuation scaling term b for α 2 of orer /α if α < 2 an log /2 if α = 2. EΣ β From these rough estimates, we woul conclue that with c EΣβ α the leaing orer in c Cα /α β α Cα /α β = O if α,, up to

9 PARETO GEEALOGIES I A POISSO EVOLUTIO MODEL WITH SELECTIO 9 log β c log β log if α = c µβ α µ β µ α α if α, 2. Depening on the values of α, we therefore anticipate α, : Because in that case c is asymptotic to a constant, this suggests a limiting iscrete-time coalescent. We will show below that it is not a iscrete-time Λ coalescent, rather it is a Ξ coalescent of the Poisson-Dirichlet type with two parameters α, β. Ξ coalescents were first introuce in [3] an further stuie in [37]. In sharp contrast with Λ coalescents, multiple collisions can occur simultaneously at the same transition time. In their block-counting version, they are characterize by the set of numbers φ j i,.., i j efining the probabilities of an i,.., i j merger i i j = i, resulting when the Ξ coalescent is iscrete, in an i to j i transition with probability P i,j = i j! i +...+i j=i i φj...i j i,.., i j. The φ j i,.., i j can be written as j φ j i,.., i j = u i l 2 l Λ j u,.., u j, j l= for some finite measures Λ j with ensity on the j + simplex { } j = u,.., u j [, ] j : u u j. The set of measures Λ j, j, characterize by their moments φ j, with values over the simplices j, completely characterize the Ξ coalescent, [3]. In the simplest cases, with u, u := l u2 l, it is also convenient see [37] to rewrite the φ j s as where = φ j i,.., i j = j k,...,k j l= all istinct u i l k l Ξ u u, u, { u:= u,.., u l,.. : u.. u l.. : l u l } an Ξ a finite measure concentrate on the subset of consisting of those us exactly summing to. Letting ν u := Ξ u / u, u, the measure ν on the infinite simplex is such that ν <. α = : A Λ coalescent with logarithmic effective population size where Λ is a beta, β probability measure with β < as in [7] reucing to the Bolthausen- Sznitman coalescent when avoiing size-biasing corresponing to β =. α, 2: A Λ coalescent with e α where Λ is a beta2 α, α β measure with β < α. In the latter case, β = leas to the stanar beta2 α, α coalescent.

10 THIERRY E. HUILLET α 2 : In the range α > 2, respectively α = 2, e respectively e / log an the obtaine scale limiting coalescent is a Kingman coalescent, as a result of Λ approaching δ. In this latter case, only binary mergers occur in the large limit. Remark: The Kingman coalescent also occurs when ealing with sampling from some alternative ranom partition. For instance, woul sampling be efine from a ranom partition of unity given by S n := X n / X n, n =,..., where X now obeys the following gammaθ ensity: f X x = Γ θ x θ e x, θ, x >, then the law of Σ = X X is f Σ x = Γ θ x θ e x, inepenent of S an P i, = E Σ β Si n = E Σ β = Γ θ Γ θ E E Γ i + θ Γ θ + i Σ β Si Σ β = E S i i Γ i + θ θ i. Γ θ Thus c = P 2, = +θ θ together with = P 3, = +θ2+θ. Because triple mergers are asymptotically negligible compare to binary ones /c 2 θ 2, the time-scale limiting coalescent using c = is a Kingman coalescent. The prefactor +θ θ appearing in front of c is the ratio ρ/µ 2 where ρ := E X 2 = θ θ + an µ := E X = θ an c is inepenent of β. 4. Large asymptotic estimation of E Σ β an consequences +θ θ In this Section, we compute the asymptotic behavior of the β moments of Σ in the cases α,, α, 2 an α =, an α 2, making the previous conclusions base on the GCLT consistent 6. We start with the case α,. Theorem 3. When α,, as, x k = lim x k iscrete-time Poisson-Dirichletα, β Ξ coalescent. exists an is a Proof: For the values of β for which it makes sense, we have E Σ β = λ λ β E e λσ = λ λ β E e λx. Γ β Γ β When is large, only the small λ approximation of E e λx to the latter integral contributes. For small λ, we have 8 E e λx = α α x α+ e λx x = α x α+ e λx x x α+ e λx x Γ α λ α e Γ αλα. 6 The technique we use is inspire from the one use in [7] in a particular case. The author is inebte to B. Derria for pointing this out to him.

