On the number of collisions in beta(2, $b$)-coalescents

Abstract

Expansions are provided for the moments of the number of collisions $X_n$ in the $\beta (2,b)$-coalescent restricted to the set $\{1,...,n\}$. We verify that $X_n/\mathbb{E}{X}_n$ converges almost surely to one and that $X_n$, properly normalized, weakly converges to the standard normal law. These results complement previously known facts concerning the number of collisions in $\beta (a,b)$-coalescents with $a\in (0,2)$ and $b=1$, and $a>2$ and $b>0$. The case $a=2$ is a kind of `border situation' which seems not to be amenable to approaches used for $a\ne 2$.

Published in at http://dx.doi.org/10.3150/09-BEJ192 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

References in corpus (1)

• Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws

Cited by in corpus (1)

• Lambda-coalescents with dust component