arXiv subjects

Iulia Dahmer

Publications and source records attributed to Iulia Dahmer.

5 recordsLinked to original sources

arXiv2024

On the fixation probability of an advantageous allele in a population with skewed offspring distribution

Consider an advantageous allele that arises in a haploid population of size $N$ evolving in continuous time according to a skewed reproduction mechanism, which generates under neutrality genealogies lying in the domain of attraction of a Beta$(2-α, α)$-coalescent for $α\in (1,2)$. We prove in a setting of moderate selection that the fixation probability $π_N$ of the advantageous allele is asymptotically equal to $α^{1/(α-1)} s_N^{1/(α-1)} $ , where $s_N$ is the selection strength of the advantageous allele. Our proof uses duality with a suitable $Λ$-ancestral selection graph.

math.PR↗

arXiv2022

The joint fluctuations of the lengths of the Beta$(2-α, α)$-coalescents

We consider Beta$(2-α, α)$-coalescents with parameter range $1 <α<2$ starting from $n$ leaves. The length $\ell^{(n)}_r$ of order $r$ in the $n$-Beta$(2-α, α)$-coalescent tree is defined as the sum of the lengths of all branches that carry a subtree with $r$ leaves. We show that for any $s \in \mathbb N$ the vector of suitably centered and rescaled lengths of orders $1\le r \le s$ converges in distribution to a multivariate stable distribution as the number of leaves tends to infinity.

math.PR↗

arXiv2016

The total external length of the evolving Kingman coalescent

The evolving Kingman coalescent is the tree-valued process which records the time evolution undergone by the genealogies of Moran populations. We consider the associated process of total external tree length of the evolving Kingman coalescent and its asymptotic behaviour when the number of leaves of the tree tends to infinity. We show that on the time-scale of the Moran model slowed down by a factor equal to the population size, the (centred and rescaled) external length process converges to a stationary Gaussian process with almost surely continuous paths and covariance function $c(s,t)=\Big( \frac 2 {2+|s-t|} \Big)^2$. A key role in the evolution of the external length is played by the internal lengths of finite orders in the coalescent at a fixed time which behave asymptotically in a multivariate Gaussian manner (see Dahmer and Kersting (2015)). A coupling of the Moran model with a critical branching process is used. We also derive a central limit result for normally distributed sums endowed with independent random coefficients.

math.PR↗

arXiv2015

The internal branch lengths of the Kingman coalescent

In the Kingman coalescent tree the length of order $r$ is defined as the sum of the lengths of all branches that support $r$ leaves. For $r=1$ these branches are external, while for $r\ge2$ they are internal and carry a subtree with $r$ leaves. In this paper we prove that for any $s\in\mathbb{N}$ the vector of rescaled lengths of orders $1\le r\le s$ converges to the multivariate standard normal distribution as the number of leaves of the Kingman coalescent tends to infinity. To this end we use a coupling argument which shows that for any $r\ge2$ the (internal) length of order $r$ behaves asymptotically in the same way as the length of order 1 (i.e., the external length).

math.PR↗

arXiv2015

The Kingman tree length process has infinite quadratic variation

In the case of neutral populations of fixed sizes in equilibrium whose genealogies are described by the Kingman $N$-coalescent back from time $t$ consider the associated processes of total tree length as $t$ increases. We show that the (càdlàg) process to which the sequence of compensated tree length processes converges as $N$ tends to infinity is a process of infinite quadratic variation; therefore this process cannot be a semimartingale. This answers a question posed in Pfaffelhuber et al. (2011).

math.PR↗