2008/07/17 by Alexander Gnedin, Alex Iksanov, Gnedin, Alexander +4
Biochemistry, Genetics and Molecular Biology · Mathematics · #60J10 #60K15 #Diffusion and Search Dynamics #FOS: Mathematics #Probability (math.PR) #Stochastic processes and statistical mechanics #math.PR #msc:60J10 #msc:60K15
paper · pdf · doi:10.48550/arxiv.0807.2742
arxiv created 2008/07/17 · openalex publication_date 2008/07/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study the number of collisions Xn of an exchangeable coalescent with multiple collisions (Λ-coalescent) which starts with n particles and is driven by rates determined by a finite characteristic measure ν(\rm dx)=x-2Λ(\rm dx). Via a coupling technique we derive limiting laws of Xn, using previous results on regenerative compositions derived from stick-breaking partitions of the unit interval. The possible limiting laws of Xn include normal, stable with index 1≤α<2 and Mittag-Leffler distributions. The results apply, in particular, to the case when ν is a beta(a-2,b) distribution with parameters a>2 and b>0. The approach taken allows to derive asymptotics of three other functionals of the coalescent, the absorption time, the length of an external branch chosen at random from the n external branches, and the number of collision events that occur before the randomly selected external branch coalesces with one of its neighbours.