Almost sure asymptotics for the number of types for simple $\Xi$-coalescents

@article{Freund2012AlmostSA,
  title={Almost sure asymptotics for the number of types for simple \$\Xi\$-coalescents},
  author={Fabian Freund},
  journal={Electronic Communications in Probability},
  year={2012},
  volume={17},
  pages={1-11}
}
  • F. Freund
  • Published 1 June 2012
  • Mathematics
  • Electronic Communications in Probability
Let $K_n$ be the number of types in the sample $\left\{1,\ldots, n\right\}$ of a $\Xi$-coalescent $\Pi=(\Pi_t)_{t\geq0}$ with mutation and mutation rate $r>0$. Let $\Pi^{(n)}$ be the restriction of $\Pi$ to the sample. It is shown that $M_n/n$, the fraction of external branches of $\Pi^{(n)}$ which are affected by at least one mutation, converges almost surely and in $L^p$ ($p\geq 1$) to $M:=\int^{\infty}_0 re^{-rt}S_t dt$, where $S_t$ is the fraction of singleton blocks of $\Pi_t$. Since for… 
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