11 PARETO GEEALOGIES I A POISSO EVOLUTIO MODEL WITH SELECTIO ote that, when λ is small E Xe i λx = i i λ i E e λx αγ i α λ α i. Thus, E Σ β λ λ β e Γ αλα. Γ β With the change of variables u = Γ α λ α, with β < α, we get E Σ β β/α Γ αβ/α u u β/α e u = β/α Γ αβ/α Γ β/α. αγ β Γ β Finally, using the above large estimate of E Σ β, the ientity Γ x + = xγ x an 4 with i = 2, when β < α <, we obtain E Σ β α Γ β β α c = α α = E Σ β Γ 2 α Γ α β Γ 2 β Γ β α Γ α Γ β Γ 2 α Γ α β = α Γ α β Γ α Γ 2 β β =: c. Γ 2 α Γ α β Γ 2 β Thus the coalescence probability c converges to c,. When α,, using again 4, we obtain more generally P Γ β Γ i α i, P i, = Γ α Γ i β, which are the probabilities to merge all i particles in one step in the limiting iscretetime coalescent. To erive the full transition probabilities of the limiting iscrete-time-k coalescent = lim x k, recalling that x k we use 9 E j Σ β S i l l l= Σ β l= P i,j = = α j Γ i β i E j i i...i j +...+i j=i Γ i β j Γ i l α l= λ λ i β j l= Σ β j l= Si l l, E Σ β E X i l l e λx l E e λx j λ λ i β λ αj i e Γ αλα. Performing again the change of variables u = Γ α λ α, with β < α, we get j E S i l l α j β/α j Γ α β/α Γ j β/α j Γ i l α. Γ i β Using j j /j! for large, we finally get P i,j P i,j where j i P i,j = i! Γ β Γ j β/α αj j! Γ β/α Γ i β j i +...+i j=i l= l= Γ i l α Γ α i l!.

12 2 THIERRY E. HUILLET These are the full transition probabilities of the limiting iscrete-time-k coalescent x k in the regime α, an β < α satisfying i j= P i,j =. ote that P i, are the probabilities obtaine previously an that the iagonal terms the eigenvalues of P rea i Γ β Γ i β/α P i,i = α Γ β/α Γ i β. Clearly this iscrete-time coalescent is not a iscrete Λ coalescent as simultaneous multiple collisions can occur it is a Ξ coalescent. Clearly P i,j is also P i,j = i φ j! i...i j i,.., i j, j i +...+i j=i where the φ j i,.., i j s efine the probabilities of a i,.., i j merger i i j = i. These φ j i,.., i j, which can be rea from, may be written uner the alternative form φ j i,.., i j : j Γ β Γ j β/α j Γ i l α = α Γ β/α Γ i β Γ α where Thus is also = c j,α,β Γ αj β Γ i β c j,α,β := j l= i! Γ αj β P i,j = c j,α,β j! Γ i β j Γ i l α, l= Γ l α + β Γ α Γ lα β. j i +...+i j=i l= l= Γ i l α. i l! Defining the finite Dirichlet measures Λ j with ensity on the j + simplex j given by: we get Λ j u,.., u j = c j,α,β c φ j i,.., i j = c j l= u α l u l j j l= αj β j u l, l= u i l 2 l Λ j u,.., u j. The set of finite Dirichlet measures Λ j with parameters θ = 2 α,..., θ j = 2 α, θ j+ = αj β on the simplices j completely characterize this limiting iscrete-time coalescent. ote that Λ is a beta2 α, α β probability measure. One may rewrite the φ j s as see [37] an [3] φ j i,.., i j = j k,...,k j l= all istinct u i l k l Ξ u u, u,

13 PARETO GEEALOGIES I A POISSO EVOLUTIO MODEL WITH SELECTIO 3 where = { u:= u,.., u l,.. : u.. u l.. : } l u l an Ξ a measure on. Letting ν u := Ξ u / u, u, the measure ν on the infinite simplex can be ientifie see [3] to the two-parameter Poisson-Dirichletα, β measure, with α [, an β < α. It hols that ν =. Poisson-Dirichlet measures enjoy many remarkable properties incluing a stick-breaking property, [35]. Remarks: i From, the limiting situation α = also makes sense, leaing to the oneparameter Poisson-Dirichlet, β measure with β < α =. In this case, from P i,j = βj Γ β s i,j, Γ i β where s i,j := i! j! i +...+i j=i j l= i l are the absolute first kin Stirling numbers. ii Finally, avoiing size-biasing β = gives rise to the iscrete Poisson-Dirichletα, coalescent with one-parameter α,, appearing in [39]. The case α, 2. Lemma 4. When α, 2, with µ := α α, for all β < α 2 c := P 2, αµ α B 2 α, α β α. Proof: In this case, efining a := α,, after an integration by parts an using the previous estimate 8 substituting a to α E e λx = e λ λ a a x a+ e λx x e λ λ a Γ a λa. Thus, for small λ, with µ := α α 3 E e λx λµ Γ α λ α e λµ. Thus, consistently with the GCLT approach, to the ominant orer E Σ β = λ λ β E e λx µ β, Γ β so that E c := α E This suggests that Σ β α Σ β B 2 α, α β αµ α B 2 α, α β α. Proposition 5. When α, 2, upon scaling time using an effective population size e = c with c as in 2, we obtain a limiting continuous-time-t Λ coalescent: x t = lim x [t/c ], with Λ a beta2 α, α β probability measure, β < α. If β =, we get the stanar beta2 α, α coalescent.

