# The size of the last merger and time reversal in $\Lambda$-coalescents

@article{Kersting2018TheSO,
title={The size of the last merger and time reversal in \$\Lambda\$-coalescents},
author={G{\"o}tz Kersting and Jason Schweinsberg and A. Wakolbinger},
journal={Annales de l'Institut Henri Poincar{\'e}, Probabilit{\'e}s et Statistiques},
year={2018}
}
• Published 2 January 2017
• Mathematics
• Annales de l'Institut Henri Poincaré, Probabilités et Statistiques
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 conditions under which, as $n \to \infty$, the sequence of these random variables a) is tight, b) converges in distribution to a finite random variable or c) converges to infinity in probability. Our conditions are optimal for $\Lambda$-coalescents that have a dust component. For general $\Lambda… 8 Citations The collision spectrum of$\Lambda$-coalescents • Physics, Mathematics The Annals of Applied Probability • 2018$\Lambda$-coalescents model the evolution of a coalescing system in which any number of blocks randomly sampled from the whole may merge into a larger block. For the coalescent restricted to Probabilistic aspects of$\Lambda$-coalescents in equilibrium and in evolution • Physics • 2020 We present approximation methods which lead to law of large numbers and fluctuation results for functionals of$\Lambda$-coalescents, both in the dust-free case and in the case with a dust component. On the size of the block of 1 for$\varXi$-coalescents with dust • Mathematics • 2017 We study the frequency process$f_1$of the block of 1 for a$\varXi$-coalescent$\varPi$with dust. If$\varPi$stays infinite,$f_1$is a jump-hold process which can be expressed as a sum of broken External branch lengths of$\Lambda $-coalescents without a dust component • Mathematics Electronic Journal of Probability • 2019$\Lambda$-coalescents model genealogies of samples of individuals from a large population by means of a family tree whose branches have lengths. The tree's leaves represent the individuals, and the THE COLLISION SPECTRUM OF -COALESCENTS1 • Physics, Mathematics • 2018 -coalescents model the evolution of a coalescing system in which any number of blocks randomly sampled from the whole may merge into a larger block. For the coalescent restricted to initially n On the time to absorption in$\Lambda$-coalescents • Mathematics • 2017 We present a law of large numbers and a central limit theorem for the time to absorption of$\Lambda$-coalescents, started from$n$blocks, as$n \to \infty$. The proofs rely on an approximation of External branch lengths of Λ-coalescents without a dust component • Mathematics • 2019 Λ-coalescents model genealogies of samples of individuals from a large population by means of a family tree whose branches have lengths. The tree’s leaves represent the individuals, and the lengths The time to absorption in Λ-coalescents • Mathematics Advances in Applied Probability • 2018 Abstract We present a law of large numbers and a central limit theorem for the time to absorption of Λ-coalescents with dust started from n blocks, as n→∞. The proofs rely on an approximation of the ## References SHOWING 1-10 OF 17 REFERENCES Random Recursive Trees and the Bolthausen-Sznitman Coalesent • Mathematics • 2005 We describe a representation of the Bolthausen-Sznitman coalescent in terms of the cutting of random recursive trees. Using this representation, we prove results concerning the final collision of the$\beta\$-coalescents and stable Galton-Watson trees
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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
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