Limit theorems for supercritical age-dependent branching processes with neutral immigration
LLIMIT THEOREMS FOR SUPERCRITICAL AGE-DEPENDENTBRANCHING PROCESSES WITH NEUTRAL IMMIGRATION
MATHIEU RICHARD
Abstract.
We consider a branching process with Poissonian immigration where individualshave inheritable types. At rate θ , new individuals singly enter the total population and start anew population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagersprocess: individuals have i.i.d. lifetimes durations (non necessarily exponential) during whichthey give birth independently at constant rate b . First, using spine decomposition, we relaxpreviously known assumptions required for a.s. convergence of total population size. Then,we consider three models of structured populations: either all immigrants have a differenttype, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model,the vector ( P , P , . . . ) of relative abundances of surviving families converges a.s. In the firstmodel, the limit is the GEM distribution with parameter θ/b . Introduction
We want to study and give some properties about several birth-death and immigrationmodels where immigration is structured. All individuals behave independently of the others,their lifetimes are i.i.d. but non-necessarily exponential and each individual gives birth atconstant rate b during her life. We will consider the supercritical case i.e., the mean numberof children of an individual is greater than 1. In the absence of immigration, if X ( t ) denotesthe number of extant individuals at time t , the process ( X ( t ) , t ≥
0) is a particular case ofCrump-Mode-Jagers (or CMJ) processes [12], also called general branching processes. Here, X is a binary (births arrive singly) and homogeneous (constant birth rate) CMJ process.Now, we assume that at each arrival time of a Poisson process with rate θ , a new individualenters the population and starts a new population independently of the previously arrivedones. This immigration model extends to general lifetimes the mainland-island model of S.Karlin and J. McGregor [14]. In that case, the total population process X is a linear birth-and-death process with immigration. For more properties about this process, see [23] or [26]and references therein. In the context of ecology [4], this model can be used as a null modelof species diversity, in the framework of the neutral theory of biodiversity [11].We first give the asymptotic behavior of the process ( I ( t ) , t ≥
0) representing the totalnumber of extant individuals on the island at time t . Specifically, there exists η > X ) such that e − ηt I ( t ) convergesalmost surely. S. Tavar´e [22] proved this result in the case of a linear birth process withimmigration. The case of general CMJ processes was treated by P. Jagers [12] under thehypothesis that the variance of the number of children per individual is finite. We manageto relax this assumption in the case of homogeneous (binary) CMJ-processes thanks to spine Mathematics Subject Classification.
Primary: 60J80; Secondary: 60G55, 92D25, 60J85, 60F15, 92D40.
Key words and phrases. splitting tree, Crump-Mode-Jagers process, spine decomposition, immigration,structured population, GEM distribution, biogeography, almost-sure limit theorem . a r X i v : . [ m a t h . P R ] D ec MATHIEU RICHARD decomposition of splitting trees [9, 16, 17] which are the genealogical trees generated bythose branching processes. In passing, we obtain technical results on the log-integrability ofsup t e − ηt X ( t ).Then, we consider models where individuals bear clonally inherited types. They intendto model a metacommunity (or mainland) which delivers immigrants to the island as inthe theory of island biogeography [19]. However, we made specific assumptions about thespectrum of abundances in the metacommunity. In Model I, there is a discrete spectrum withzero macroscopic relative abundances: when an immigrant enters the population, it is eachtime of a new type. In Model II, we consider a discrete spectrum with nonzero macroscopicrelative abundances: the type of each new immigrant is chosen according to some probability( p i , i ≥ x, x + dx )is chosen with probability xf ( x ) θ dx (where f is a positive function representing the abundancedensity and such that θ := (cid:82) ∞ xf ( x ) dx < ∞ ) and it starts an immigration process withimmigration rate x . The particular case of abundance density f ( x ) = e − ax x appears in manypapers. I. Volkov et al. [24] and G. Watterson [25] consider it as a continuous equivalent ofthe logarithmic series distribution proposed by R. Fisher et al. [8] as a species abundancedistribution. In this particular case, species with small abundances are often drawn but theywill have a small immigration rate.In the three models, we get results for the abundances P , P , . . . of different types as time t goes to infinity: the vector ( P , P , . . . ) rescaled by the total population size converges almostsurely. More precisely, in Model I which is an extension of S. Tavar´e’s result [22] to generallifetimes, we consider the abundances of the surviving families ranked by decreasing ages andthe limit follows a GEM distribution with parameter θ/b . This distribution appears in othercontexts: P. Donnelly and S. Tavar´e [5] proved that for a sample of size n whose genealogy isdescribed by a Kingman coalescent with mutation rate θ , the frequencies of the oldest, secondoldest, etc. alleles converge in distribution as n → ∞ to the GEM distribution with parameter θ ; S. Ethier [7] showed that it is also the distribution of the frequencies of the alleles rankedby decreasing ages in the stationary infinitely-many-neutral-alleles diffusion model.In a sense, the surviving families that we consider in our immigration model are ”large”families because their abundances are of the same order as the population size. A. Lambert[18] considered ”small” families: he gave the joint law of the number of species containing k individuals, k = 1 , , . . . In Section 2, we describe the models we consider and state results we prove in other sections.Section 3 is devoted to proving a result about the process X , Section 4 to proving a propertyof the immigration process ( I ( t ) , t ≥
0) while in Section 5, we prove theorems concerning therelative abundances of types in the three models.2.
Preliminaries and statement of results
We first define splitting trees which are random trees satisfying: • individuals behave independently from one another and have i.i.d. lifetime durations, • conditional on her birthdate α and her lifespan ζ , each individual reproduces accordingto a Poisson point process on ( α, α + ζ ) with intensity b , • births arrive singly. IMIT THEOREMS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION 3
We denote the common distribution of lifespan ζ by Λ( · ) /b where Λ is a positive measure on(0 , ∞ ) with mass b called the lifespan measure [17].The total population process ( X ( t ) , t ≥
0) belongs to a large class of processes calledCrump-Mode-Jagers or CMJ processes. In these processes, also called general branchingprocesses [12, ch.6], a typical individual reproduces at ages according to a random pointprocess ξ on [0 , ∞ ) (denote by µ := E [ ξ ] its intensity measure) and it is alive during a randomtime ζ . Then, the CMJ-process is defined as X ( t ) = (cid:88) x { α x ≤ t<α x + ζ x } , t ≥ x , α x is her birth time and ζ x is her lifespan. In this work, theprocess X is a homogeneous (constant birth rate) and binary CMJ-process and we get µ ( dx ) = dx (cid:90) [ x, ∞ ) Λ( dr ) . We assume that the mean number of children per individual m := (cid:82) (0 , ∞ ) r Λ( dr ) is greaterthan 1 (supercritical case).For λ ≥
0, define ψ ( λ ) := λ − (cid:82) (0 , ∞ ) (1 − e − λr )Λ( dr ). The function ψ is convex, differentiableon (0 , ∞ ), ψ (0 + ) = 0 and ψ (cid:48) (0 + ) = 1 − (cid:82) ∞ r Λ( dr ) <
0. Then there exists a unique positivereal number η such that ψ ( η ) = 0. It is seen by direct computation that this real number isa Malthusian parameter [12, p.10], i.e. it is the finite positive solution of (cid:82) ∞ e − ηr µ ( dr ) = 1and is such that X ( t ) grows like e ηt on the survival event (see forthcoming Proposition 2.1).From now on, we define c := ψ (cid:48) ( η )which is positive because ψ is convex.Another branching process appears in splitting trees: if we denote by Z n the number ofindividuals belonging to generation n of the tree, then ( Z n , n ≥
0) is a Bienaym´e-Galton-Watson process started at 1 with offspring generating function f ( s ) := (cid:90) (0 , ∞ ) b − Λ( dr ) e − br (1 − s ) ≤ s ≤ . To get results about splitting trees and CMJ-processes, A. Lambert [16, 17] used treecontour techniques. He proved that the contour process Y of a splitting tree was a spectrallypositive (i.e. with no negative jumps) L´evy process whose Laplace exponent is ψ . Lambertobtained result about the law of the population in a splitting tree alive at time t . If ˜ P x denotesthe law of the process ( X ( t ) , t ≥
0) conditioned to start with a single ancestor living x unitsof time,(1) ˜ P x ( X ( t ) = 0) = W ( t − x ) /W ( t )and conditional on being nonzero, X ( t ) has a geometric distribution with success probability1 /W ( t ) i.e. for n ∈ N ∗ ,(2) ˜ P x ( X ( t ) = n ) = (cid:18) − W ( t − x ) W ( t ) (cid:19) (cid:18) − W ( t ) (cid:19) n − W ( t ) MATHIEU RICHARD where W is the scale function [2, ch.VII] associated with Y : this is the unique absolutelycontinuous increasing function W : [0 , ∞ ] → [0 , ∞ ] satisfying(3) (cid:90) ∞ e − λx W ( x ) dx = 1 ψ ( λ ) λ > η. The two-sided exit problem can be solved thanks to this scale function:(4) P (cid:0) T < T ( a, + ∞ ) | Y = x (cid:1) = W ( a − x ) /W ( a ) , < x < a where for a Borel set B of R , T B = inf { t ≥ , Y t ∈ B } .We now give some properties, including asymptotic behavior, about the CMJ-process X . Proposition 2.1.
