On the length of an external branch in the Beta-coalescent
Jean-Stephane Dhersin, Fabian Freund, Arno Siri-Jegousse, Linglong Yuan
Abstract
In this paper, we consider Beta
(2−α,α)
(with
1<α<2
) and related
Λ
-coalescents. If
T
(n)
denotes the length of an external branch of the
n
-coalescent, we prove the convergence of
n
α−1
T
(n)
when
n
tends to
∞
, and give the limit. To this aim, we give asymptotics for the number
σ
(n)
of collisions which occur in the
n
-coalescent until the end of the chosen external branch, and for the block counting process associated with the
n
-coalescent.