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

Dublin Core		PKP Metadata Items	Metadata for this Document
1.	Title	Title of document	Almost sure asymptotics for the number of types for simple $\Xi$-coalescents
2.	Creator	Author's name, affiliation, country	Fabian Freund; University of Hohenheim; Germany
3.	Subject	Discipline(s)
3.	Subject	Keyword(s)	almost sure convergence; coalescent; external branches; mutation
3.	Subject	Subject classification	60F15; 05C05; 60F25; 92D15
4.	Description	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.
5.	Publisher	Organizing agency, location
6.	Contributor	Sponsor(s)
7.	Date	(YYYY-MM-DD)	2012-01-06
8.	Type	Status & genre	Peer-reviewed Article
8.	Type	Type
9.	Format	File format	PDF
10.	Identifier	Uniform Resource Identifier	http://ecp.ejpecp.org/article/view/1704
10.	Identifier	Digital Object Identifier	10.1214/ECP.v17-1704
11.	Source	Journal/conference title; vol., no. (year)	Electronic Communications in Probability; Vol 17
12.	Language	English=en	en
14.	Coverage	Geo-spatial location, chronological period, research sample (gender, age, etc.)
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