We denote by
Ext the event (cid:110) lim t →∞ X ( t ) = 0 (cid:111) .(i) We have (5) P (Ext) = 1 − η/b and conditional on Ext c , (6) e − ηt X ( t ) a.s. −→ t →∞ E where E is an exponential random variable with parameter c .(ii) If, for x > , log + x := log x ∨ , (7) E (cid:34) (cid:18) log + sup t ≥ (cid:0) e − ηt X ( t ) (cid:1)(cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Ext c (cid:35) < ∞ We remind that c = ψ (cid:48) ( η ) and then depends on the measure Λ( · ). The proof of the last as-sertion requires involved arguments using spine decomposition of splitting trees. Proposition2.1, which will be proved in Section 3, is known in a particular case: if the lifetime Λ( · ) /b has an exponential density with parameter d , ( X ( t ) , t ≥
0) is a Markovian birth and deathprocess with birth rate b and death rate d < b . In that case, η = b − d , c = 1 − d/b = P (Ext c )and integrability of sup t ≥ ( e − ηt X ( t )) stems from Doob’s maximal inequality.We now define the immigration model: let θ be a positive number and 0 = T < T 0) the i th oldest family (thefamily which was started at T i ) then Z i ( t ) = X i ( t − T i ) { t ≥ T i } where X , X , . . . are copies of X and ( X i , i ≥ 1) and ( T i , i ≥ 1) are independent. This immigration model is a generalizationof S. Karlin and J. McGregor’s model [14] in the case of general lifetimes.For i ≥ 1, define ( Z ( i ) ( t ) , t ≥ 0) as the i th oldest family among the surviving populationsand T ( i ) its birthdate. In particular, by thinning of Poisson point process, ( T ( i ) , i ≥ 1) is aPoisson point process with parameter θη/b thanks to (5).We are now interested in the joint behavior of the surviving families ( Z ( i ) ( t ) , t ≥ 0) for i ≥ e − ηt Z ( i ) ( t ) = e − ηT ( i ) e − η ( t − T ( i ) ) Z ( i ) ( t ) ( d ) = e − ηT ( i ) e − η ( t − T ( i ) ) X ( i ) ( t − T ( i ) ) { T ( i ) ≤ t } IMIT THEOREMS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION 5 t T T T t I ( t ) = 5 Z (1) ( t ) = 1 Z (2) ( t ) = 2 Z (3) ( t ) = 2 Figure 1. splitting trees with immigration. The vertical axis is time, hori-zontal axis shows filiation. At time t , three populations are extant.As in (6), denote by E i := lim t →∞ e − ηt X ( i ) ( t ) for i ≥ 1. Thus E , E , . . . are i.i.d. exponentialr.v. with parameter c . Moreover, the sequences ( E i , i ≥ 1) and ( T ( i ) , i ≥ 1) are independent.It follows that e − ηt Z ( i ) ( t ) → e − ηT ( i ) E i a.s. as t → ∞ . We record this in the following Proposition 2.2. e − ηt ( Z (1) ( t ) , Z (2) ( t ) , . . . ) −→ t →∞ ( e − ηT (1) E , e − ηT (2) E , . . . ) a.s.where the E i ’s are independent copies of E and independent of the T ( i ) ’s. For t ≥ 0, let I ( t ) be the size of the total population at time tI ( t ) = (cid:88) i ≥ Z i ( t ) . The process ( I ( t ) , t ≥ 0) is a non-Markovian continuous-time branching process with immi-gration. Theorem 2.3. (i) For t positive, I ( t ) has a negative binomial distribution with parameters − W ( t ) − and θ/b . i.e. for s ∈ [0 , , its generating function is G t ( s ) := E (cid:104) s I ( t ) (cid:105) = (cid:18) W ( t ) − − s (1 − W ( t ) − ) (cid:19) θ/b . (ii) We have I := lim t →∞ e − ηt I ( t ) = (cid:88) i ≥ e − ηT ( i ) E i a.s.and I has a Gamma distribution Γ( θ/b, c ) i.e the density of I with respect to Lebesguemeasure is g ( x ) = c θ/b x θ/b − e − cx Γ( θ/b ) , x > . MATHIEU RICHARD The result (i) is a generalization of a result by D.G. Kendall [15] which was the particularMarkovian case of a birth, death and immigration process. The proof we give in Section 4uses equations (1) and (2) about the law of X ( t ).There exist other proofs of the almost sure convergence in (ii), but they require strongerassumptions. For example, P. Jagers [12] gives a proof for the convergence of general branchingprocesses with immigration under the hypothesis that the variance of the number of childrenper individual ξ ( ∞ ) is finite. In our case, this is only true if (cid:82) (0 , ∞ ) r Λ( dr ) < ∞ . In theparticular Markovian case described previously, the proof is also easier since ( e − ηt X ( t ) , t ≥ e − ηt I ( t ) , t ≥ 0) is a non-negative submartingale andboth converge a.s. In the proof we give in Section 4, the only assumption we use about themeasure Λ is that its mass is finite. The proof is based on Proposition 2.1(ii).In the following, we will consider different kinds of metacommunity where immigrants arechosen and will give results about abundances of surviving populations. In Model I, there is adiscrete spectrum with zero macroscopic relative abundances: when a new family is initiated,it is of a type different from those of previous families. The following theorem yields theasymptotic behaviors of the fractions of the surviving subpopulations ranked by decreasingages in the total population: Theorem 2.4 (Model I) . lim t →∞ I ( t ) − ( Z (1) ( t ) , Z (2) ( t ) , . . . ) = ( P , P , . . . ) a.s.where the law of ( P , P , . . . ) is a GEM distribution with parameter θ/b . In other words, for i ≥ P i ( d ) = B i i − (cid:89) j =1 (1 − B j ) and ( B i ) i ≥ is a sequence of i.i.d. random variables with law Beta( , θ/b ) whose density withrespect to Lebesgue measure is θb (1 − x ) θ/b − [0 , ( x ) . This result was proved by S. Tavar´e [22] in the case where Λ( dr ) = δ ∞ ( dr ) (pure birthprocess); it is the exponential case defined previously with b = 1 and d = 0. His result isrobust because we see that in our more general case, the limit distribution does not dependon the lifespan distribution but only on the immigration-to-birth ratio θ/b . In biogeography,a typical question is to recover data about population dynamics (immigration times, law oflifespan duration) from the observed diversity patterns. In this model, we see that there isa loss of information about the lifespan duration. However, the ratio θ/b can be