Electronic Journal of Probability

Trees within trees: simple nested coalescents

Airam Blancas, Jean-Jil Duchamps, Amaury Lambert, and Arno Siri-Jégousse

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Abstract

We consider the compact space of pairs of nested partitions of $\mathbb{N} $, where by analogy with models used in molecular evolution, we call “gene partition” the finer partition and “species partition” the coarser one. We introduce the class of nondecreasing processes valued in nested partitions, assumed Markovian and with exchangeable semigroup. These processes are said simple when each partition only undergoes one coalescence event at a time (but possibly the same time). Simple nested exchangeable coalescent (SNEC) processes can be seen as the extension of $\Lambda $-coalescents to nested partitions. We characterize the law of SNEC processes as follows. In the absence of gene coalescences, species blocks undergo $\Lambda $-coalescent type events and in the absence of species coalescences, gene blocks lying in the same species block undergo i.i.d. $\Lambda $-coalescents. Simultaneous coalescence of the gene and species partitions are governed by an intensity measure $\nu _s$ on $(0,1]\times{\mathcal M} _1 ([0,1])$ providing the frequency of species merging and the law in which are drawn (independently) the frequencies of genes merging in each coalescing species block. As an application, we also study the conditions under which a SNEC process comes down from infinity.

Article information

Source
Electron. J. Probab., Volume 23 (2018), paper no. 94, 27 pp.

Dates
Received: 6 March 2018
Accepted: 3 September 2018
First available in Project Euclid: 18 September 2018

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1537257886

Digital Object Identifier
doi:10.1214/18-EJP219

Mathematical Reviews number (MathSciNet)
MR3858922

Zentralblatt MATH identifier
06964788

Subjects
Primary: 60G09: Exchangeability 60G57: Random measures 60J35: Transition functions, generators and resolvents [See also 47D03, 47D07] 60J75: Jump processes 92D10: Genetics {For genetic algebras, see 17D92} 92D15: Problems related to evolution

Keywords
lambda-coalescent exchangeable partition coming down from infinity random tree gene tree population genetics species tree phylogenetics evolution

Rights
Creative Commons Attribution 4.0 International License.

Citation

Blancas, Airam; Duchamps, Jean-Jil; Lambert, Amaury; Siri-Jégousse, Arno. Trees within trees: simple nested coalescents. Electron. J. Probab. 23 (2018), paper no. 94, 27 pp. doi:10.1214/18-EJP219. https://projecteuclid.org/euclid.ejp/1537257886


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