In this paper we define and study self-similar ranked fragmentations. We first show that any ranked fragmentation is the image of some partition-valued fragmentation, and that there is in fact a one-to-one correspondence between the laws of these two types of fragmentations. We then give an explicit construction of homogeneous ranked fragmentations in terms of Poisson point processes. Finally we use this construction and classical results on records of Poisson point processes to study the small-time...
In this paper we define and study self-similar ranked
fragmentations. We first show that any ranked fragmentation is the
image of some partition-valued fragmentation, and that there is in
fact a one-to-one correspondence between the laws of these two
types of fragmentations. We then give an explicit construction of
homogeneous ranked fragmentations in terms of Poisson point
processes. Finally we use this construction and classical results
on records of Poisson point processes to study the small-time
behavior...
For a finite measure on [0, 1], the -coalescent is a coalescent process such that, whenever there are clusters, each -tuple of clusters merges into one at rate
(1−)
(d). It has recently been shown that if 1<<2, the -coalescent in which is the Beta (2−, ) distribution can be used to describe the genealogy of a continuous-state branching process (CSBP) with an -stable branching mechanism. Here we use facts...
We present a robust method which translates information on the speed of coming down from infinity of a genealogical tree into sampling formulae for the underlying population. We apply these results to population dynamics where the genealogy is given by a -coalescent. This allows us to derive an exact formula for the asymptotic behavior of the site and allele frequency spectrum and the number of segregating sites, as the sample size tends to . Some of our results hold in the case of a general -coalescent...
Download Results (CSV)