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December 2018 The collision spectrum of $\Lambda$-coalescents
Alexander Gnedin, Alexander Iksanov, Alexander Marynych, Martin Möhle
Ann. Appl. Probab. 28(6): 3857-3883 (December 2018). DOI: 10.1214/18-AAP1409

Abstract

$\Lambda$-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 initially $n$ singletons, we study the collision spectrum $(X_{n,k}:2\le k\le n)$, where $X_{n,k}$ counts, throughout the history of the process, the number of collisions involving exactly $k$ blocks. Our focus is on the large $n$ asymptotics of the joint distribution of the $X_{n,k}$’s, as well as on functional limits for the bulk of the spectrum for simple coalescents. Similar to the previous studies of the total number of collisions, the asymptotics of the collision spectrum largely depends on the behaviour of the measure $\Lambda$ in the vicinity of $0$. In particular, for beta$(a,b)$-coalescents different types of limit distributions occur depending on whether $0<a\leq1$, $1<a<2$, $a=2$ or $a>2$.

Citation

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Alexander Gnedin. Alexander Iksanov. Alexander Marynych. Martin Möhle. "The collision spectrum of $\Lambda$-coalescents." Ann. Appl. Probab. 28 (6) 3857 - 3883, December 2018. https://doi.org/10.1214/18-AAP1409

Information

Received: 1 August 2017; Revised: 1 April 2018; Published: December 2018
First available in Project Euclid: 8 October 2018

zbMATH: 06994408
MathSciNet: MR3861828
Digital Object Identifier: 10.1214/18-AAP1409

Subjects:
Primary: 60F17, 60J25
Secondary: 60C05, 60G09

Rights: Copyright © 2018 Institute of Mathematical Statistics

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Vol.28 • No. 6 • December 2018
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