The Annals of Applied Probability

The collision spectrum of $\Lambda$-coalescents

Alexander Gnedin, Alexander Iksanov, Alexander Marynych, and Martin Möhle

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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 $a>2$.

Article information

Source
Ann. Appl. Probab., Volume 28, Number 6 (2018), 3857-3883.

Dates
Received: August 2017
Revised: April 2018
First available in Project Euclid: 8 October 2018

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1538985637

Digital Object Identifier
doi:10.1214/18-AAP1409

Mathematical Reviews number (MathSciNet)
MR3861828

Subjects
Primary: 60J25: Continuous-time Markov processes on general state spaces 60F17: Functional limit theorems; invariance principles
Secondary: 60C05: Combinatorial probability 60G09: Exchangeability

Keywords
Collision spectrum coupling exchangeable coalescent functional approximation

Citation

Gnedin, Alexander; Iksanov, Alexander; Marynych, Alexander; Möhle, Martin. The collision spectrum of $\Lambda$-coalescents. Ann. Appl. Probab. 28 (2018), no. 6, 3857--3883. doi:10.1214/18-AAP1409. https://projecteuclid.org/euclid.aoap/1538985637


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