14 4 THIERRY E. HUILLET Proof: To confirm this point, we will first evaluate a large estimate of P i, efine in 2, in the range α, 2. Using the above small λ estimate of E e λx in the parameter range uner concern, for i 2, we get E X i e λx = i Thus, using 9 E Σ β Si = Γ i β Γ i α α Γ i β Finally, we obtain P i λ i E e λx αγ i α λ α i. λ λ i β E X i e λx E e λx λ λ i β λ α i e µλ Γ i α Γ α β = α Γ i β µ α β. i, α α Γ α β Γ i α µ α, Γ i β showing that, with c = P 2, α, for each i, lim c an are strictly positive constants. More precisely, P i, exist c P Γ α β Γ i α i, φ i = = B 2 α, α β Γ i β where Λ = Λ is a beta2 α, α β probability measure. u i 2 Λ u, To eal with the higher orer terms, with m := {l {,..., j} : i l 2}, assuming j < i, let us write 3 as j P j i E Σ β m l= Si l j l l=m+ l i,j = S, j m i m= i...i m +...+i m=i j+m E Σ β where the ouble-star sum is now over the i l, l =,..., m satisfying i l 2. Proceeing similarly with higher orer terms, it clearly hols that for all j 2 an i l, l =,..., j, satisfying i l 2 for at least two l in the list, lim c E j Σ β S i l l l= /E Σ β =, so that simultaneous multiple collisions cannot occur in the limit actually the contribution of these simultaneous multiple collisions terms in P i,j is of orer O c m. In the latter expression of P i,j therefore, only the term corresponing to m = will contribute to the O c orer. Because, E X e λx = λ E e λx µ an E X i αγ e λx i α λ α i with i = i j + 2, using 9, we get E Σ β Si E Σ β j l=2 S l = αµj Γ i α E Γ i β Σ β λ λ i β E X i e λx j l=2 E X l e λxl E e λx j E Σ β λ λ i β λ α i e µλ = αµj Γ i α E Γ i β Σ β Γ i β Γ α β + j µ α β+j.

15 PARETO GEEALOGIES I A POISSO EVOLUTIO MODEL WITH SELECTIO 5 This shows, using j j /j!, 2 an after some elementary algebra, that for j =,..., i, c P i,j λ i,j = = i B i j + α, α β + j j B 2 α, α β i u i j u j Λ u, j where Λ = Λ beta2 α, α β. As a result, xt = lim x [t/c ] is a continuoustime-t pure eath coalescent process on with infinitesimal transition rates λ i,j. The case α =. We view it as a limiting case of the previous analysis eriving from 3 when α +. Proposition 6. When α =, upon scaling time using a logarithmic effective population size e = c log, the limiting process x t = lim x [t/c ] exists an is a continuous-time-t Λ coalescent with Λ a beta, β probability measure, β <. If β =, we get the stanar Bolthausen-Sznitman coalescent with Λ uniform. Proof: Put inee α = + ε in the previous small-λ estimate 3 of E e λx in the parameter range α, 2, with ε > small. Then µ + /ε an because Γ α = Γ ε + /ε, 4 E e λx λ ε Thus, E Σ β Γ β λ λ +ε λ + λ log λ =: I λ. λ λ β I λ. The leaing contribution of I λ is, when λ is small, of orer / log. Putting λ = u/ log I λ e u + u log u log log. log Thus, with β < E Σ β log β Γ β u u β e u + u log u log log log 5 log β + β ψ β log log, log where ψ x = Γ x /Γ x is the igamma function. In the latter estimate, we use that ifferentiating u u β u θ e u = Γ β + θ with respect to the extra parameter θ an then putting θ = gives u u β log u e u = Γ β. Finally, consistently with the GCLT approach E Σ β c = E Σ β log.

