arXiv subjects

Martin Möhle

Publications and source records attributed to Martin Möhle.

At least 19 recordsLinked to original sources

arXiv2026

On a $Λ$-mutation model and the harmonic model

We introduce a continuous-time mutation model with two types determined by a finite measure $Λ$ on the unit interval. The model satisfies a certain consistency property known from mathematical population genetics and includes so called harmonic models being of interest in mathematical statistical physics. We mainly focus on the situation when the number of particles is equal to some constant $N$. Duality results and scaling limits as $N\to\infty$ for the forward and backward processes are provided leading to a commutative diagram. The stationary distribution of the forward process is studied with an emphasis on the case when $Λ$ is a beta distribution. The work bridges particle models from mathematical statistical physics and mutation models from mathematical population genetics.

math.PR↗

arXiv2025

On the genealogy of multi-type Cannings models and their limiting exchangeable coalescents

We study the multi-type Cannings population model. Each individual has a type belonging to a given at most countable type space $E$. The population is hence divided into $|E|$ subpopulations. The subpopulation sizes are assumed to be constant over the generations, whereas the number of offspring of type $\ell\in E$ of all individuals of type $k\in E$ is allowed to be random. Under a joint exchangeability assumption on the offspring numbers, the transition probabilities of the ancestral process of a sample of individuals satisfy a multi-type consistency property, paving a way to prove in the limit for large subpopulation sizes the existence of multi-type exchangeable coalescent processes via Kolmogorov's extension theorem. Integral representations for the infinitesimal rates of these multi-type exchangeable coalescents and some of their properties are studied. Examples are provided, among them multi-type Wright-Fisher models and multi-type pure mutation models. The results contribute to the foundations of multi-type coalescent theory and provide new insights into (the existence of) multi-type exchangeable coalescents.

math.PR↗

arXiv2023

On Bernoulli trials with unequal harmonic success probabilities

A Bernoulli scheme with unequal harmonic success probabilities is investigated, together with some of its natural extensions. The study includes the number of successes over some time window, the times to (between) successive successes and the time to the first success. Large sample asymptotics, statistical parameter estimation, and relations to Sibuya distributions and Yule-Simon distributions are discussed. This toy model is relevant in several applications including reliability, species sampling problems, record values breaking and random walks with disasters.

math.PR↗

arXiv2023

On multi-type Cannings models and multi-type exchangeable coalescents

A multi-type neutral Cannings population model with mutation and fixed subpopulation sizes is analyzed. Under appropriate conditions, as all subpopulation sizes tend to infinity, the ancestral process, properly time-scaled, converges to a multi-type exchangeable coalescent with mutation sharing the exchangeability and consistency property. The proof gains from coalescent theory for single-type Cannings models and from decompositions into reproductive and mutational parts. The second part deals with a different but closely related multi-type Cannings model with mutation and fixed total population size but stochastically varying subpopulation sizes. The latter model is analyzed forward and backward in time with an emphasis on its behaviour as the total population size tends to infinity. Forward in time, multitype limiting branching processes arise for large population size. Its backward structure and related open problems are briefly discussed.

math.PR↗

arXiv2022

Scaling limits for a class of regular $Ξ$-coalescents

The block counting process with initial state $n$ counts the number of blocks of an exchangeable coalescent ($Ξ$-coalescent) restricted to a sample of size $n$. This work provides scaling limits for the block counting process of regular $Ξ$-coalescents that stay infinite, including $Ξ$-coalescents with dust and a large class of dust-free $Ξ$-coalescents. The main convergence result states that the block counting process, properly logarithmically scaled, converges in the Skorohod space to an Ornstein--Uhlenbeck type process as $n$ tends to infinity. The existence of such a scaling depends on a sort of curvature condition of a particular function well-known from the literature. This curvature condition is intrinsically related to the behavior of the measure $Ξ$ near the origin. The method of proof is to show the uniform convergence of the associated generators. Via Siegmund duality an analogous result for the fixation line is proven. Several examples are studied.

math.PR↗

arXiv2021

Asymptotic genealogies for a class of generalized Wright-Fisher models

We study a class of Cannings models with population size $N$ having a mixed multinomial offspring distribution with random success probabilities $W_1,\ldots,W_N$ induced by independent and identically distributed positive random variables $X_1,X_2,\ldots$ via $W_i:=X_i/S_N$, $i\in\{1,\ldots,N\}$, where $S_N:=X_1+\cdots+X_N$. The ancestral lineages are hence based on a sampling with replacement strategy from a random partition of the unit interval into $N$ subintervals of lengths $W_1,\ldots,W_N$. Convergence results for the genealogy of these Cannings models are provided under regularly varying assumptions on the tail distribution of $X_1$. In the limit several coalescent processes with multiple and simultaneous multiple collisions occur. The results extend those obtained by Huillet (2014) for the case when $X_1$ is Pareto distributed and complement those obtained by Schweinsberg (2003) for models where one samples without replacement from a supercritical branching process.

