The minimal observable clade size of exchangeable coalescents

@article{Freund2019TheMO,
  title={The minimal observable clade size of exchangeable coalescents},
  author={Fabian Freund and Arno Siri-J'egousse},
  journal={arXiv: Probability},
  year={2019}
}
For $\Lambda$-$n$-coalescents with mutation, we analyse the size $O_n$ of the partition block of $i\in\{1,\ldots,n\}$ at the time where the first mutation appears on the tree that affects $i$ and is shared with any other $j\in\{1,\ldots,n\}$. We provide asymptotics of $O_n$ for $n\to\infty$ and a recursion for all moments of $O_n$ for finite $n$. This variable gives an upper bound for the minimal clade size [2], which is not observable in real data. In applications to genetics, it has been… 
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References

SHOWING 1-10 OF 47 REFERENCES
On the size of the block of 1 for $\varXi$-coalescents with dust
We study the frequency process $f_1$ of the block of 1 for a $\varXi$-coalescent $\varPi$ with dust. If $\varPi$ stays infinite, $f_1$ is a jump-hold process which can be expressed as a sum of broken
Cannings models, population size changes and multiple-merger coalescents
  • F. Freund
  • Mathematics
    Journal of mathematical biology
  • 2020
TLDR
This article gives a more general construction of time-changed Λ - n -coalescents as limits of specific Cannings models with rather arbitrary time changes.
Minimal Clade Size in the Bolthausen-Sznitman Coalescent
TLDR
The asymptotics of distribution and moments of the size $X_n$ of the minimal clade of a randomly chosen individual in a Bolthausen-Sznitman $n-coalescent for $n\to\infty$ are shown.
Random Discrete Distributions Derived from Self-Similar Random Sets
A model is proposed for a decreasing sequence of random variables $(V_1, V_2, \cdots)$ with $\sum_n V_n = 1$, which generalizes the Poisson-Dirichlet distribution and the distribution of ranked
Asymptotics of the Minimal Clade Size and Related Functionals of Certain Beta-Coalescents
The Beta(2−α,α) n-coalescent with 1<α<2 is a Markov process taking values in the set of partitions of {1,…,n}. It evolves from the initial value {1},…,{n} by merging (coalescing) blocks together into
Minimal clade size and external branch length under the neutral coalescent
Given a sample of genes taken from a large population, we consider the neutral coalescent genealogy and study the theoretical and empirical distributions of the size of the smallest clade containing
Random Recursive Trees and the Bolthausen-Sznitman Coalesent
We describe a representation of the Bolthausen-Sznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the
The Site Frequency Spectrum for General Coalescents
TLDR
This work derives a new formula for the expected SFS for general Λ- and Ξ-coalescents, which leads to an efficient algorithm and obtains general theoretical results for the identifiability of the Λ measure when ζ is a constant function, as well as for the identity of the function ζ under a fixed Ξ measure.
Random Discrete Distributions
...
...