arXiv subjects

Götz Kersting

Publications and source records attributed to Götz Kersting.

At least 19 recordsLinked to original sources

arXiv2024

Haldane's asymptotics for supercritical branching processes in an iid random environment

Branching processes in a random environment are natural generalisations of Galton-Watson processes. In this paper we analyse the asymptotic decay of the survival probability for a sequence of slightly supercritical branching processes in an iid random environment, where the offspring expectation converges from above to $1$. We prove that Haldane's asymptotics, known from classical Galton-Watson processes, turns up again in the random environment case, provided that one stays away from the critical/subcritical regime. A central building block is a connection to and a limit theorem for perpetuities with asymptotically vanishing interest rates.

math.PR↗

arXiv2024

Fubini's Theorem for Daniell Integrals

We show that in the theory of Daniell integration iterated integrals may always be formed, and the order of integration may always be interchanged. By this means, we discuss product integrals and show that the related Fubini theorem holds in full generality. The results build on a density theorem on Riesz tensor products due to Fremlin, and on the Fubini-Stone Theorem.

math.FA↗

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↗

arXiv2022

On the genealogical structure of critical branching processes in a varying environment

Critical branching processes in a varying environment behave much the same as critical Galton-Watson processes. In this note we like to confirm this finding with regard to the underlying genealogical structures. In particular, we consider the most recent common ancestor given survival and the corresponding reduced branching processes, in the spirit of Zubkov (1975) and Fleischmann and Siegmund-Schultze (1977).

math.PR↗

arXiv2021

Defective Galton-Watson processes in a varying environment

We study an extension of the so-called defective Galton-Watson processes obtained by allowing the offspring distribution to change over the generations. Thus, in these processes, the individuals reproduce independently of the others and in accordance to some possibly defective offspring distribution depending on the generation. Moreover, the defect $1-f_n(1)$ of the offspring distribution at generation $n$ represents the probability that the process hits an absorbing state $Δ$ at that generation. We focus on the asymptotic behaviour of these processes. We establish the almost sure convergence of the process to a random variable with values in $\mathbb{N}_0\cup\{Δ\}$ and we provide two characterisations of the duality extinction-absorption at $Δ$. We also state some results on the absorption time and the properties of the process conditioned upon its non-absorption, some of which require us to introduce the notion of defective branching trees in varying environment.

math.PR↗

arXiv2020

Probabilistic aspects of $Λ$-coalescents in equilibrium and in evolution

We present approximation methods which lead to law of large numbers and fluctuation results for functionals of $Λ$-coalescents, both in the dust-free case and in the case with a dust component. Our focus is on the tree length (or total branch length) and the total external branch length, as well as the time to the most recent common ancestor and the size of the last merger. In the second part we discuss evolving coalescents. For certain Beta-coalescents we analyse fluctuations of a class of functionals in appropriate time scales. Finally we review results of Gufler on the representation of evolving $Λ$-coalescents in terms of the lookdown space.

math.PR↗

arXiv2019

A unifying approach to branching processes in varying environments

Branching processes $(Z_n)_{n \ge 0}$ in a varying environment generalize the Galton-Watson process, in that they allow time-dependence of the offspring distribution. Our main results concern general criteria for a.s. extinction, square-integrability of the martingale $(Z_n/\mathbf E[Z_n])_{n \ge 0}$, properties of the martingale limit $W$ and a Yaglom type result stating convergence to an exponential limit distribution of the suitably normalized population size $Z_n$, conditioned on the event $Z_n >0$. The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.

math.PR↗

arXiv2019

Site Frequency Spectrum of the Bolthausen-Sznitman Coalescent

We derive explicit formulas for the two first moments of he site frequency spectrum $(SFS_{n,b})_{1\leq b\leq n-1}$ of the Bolthausen-Sznitman coalescent along with some precise and efficient approximations, even for small sample sizes $n$. These results provide new $L_2$-asymptotics for some values of $b=o(n)$. We also study the length of internal branches carrying $b>n/2$ individuals. In this case we obtain the distribution function and a convergence in law. Our results rely on the random recursive tree construction of the Bolthausen-Sznitman coalescent.

math.PR↗

arXiv2019

External branch lengths of $Λ$-coalescents without a dust component

$Λ$-coalescents model genealogies of samples of individuals from a large population by means of a family tree whose branches have lengths. The tree's leaves represent the individuals, and the lengths of the adjacent edges indicate the individuals' time durations up to some common ancestor. These edges are called external branches. We consider typical external branches under the broad assumption that the coalescent has no dust component, and maximal external branches under further regularity assumptions. As it transpires, the crucial characteristic is the coalescent's rate of decrease $μ(b)$, $b\geq 2$. The magnitude of a typical external branch is asymptotically given by $n/μ(n)$, where $n$ denotes the sample size. This result, in addition to the asymptotic independence of several typical external lengths hold in full generality, while convergence in distribution of the scaled external lengths requires that $μ(n)$ is regularly varying at infinity. For the maximal lengths, we distinguish two cases. Firstly, we analyze a class of $Λ$-coalescents coming down from infinity and with regularly varying $μ$. Here the scaled external lengths behave as the maximal values of $n$ i.i.d. random variables, and their limit is captured by a Poisson point process on the positive real line. Secondly, we turn to the Bolthausen-Sznitman coalescent, where the picture changes. Now the limiting behavior of the normalized external lengths is given by a Cox point process, which can be expressed by a randomly shifted Poisson point process.

