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

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

2009-09-04

arxiv.org/abs/0909.0870v1

Bernoulli 2009, Vol. 15, No. 3, 829-845

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.3150/09-BEJ192 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statisti

Scientific paper

10.3150/09-BEJ192

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 $a\neq2$.

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