16 6 THIERRY E. HUILLET Clearly, with Λ a beta, β probability measure an β < α = c P i, u i 2 Λ u = Γ 2 β Γ i Γ i β = λ i,. Using similar arguments on higher orer terms, showing that simultaneous multiple collisions o not contribute in the limit, one can easily show 6 c P i,j = B i j, j β B, β u i j u j Λ u =: λ i,j, j =,..., i, where Λ = Λ beta, β. This confirms that, when α =, upon scaling time using an effective logarithmic population size e = c log, we get a limiting continuous-time-t Λ coalescent x t = lim x [t/c ], where Λ is a beta, β probability measure, β <. The case α > 2. α α Proposition 7. When α > 2, for any value of β, with µ := > an ρ := >, upon scaling time using a linear effective population size e = c α α 2 = µ 2 /ρ, the limiting process x t = lim x [t/c ] exists an is the continuous-timet Kingman coalescent. Proof: If α > 2, Σ is in the omain of attraction of the normal law. As a result, for small λ, with µ := E X = α α an ρ := E X 2 = α α 2 7 E e λx λµ + 2 ρλ2 e λµ. From 7, for small λ, E X e λx = λ E e λx µ an E X 2 e λx = 2 λ 2 E e λx ρ. We have c := P 2, = EΣβ S2 E Σ β S2 E Σ β EΣ β ρ Γ 2 β Γ β with, from 9 λ λ β e λµ = ρ µ β 2 λ λ β e λµ = µ β. Thus, E Σ β S2 c = ρ E Σ β µ 2 goes to as claime. As in Proposition 5, simultaneous multiple collisions cannot contribute in the limit. Only the term in the expression of P i,j corresponing to m = will contribute to the O c orer an so we nee to focus on a collision with a single i 2. ow, from 7, E X e λx = λ E e λx µ an E X i e λx ρδi,2 with i = i j + 2. Proceeing again as in Proposition 5 therefore, only

17 PARETO GEEALOGIES I A POISSO EVOLUTIO MODEL WITH SELECTIO 7 effective transitions from i to j < i with j = i i + = i are seen in the limit, corresponing to the binary mergers of a Kingman coalescent. α = 2. To complete the picture, it remains to stuy the limiting critical case α = 2. Lemma 8. When α = 2, for all values of β, the coalescence probability goes to as like c log 2. Proof: We view the case α = 2 as a limiting case of 7 as α 2 +. When λ is small an for α > 2, we have E e λx λµ + 2 ρλ2 Γ α λ α. Putting α = 2 + ε, for ε > small, using Γ ε + λ is small, we have /ε, with ρ 2/ε, when 8 E e λx 2λ + ε λ2 ε λ2+ε 2λ λ 2 log λ e 2λ. Because E X e λx = λ E e λx 2 an E X 2 e λx = 2 E e λ 2 λx 2 log λ 3, we have E Σ β S2 λ λ β E X 2 e λx E e λx Γ 2 β λ λ β 2 log λ + 3 e 2λ Γ 2 β E Σ β λ λ β e 2λ = 2 β. Γ β The Euler integral with a logarithmic term insie appearing in the expression of E Σ β S2 can be obtaine while taking the erivative of λ λ β λ θ e 2λ with respect to the extra parameter θ an then putting θ = in the obtaine expression. Observing therefore λ λ β log λ e 2λ = Γ 2 β 2 2 β Γ 2 β 2 2 β log 2, to the ominant orer in, E Σ β S β log, leaing to E c := P 2, = E Σ β S2 Σ β 2 log. Remark: A similar scaling behavior for c was recently obtaine in Theorem 2.4 of [2], ealing with coalescents arising from compoun Poisson iscrete reprouction moels, in the critical case.

18 8 THIERRY E. HUILLET Proposition 9. When α = 2, upon scaling time using an effective population size e = c = 2 / log, the limiting process x t = lim x [t/c ] is the continuous-time-t Kingman coalescent. Proof: Using 8, for small λ, we have i E Xe i λx = i λ i E e λx 2Γ i 2 λ i 2, i 3, to which one shoul a E X e λx 2 an E X 2 e λx 2 log λ + 3. For i 3, we get E Σ β Si = λ λ i β E X Γ i β e i λx E e λx 2Γ i 2 λ λ β e 2λ B i 2, 2 β = 2 Γ i β 2 2 β. Thus, for all i 3, as E Σ β c P i, = c Si B i 2, 2 β. E Σ β log Therefore, ue to the extra factor log appearing in c, the transitions from i 3 to cannot be seen in the limit, nor for the same reason the transitions involving a single multiple collision with i 3, nor transitions involving simultaneous multiple collisions of any orer. In fact, only the events involving a single multiple collision with i = i j + = 2 corresponing to transitions from i to j = i will contribute in the limit. Inee, E Σ β j S2 l=2 S l = E Σ β 2 j E Σ β Γ i β This shows, using P i,j λ λ i β E X 2 e λx j j l=2 E X l e λxl E e λx j E Σ β Γ i β λ λ i β 2 log λ + 3 e 2λ 2 i log. j i E 2... Σ β j S2 l=2 S l E Σ β with j = i, j j /j!, an after some elementary algebra, that Remarks: c P i,i λ i,i = i. 2 i Results of a similar flavor can be foun in Schweinsberg s work [39]. However, his moel an techniques are ifferent from ours because he consiers large limiting coalescents obtaine while sampling without replacement from a iscrete super-critical Galton-Watson branching process, assuming the reprouction law of each offspring to exhibit power-law Zipf tails of inex α a in his notations. ote

