Asymptotics of the Minimal Clade Size and Related Functionals of Certain Beta-Coalescents

作者：Siri Jegousse Arno; Yuan Linglong^*

来源：Acta Applicandae Mathematicae, 2016, 142(1): 127-148.

DOI：10.1007/s10440-015-0020-7

• Beta-coalescent
• Ranked Lambda-coalescent
• Kingman's paintbox construction
• Minimal clade size
• External branch lengths
• Block-counting process

The Beta(2-alpha,alpha) n-coalescent with 1 <alpha < 2 is a Markov process taking values in the set of partitions of {1,aEuro broken vertical bar,n}. It evolves from the initial value {1},aEuro broken vertical bar,{n} by merging (coalescing) blocks together into one and finally reaching the absorbing state {1,aEuro broken vertical bar,n}. This article aims to give the asymptotic distribution of the size of the minimal clade of 1, which is the block containing 1 at the time of coalescence of the singleton {1}. To this, we express it as a function of the coalescence time of {1}, the number of blocks involved and their sizes. The asymptotic behaviours of those related functionals are therefore also studied.

• 出版日期2016-4