Λ-coalescents: a survey

Gnedin, Alexander; Iksanov, Alexander; Marynych, Alexander

doi:10.1017/s0021900200021161

Cited by 6 publications

(6 citation statements)

References 44 publications

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“…From Eq. (15) we see that f 1 only takes values in M p , since P k = p for all k ∈ N and C < ∞ almost surely. Recall the definition of the singleton sets S i and their properties from the proof of Proposition 3.…”

Section: The Block Of 1 In Simple λ-Coalescents -Proofs and Remarksmentioning

confidence: 76%

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On the size of the block of 1 for Ξ-coalescents with dust

Freund¹,

Möhle²

2017

Modern Stoch. Theory Appl.

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We study the frequency process f1 of the block of 1 for a Ξ-coalescent Π with dust. If Π stays infinite, f1 is a jump-hold process which can be expressed as a sum of broken parts from a stick-breaking procedure with uncorrelated, but in general non-independent, stick lengths with common mean. For Dirac-Λ-coalescents with Λ = δp, p ∈ [ 1 2 , 1), f1 is not Markovian, whereas its jump chain is Markovian. For simple Λ-coalescents the distribution of f1 at its first jump, the asymptotic frequency of the minimal clade of 1, is expressed via conditionally independent shifted geometric distributions.

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Section: The Block Of 1 In Simple λ-Coalescents -Proofs and Remarksmentioning

confidence: 76%

“…The family of Ξ-coalescents is a diverse class of processes with very different properties, see e.g. the review [15] for Λ-coalescents. We will focus on Ξ-coalescents with dust, i.e.…”

Section: Introduction and Resultsmentioning

confidence: 99%

On the size of the block of 1 for Ξ-coalescents with dust

Freund¹,

Möhle²

2017

Modern Stoch. Theory Appl.

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“…The random variable T arises in different areas of applied probability and also in varying disguises, for example, the number of blocks in regenerative compositions [7], [8], [10], the number of collisions in exchangeable coalescents [11], [15], [23], [26], the number of positive jumps in a random walk with a barrier [17], [22], or the number of cuts needed to isolate the root of a random recursive tree [16].…”

Section: Bibliographic Notesmentioning

confidence: 99%

“…We refer the reader to the seminal papers [23] and [26] for the construction and basic properties of -coalescents, and to [2] for a survey and further references. More information on the limiting behavior of C n as well as other functionals of the -coalescent can be found in [11].…”

Section: Motivating Examplesmentioning

confidence: 99%

Renewal approximation for the absorption time of a decreasing Markov chain

Alsmeyer

Marynych

2016

J. Appl. Probab.

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We consider a Markov chain (M n ) n≥0 on the set N 0 of nonnegative integers which is eventually decreasing, i.e. P{M n+1 < M n |M n ≥ a} = 1 for some a ∈ N and all n ≥ 0. We are interested in the asymptotic behaviour of the law of the stopping time T = T (a) := inf{k ∈ N 0 : M k < a} under P n := P(·|M 0 = n) as n → ∞. Assuming that the decrements of (M n ) n≥0 given M 0 = n possess a kind of stationarity for large n, we derive sufficient conditions for the convergence in minimal L p -distance of P n ((T − a n )/b n ∈ ·) to some non-degenerate, proper law and give an explicit form of the constants a n and b n .

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“…Among the functionals characterising the speed of coalescence, the total number of collisions X n (transitions of Π n until absorption) has attracted the most attention, see [16,19,23,24,26] and a survey paper [17]. Here, we are interested in more delicate properties of the coalescent process by distinguishing mergers of various sizes, that is decomposing the total number of collisions as X n = n k=2 X n,k , 2 ≤ k ≤ n, where X n,k is the number of collisions resulting in a merger of k blocks.…”

Section: Introductionmentioning

confidence: 99%

The collision spectrum of $\Lambda$-coalescents

Gnedin¹,

Iksanov²,

Marynych³

et al. 2018

Ann. Appl. Probab.

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Λ-coalescents model the evolution of a coalescing system in which any number of blocks randomly sampled from the whole may merge into a larger block. For the coalescent restricted to initially n singletons we study the collision spectrum (X n,k : 2 ≤ k ≤ n), where X n,k counts, throughout the history of the process, the number of collisions involving exactly k blocks. Our focus is on the large n asymptotics of the joint distribution of the X n,k 's, as well as on functional limits for the bulk of the spectrum for simple coalescents. Similarly to the previous studies of the total number of collisions, the asymptotics of the collision spectrum largely depends on the behaviour of the measure Λ in the vicinity of 0. In particular, for beta(a, b)-coalescents different types of limit distributions occur depending on whether 0 < a ≤ 1, 1 < a < 2, a = 2 or a > 2.

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Λ-coalescents: a survey

Cited by 6 publications

References 44 publications

On the size of the block of 1 for Ξ-coalescents with dust

On the size of the block of 1 for Ξ-coalescents with dust

Renewal approximation for the absorption time of a decreasing Markov chain

The collision spectrum of $\Lambda$-coalescents

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