| Título: | Almost sure asymptotics for the number of types for simple $\Xi$-coalescents |
| Autores: | Freund, Fabian; University of Hohenheim |
| Fecha: | 2012-01-02 |
| Publicador: | Electronic communications in probability |
| Fuente: |
 |
| Tipo: | Peer-reviewed Article |
| Tema: |
almost sure convergence; coalescent; external branches; mutation 60F15; 05C05; 60F25; 92D15 |
| Descripción: | 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. |
| Idioma: | Inglés |
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