Mutations on a random binary tree with measured boundary

Duchamps, Jean-Jil; Lambert, Amaury

doi:10.1214/17-aap1353

Cited by 5 publications

(9 citation statements)

References 25 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let (d ij ; i, j ≥ 1) and (Y i ; i ≥ 1) be an exchangeable marked ultrametric matrix. There exists a random marked probability UMS [U, d, U , µ] (that is, µ(U ) = 1 a.s.) such that the exchangeable marked ultrametric matrix obtained by sampling from it as in (11) is distributed as (d ij ; i, j ≥ 1) and (Y i ; i ≥ 1). Moreover this marked UMS is unique in distribution.…”

Section: An Ultrametric On N;mentioning

confidence: 99%

Convergence of genealogies through spinal decomposition with an application to population genetics

Foutel-Rodier¹,

Schertzer²

2022

Preprint

View full text Add to dashboard Cite

Consider a branching Markov process with values in some general type space. Conditional on survival up to generation N , the genealogy of the extant population defines a random marked metric measure space, where individuals are marked by their type and pairwise distances are measured by the time to the most recent common ancestor. In the present manuscript, we devise a general method of moments to prove convergence of such genealogies in the Gromov-weak topology when N → ∞.Informally, the moment of order k of the population is obtained by observing the genealogy of k individuals chosen uniformly at random after size-biasing the population at time N by its k-th factorial moment. We show that the sampled genealogy can be expressed in terms of a k-spine decomposition of the original branching process, and that convergence reduces to the convergence of the underlying k-spines.As an illustration of our framework, we analyse the large-time behavior of a branching approximation of the biparental Wright-Fisher model with recombination. The model exhibits some interesting mathematical features. It starts in a supercritical state but is naturally driven to criticality. We show that the limiting behavior exhibits both critical and supercritical characteristics.

show abstract

Section: An Ultrametric On N;mentioning

confidence: 99%

Convergence of genealogies through spinal decomposition with an application to population genetics

Foutel-Rodier¹,

Schertzer²

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…It can be shown [8] that the boundary of (τ f ,d f ) is indeed (Ī,d f ), which explains why we keep the same notation for the two distances. Also that for each t ∈ I, L t = ρ, α t , where α t ∈Ī is equal to (t, r).…”

Section: The Comb Metricmentioning

confidence: 99%

“…Proof. We know from [8] that ϕ(t) is the genealogy of a reversed (i.e. time flows from T to 0) pure-birth tree with birth rate β = β • ϕ −1 .…”

Section: The Boundary Of a Pure-birth Processmentioning

confidence: 99%

“…In particular, we can define µ(t) := µ([0, t]). Then it can be seen [8] that if 0 µ(x)ν(dx) < ∞ the total number of mutations is finite a.s. whereas if 0 µ(x)ν(dx) = ∞ then the number of mutations in any clade (set of all descendants of a point) is infinite a.s.…”

Section: Cpp With Mutationsmentioning

confidence: 99%

“…where T is here to recall the dependence on the height of the CPP, and the measure Λ T is defined on N when ν is finite [20] and on (0, ∞) when ν is infinite [8]. In addition, the following convergence holds weakly on [0, ∞)…”

Section: Cpp With Mutationsmentioning

confidence: 99%

See 2 more Smart Citations

Random ultrametric trees and applications

Lambert¹

2017

ESAIM: Procs

Self Cite

View full text Add to dashboard Cite

Abstract. Ultrametric trees are trees whose leaves lie at the same distance from the root. They are used to model the genealogy of a population of particles co-existing at the same point in time. We show how the boundary of an ultrametric tree, like any compact ultrametric space, can be represented in a simple way via the so-called comb metric. We display a variety of examples of random combs and explain how they can be used in applications. In particular, we review some old and recent results regarding the genetic structure of the population when throwing neutral mutations on the skeleton of the tree.Résumé. Les arbres ultramétriques sont les arbres dont les feuilles se trouvent toutes à la même distance de la racine. Ces arbres sont utilisés pour modéliser la généalogie d'une population de particules qui co-existent à un temps donné. Nous montrons que la frontière d'un arbre ultramétrique, comme tout espace ultramétrique compact, peut être représentée simplement via ce que nous appelons la distance de peigne. Nous examinons plusieurs exemples de peignes aléatoires et nous expliquons comme ils peuvent être utilisés dans certaines applications. En particulier, nous évoquons quelques résultats anciens ou plus récents concernant la structure génétique de la population lorsque l'on jette des mutations neutres sur le squelette de l'arbre.

show abstract

Convergence of genealogies through spinal decomposition with an application to population genetics

Foutel-Rodier,

Schertzer

2023

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

Consider a branching Markov process with values in some general type space. Conditional on survival up to generation N, the genealogy of the extant population defines a random marked metric measure space, where individuals are marked by their type and pairwise distances are measured by the time to the most recent common ancestor. In the present manuscript, we devise a general method of moments to prove convergence of such genealogies in the Gromov-weak topology when $$N \rightarrow \infty $$ N → ∞ . Informally, the moment of order k of the population is obtained by observing the genealogy of k individuals chosen uniformly at random after size-biasing the population at time N by its kth factorial moment. We show that the sampled genealogy can be expressed in terms of a k-spine decomposition of the original branching process, and that convergence reduces to the convergence of the underlying k-spines. As an illustration of our framework, we analyse the large-time behavior of a branching approximation of the biparental Wright–Fisher model with recombination. The model exhibits some interesting mathematical features. It starts in a supercritical state but is naturally driven to criticality. We show that the limiting behavior exhibits both critical and supercritical characteristics.

show abstract

Mutations on a random binary tree with measured boundary

Cited by 5 publications

References 25 publications

Convergence of genealogies through spinal decomposition with an application to population genetics

Convergence of genealogies through spinal decomposition with an application to population genetics

Random ultrametric trees and applications

Convergence of genealogies through spinal decomposition with an application to population genetics

Contact Info

Product

Resources

About