arXiv subjects

Götz Kersting

Publications and source records attributed to Götz Kersting.

24 records · Page 2Linked to original sources

arXiv2012

The total external branch length of Beta-coalescents

For $1<α<2$ we derive the asymptotic distribution of the total length of {\em external} branches of a Beta$(2-α, α)$-coalescent as the number $n$ of leaves becomes large. It turns out the fluctuations of the external branch length follow those of $τ_n^{2-α}$ over the entire parameter regime, where $τ_n$ denotes the random number of coalescences that bring the $n$ lineages down to one. This is in contrast to the fluctuation behavior of the total branch length, which exhibits a transition at $α_0 = (1+\sqrt 5)/2$.

math.PR↗

arXiv2012

The asymptotic distribution of the length of Beta-coalescent trees

We derive the asymptotic distribution of the total length $L_n$ of a $\operatorname {Beta}(2-α,α)$-coalescent tree for $1<α<2$, starting from $n$ individuals. There are two regimes: If $α\le1/2(1+\sqrt{5})$, then $L_n$ suitably rescaled has a stable limit distribution of index $α$. Otherwise $L_n$ just has to be shifted by a constant (depending on $n$) to get convergence to a nondegenerate limit distribution. As a consequence, we obtain the limit distribution of the number $S_n$ of segregation sites. These are points (mutations), which are placed on the tree's branches according to a Poisson point process with constant rate.

math.PR↗

arXiv2012

Simulations and a conditional limit theorem for intermediately subcritical branching processes in random environment

Intermediately subcritical branching processes in random environment are at the borderline between two subcritical regimes and exhibit a particularly rich behavior. In this paper, we prove a functional limit theorem for these processes. It is discussed together with two other recently proved limit theorems for the intermediately subcritical case and illustrated by several computer simulations.

math.PR↗

arXiv2011

Conditional limit theorems for intermediately subcritical branching processes in random environment

For a branching process in random environment it is assumed that the offspring distribution of the individuals varies in a random fashion, independently from one generation to the other. For the subcritical regime a kind of phase transition appears. In this paper we study the intermediately subcritical case, which constitutes the borderline within this phase transition. We study the asymptotic behavior of the survival probability. Next the size of the population and the shape of the random environment conditioned on non-extinction is examined. Finally we show that conditioned on non-extinction periods of small and large population sizes alternate. This kind of 'bottleneck' behavior appears under the annealed approach only in the intermediately subcritical case.

math.PR↗

arXiv2011

On the Height Profile of a Conditioned Galton-Watson Tree

Drmota and Gittenberger (1997) proved a conjecture due to Aldous (1991) on the height profile of a Galton-Watson tree with an offspring distribution of finite variance, conditioned on a total size of $n$ individuals. The conjecture states that in distribution its shape, more precisely its scaled height profile coincides asymptotically with the local time process of a Brownian excursion of duration 1. We give a proof of the result, which extends to the case of an infinite variance offspring distribution. This requires a different strategy, since in the infinite variance case there is no longer a relationship to the local time of Brownian resp. Lévy excursions.

math.PR↗

arXiv2011

The external lengths in Kingman's coalescent

In this paper we prove asymptotic normality of the total length of external branches in Kingman's coalescent. The proof uses an embedded Markov chain, which can be descriped as follows: Take an urn with n black balls. Empty it in n steps according to the rule: In each step remove a randomly chosen pair of balls and replace it by one red ball. Finally remove the last remaining ball. Then the numbers U_k, 0 \leq k \leq n, of red balls after k steps exhibits an unexpected property: (U_0,...,U_n) and (U_n,..., U_0) are equal in distribution.

math.PR↗