estimatedthanks to the species abundance distribution. We will prove Theorem 2.4 in Subsection 5.1.In Model II, we consider a discrete spectrum with nonzero macroscopic relative abundances.Contrary to Model I where types were always new, they are now given a priori and types ofimmigrants are independently drawn according to some probability p = ( p i , i ≥ θ -Poisson point process), it is of type i withprobability p i > Theorem 2.5 (Model II) . For i ≥ , denote by I i ( t ) the number of individuals of type i attime t and set θ i := θp i b . Then lim t →∞ I ( t ) − ( I ( t ) , I ( t ) , . . . ) = ( P (cid:48) , P (cid:48) , . . . ) a.s. IMIT THEOREMS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION 7 where for i ≥ P (cid:48) i ( d ) = B (cid:48) i i − (cid:89) j =1 (1 − B (cid:48) j ) and ( B (cid:48) i ) i ≥ is a sequence of independent random variables such that B (cid:48) i ∼ Beta θ i , θb (cid:88) j ≥ i +1 p j . In particular, for i ≥ , P (cid:48) i has a Beta distribution B ( θ i , θ/b − θ i ) . The proof of this theorem will be done in Subsection 5.2. In this model, the limit onlydepend on θ/b and the metacommunity spectrum ( p i , i ≥ Remark 2.6. If the number of possible types n is finite, then n (cid:88) i =1 P (cid:48) i = n (cid:88) i =1 I i I = 1 , n (cid:88) i =1 θ i = θb and the joint density of ( P (cid:48) , . . . , P (cid:48) n ) is Γ( θ/b ) (cid:81) ni =1 Γ( θ i ) (cid:32) n − (cid:89) i =1 x θ i − i { x i > } (cid:33) (1 − x − · · · − x n − ) θ n − { x + ··· + x n − < } . This is the joint density of a Dirichlet distribution Dir ( θ , . . . , θ n ) . In Model III, we consider a continuous spectrum of possible types and we slightly modify theimmigration process: when an individual arrives on the island, it starts a new population withan immigration rate proportional to its abundance on the metacommunity. More precisely,let Π be a Poisson point process on R + × R + with intensity dt ⊗ xf ( x ) dx where f is anonnegative function such that θ := (cid:82) ∞ xf ( x ) dx is finite. Then, write Π := (( T i , ∆ i ) , i ≥ T < T < · · · are the times of a θ -linear Poisson point process and (∆ i , i ≥ 1) isa sequence of i.i.d random variables whose density is θ − xf ( x ) dx which is independent of( T i , i ≥ T i , a new population starts out and it evolves like the continuous-timebranching process with immigration defined for the two previous models with an immigrationrate ∆ i .The interpretation of this model is as follows: for x > f ( x ) represents the density ofspecies with abundance x in the metacommunity and at each immigration time, an individualof a species with abundance in ( x, x + dx ) is chosen with probability xf ( x ) θ dx proportional toits abundance in the metacommunity.If ( Z i ( t ) , t ≥ 0) is the i -th oldest family, Z i ( t ) = I i ∆ i ( t − T i ) { t ≥ T i } , t ≥ I i ∆ i ’s are independent copies of I ∆ , which, conditional on ∆, evolves like theimmigration process of the first two models with an immigration rate ∆. According toTheorem 2.3(ii), we know that e − ηt I ∆ ( t ) −→ t →∞ G a.s.where conditional on ∆, G ∼ Gamma(∆ /b, c ). We denote by F its distribution tail F ( v ) := P ( G ≥ v ) = (cid:90) ∞ dx xf ( x ) θ (cid:90) ∞ v e − ct t x/b − c x/b Γ( x/b ) dt. MATHIEU RICHARD Hence, we also have Proposition 2.7. For i ≥ , e − ηt Z i ( t ) −→ t →∞ e − ηT i G i a.s.where ( G i , i ≥ is a sequence of i.i.d. r.v. with the same distribution as G and independentof ( T i , i ≥ . We again denote by I ( t ) the total population at time t : I ( t ) := (cid:88) i ≥ Z i ( t ) and we obtain aresult similar to Theorem 2.3 concerning the asymptotic behavior of I ( t ). Proposition 2.8. If (cid:82) ∞ x f ( x ) dx < ∞ we have e − ηt I ( t ) a.s. −→ t →∞ (cid:88) i ≥ e − ηT i G i and the Laplace transform of σ := (cid:80) i ≥ e − ηT i G i is E (cid:2) e − sσ (cid:3) = exp (cid:18) − θη (cid:90) ∞ F ( v ) v (cid:0) − e − sv (cid:1) dv (cid:19) . Moreover, E [ σ ] = 1 ηbc (cid:90) ∞ x f ( x ) dx < ∞ . We also have a result about abundances of different types Theorem 2.9 (Model III) . We have (cid:18) Z ( t ) I ( t ) , Z ( t ) I ( t ) , . . . (cid:19) −→ t →∞ (cid:16) σ σ , σ σ , . . . (cid:17) a.s.where ( σ i , i ≥ are the points of a non-homogeneous Poisson point process on (0 , ∞ ) withintensity measure θη F ( y ) y dy and σ = (cid:80) i ≥ σ i . The proofs of these two results will be done in Subsection 5.3. Notice that in this model, thelimit only depends on the lifespan measure via the Malthusian parameter η and the constant c . 3. Proof of Proposition 2.1 Some useful, technical lemmas. Thereafter, we state some lemmas that will be usefulin subsequent proofs. Lemma 3.1. Let Y , Y , . . . be a sequence of i.i.d random variables with finite expectation.Then, if S := sup n ≥ (cid:0) n (cid:80) ni =1 Y i (cid:1) , E (cid:104) (log + S ) k (cid:105) < ∞ , k ≥ . Proof. According to Kallenberg [13, p.184], for r > r P ( S ≥ r ) ≤ E [ Y ; Y ≥ r ] . Hence, choosing r = e s / 2, we have for s ≥ P (log + S ≥ s ) ≤ e − s E [ Y ] IMIT THEOREMS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION 9 and E [(log + S ) k ] = (cid:90) ∞ ks k − P (log + S ≥ s ) ds ≤ E [ Y ] k (cid:90) ∞ s k − e − s ds < ∞ . This completes the proof. (cid:3) Lemma 3.2. Let A be a homogeneous Poisson process with parameter ρ . If S := sup t> ( A t /t ) ,then for a > , P ( S > a ) = ρa ∨ . In particular, ∀ k ≥ , E (cid:104) (log + S ) k (cid:105) < ∞ . Proof. If a < ρ , since lim t →∞ A t /t = ρ , P ( S > a ) = 1. Let now a be a real number greaterthan ρ . Then P ( S ≤ a ) = P ( ∀ t ≥ , at − A t ≥ . According to Bertoin [2, chap.VII], since ( at − A t , t > 0) is a L´evy process with no positivejumps and with Laplace exponent φ ( λ ) = λa − ρ (1 − e − λ ), we have as in (4) P ( S ≤ a ) = H (0) H ( ∞ )where H is the scale function associated with ( at − A t , t > 0) . We compute