Coalescents with Simultaneous Multiple Collisions

By Jason Schweinsberg. 2000

217 citations recorded by the source corpus.

We study a family of coalescent processes that undergo ``simultaneous multiple collisions,'' meaning that many clusters of particles can merge into a single cluster at one time, and many such mergers can occur simultaneously. This family of processes, which we obtain from simple assumptions about the rates of different types of mergers, essentially coincides with a family of processes that Mohle and Sagitov obtain as a limit of scaled ancestral processes in a population model with exchangeable family sizes. We characterize the possible merger rates in terms of a single measure, show how these coalescents can be constructed from a Poisson process, and discuss some basic properties of these processes. This work generalizes some work of Pitman, who provides similar analysis for a family of coalescent processes in which many clusters can coalesce into a single cluster, but almost surely no two such mergers occur simultaneously.

Subfields: Artificial Intelligence, Mathematical Physics, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/c533f380eed89ee13b5fa0b51bb53c1d7bad2df1

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

In Mathematical Physics

In Molecular Biology

Citations in the corpus

A Classification of Coalescent Processes for Haploid Exchangeable Population Models (2001, unknown): “Pitman (1999) and Schweinsberg (2000a) study directly the class of coalescent processes with generator (9) without having a limit from a discrete population model in the background.”
Homogeneous fragmentation processes (2001, background): “This can also be viewed as an analogue of Pitman's description [13] of coalescent processes with multiple collisions (see also Schweinsberg [15]).”
CONVERGENCE TO THE COALESCENT WITH SIMULTANEOUS MULTIPLE MERGERS (2003, unknown): “The coalescent with simultaneous multiple mergers was further investigated in [17] by extending the Poisson process construction (cid:1) (cid:2) = (x , x , . . . ) : x ≥ x ≥ · · · ≥ , i = 1 x ≤ .”
Genetic Draft, Selective Interference, and Population Genetics of Rapid Adaptation (2013, background): “This type of coalescent process has been described in Schweinsberg (2000).”
Exchangeable Fragmentation-Coalescence Processes and their Equilibrium Measures (2004, background): “The existence of such a construction is assured by the condition ck > 0.”
A coalescent model for the effect of advantageous mutations on the genealogy of a population (2004, background): “ar X iv :m at h/ 04 11 07 1v 2 [ m at h. PR ] 2”
Stochastic flows associated to coalescent processes (2003, background): “Necessary and sufficient conditions for the proper case to hold can be found in Schweinsberg [18]; see also Theorem 8 in Pitman [15] for the case of Λ-coalescents.”
Combinatorial Stochastic Processes (2006, methodology): “For instance there is a representation of both coagulation and fragmentation semigroups, analogous to a Lévy-Khintchine representation, as has been shown by Schweinsberg [396] and Bertoin [45].”
The Λ-coalescent speed of coming down from infinity (2008, background): “For a generalization of -coalescents where multiple mergers are possible, see Schweinsberg [29].”
A New Model for Evolution in a Spatial Continuum (2009, background): “Independently, Schweinsberg (2000) obtained the same class of coalescents and characterised the possible rates of mergers in terms of a single measure Ξ on an infinite simplex.”
How to use MIGRATE or why are Markov chain Monte Carlo programs difficult to use (2009, methodology): “Additions to the coalescence theory by Pitman (1999), Möhle (2000), Schweinsberg (2000), Möhle and Sagitov (2003) and Fu (2006) allow for situations in which more than two lineages merge in the same generation and therefore, for a less restrictive ratio of sample size and population size.”
Stochastic flows associated to coalescent processes II: Stochastic differential equations (2005, background): “Exchangeable coalescents, also called coalescents with simultaneous multiple collisions by Schweinsberg [14], are processes taking values in the set P of all partitions of N, which appear as asymptotic models for phenomena of coagulation that occur when studying the genealogy of large populations.”
Recent progress in coalescent theory (2009, unknown): “(Processes without this last restriction have also been studied, most notably in [143].”
Total variation distances and rates of convergence for ancestral coalescent processes in exchangeable population models (2000, background): “Properties of the corresponding processes with simultaneous and multiple collisions were studied very recently by Schweinsberg [13, 14].”
The asymptotic behavior of fragmentation processes (2003, methodology): “We first recall some features lifted from [4,6], and also refer to Pitman [25] and Schweinsberg [26] for related works.”