A Multi-Type Λ-Coalescent

Griffiths, R. C.

doi:10.1007/978-3-319-31641-3_2

Cited by 3 publications

(3 citation statements)

References 8 publications

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“…Theorem 1.3 follows quickly from Theorem 1.4. Sketching the key steps here, for each i the array (λ b,k→i : k ≤ b ∈ Z d ≥0 ) of type-i-terminal merger rates satisfies the recursion (8) and is defined for all k ≤ b outside of the box B e i = {0, e i }. In particular, we are in the setting of Theorem 1.4 so that for each i, there exists a function ρ i and a measure J i such that…”

Section: Introduction and Statements Of Main Resultsmentioning

confidence: 99%

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Multitype $Λ$-coalescents

Johnston¹,

Kyprianou²,

Rogers³

2021

Preprint

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Consider a multitype coalescent process in which each block has a colour in {1, . . . , d}. Individual blocks may change colour, and some number of blocks of various colours may merge to form a new block of some colour. We show that if the law of a multitype coalescent process is invariant under permutations of blocks of the same colour, has consistent Markovian projections, and has asychronous mergers, then it is a multitype Λ-coalescent: a process in which single blocks may change colour, two blocks of like colour may merge to form a single block of that colour, or large mergers across various colours happen at rates governed by a d-tuple of measures on the unit cube [0, 1] d . We go on to identify when such processes come down from infinity. Our framework generalises Pitman's celebrated classification theorem for singletype coalescent processes, and provides a unifying setting for numerous examples that have appeared in the literature including the seed-bank model, the island model and the coalescent structure of continuous-state branching processes.

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Section: Introduction and Statements Of Main Resultsmentioning

confidence: 99%

“…Griffiths [8] shows how general forms of mutlitype coalescent arise in the context of coalescent duals to multitype Λ-fleming viot processes, showing that processes with arbitrary Q →i measures (but with ρ ii→i and ρ j→i zero) arise naturally in these settings.…”

Section: 4mentioning

confidence: 99%

Multitype $Λ$-coalescents

Johnston¹,

Kyprianou²,

Rogers³

2021

Preprint

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“…The ancestral structure of seed bank models has gained some interest in the literature (see, for example, Blath et al [5,6] or González Casanova et al [17]). At this point we also would like to refer the reader to the work of Griffiths [22], where the notion of a 'multi-type Λ-coalescent' seems to appear for the first time. In different mathematical context, the notion of a 'multi-type coalescent point process' also appears in the work of Popovic and Rivas [37], which is mentioned for completeness here.…”

Section: Multi-type Coalescent With Mutationmentioning

confidence: 99%

The rate of convergence of the block counting process of exchangeable coalescents with dust

Möhle¹

2021

ALEA

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Exchangeable coalescents with dust are studied. The rate of convergence as the sample size tends to infinity of the scaled block counting process to the frequency of singleton process is determined. This rate is expressed in terms of a certain Bernstein function. The proofs are based on Taylor expansions of the infinitesimal generators and semigroups and involve a particular concentration inequality arising in the context of Karlin's infinite urn model. The rate of convergence is calculated for several examples of coalescents.

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