H (0) and H ( ∞ )using Tauberian theorems [2, p.10]: • φ ( λ ) ∼ λ ( a − ρ ) then H ( x ) ∼ ∞ ( a − ρ ) − • φ ( λ ) ∼ ∞ λa then H ( x ) ∼ a − Hence, P ( S ≤ a ) = a − ρa = 1 − ρa , a > ρ. Then, E [(log + S ) k ] = (cid:90) ∞ kr k − P (log + S ≥ r ) dr ≤ (cid:90) log + ρ kr k − dr + (cid:90) ∞ log + ρ kr k − ρe r dr < ∞ and the proof is completed. (cid:3) Proof of Proposition 2.1(i). A. Lambert proved in [16] that P (Ext) = 1 − η/b and in[17] that, conditional on Ext c , e − ηt X ( t ) L −→ t →∞ E where E is an exponential random variable with parameter c . To obtain a.s. convergence,we use [20, Thm 5.4] where O. Nerman gives sufficient conditions for convergence of CMJprocesses to hold almost surely. Here, the two conditions of his theorem are satisfied. Indeed,the second one holds if there exists on [0 , ∞ ) an integrable, bounded, non-increasing positivefunction h such that E (cid:34) sup t ≥ (cid:32) e − ηt { t<ζ } h ( t ) (cid:33)(cid:35) < ∞ where we recall that ζ is the lifespan duration of a typical individual in the CMJ-process X .Then, choosing h ( t ) = e − ηt , this condition is trivially satisfied.The first one holds if there exists a non-increasing Lebesgue integrable positive function g such that(8) (cid:90) ∞ g ( t ) e − ηt µ ( dt ) < ∞ . Taking g ( t ) = e − βt with η > β > µ ( dt ) = (cid:82) ( t, ∞ ) Λ( dr ) dt , we have (cid:90) ∞ g ( t ) e − ηt µ ( dt ) = (cid:90) ∞ e ( β − η ) t (cid:90) ∞ t Λ( dr ) dt = (cid:90) (0 , ∞ ) Λ( dr ) (cid:90) r e ( β − η ) t dt ≤ C (cid:90) (0 , ∞ ) Λ( dr ) = Cb < ∞ and condition (8) is fulfilled.3.3. Proof of Proposition 2.1(ii). We will say that a process G satisfies condition (C) if E (cid:34)(cid:18) log + sup t ≥ e − ηt G t (cid:19) (cid:35) < ∞ and our aim is to prove that, conditional on non-extinction, the homogeneous CMJ-process( X ( t ) , t ≥ 0) satisfies it.According to Theorem 4.4.1.1 in [16], conditional on non-extinction of ( X ( t ) , t ≥ X ( t ) = X ∞ t + X dt + X gt where • X ∞ t is the number of individuals alive at time t and whose descendance is infinite. Inparticular, ( X ∞ t , t ≥ 0) is a Yule process with rate η . • X dt is the number of individuals alive at time t descending from trees grafted on theright hand side of the Yule tree (right refers to the order of the contour of the planarsplitting tree) X dt := ˜ N t (cid:88) i =1 ˜ X i ( t − ˜ T i )where – ( ˜ X i , i ≥ 1) is a sequence of i.i.d. splitting trees conditional on extinction andindependent of X ∞ . We know that such trees have the same distribution assubcritical splitting trees with lifespan measure ˜Λ( dr ) = e − ηr Λ( dr ) (cf. [16]). – Conditionally on ( X ∞ t , t ≥ N t , t ≥ 0) is an non-homogeneous Poisson processwith mean measure ( b − η ) X ∞ t dt and independent of ( ˜ X i ) i . We denote its arrivaltimes by ˜ T , ˜ T , . . . • X gt is the number of individuals alive at time t descending from trees grafted on theleft hand side of the Yule tree (left also refers to the contour order).More specifically, let ( A, R ) be a couple of random variables with joint law givenby(9) P ( A + R ∈ dz, R ∈ dr ) = e − ηr dr Λ( dz ) , < r < z IMIT THEOREMS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION 11 and let (( A i,j , R i,j ) , i ≥ , j ≥ 1) be i.i.d random variables distributed as ( A, R ). Weconsider the arrival times T i,j = τ i + A i, + A i, + · · · + A i,j , i ≥ , j ≥ τ < τ < τ < · · · are the splitting times of the Yule tree, that is, on eachnew infinite branch, we start a new A -renewal process independent of the others. Wedefine for t ≥ X gt := (cid:88) i,j ˆ X i,j ( t − T i,j ) { T i,j ≥ t } . where ( ˆ X i,j , i ≥ , j ≥ 1) is a sequence of i.i.d. splitting trees independent of X ∞ ,conditioned on extinction and such that the unique ancestor of ˆ X i,j has lifetime R i,j .We denote by ˆ N t := { ( i, j ) , T i,j ≤ t } the number of graft times before t . t T , T , R R T , R ˜ T ˜ T τ τ Figure 2. Spine decomposition of a splitting tree. In bold, the Yule tree X ∞ on which we graft on the left (at times T , , . . . ) the trees conditioned onextinction whose ancestors have lifetime durations distributed as R and on theright (at time ˜ T , . . . ) the trees conditioned on extinction.In the following, we will prove that ( X t , t ≥ 0) satisfies condition (C). To do this, usingMinkowski inequality and the inequality ∀ x, y ≥ , log + ( x + y ) ≤ log + x + log + y + log 2 , we only have to check that the three processes X ∞ , X g and X d satisfy (C).3.3.1. Proof of condition (C) for X ∞ . Since ( X ∞ t , t ≥ 0) is a η -Yule process, ( e − ηt X ∞ t , t ≥ 0) is a non-negative martingale [1, p.111]and so by Doob’s inequality [21],(10) E (cid:20) sup t ≥ ( e − ηt X ∞ t ) (cid:21) ≤ t ≥ E [( e − ηt X ∞ t ) ] Moreover, E [( X ∞ t ) ] = 2 (cid:0) e ηt − e ηt (cid:1) (again in [1]) and E [( e − ηt X ∞ t ) ] = 2 e − ηt (cid:0) e ηt − e ηt (cid:1) −→ t →∞ . Hence, the supremum in the right hand side of (10) is finite and E (cid:34)(cid:18) sup t ≥ (cid:0) e − ηt X ∞ t (cid:1)(cid:19) (cid:35) < ∞ . From now on, we will set M := sup t ≥ e − ηt X ∞ t . Since E [ M ] < ∞ , (C) is trivially satisfiedby X ∞ .3.3.2. Proof of condition (C) for X d . We recall that X dt = ˜ N t (cid:88) i =1 ˜ X i ( t − ˜ T i ) . Denote by Y i the total progeny of the conditioned splitting tree ˜ X i , that is, the total number ofdescendants of the ancestor plus one. Then, a.s for all t ≥ i ≥ 1, we have ˜ X i ( t − ˜ T i ) ≤ Y i and X dt ≤ ˜ N t (cid:88) i =1 Y i a.s. t ≥ . Hence, almost surely for all t , e − ηt X dt ≤ e − ηt ˜ N t (cid:88) i =1 Y i = (cid:16) e − ηt ˜ N t (cid:17) N t ˜ N t (cid:88) i =1 Y i and, thanks to Minkowski’s inequality, E (cid:34)(cid:18) log + sup t ≥ (cid:16) e − ηt X dt (cid:17)(cid:19) (cid:35) / ≤ E (cid:34)(cid:18) log + sup t ≥ (cid:16) e − ηt ˜ N t (cid:17)(cid:19) (cid:35) / + E log + sup t> N t ˜ N t (cid:88) i =1 Y