math.PR↗

arXiv2021

Scaling limits for the block counting process and the fixation line of a class of $Λ$-coalescents

We provide scaling limits for the block counting process and the fixation line of $Λ$-coalescents as the initial state $n$ tends to infinity under the assumption that the measure $Λ$ on $[0,1]$ satisfies $\int_{[0,1]}u^{-1}(Λ-bλ)({\rm d}u)<\infty$ for some $b>0$. Here $λ$ denotes the Lebesgue measure. The main result states that the block counting process, properly logarithmically scaled, converges in the Skorohod space to an Ornstein--Uhlenbeck type process as $n$ tends to infinity. The result is applied to beta coalescents with parameters $1$ and $b>0$. We split the generators into two parts by additively decomposing Lambda and then prove the uniform convergence of both parts separately.

math.PR↗

arXiv2020

Asymptotic behaviour of ancestral lineages in subcritical continuous-state branching populations

Consider the population model with infinite size associated to subcritical continuous-state branching processes (CSBP). Individuals reproduce independently according to the same subcritical offspring distribution. We study the long-term behaviour of the ancestral lineages as time goes to the past and show that the flow of ancestral lineages, properly renormalized, converges almost surely to the inverse of a drift-free subordinator whose Laplace exponent is explicit in terms of the branching mechanism. We provide an interpretation in terms of the genealogy of the population. In particular, we show that the inverse subordinator is partitioning the current population into ancestral families with distinct common ancestors. When Grey's condition is satisfied, the population comes from a discrete set of ancestors and the ancestral families are i.i.d and distributed according to the quasi-stationary distribution of the CSBP conditioned on non-extinction. When Grey's condition is not satisfied, the population comes from a continuum of ancestors which is described as the set of increase points $\mathscr{S}$ of the limiting inverse subordinator. The Hausdorff dimension of $\mathscr{S}$ is given. The proof is based on a general result for stochastically monotone processes of independent interest, which relates $θ$-invariant measures and $θ$-invariant functions for a process and its Siegmund dual.

math.PR↗

arXiv2020

Asymptotics of continuous-time discrete state space branching processes for large initial state

Scaling limits for continuous-time branching processes with discrete state space are provided as the initial state tends to infinity. Depending on the finiteness or non-finiteness of the mean and/or the variance of the offspring distribution, the limits are in general time-inhomogeneous Gaussian processes, time-inhomogeneous generalized Ornstein-Uhlenbeck type processes or continuous-state branching processes. We also provide transfer results showing how specific asymptotic relations for the probability generating function of the offspring distribution carry over to those of the one-dimensional distributions of the branching process.

math.PR↗

arXiv2018

On the stationary distribution of the block counting process for population models with mutation and selection

We consider two population models subject to the evolutionary forces of selection and mutation, the Moran model and the $Λ$-Wright-Fisher model. In such models the block counting process traces back the number of potential ancestors of a sample of the population at present. Under some conditions the block counting process is positive recurrent and its stationary distribution is described via a linear system of equations. In this work, we first characterise the measures $Λ$ leading to a geometric stationary distribution, the Bolthausen-Sznitman model being the most prominent example having this feature. Next, we solve the linear system of equations corresponding to the Moran model. For the $Λ$-Wright-Fisher model we show that the probability generating function associated to the stationary distribution of the block counting process satisfies an integro differential equation. We solve the latter for the Kingman model and the star-shaped model.

math.PR↗

arXiv2018

On the size of the block of 1 for $\varXi$-coalescents with dust

We study the frequency process $f_1$ of the block of 1 for a $\varXi$-coalescent $\varPi$ with dust. If $\varPi$ stays infinite, $f_1$ is a jump-hold process which can be expressed as a sum of broken parts from a stick-breaking procedure with uncorrelated, but in general non-independent, stick lengths with common mean. For Dirac-$\varLambda$-coalescents with $\varLambda=δ_p$, $p\in[\frac{1}{2},1)$, $f_1$ is not Markovian, whereas its jump chain is Markovian. For simple $\varLambda$-coalescents the distribution of $f_1$ at its first jump, the asymptotic frequency of the minimal clade of 1, is expressed via conditionally independent shifted geometric distributions.

math.PR↗

arXiv2017

The collision spectrum of $Λ$-coalescents

$Λ$-coalescents model the evolution of a coalescing system in which any number of blocks randomly sampled from the whole may merge into a larger block. For the coalescent restricted to initially $n$ singletons we study the collision spectrum $(X_{n,k}:2\le k\le n)$, where $X_{n,k}$ counts, throughout the history of the process, the number of collisions involving exactly $k$ blocks. Our focus is on the large $n$ asymptotics of the joint distribution of the $X_{n,k}$'s, as well as on functional limits for the bulk of the spectrum for simple coalescents. Similarly to the previous studies of the total number of collisions, the asymptotics of the collision spectrum largely depends on the behaviour of the measure $Λ$ in the vicinity of $0$. In particular, for beta$(a,b)$-coalescents different types of limit distributions occur depending on whether $0 2$.