19 PARETO GEEALOGIES I A POISSO EVOLUTIO MODEL WITH SELECTIO 9 that there is no parameter β in the construction [39]. In the same spirit, results can also be foun in Huillet-Möhle [9] where, following [3], Λ coalescents are obtaine as scaling limits of iscrete extene Moran moels, the skewe reprouction law of which isplaying occasional extreme events with one iniviual allowe to prouce a large amount of offspring. Whenever the reprouction law isplays systematic extreme events, iscrete coalescents were even shown to emerge in the large limit, but in this Moran context, they are only Λ coalescents with multiple but no simultaneous collisions, [9]. This contrasts with the occurrence in our present work of iscrete Poisson-Dirichlet Ξ coalescents. In contrast also with our current work where coalescents are erive from a sampling proceure in the continuum, the coalescents consiere in [39] an [9] were built from iscrete reprouction laws at fixe population size an looking at their scaling limits. We refer to these two works an to [7] for aitional backgroun on Λ coalescent processes aapte to our purposes. ii When α 2, neither the large estimate of the scaling constant c nor the limiting Kingman coalescent epen on the bias parameter β. iii We observe that the probability c := P 2, that two iniviuals chosen at ranom share the same common ancestor, efining the time-scale to erive the large- limits of the Pareto-coalescents all have the same large- orer of magnitue as the length l of an external branch chosen at ranom in the limiting Λ coalescents x t α [, 2 or Ξ coalescent x k α,, or Kingman coalescent x t α > 2, starte at x =. This curious fact is unexplaine so far. 5. Forwar in time selection moel an genealogies In this Section, in the spirit of [7], we inicate that the coalescent processes just iscusse may be viewe as the genealogical processes of some forwar in time evolving branching population moels with selection. As it is often the case in population genetics, the process we are intereste in is in the class of branching processes conitione on having a fixe population size over each generation, in the spirit of [23]. 5.. A Poisson-point process moel with selection. Start with iniviuals at generation t = an assume that each iniviual has an initial fitness x n >, n =,...,. To escribe the state of the population at the next generation, assume first that, inepenently of one another, each iniviual potentially generates an infinite number of offspring along a Poisson point process PPP with intensity or occupation ensity 9 π xn x = π x n x = αx n α x α+, 7 7 The symbol inicates erivative with respect to x.

20 2 THIERRY E. HUILLET where π xn x := x/x n α, x >, α >, n =,..,. We observe that, with π x := x α πxn x = π x π x n an if we let π x = π x = αx α+, then π xn x = x n α π x. So the fitnesses of the offspring of each iniviual is generate accoring to a PPP epening on the fitness of its parent. From these simple assumptions, an as conventional wisom suggests, we get: Proposition. In a PPP moel for fitness-epenent offspring reprouction with occupation ensity 9, the fitnesses of the offspring of some parental iniviual with fitness x n are x n times the fitnesses of the offspring of some canonical iniviual with unit fitness: In this sense, the larger the fitness x n of some iniviual is, the more he will, proportionally, prouce offspring with large fitness. Proof: Let τ n ; n be the points of a stanar homogeneous Poisson point process PPP on the half-line with rate, an let π s = s /α be the ecreasing inverse of π. Then, with π x s = x n n s /α, π x τ n n ; n are the orere points of the offspring PPP on the positive half-line or here the fitness space with occupation ensity π xn. So, the fitter the iniviuals, the fitter their offspring, proportionally to the parental fitness. Let us also briefly emphasize that, avoiing the fitness epenence on the parent of the PPP, π τ n ; n are just the orere points of a PPP on the half-line with occupation ensity π. Whenever, as in our case stuy here, the rate function π is not integrable up to x =, there are infinitely many such points, with as an accumulation point whereas there is of course a finite Poisson with mean π ε number of them above some threshol ε >. It is well-known that when α,, with π x := x α an π s = s /α, the positive cumulative rv 2 χ = n π τ n is a one-sie α stable rv on, with LST E e λχ = e κλα, κ = Γ α >, λ. When α, 2, the law of χ is the one of a positive Lamperti rv with LST E e λχ = e cλ+κλα, κ = Γ α >, c := E χ >, [27]. When α =, the law of χ is the one of a positive eveu rv with LST E e λχ = e λ log λ see [32]; the latter may be viewe as a Lamperti rv in the limit α +, [8]. Typically inee, by the Lévy-Khintchine formula, π x x stans for the Lévy measure for the non-negative jumps of χ which is the value at time t = of an infinitely ivisible suborinator χ t ; t with LST E e λχ t, [2]. In 2, the terms π τ n are thus the ranke jumps of χ in its Lévy ecomposition see [33] for example.