i / (11)We first consider the second term in the right hand side of (11): E log + sup t> N t ˜ N t (cid:88) i =1 Y i ≤ E (cid:32) log + sup n ≥ (cid:32) n n (cid:88) i =1 Y i (cid:33)(cid:33) since ( ˜ N t , t ≥ 0) is integer-valued. Thanks to Lemma 3.1, this term is finite because E [ Y ] isfinite. Indeed, Y is the total progeny of a subcritical branching process and it is known [10]that its mean is finite. IMIT THEOREMS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION 13 We are now interested in the first term in the r.h.s. of (11). We work conditionallyon X ∞ =: ( f ( t ) , t ≥ e − ηt (cid:82) t f ( s ) ds ≤ M using the supremum M of( e − ηt X ∞ t , t ≥ e − ηt ˜ N t = e − ηt (cid:90) t f ( s ) ds ˜ N t (cid:82) t f ( s ) ds ≤ M ˜ N t (cid:82) t f ( s ) ds Moreover, (cid:16) ˜ N t , t ≥ (cid:17) ( d ) = (cid:16) N (cid:48) (cid:82) t f ( s ) ds , t ≥ (cid:17) where N (cid:48) is a homogeneous Poisson process with parameter b − η .Hence, using Minkowski’s inequality, E (cid:34)(cid:18) log + sup t ≥ (cid:16) e − ηt ˜ N t (cid:17)(cid:19) (cid:35) / ≤ log + M + E log + sup t> N (cid:48) (cid:82) t f ( s ) ds (cid:82) t f ( s ) ds / = log + M + E (cid:34)(cid:18) log + sup t> (cid:18) N (cid:48) t t (cid:19)(cid:19) (cid:35) / and the second term of the r.h.s. is finite using Lemma 3.2.Hence, ( ˜ N t , t ≥ 0) satisfies (C) since E [ M ] < ∞ and X d as well, which ends this paragraph.3.3.3. Proof of condition (C) for X g . We have X gt = ˆ N t (cid:88) i =1 ˆ X i ( t − ˆ T i )As in the previous section, e − ηt X gt ≤ (cid:16) e − ηt ˆ N t (cid:17) N t ˆ N t (cid:88) i =1 ˆ Y i a.s.where ˆ Y i is the total progeny of the conditioned CMJ-process ( ˆ X i ( t ) , t ≥ E (cid:34)(cid:18) log + sup t ≥ (cid:0) e − ηt X gt (cid:1)(cid:19) (cid:35) / ≤ E (cid:34)(cid:18) log + sup t ≥ (cid:16) e − ηt ˆ N t (cid:17)(cid:19) (cid:35) / + E (cid:32) log + sup n> (cid:32) n n (cid:88) i =1 ˆ Y i (cid:33)(cid:33) / (12)We first prove that the second term in the r.h.s. is finite using Lemma 3.1. We only haveto check that E [ ˆ Y ] is finite. We recall that ˆ Y is the total progeny of a splitting tree whoseancestor has random lifespan R and conditioned on extinction. Conditioning on R , it is alsothe total progeny of a subcritical Bienaym´e-Galton-Watson process starting from a Poisson random variable with mean R . Hence, E [ ˆ Y ] is finite if E [ R ] is finite. As a consequence of(9), we have that P ( R ∈ dr ) = e − ηr (cid:82) ∞ r Λ( dz ) dr and E [ R ] = (cid:90) (0 , ∞ ) Λ( dz ) (cid:90) z re − ηr dr = − (cid:90) (0 , ∞ ) Λ( dz ) ze − ηz η + (cid:90) (0 , ∞ ) Λ( dz ) 1 − e − ηz η = ψ (cid:48) ( η ) − η + 1 η = ψ (cid:48) ( η ) η < ∞ . We are now interested in the first term of the r.h.s. of (12). We need to make calculationson ˆ N t which is the total number of times of graftings T i,j less than or equal to t . Recall thatfor i ≥ , j ≥ T i,j = τ i + A i,j where A i,j := A i, + · · · + A i,j and that τ i is the birth timeof individual i and that ˆ N t is the sum of the numbers of graftings before t on each of the X ∞ t branches. For i ≥ 0, denote by α i , α i , . . . , the birth times of the daughters of individual i and α i = τ i . For k ≥ 1, denote by ˜ τ ik := α ik − α ik − the interbirth times. In particular,(˜ τ ik , i ≥ , k ≥ 1) are i.i.d. exponential r.v. with parameter η since we consider a η -Yule tree. A A A α α A A A A A A A α A A Figure 3. Construction of the renewal process ( A j , j ≥ 1) by concatenationof the renewal processes A k .We enlarge the probability space by redefining the renewal processes ( A i,j , j ≥ 1) from adoubly indexed sequence of i.i.d. A -renewal processes ( A i,k , i ≥ , k ≥ A i,j , j ≥ 1) recursively by concatenation of the A i,k ’s as in Figure 3. To simplifynotation, we only define ( A ,j , j ≥ 1) which will be denoted by ( A j , j ≥ A (cid:48) := inf { t > α |A ∩ [ t, + ∞ ) (cid:54) = ∅} , C := A ∩ [0 , α ] + 1 , and A j := inf { t ≥ | A ∩ [0 , t ] = j } , j = 1 , . . . , C + 1 . Moreover, for l ≥ 1, if one knows C l and A (cid:48) l , let r l be the unique integer such that A (cid:48) l belongsto ] α r l , α r l +1 ] and define A (cid:48) l +1 := A (cid:48) l + inf { t > α r l +1 − A (cid:48) l |A r l +1 ∩ [ t, + ∞ ) (cid:54) = ∅} ,C l +1 := A r l +1 ∩ [0 , α r l +1 − A (cid:48) l ] + 1 , IMIT THEOREMS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION 15 and A C + ··· + C l + j := A (cid:48) l + inf { t ≥ | A r l +1 ∩ [0 , t ] = j } , j = 1 , . . . , C l +1 . Then, ( A j , j ≥ 1) is a A -renewal process because we concatenated the independent renewalprocesses ( A k ) stopped at the first renewal time after a given time. Indeed, one can see arenewal process as the range of a compound Poisson process whose jumps are distributed as A . The renewal process stopped at the first point after t is then the range of a compoundPoisson process stopped at the first hitting time T of [ t, ∞ ), which is a stopping time. Then( A j , j ≥ A j , j ≥ 1) is a A -renewal process.According to previous computations, we haveˆ N t ≤ (cid:88) i ≥ ,l ≥ C il { α irl,i ≤ t } where C il and r il,i are the analogous notations as C l and r l for individual i . Then if D ik := A i,k ∩ [0 , ˜ τ ik ] + 1 k ≥ , i ≥ , since α ir l,i +1 − A (cid:48) l,i ≤ α ir l,i +1 − α ir l,i = ˜ τ ir l,i , then C il ≤ D ir l,i and(13) ˆ N t ≤ (cid:88) i ≥ ,l ≥ D ir l,i { α irl,i ≤ t } ≤ (cid:88) i ≥ ,k ≥ D ik { α ik ≤ t } a.s.Moreover, the random variables ( D ik , i ≥ , k ≥ 1) are independent and identically dis-tributed as 1 plus the value of a A -renewal process at an independent exponential time E with parameter η D := sup { j ≥ , A + · · · + A j ≤ E } + 1 . The sum of the r.h.s. of (13) has 2 X ∞ t − n i terms in the sum where n i is the number of daughters of individual i born before t . Then, there is X ∞ t + X ∞ t − (cid:88) i =0 n i terms in the sum and (cid:80) i