On the measure division construction of $\Lambda$-coalescents

#### Linglong Yuan

2014, v.20, №2, 229-264

ABSTRACT

This paper provides a new construction of $\Lambda$-coalescents called "measure division construction". This construction is pathwise and consists of dividing the characteristic measure $\Lambda$ into several parts and adding them one by one to have a whole process. Using this construction, a "universal" normalization factor $\mu^{(n)}$ for the randomly chosen external branch length $T^{(n)}$ has been discovered for a class of coalescents. This class of coalescents covers processes similar to Bolthausen - Sznitman coalescent, the coalescents without proper frequencies, and also others. A conjecture is also made to extend the universality of $\mu^{(n)}$ to more processes.

Keywords: $\Lambda$-coalescent,two-type $\Lambda$-coalescent,branchlength,measure division construction,noise measure,main measure