21 PARETO GEEALOGIES I A POISSO EVOLUTIO MODEL WITH SELECTIO 2 Reprouction step. Because we consier the offspring of all the initial iniviuals with fitnesses x n, n =,...,, the step state of the whole population is thus obtaine from a PPP with global equivalent occupation ensity 2 π x,α x = αx α+ x n α = αx,α α x α+, where n= x,α := x α n n= /α is the global equivalent initial fitness of the whole population at generation to consier. We get Proposition. In a PPP with occupation ensity 9 for the escent of each iniviual with fitness x n, n =,...,, the occupation ensity of the population /α as a whole is given by 2, where x,α := n= xα n is the global equivalent fitness. Proof: This follows from the superposition principle of Poisson point processes see [25] p. 6. The fact that the intensity of the superpose PPP is in the same class as the one of a single PPP escening from x n is a remarkable scaling property of π xn x. In this setup therefore, π x,α x := π x,α x stans for the occupation ensity that there is a point at x escening from any of the initial iniviuals with fitnesses x n, n =,...,. ote that the cumulate fitness of all first-generation offspring is π x,α τ n = x,α χ, where χ is given by 2. Selection step. n In orer to moel a population with fixe size over the generations, the final state of the population at time is obtaine while selecting the iniviuals of the whole population whose fitnesses are the largest the selection step, truncating therefore the latter sum to its first terms. The whole process incluing reprouction an selection steps is then iterate inepenently over the next generations. From this efinition of the process, if the x n ks are the fitnesses of the fittest iniviuals at generation k, the orere ones x n k + at generation k+ x >... > x escening from the whole population at step k are given by x n = π x,α k τ n = x,α k τ /α n. Here the τ n s are the orere points of a stanar Poisson process on the half-line with τ <... < τ n <... < τ. The law of τ n is thus

22 22 THIERRY E. HUILLET the one of an Erlang gamman rv with ensity f τ n s = s n e s /Γ n. Further, given τ + = s, the probability ensity of τ,..., τ is an so f τ,...,τ s,.., s τ + = s =! s <s <...<s <s f τ,...,τ s,.., s = <s<...<s s se s = e s <s<...<s. The joint law of the orere x n k + s is thus the one of the images π x,α k τ ns, namely f x k+,...,x k+ x,.., x = e π x,α kx π x,α k x n x>...>x. Clearly also with the two terms in the right-han sie term mutually inepenent, n= x + k + = x,α k x + k +, where x + k + is the + st largest point of a PPP with occupation ensity π x = αx α+, x, α >. By the image measure theorem, the ensity of x + k + = π τ + is obtaine as a power-gamma ensity 22 f x + k+ x = α! x +α+ e x α, x >. ext, the conitional ensity of each x n k + given x + k + = x is f xnk+ xn x + k + = x = π x n π x x n>x, showing that with the two terms in the right-han sie term mutually inepenent, x n k + = x + k + X n k + where X n k + is a Paretoα istribute rv with ensity f x = αx α+ on, 8. Putting all this together, we obtaine Proposition 2. Inepenently for each k, with x + k + having the power-gamma istribution 22 an X n k + being Paretoα istribute an with the three right-han sie terms being mutually inepenent, 23 x n k + = x,α k x + k + X n k +, n =,...,, inicates how to upate multiplicatively the fitness of the n th iniviual at generation k+, when the fitnesses of the previous generation are summarize in x,α k. We also clearly have an upate of the global equivalent fitness x,α from step k to step k + as: Corollary 3. With x,α k := n= x n k α /α the global equivalent fitness of the whole population at generation k, the following recursion hols /α 24 x,α k + = x,α k x + k + X n k + α. 8 We use the scaling property of Paretoα rvs X on, stating that X X > a = ax. n=

