@article{ECP1704,
author = {Fabian Freund},
title = {Almost sure asymptotics for the number of types for simple $\Xi$-coalescents},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {17},
year = {2012},
keywords = {almost sure convergence; coalescent; external branches; mutation},
abstract = {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 coalescents without proper frequencies, the effects of mutations on non-external branches is neglectible for the asymptotics of $K_n/n$, it is shown that $K_n/n\rightarrow M$ for $n\rightarrow\infty$ in $L^p$ $(p\geq 1)$. For simple coalescents, this convergence is shown to hold almost surely. The almost sure results are based on a combination of the Kingman correspondence for random partitions and strong laws of large numbers for weighted i.i.d. or exchangeable random variables.},
pages = {no. 3, 1-11},
issn = {1083-589X},
doi = {10.1214/ECP.v17-1704},
url = {http://ecp.ejpecp.org/article/view/1704}}