# On the size of the block of 1 for $\varXi$-coalescents with dust

@inproceedings{Freund2017OnTS,
title={On the size of the block of 1 for \$\varXi\$-coalescents with dust},
author={Fabian Freund and Martin Mohle},
year={2017}
}
• Published 17 March 2017
• Mathematics
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 parts from a stick-breaking procedure with uncorrelated, but in general non-independent, stick lengths with common mean. For Dirac-$\varLambda$-coalescents with $\varLambda=\delta_p$, $p\in[\frac{1}{2},1)$, $f_1$ is not Markovian, whereas its jump chain is Markovian. For simple $\varLambda… 2 Citations The minimal observable clade size of exchangeable coalescents • Mathematics • 2019 This variable gives an upper bound for the minimal clade size [2], which is not observable in real data but has been shown to be useful to lower classification errors in genealogical model selection. Genealogical Properties of Subsamples in Highly Fecund Populations • Mathematics bioRxiv • 2017 The results indicate how ’informative’ a subsample is about the properties of the larger complete sample, how much information is gained by increasing the sample size, and how the ‘informativeness’ of the subsample varies between different coalescent processes. ## References SHOWING 1-10 OF 38 REFERENCES Almost sure asymptotics for the number of types for simple$\Xi$-coalescents Let$K_n$be the number of types in the sample$\left\{1,\ldots, n\right\}$of a$\Xi$-coalescent$\Pi=(\Pi_t)_{t\geq0}$with mutation and mutation rate$r>0$. Let$\Pi^{(n)}$be the restriction of$\beta$-coalescents and stable Galton-Watson trees • Mathematics • 2013 Representation of coalescent process using pruning of trees has been used by Goldschmidt and Martin for the Bolthausen-Sznitman coalescent and by Abraham and Delmas for the The size of the last merger and time reversal in$\Lambda$-coalescents • Mathematics Annales de l'Institut Henri Poincaré, Probabilités et Statistiques • 2018 Author(s): Kersting, Goetz; Schweinsberg, Jason; Wakolbinger, Anton | Abstract: We consider the number of blocks involved in the last merger of a$\Lambda$-coalescent started with$n$blocks. We give A Necessary and Sufficient Condition for the$\Lambda$-Coalescent to Come Down from Infinity. Let$\Pi_{\infty}$be the standard$\Lambda$-coalescent of Pitman, which is defined so that$\Pi_{\infty}(0)$is the partition of the positive integers into singletons, and, if$\Pi_n$denotes the On the Speed of Coming Down from Infinity for$\Xi$-Coalescent Processes The$\Xi$-coalescent processes were initially studied by Mohle and Sagitov (2001), and introduced by Schweinsberg (2000) in their full generality. They arise in the mathematical population genetics The fixation line in the${\Lambda}$-coalescent We define a Markov process in a forward population model with backward genealogy given by the$\Lambda$-coalescent. This Markov process, called the fixation line, is related to the block counting Random Recursive Trees and the Bolthausen-Sznitman Coalesent • Mathematics • 2005 We describe a representation of the Bolthausen-Sznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the Minimal Clade Size in the Bolthausen-Sznitman Coalescent • Mathematics Journal of Applied Probability • 2014 The asymptotics of distribution and moments of the size$X_n$of the minimal clade of a randomly chosen individual in a Bolthausen-Sznitman$n-coalescent for $n\to\infty$ are shown.
Small-time behavior of beta coalescents
• Mathematics
• 2008
For a finite measureon (0,1), the �-coalescent is a coalescent process such that, whenever there are b clusters, each k-tuple of clusters merges into one at rate R 1 0 x k 2 (1 x) b k �(dx). It has
-coalescents and stable Galton-Watson trees
• Mathematics
• 2013
. Representation of coalescent process using pruning of trees has been used by Goldschmidt and Martin for the Bolthausen-Sznitman coalescent and by Abraham and Delmas for the β (3 / 2 , 1 /