23 PARETO GEEALOGIES I A POISSO EVOLUTIO MODEL WITH SELECTIO 23 In the latter sum term, each Xn α is thus a Pareto istribute rv with ensity f x = x 2 on, an we nee to sum of them inepenently which is reminiscent of 5. Large asymptotics of the α mean fitness. Defining /α x,α k := x α n k, to be the generalize Höler α mean of the fitnesses x n, n =,...,, at generation k, it follows from 23 that /α 25 x,α k + = x,α k x + k + X n k + α. By Jensen inequality, the Höler α means x,α k are non-ecreasing functions of α. Corollary 4. The β moments of x,α at generation k are given by n= n= E x,α k β = x,α β E x β + E n= X α n β/α Proof: This follows from the inepenence of the x + s an the Xα n an their i.i.. character within each generation k an from 25. Proposition 5. For large, with v := log log k log x a.s.,α k k α v. With F β := β β log log ψ β/α log log α α log an f a, a <, its Legenre transform, the large eviation regime is given by k log P k log x,α k a f a. k Proof: Observing that E x β + = Γ + β/α /Γ + β/α an applying 5 giving the moments of a partial sum of i.i.. Pareto istribute rvs, it follows that, with β < α E n= X α n β/α As a result, we get E x,α k β x,α β β log β/α + ψ β/α log log. large α log log βk/α + large k β ψ β/α log log. α log k.

24 24 THIERRY E. HUILLET Thus, for large, with F β = β β log log ψ β/α log log α α log efining the concave thermoynamical pressure, k log E x,α k β k F β. Thus, with a = F β <, by the large eviation principle 26 k log P k log x,α k a f a, k where f a = inf β<α aβ F β is the concave Legenre transform of F, giving the large eviation rate function of log x,α k /k. In particular, for large, with v := log log 27 k log x,α k a.s. k F α v, gives the limiting right shift of the Höler α mean fitness inuce by selection effects. Remarks: i The limiting right-han-sie term in 27, although increasing very slowly with, oes not stabilize to a limit, in contrast to other similar moels [] of branching with selection where, in each generation, each iniviual prouces only two offspring with ranomly shifte fitnesses. ii With α >, let f x := x α > efine some increasing output map of the iniviuals fitnesses x, with x >. Defining f x k := f x n k = n= x n k α to be the mean output fitness in generation k of the whole population, then, whatever α, 27 is also k log f x k a.s. k v, interpreting the spee v itself. This suggests that it is of interest to work not only on the fitnesses x n themselves an their α mean x,α but rather on some eforme version of the fitnesses x α n an their stanar mean f x. Clearly f x itself obeys the recursion n= f x k + = f x k x + k + α X n k + α, eriving again from 23 an the efinition of the equivalent global fitness x,α k. n=

25 PARETO GEEALOGIES I A POISSO EVOLUTIO MODEL WITH SELECTIO 25 iii ote finally that, given x,α k, the cumulative istribution function cf of the fitness x k + of the fittest iniviual among the iniviuals at generation k + is given by P x,α k x k + x = P π x,α k τ x = e x/x,αk α, where τ is exp istribute. Thus the fitness of the fittest iniviual obeys x k + = x,α k Y k +, where Y k, k is a sequence of i.i.. Fréchet rvs with cf P Y x = e x α. The conitional mean given x,α k of x k + is x,α k Γ /α an its meian value x,α k log 2 /α. More generally, when is large an unconitionally, ue to the recursion 24 on the x,α ks: E x k + β x,α β Γ β/α e kf β Genealogies. ow we turn to the genealogies of this branching process with selection. The beta, β an Bolthausen-Sznitman coalescents. Recall that π x,α k x := π x,α k x is the occupation ensity that there woul be a point an offspring of the PPP at position with fitness x at generation k +, given a global population state x,α k. Proposition 6. Looking backwar in time, upon scaling time using c / log, the genealogy of the branching moel with selection is a beta, β coalescent, reucing to the Bolthausen-Sznitman coalescent if β =. Proof: Suppose first β =. Given there is an offspring at x at generation k +, the sampling probability that it woul be an offspring of the iniviual with fitness x n k is thus π xnk x π x,α k x = αx n k α x α+ αx,α k α x α+ = x n k α x,α k α, which is inepenent of x. Observing from 23 an 24 that x n k = x,α k x + k X n k, n =,...,, /α x,α k = x,α k x + k X n k α, this ranom probability is also X n k α n= X n k α, where, for each k inepenently, the X n s are i.i.. Paretoα istribute on,. Because in each generation, parents generate offspring inepenently an inepenently of one another, upon averaging, the probability that, at generation k +, i n=

