Coalescents with multiple collisions

@article{Pitman1999CoalescentsWM,
  title={Coalescents with multiple collisions},
  author={Jim Pitman},
  journal={Annals of Probability},
  year={1999},
  volume={27},
  pages={1870-1902}
}
  • J. Pitman
  • Published 1 October 1999
  • Mathematics
  • Annals of Probability
k−2 � 1 − xb−k � � dx� . Call this process a � -coalescent. Discrete measure-valued processes derived from the � -coalescent model a system of masses undergoing coalescent collisions. Kingman's coalescent, which has numerous applications in population genetics, is the δ0-coalescent for δ0 a unit mass at 0. The coalescent recently derived by Bolthausen and Sznit- man from Ruelle's probability cascades, in the context of the Sherrington- Kirkpatrick spin glass model in mathematical physics, is… 
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References

SHOWING 1-10 OF 48 REFERENCES
Construction of markovian coalescents
Stochastic Coalescence
TLDR
The general stochastic coalescent, whose state space is the innnite-dimensional simplex, is a model of coalescence that evolves by clusters of masses x i and x j coalescing at rate K(x i ; x j).
The standard additive coalescent
Regard an element of the set Δ := {(x 1 , x 2 , . . .): x 1 ≥ x 2 ≥ ⋯ ≥ 0, ∑ i x i = 1} as a fragmentation of unit mass into clusters of masses x i . The additive coalescent of Evans and Pitman is
The Entrance Boundary of the Multiplicative Coalescent
The multiplicative coalescent $X(t)$ is a $l^2$-valued Markov process representing coalescence of clusters of mass, where each pair of clusters merges at rate proportional to product of masses. From
Brownian excursions, critical random graphs and the multiplicative coalescent
Let (B t (s), 0 ≤ s < ∞) be reflecting inhomogeneous Brownian motion with drift t - s at time s, started with B t (0) = 0. Consider the random graph script G sign(n, n -1 + tn -4/3 ), whose largest
The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator
The two-parameter Poisson-Dirichlet distribution, denoted PD(α,θ), is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with
The general coalescent with asynchronous mergers of ancestral lines
  • S. Sagitov
  • Mathematics
    Journal of Applied Probability
  • 1999
Take a sample of individuals in the fixed-size population model with exchangeable family sizes. Follow the ancestral lines for the sampled individuals backwards in time to observe the ancestral
THE COALESCENT
Coagulation in finite systems
Partition structures derived from Brownian motion and stable subordinators
Explicit formulae are obtained for the distribution of various random partitions of a positive integer n, both ordered and unordered, derived from the zero set M of a Brownian motion by the following
...
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