n i is the number ofdescendants of the individual 0 born before t which equals X ∞ t − E (cid:34)(cid:18) log + sup t ≥ (cid:16) e − ηt ˆ N t (cid:17)(cid:19) (cid:35) / ≤ E (cid:34)(cid:18) log + sup t ≥ (cid:0) e − ηt (2 X ∞ t − (cid:1)(cid:19) (cid:35) / + E (cid:32) log + sup n> (cid:32) n n (cid:88) i =1 D i (cid:33)(cid:33) / (14)where ( D i , i ≥ 1) are i.i.d. r.v. distributed as D . The first term of (14) is smaller than E [(log + (2 M )) ] / and the second term is finite by another use of Lemma 3.1 if E [ D ] < ∞ .Denote by ( C t , t ≥ 0) the renewal process whose arrival times are distributed as A . A littlecalculation from (9) gives us that E [ A ] = m − η > 0. Thus, according to Theorem 2.3 ofChapter 5 in [6], we have lim t →∞ E [ C t ] t = ηm − and so there exists κ > E [ C t ] ≤ κt for t ≥ 0. Then, E [ D ] = 1 + E [ C E ] = 1 + (cid:90) ∞ ηe − ηt dt E [ C t ] ≤ κ (cid:90) ∞ ηe − ηt dt = 1 + κη < ∞ . Finally, the r.h.s. of (14) and (12) are finite and X g satisfies condition (C).4. Proof of Theorem 2.3 Some preliminary lemmas. We start with some properties about W the scale functionassociated with ψ and defined by (3). Lemma 4.1. (i) W (0) = 1 ,(ii) e − ηt W ( t ) −→ c − as t → ∞ ,(iii) W is differentiable and W (cid:63) Λ = bW − W (cid:48) where (cid:63) is convolution product.Proof. (i) We have (cid:90) ∞ e − λt W ( t ) dt = 1 ψ ( λ ) ∼ λ →∞ λ because ψ ( λ ) = λ − b + (cid:82) (0 , ∞ ) e − λr Λ( dr ). Then, by a Tauberian theorem [2, p10],lim t → W ( t ) = 1.(ii) (From [17]) For λ > 0, using a Taylor expansion and ψ ( η ) = 0, ψ ( λ + η ) ∼ λ → λψ (cid:48) ( η ) = λc. Then (cid:90) ∞ W ( t ) e − ηt e − λt dt ∼ λ → λc and another Tauberian theorem entails that W ( t ) e − ηt converges to 1 /c as t → ∞ .(iii) We first compute the Laplace transform of W (cid:63) Λ. Let λ > η (cid:90) ∞ e − λt W (cid:63) Λ( t ) dt = (cid:90) ∞ e − λt W ( t ) dt (cid:90) (0 , ∞ ) e − λr Λ( dr )= 1 ψ ( λ ) ( ψ ( λ ) − λ + b )Integrating by parts and using (i) and (ii), (cid:90) ∞ e − λt W (cid:48) ( t ) dt = (cid:104) e − λt W ( t ) (cid:105) ∞ + λ (cid:90) ∞ e − λt W ( t ) dt = − λψ ( λ )and so the Laplace transform of bW − W (cid:48) is b/ψ ( λ ) + 1 − λ/ψ ( λ ) which equals that of W (cid:63) Λ. This completes the proof. (cid:3) The following lemma deals with the convergence of random series: Lemma 4.2. Let ( ζ i , i ≥ be a sequence of i.i.d. positive random variables such that E [log + ζ ] is finite and let ( τ i , i ≥ be the arrival times of a Poisson point process withparameter ρ independent of ( ζ i , i ≥ . Then for any r > , the series (cid:80) i ≥ e − rτ i ζ i convergesa.s. IMIT THEOREMS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION 17 Proof. We have(15) (cid:88) i ≥ e − rτ i ζ i ≤ (cid:88) i ≥ exp (cid:18) − i (cid:20) r τ i i − log + ζ i i (cid:21)(cid:19) We use the following consequence of Borel-Cantelli’s lemma: if ξ , ξ , . . . are i.i.d. non-negative random variables, lim sup n →∞ ξ n n = 0 or ∞ a.s.according to whether E [ ξ ] is finite or not. We use it with ξ i = log + ζ i . Hence, since E [log + ζ ]is finite, lim i →∞ log + ζ i /i = 0 a.s. Moreover, by the strong law of large numbers, τ i /i convergesalmost surely to 1 /ρ as i goes to infinity. Then, r τ i i − log + ζ i i −→ i →∞ rρ > (cid:3) Proof of Theorem 2.3(i). In order to find the law of I ( t ) the total population attime t , we use the fact that it is the sum of a Poissonian number of population sizes. Morespecifically, if we denote by N t the number of populations at time t , ( N t , t ≥ 0) is a Poissonprocess with parameter θ and conditionally on { N t = k } , the k -tuple ( T , . . . , T k ) has thesame distribution as (cid:0) U (1) , . . . , U ( k ) (cid:1) which is the reordered k -tuple of k independent uniformrandom variables on [0 , t ]. Hence, conditionally on { N t = k } , I ( t ) ( d ) = k (cid:88) i =1 X i (cid:0) t − U ( i ) (cid:1) ( d ) = k (cid:88) i =1 X i ( t − U i ) ( d ) = k (cid:88) i =1 X i ( U i )since all U ( i ) ’s appear in the sum and the U i ’s are independent from the X i ’s. Hence,conditionally on { N t = k } , I ( t ) has the same distribution as a sum of k i.i.d. r.v. with law X ( U ). Then, G t ( s ) = (cid:88) k ≥ E (cid:104) s I ( t ) (cid:12)(cid:12)(cid:12) N t = k (cid:105) P ( N t = k )= (cid:88) k ≥ E (cid:104) s X ( U ) (cid:105) k ( tθ ) k k ! e − θt = e − θt exp (cid:16) tθ E (cid:104) s X ( U ) (cid:105)(cid:17) (16)We now compute the law of X ( t ) for t > X ( U ).Using (1), (2) and Lemma 4.1(iii), we have P ( X ( t ) = 0) = (cid:90) (0 , ∞ ) ˜ P r ( X ( t ) = 0) Λ( dr ) b = (cid:90) (0 , ∞ ) W ( t − x ) W ( t ) Λ( dr ) b = 1 bW ( t ) W (cid:63) Λ( t ) = 1 − W (cid:48) ( t ) bW ( t ) and for n ∈ N ∗ , P ( X ( t ) = n ) = (cid:90) (0 , ∞ ) ˜ P r ( X ( t ) = n ) Λ( dr ) b = 1 bW ( t ) (cid:18) − W ( t ) (cid:19) n − (cid:16) b − W ( t ) − W (cid:63) Λ( t ) (cid:17) = (cid:18) − W ( t ) (cid:19) n − W (cid:48) ( t ) bW ( t ) . We now compute P ( X ( U ) = n ) P ( X ( U ) = 0) = 1 t (cid:90) t P ( X ( u ) = 0) du = 1 − tb (cid:90) t W (cid:48) ( u ) W ( u ) du = 1 − log W ( t ) tb because W (0) = 1. For n > P ( X ( U ) = n ) = 1 t (cid:90) t P ( X ( u ) = n ) du = 1 t (cid:90) t (cid:18) − W ( u ) (cid:19) n − W (cid:48) ( u ) bW ( u ) du = 1 bt (cid:90) W ( t ) − (1 − u ) n − n du = (1 − /W ( t )) n btn . We are now able to compute the generating function of X ( U ). For s, t > E (cid:104) s X ( U ) (cid:105) = 1 bt (cid:88) n ≥ s n n (cid:18) − W ( t ) (cid:19) n + 1 − log W ( t ) tb = 1 − bt (cid:104) log (cid:16) − s (1 − /W ( t )) (cid:17) + log W ( t ) (cid:105) = 1 − bt log (cid:16) W ( t ) + s (1 − W ( t )) (cid:17) . Finally for t, s > 0, according to (16), G t ( s ) = e − θt exp (cid:16) tθ E (cid:104) s X ( U ) (cid:105)(cid:17) = ( W ( t ) + s (1 − W ( t ))) − θ/b . which is the p.g.f. of a negative binomial distribution with