math.PR↗

arXiv2016

On the block counting process and the fixation line of the Bolthausen-Sznitman coalescent

The block counting process and the fixation line of the Bolthausen-Sznitman coalescent are analyzed. Spectral decompositions for their generators and transition probabilities are provided leading to explicit expressions for functionals such as hitting probabilities and absorption times. It is furthermore shown that the block counting process and the fixation line of the Bolthausen-Sznitman $n$-coalescent, properly scaled, converge in the Skorohod topology to the Mittag-Leffler process and Neveu's continuous-state branching process respectively as the sample size $n$ tends to infinity. Strong relations to Siegmund duality and to Mehler semigroups and self-decomposability are pointed out.

math.PR↗

arXiv2016

On the block counting process and the fixation line of exchangeable coalescents

We study the block counting process and the fixation line of exchangeable coalescents. Formulas for the infinitesimal rates of both processes are provided. It is shown that the block counting process is Siegmund dual to the fixation line. For exchangeable coalescents restricted to a sample of size n and with dust we provide a convergence result for the block counting process as n tends to infinity. The associated limiting process is related to the frequencies of singletons of the coalescent. Via duality we obtain an analog convergence result for the fixation line of exchangeable coalescents with dust. The Dirichlet coalescent and the Poisson-Dirichlet coalescent are studied in detail.

math.PR↗

arXiv2015

Absorption time and tree length of the Kingman coalescent and the Gumbel distribution

Formulas are provided for the cumulants and the moments of the time $T$ back to the most recent common ancestor of the Kingman coalescent. It is shown that both the $j$th cumulant and the $j$th moment of $T$ are linear combinations of the values $ζ(2m)$, $m\in\{0,\ldots,\lfloor j/2\rfloor\}$, of the Riemann zeta function $ζ$ with integer coefficients. The proof is based on a solution of a two-dimensional recursion with countably many initial values. A closely related strong convergence result for the tree length $L_n$ of the Kingman coalescent restricted to a sample of size $n$ is derived. The results give reason to revisit the moments and central moments of the classical Gumbel distribution.

math.PR↗

arXiv2014

The Mittag-Leffler process and a scaling limit for the block counting process of the Bolthausen-Sznitman coalescent

The Mittag-Leffler process $X=(X_t)_{t\ge 0}$ is introduced. This Markov process has the property that its marginal random variables $X_t$ are Mittag-Leffler distributed with parameter $e^{-t}$, $t\in [0,\infty)$, and the semigroup $(T_t)_{t\ge 0}$ of $X$ satisfies $T_tf(x)={\mathbb E}(f(x^{e^{-t}}X_t))$ for all $x\ge 0$ and all bounded measurable functions $f:[0,\infty)\to{\mathbb R}$. Further characteristics of the process $X$ are derived, for example an explicit formula for the joint moments of its finite dimensional distributions. The main result states that the block counting process of the Bolthausen-Sznitman $n$-coalescent, properly scaled, converges in the Skorohod topology to the Mittag-Leffler process $X$ as the sample size $n$ tends to infinity.

math.PR↗

arXiv2009

On the number of collisions in beta(2, $b$)-coalescents

Expansions are provided for the moments of the number of collisions $X_n$ in the $β(2,b)$-coalescent restricted to the set $\{1,...,n\}$. We verify that $X_n/\mathbb{E}X_n$ converges almost surely to one and that $X_n$, properly normalized, weakly converges to the standard normal law. These results complement previously known facts concerning the number of collisions in $β(a,b)$-coalescents with $a\in(0,2)$ and $b=1$, and $a>2$ and $b>0$. The case $a=2$ is a kind of `border situation' which seems not to be amenable to approaches used for $a\neq2$.

math.ST↗

arXiv2008

On the number of allelic types for samples taken from exchangeable coalescents with mutation

Let $K_n$ denote the number of types of a sample of size $n$ taken from an exchangeable coalescent process ($Ξ$-coalescent) with mutation. A distributional recursion for the sequence $(K_n)_{n\in{\mathbb N}}$ is derived. If the coalescent does not have proper frequencies, i.e., if the characterizing measure $Ξ$ on the infinite simplex $Δ$ does not have mass at zero and satisfies $\int_Δ|x|Ξ(dx)/(x,x)<\infty$, where $|x|:=\sum_{i=1}^\infty x_i$ and $(x,x):=\sum_{i=1}^\infty x_i^2$ for $x=(x_1,x_2,...)\inΔ$, then $K_n/n$ converges weakly as $n\to\infty$ to a limiting variable $K$ which is characterized by an exponential integral of the subordinator associated with the coalescent process. For so-called simple measures $Ξ$ satisfying $\int_ΔΞ(dx)/(x,x)<\infty$ we characterize the distribution of $K$ via a fixed-point equation.

math.PR↗