## The Annals of Applied Probability

### The collision spectrum of $\Lambda$-coalescents

#### 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
Revised: April 2018
First available in Project Euclid: 8 October 2018

https://projecteuclid.org/euclid.aoap/1538985637

Digital Object Identifier
doi:10.1214/18-AAP1409

Mathematical Reviews number (MathSciNet)
MR3861828

Zentralblatt MATH identifier
06994408

#### 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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