math.PR↗

arXiv2019

Tree lengths for general $Λ$-coalescents and the asymptotic site frequency spectrum around the Bolthausen-Sznitman coalescent

We study tree lengths in $Λ$-coalescents without a dust component from a sample of $n$ individuals. For the total length of all branches and the total length of all external branches we present laws of large numbers in full generality. The other results treat regularly varying coalescents with exponent 1, which cover the Bolthausen-Sznitman coalescent. The theorems contain laws of large numbers for the total length of all internal branches and of internal branches of order $a$ (i.e. branches carrying $a$ individuals out of the sample). These results transform immediately to sampling formulas in the infinite sites model. In particular, we obtain the asymptotic site frequency spectrum of the Bolthausen-Sznitman coalescent. The proofs rely on a new technique to obtain laws of large numbers for certain functionals of decreasing Markov chains.

math.PR↗

arXiv2017

On the time to absorption in $Λ$-coalescents

We present a law of large numbers and a central limit theorem for the time to absorption of $Λ$-coalescents, started from $n$ blocks, as $n \to \infty$. The proofs rely on an approximation of the logarithm of the block-counting process of $Λ$-coalescents with a dust component by means of a drifted subordinator.

math.PR↗

arXiv2017

The size of the last merger and time reversal in $Λ$-coalescents

We consider the number of blocks involved in the last merger of a $Λ$-coalescent started with $n$ blocks. We give conditions under which, as $n \to \infty$, the sequence of these random variables a) is tight, b) converges in distribution to a finite random variable or c) converges to infinity in probability. Our conditions are optimal for $Λ$-coalescents that have a dust component. For general $Λ$, we relate the three cases to the existence, uniqueness and non-existence of quasi-invariant measures for the dynamics of the block-counting process, and in case b) investigate the time-reversal of the block-counting process back from the time of the last merger.

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↗

arXiv2016

On recurrence and transience of multivariate near-critical stochastic processes

We obtain complementary recurrence and transience criteria for processes $X=(X_n)_{n \ge 0}$ with values in $\mathbb R^d_+$ fulfilling a non-linear equation $X_{n+1}=MX_n+g(X_n)+ ξ_{n+1}$. Here $M$ denotes a primitive matrix having Perron-Frobenius eigenvalue 1, and $g$ denotes some function. The conditional expectation and variance of the noise $(ξ_{n+1})_{n \ge 0}$ are such that $X$ obeys a weak form of the Markov property. The results generalize criteria for the 1-dimensional case in [5].

math.PR↗

arXiv2016

Recurrence and transience of near-critical multivariate growth models: criteria and examples

We discuss complementary recurrence and transience criteria for stochastic processes $(X_n)_{n \ge 0}$ with values in the $d$-dimensional orthant $\mathbb R^d_+$ fulfilling a non-linear stochastic equation of the form $X_{n+1}=MX_n+g(X_n)+ ξ_n$ with a primitive matrix $M$ and random noise $ξ_n$ and obeying a weak Markov property. As examples we discuss bisexual Galton-Watson processes and multivariate Galton-Watson processes, which both may be population size dependent.

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↗

arXiv2014

The evolving beta coalescent

In mathematical population genetics, it is well known that one can represent the genealogy of a population by a tree, which indicates how the ancestral lines of individuals in the population coalesce as they are traced back in time. As the population evolves over time, the tree that represents the genealogy of the population also changes, leading to a tree-valued stochastic process known as the evolving coalescent. Here we will consider the evolving coalescent for populations whose genealogy can be described by a beta coalescent, which is known to give the genealogy of populations with very large family sizes. We show that as the size of the population tends to infinity, the evolution of certain functionals of the beta coalescent, such as the total number of mergers, the total branch length, and the total length of external branches, converges to a stationary stable process. Our methods also lead to new proofs of known asymptotic results for certain functionals of the non-evolving beta coalescent.

math.PR↗

arXiv2013

Total internal and external lengths of the Bolthausen-Sznitman coalescent

In this paper, we study a weak law of large numbers for the total internal length of the Bolthausen-Szmitman coalescent. As a consequence, we obtain the weak limit law of the centered and rescaled total external length. The latter extends results obtained by Dhersin & Möhle \cite{DM12}. An application to population genetics dealing with the total number of mutations in the genealogical tree is also given.

math.PR↗