26 26 THIERRY E. HUILLET iniviuals share the same common ancestor is inepenent of k, with P i, := E X n k α i n= X n k α = E X k α i n= X n k α. n= The X n ks being i.i.. Paretoα istribute rvs, the X n k α s are i.i.. Pareto istribute rvs an we are thus back to the results of Proposition 6 with β =, stating that with c / log, c P i, u i 2 Λ u = Γ 2 Γ i Γ i = i, where Λ beta,, uniform. We can procee similarly to erive the probabilities P i,j that i iniviuals have j < i parents, behaving consistently with 6 with β =. We conclue that, whatever α, in the large limit, the time-scale genealogy of the branching moel with selection is a Bolthausen-Sznitman coalescent process, obtaine while sampling from Pareto i.i.. rvs. Woul the sampling probabilities P i,j inclue a β size biasing effect on total length Σ := n= Xα n, the genealogy of this branching moel with selection woul be a full beta, β coalescent, provie β <. While aopting this sampling point of view to compute the coalescence an merging probabilities, we therefore obtain a limiting genealogical coalescent process which is inepenent of α. Genealogies from the output PPP. What now if we set that the occupation ensity that, at generation k +, there is an offspring at x escening from some iniviual with fitness x n k at generation k, is instea given by 28 π xnk x := x n k x 2, while istorting the original occupation intensity π xnk x = αx n k α x α+? Then the occupation ensity that, at generation k +, there is an offspring at x escening from any iniviual of the whole population woul take the form π x, k x := x, k x 2, where x, k := n= x n k is the cumulative fitness in generation k. If this were to be the case, given there is an offspring at x at time k +, the sampling probability that it is an offspring of the iniviual with fitness x n k woul be, thanks to 23 an 24 with α = : 29 π xnk x π x, k x = x n k x 2 x, k x 2 = x n k x, k = X n k n= X n k, again inepenently of x. This probability now involves a normalize sum of the X n s, which are i.i.. Paretoα istribute an the strategy to compute the merging probabilities of the ancestral process will be moifie.

27 PARETO GEEALOGIES I A POISSO EVOLUTIO MODEL WITH SELECTIO 27 Uner this hypothesis inee, the probability that, at generation k+, i iniviuals share the same common ancestor reas i i P i, := E X n k n= X = E X k n k n= X =: E S k i, n k n= where S is the normalize segment size now obtaine from i.i.. Paretoα istribute rvs, normalize by their sum Σ k := n= X n k. We can procee similarly to erive the probabilities P i,j that i iniviuals have j < i parents an we are back to the stuies of Sections 2 4. An we can as well β size-bias these sampling probabilities on the total lengths Σ. We call this sampling proceure the istorte sampling proceure. Proposition 7. Looking backwar in time, using a istorte size-biase sampling proceure, the genealogy of the branching moel with selection is - a continuous-time Kingman coalescent if α 2 upon scaling time with c / if α > 2 or c log / if α = 2. - a continuous-time beta2 α, α β coalescent if α, 2 an β < α upon scaling time with c α. - a continuous-time beta, β coalescent if α = an β < upon scaling time with c / log. - a iscrete-time Poisson-Dirichletα, β coalescent if α, an β < α. Proof: It remains to interpret the istorte size-biase sampling proceure which is propose to compute the merging probabilities of the ancestral process: Assume that the fitness epenent PPP escribing the escent of an iniviual with fitness x n k is now the output image of the original one, given by the canonical application f xnk x = x n k x/x n k α, f xnk : R + R +. Its gives rise to a new PPP with istorte intensity π fxnk x = x n k x 2 x, the image measure of π xnk x = π xnk x x = αx n k α x α+ x by f xnk as a result of the classical Campbell formula for PPPs, see [25] p. 28. This was the starting point in 28. The skewe computation in 29 of the probability that some offspring is escening from one iniviual with fitness x n k is thus base not on the original PPP attache to x n k but rather on a eforme version of it through f xnk. We note that f xnk x, as a function of the two arguments x n k, x is homogeneous with λ b x = λ a α+bα f xnk x, λ >, leaing obviously, if λ = x n k /a, f λ a x nk b/a = an f x := f x = x α to: f xnk x = x n k f x/x n k. The function f x = x α is the output fitness function introuce in Subsection 5., Remarkii. Using this istorte size-biase sampling proceure therefore, following the introuctory arguments, the full class of the Pareto-coalescents escribe in Sections 2 4 are obtaine. Suppose for instance α, 2, β =. Base on the previous computations of Sections 2 4, we conclue that the large istorte genealogy of the branching moel with selection coincies upon scaling time corresponingly: k [t/c ] with a beta2 α, α coalescent process, obtaine while sampling from Paretoα