parameters 1 − W ( t ) − and θ/b .4.3. Proof of Theorem 2.3(ii). We first prove the almost sure convergence. Splitting I ( t )between the surviving and the non-surviving populations, we have(17) e − ηt I ( t ) = (cid:88) i ≥ e − ηt Z ( i ) ( t ) + (cid:88) i ≥ e − ηt X i ( t − T i ) { t ≥ T i }∩ Ext i where for i ≥ 1, Ext i denotes the extinction of the process X i . We will show that for eachof these two terms, we can exchange summation and limit, so that in particular, the secondterm vanishes as t → ∞ .We first treat the second term of the r.h.s. of (17). We have C t := (cid:88) i ≥ X i ( t − T i ) { t ≥ T i }∩ Ext i ≤ (cid:88) i ≥ { t ≥ T i } Y i Ext i a.s. IMIT THEOREMS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION 19 where Y i is the total progeny of the i -th population which does not survive. Moreover, E [ Y Ext ] ≤ E [ Y ], Y is the total progeny of a subcritical Bienaym´e-Galton-Watson processso its mean is finite. Hence, since the r.h.s. in the previous equation is a compound Poissonprocess with finite mean, it grows linearly and e − ηt C t vanishes as t → ∞ .To exchange summation and limit in the first term of the r.h.s. of (17), we will use thedominated convergence theorem. By Proposition 2.2, we already know that e − ηt Z ( i ) ( t ) a.s.converges as t goes to infinity to e − ηT ( i ) E i . Hence, it is sufficient to prove that(18) (cid:88) i ≥ sup t ≥ (cid:16) e − ηt Z ( i ) ( t ) (cid:17) < ∞ a.s.Since sup t ≥ (cid:16) e − ηt Z ( i ) ( t ) (cid:17) = e − ηT ( i ) sup t ≥ (cid:0) e − ηt X ( i ) ( t ) (cid:1) , we have (cid:88) i ≥ sup t ≥ (cid:16) e − ηt Z ( i ) ( t ) (cid:17) = (cid:88) i ≥ e − ηT ( i ) J i where J i := sup t ≥ (cid:0) e − ηt X ( i ) ( t ) (cid:1) for i ≥ J , J , . . . are i.i.d. Thus, using Lemmas 4.2,this series a.s. converges if E [log + J ] is finite, which is checked thanks to Proposition 2.1(ii).Then we get (18) and using the dominated convergence theorem, I := lim t →∞ e − ηt I ( t ) = (cid:88) i ≥ e − ηT ( i ) E i a.s.In order to find the law of I , we compute its Laplace transform. For a > 0, using part(i)of this theorem, E (cid:104) e − ae − ηt I ( t ) (cid:105) = G t ( e − ae − ηt ) = (cid:16) e − ae − ηt + (cid:16) − e − ae − ηt (cid:17) W ( t ) (cid:17) − θ/b and e − ae − ηt + (cid:16) − e − ae − ηt (cid:17) W ( t ) ∼ t →∞ ae − ηt W ( t ) −→ t →∞ ac using Lemma 4.1 (ii). Then, E (cid:2) e − aI (cid:3) = (cid:18) ca + c (cid:19) θ/b which is the Laplace transform of a Gamma( θ/b, c ) random variable.5. Other proofs Proof for Model I. To prove Theorem 2.4, we will follow Tavar´e’s proof [22]. We beginwith a technical lemma which will be useful in the proof of this theorem and in forthcomingproofs. Lemma 5.1. Let ( T i , i ≥ be the arrival times of a Poisson process with parameter ρ and ζ , ζ , . . . be i.i.d. r.v. independent of the T i ’s. Denote by g the density of ζ withrespect to Lebesgue measure and by F ( v ) := P ( ζ ≥ v ) its distribution tail. Then for r > , ( e − rT i ζ i , i ≥ are the points of an non-homogeneous Poisson point process on (0 , ∞ ) withintensity measure ρr F ( v ) v dv. Proof. We first study the collection Π = { ( T i , ζ i ) , i ≥ } . This is a Poisson point process on(0 , ∞ ) × (0 , ∞ ) with intensity measure ρg ( y ) dtdy . Then, ( e − rT i ζ i , i ≥ 1) is a Poisson pointprocess whose intensity measure is the image of ρg ( y ) dtdy by ( t, y ) (cid:55)→ e − rt y . We now computeit. Let h be a non-negative mapping. Changing variables, we get (cid:90) ∞ (cid:90) ∞ h ( e − rt y ) ρg ( y ) dtdy = ρr (cid:90) ∞ h ( v ) dv (cid:90) g (cid:16) vu (cid:17) duu = ρr (cid:90) ∞ h ( v ) F ( v ) v dv and the proof is completed. (cid:3) We are now able to prove Theorem 2.4. By Proposition 2.2 and Theorem 2.3, I ( t ) − ( Z (1) ( t ) , Z (2) ( t ) , . . . ) = e − ηt ( Z (1) ( t ) , Z (2) ( t ) , . . . ) e − ηt I ( t ) −→ t →∞ (cid:16) σ σ , σ σ , . . . (cid:17) a.s.where σ i := exp (cid:0) − ηT ( i ) (cid:1) E i and σ := (cid:80) i ≥ σ i .Moreover, the ( σ i ) i ≥ are the points of a non-homogeneous Poisson point process on (0 , ∞ )with intensity measure θb e − cy y dy thanks to Lemma 5.1 with ρ = θη/b , r = η and F ( v ) = e − cv because ( T ( i ) ) i ≥ is a Poisson process of rate θη/b and ( E i ) i ≥ is an independent sequence ofi.i.d. exponential variables with parameter c .According to [3, p. 89], the Poisson point process ( σ i ) i ≥ satisfies σ = (cid:80) i ≥ σ i < ∞ (actually σ has a Gamma distribution) and the vector (cid:16) σ σ , σ σ , . . . (cid:17) follows the GEM distribution with parameter θ/b and is independent of σ .5.2. Proof for Model II. We will prove Theorem 2.5. We recall that in Model II, immigrantsare of type i with probability p i . Denote by N i ( t ) the number of immigrants of type i whicharrived before time t . Then, ( N i ( t ) , t ≥ 0) is a Poisson process with parameter θp i andthe processes ( N i , i ≥ 1) are independent. Hence, I ( t ) , I ( t ) , . . . are independent and theirasymptotic behaviors are the same as I ( t ) in Theorem 2.3, replacing θ with θp i . Then wehave e − ηt I i ( t ) −→ t →∞ I i a.s. i ≥ I i ’s are independent and I i has a Gamma distribution Γ( θ i , c ) (recall that θ i = θp i /b ).Moreover, e − ηt I ( t ) −→ t →∞ I a.s.where I ∼ Γ( θ/b, c ). Therefore, for r ≥ t →∞ I ( t ) − ( I ( t ) , . . . , I r ( t )) = (cid:18) I I , . . . , I r I (cid:19) a.s.In order to investigate the law of this r -tuple, we prove that(19) I = (cid:88) i ≥ I i a.s. IMIT THEOREMS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION 21 First, by Fatou’s lemma,lim inf t →∞ e − ηt (cid:88) i ≥ I i ( t ) ≥ (cid:88) i ≥ lim inf t →∞ e − ηt I i ( t ) a.s.and so I ≥ (cid:88) i ≥ I i a.s.Second, E (cid:88) i ≥ I i = (cid:88) i ≥ θ i c = θbc = E [ I ] . The last two equations yield (19). For 1 ≤ i ≤ r , we can write I i I = I i I + · · · + I r + I ∗ where I ∗ is independent