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2012 Almost sure asymptotics for the number of types for simple $\Xi$-coalescents
Fabian Freund
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Electron. Commun. Probab. 17: 1-11 (2012). DOI: 10.1214/ECP.v17-1704


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.


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Fabian Freund. "Almost sure asymptotics for the number of types for simple $\Xi$-coalescents." Electron. Commun. Probab. 17 1 - 11, 2012.


Accepted: 6 January 2012; Published: 2012
First available in Project Euclid: 7 June 2016

zbMATH: 1255.60043
MathSciNet: MR2872572
Digital Object Identifier: 10.1214/ECP.v17-1704

Primary: 60F15
Secondary: 05C05 , 60F25 , 92D15

Keywords: Almost sure convergence , Coalescent , external branches , mutation


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