Coalescences in continuous-state branching processes

Foucart, Clément; Ma, Chunhua; Mallein, Bastien

doi:10.1214/19-ejp358

Cited by 5 publications

(6 citation statements)

References 80 publications

(70 reference statements)

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“…Relation ( 13) also provides further insight into the function γ. For example, by (13), for all k ∈ N,…”

Section: Results Concerning the Block Counting Processmentioning

confidence: 99%

Scaling limits for a class of regular $Ξ$-coalescents

Möhle¹,

Vetter²

2022

Preprint

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The block counting process with initial state n counts the number of blocks of an exchangeable coalescent (Ξ-coalescent) restricted to a sample of size n. This work provides scaling limits for the block counting process of regular Ξ-coalescents that stay infinite, including Ξ-coalescents with dust and a large class of dust-free Ξ-coalescents. The main convergence result states that the block counting process, properly logarithmically scaled, converges in the Skorohod space to an Ornstein-Uhlenbeck type process as n tends to infinity. The existence of such a scaling depends on a sort of curvature condition of a particular function well-known from the literature. This curvature condition is intrinsically related to the behavior of the measure Ξ near the origin. The method of proof is to show the uniform convergence of the associated generators. Via Siegmund duality an analogous result for the fixation line is proven. Several examples are studied.

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“…Relation ( 13) also provides further insight into the function γ. For example, by (13), for all k ∈ N,…”

Section: Results Concerning the Block Counting Processmentioning

confidence: 99%

Scaling limits for a class of regular $Ξ$-coalescents

Möhle¹,

Vetter²

2022

Preprint

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“…The next theorem characterizes the semigroup of ( Xt , t ≥ 0). Theorem 2.2 (Theorem 3.5, Proposition 3.6 in [FMM19]). For any continuous function f defined on (0, ∞) and any q > 0,…”

Section: Andmentioning

confidence: 99%

“…To the best of our knowledge, fewer works on CSBPs have been done in this direction. We refer however to Labbé [Lab14], Lambert [Lam03], Lambert and Popovic [LP13] and Foucart et al [FMM19]. The latter work initiates the study of the inverse flow ( Xs,t (x), s ≤ t, x ≥ 0) defined for s ≤ t and x ∈ [0, ∞], as (1.1) Xs,t (x) := inf{y ≥ 0 : X −t,−s (y) > x}.…”

Section: Introductionmentioning

confidence: 99%

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Asymptotic behaviour of ancestral lineages in subcritical continuous-state branching populations

Foucart¹,

Möhle²

2020

Preprint

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Consider the population model with infinite size associated to subcritical continuous-state branching processes (CSBP). Individuals reproduce independently according to the same subcritical offspring distribution. We study the long-term behaviour of the ancestral lineages as time goes to the past and show that the flow of ancestral lineages, properly renormalized, converges almost surely to the inverse of a drift-free subordinator whose Laplace exponent is explicit in terms of the branching mechanism. We provide an interpretation in terms of the genealogy of the population. In particular, we show that the inverse subordinator is partitioning the current population into ancestral families with distinct common ancestors. When Grey's condition is satisfied, the population comes from a discrete set of ancestors and the ancestral families are i.i.d and distributed according to the quasi-stationary distribution of the CSBP conditioned on non-extinction. When Grey's condition is not satisfied, the population comes from a continuum of ancestors which is described as the set of increase points S of the limiting inverse subordinator. The Hausdorff dimension of S is given. The proof is based on a general result for stochastically monotone processes of independent interest, which relates θ-invariant measures and θ-invariant functions for a process and its Siegmund dual.

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“…Other models have also been considered where the distribution of the genealogical tree or a sample of the current population can be explicitely described: linear birth-death process [34], continuous time Galton-Watson trees [22,25], Brownian tree [2] see also [1], splitting trees [31]. Some recent results on the coalescent process associated with some branching process by time-reversal can be found in [41,26,18].…”

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confidence: 99%

Some properties of stationary continuous state branching processes

Abraham,

Delmas,

2020

Preprint

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We consider the genealogical tree of a stationary continuous state branching process with immigration. For a sub-critical stable branching mechanism, we consider the genealogical tree of the extant population at some fixed time and prove that, up to a deterministic timechange, it is distributed as a continuous-time Galton-Watson process with immigration. We obtain similar results for a critical stable branching mechanism when only looking at immigrants arriving in some fixed time-interval. For a general sub-critical branching mechanism, we consider the number of individuals that give descendants in the extant population. The associated processes (forward or backward in time) are pure-death or pure-birth Markov processes, for which we compute the transition rates.

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Coalescences in continuous-state branching processes

Cited by 5 publications

References 80 publications

Scaling limits for a class of regular $Ξ$-coalescents

Scaling limits for a class of regular $Ξ$-coalescents

Asymptotic behaviour of ancestral lineages in subcritical continuous-state branching populations

Some properties of stationary continuous state branching processes

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