of ( I i , ≤ i ≤ r ) and has a Gamma distribution Γ( θ/b − θ r , c ) with θ r := (cid:80) ri =1 θ i . Hence, one can compute the joint density of the r -tuple ( I /I, . . . , I r /I ) asfollows f ( x , . . . , x r ) = Γ( θ/b )Γ( θ/b − θ r ) (cid:81) ri =1 Γ( θ i ) x θ − · · · x θ r − r (1 − x − · · · − x r ) θ/b − θ r − for x , . . . , x r > x + · · · + x r < 1. This joint density is exactly that of ( P (cid:48) , . . . , P (cid:48) r )defined in the statement of the theorem.5.3. Proofs for Model III. We first prove the almost sure convergence in Proposition 2.8.In order to do that, we use the same arguments as in the proof of Theorem 2.3(ii): we willuse the dominated convergence theorem for the sum e − ηt I ( t ) = (cid:88) i ≥ e − ηt I i ∆ i ( t − T i ) { t ≥ T i } . As in a previous proof, this sum is bounded by (cid:88) i ≥ e − ηT i sup t ≥ (cid:0) e − ηt I i ∆ i ( t ) (cid:1) which, according toLemma 4.2, is a.s. finite if E (cid:20) log + sup t ≥ (cid:0) e − ηt I ( t ) (cid:1)(cid:21) < ∞ . However, I ( t ) = (cid:88) i ≥ X i ( t − ˜ T i ) { t ≥ ˜ T i } where conditionally on ∆, ( ˜ T i , i ≥ 1) is a Poissonprocess with parameter ∆. Hence,sup t ≥ (cid:0) e − ηt I ∆ ( t ) (cid:1) ≤ (cid:88) i ≥ e − η ˜ T i sup t ≥ (cid:0) e − ηt X i ( t ) (cid:1) = (cid:88) i ≥ e − η ˜ T i J i where J , J , . . . is an i.i.d. sequence of random variables independent from ˜ T , ˜ T , . . . dis-tributed as sup t ≥ ( e − ηt X ( t )) where ( X ( t ) , t ≥ 0) is a homogeneous CMJ-process. Accordingto Proposition 2.1(ii), we know that E [(log + J ) ] < ∞ .We define for i ≥ ς i := e − η ˜ T i J i and ς := (cid:80) i ≥ ς i and we have to prove that E [log + ς ]is finite. To do that, we first work conditionally on ∆. According to Lemma 5.1, we know that ( ς i , i ≥ 1) are the points of a non-homogeneous Poisson process on (0 , ∞ ) with intensitymeasure ∆ η L ( v ) v dv where L ( v ) := P ( J ≥ v ). Then, using the inequalitylog + ( x + y ) ≤ log + x + log + y + log 2 , x, y ≥ , we have(20) E [log + ς ] ≤ log 2 + E log + (cid:88) i ≥ ς i { ς i ≤ } + E log + (cid:88) i ≥ ς i { ς i > } We first consider the second term of the r.h.s. E log + (cid:88) i ≥ ς i { ς i ≤ } ≤ E (cid:88) i ≥ ς i { ς i ≤ } = (cid:90) v ∆ η L ( v ) v dv ≤ ∆ η . (21)Then, we compute the third term of the r.h.s. of (20): if A := sup i ς i , E log + (cid:88) i ≥ ς i { ς i > } ≤ E (cid:2) log + ( A · { i ≥ | ς i > } ) (cid:3) ≤ E (cid:2) log + A (cid:3) + E (cid:2) log + { i ≥ | ς i > } (cid:3) Furthermore, the number of ς i greater than 1 has a Poisson distribution with parameter (cid:82) ∞ η L ( v ) v dv . Since E [log + J ] < ∞ , (cid:90) ∞ P (log + J ≥ s ) ds = (cid:90) ∞ P ( J ≥ e s ) ds = (cid:90) ∞ L ( v ) v dv < ∞ . Then,(22) E (cid:2) log + { i ≥ | ς i > } (cid:3) ≤ E [ { i ≥ | ς i > } ] = ∆ η (cid:90) ∞ L ( v ) v dv ≤ C ∆where C is a finite constant which does not depend on ∆.We now want to study A : P ( A ≤ x ) = P ( { i ≥ | ς i > x } = 0) = exp (cid:18) − (cid:90) ∞ x ∆ η L ( v ) v dv (cid:19) , x > . So that P ( A ∈ dx ) = ∆ η L ( x ) x exp (cid:18) − (cid:90) ∞ x ∆ η L ( v ) v dv (cid:19) dx. Then, E [log + A ] = (cid:90) ∞ log x ∆ η L ( x ) x exp (cid:18) − (cid:90) ∞ x ∆ η L ( v ) v dv (cid:19) dx ≤ ∆ η (cid:90) ∞ log x L ( x ) x dx = ∆ η (cid:90) ∞ uL ( e u ) du ≤ ∆ η (cid:90) ∞ u P (log + J ≥ u ) du ≤ C (cid:48) ∆(23)where C (cid:48) is a finite constant since E (cid:104)(cid:0) log + J (cid:1) (cid:105) < ∞ according to Proposition 2.1(ii). Hence,with (21), (22) and (23) we have IMIT THEOREMS FOR SUPERCRITICAL BRANCHING PROCESSES WITH IMMIGRATION 23 E [log + ς | ∆] ≤ log 2 + C (cid:48)(cid:48) ∆ . Then, E [log + ς ] is finite because E [∆] = θ − (cid:82) ∞ x f ( x ) dx < ∞ . and, using the dominatedconvergence theorem, e − ηt I ( t ) a.s. converges toward σ = (cid:80) i ≥ e − ηT i G i as t → ∞ .We now compute the law of the limit σ . We define σ i := exp ( − ηT i ) G i for i ≥ 1. Then,using Lemma 5.1, ( σ i ) i ≥ are the points of a non-homogeneous Poisson point process on (0 , ∞ )with intensity measure θη F ( y ) y dy where F ( y ) = P ( G ≥ y ). To compute the Laplace transformof σ , we use the exponential formula for Poisson processes: for s > E (cid:2) e − sσ (cid:3) = exp (cid:18) − θη (cid:90) ∞ F ( v ) v (cid:0) − e − sv (cid:1) dv (cid:19) . and to get the expectation of σ , we differentiate the last displayed equation at 0: E [ σ ] = θη (cid:90) ∞ F ( v ) dv = θη E [ G ]and E [ G ] = θ − (cid:90) ∞ xf ( x ) xb c dx < ∞ Hence, E [ σ ] = 1 ηbc (cid:90) ∞ x f ( x ) dx < ∞ which ends the proof of Proposition 2.8.It remains to prove Theorem 2.9 that is to show that the vector ( Z ( t ) , Z ( t ) , . . . ) /I ( t ) a.s.converges to a Poisson point process with intensity measure θη F ( y ) y dy . It is straightforwardusing previous calculations, Propositions 2.7 and 2.8. Remark 5.2. Thanks to similar calculations as in Theorem 2.3(i), we can compute thegenerating function of I ( t ) E (cid:104) s I ( t ) (cid:105) = exp (cid:32) − (cid:90) t du (cid:32) θ − (cid:90) ∞ xf ( x )( W ( u )(1 − s ) + s ) x/b dx (cid:33)(cid:33) , s ∈ [0 , . and we can deduce the law of σ in another way. Acknowledgments I want to thank my supervisor, Amaury Lambert, for his very helpful remarks. My thanksalso to the referee for his careful check and advice. References [1] Athreya, K. B. and Ney, P. E. (1972). Branching processes . Springer-Verlag, New York. DieGrundlehren der mathematischen Wissenschaften, Band 196.[2] Bertoin, J. (1996). L´evy processes vol. 121 of Cambridge Tracts in Mathematics . Cambridge UniversityPress, Cambridge.[3] Bertoin, J. (2006). Random fragmentation and coagulation processes vol. 102 of Cambridge Studies inAdvanced Mathematics . Cambridge University Press, Cambridge.[4] Caswell, H. (1976). Community structure: A neutral model analysis. 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