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We consider two-opinion voter models on dense dynamic random graphs. Our goal is to understand and describe the occurrence of consensus versus polarisation over long periods of time. The former means that all vertices have the same opinion, the latter means that the vertices split into two communities with different opinions and few disagreeing edges. We consider three models for the joint dynamics of opinions and graphs: one with a one-way feedback and two which are co-evolutionary, i.e., with a two-way feedback. In the first model only coexistence is attainable, meaning that both opinions survive, but with the presence of many disagreeing edges. In the second model only consensus prevails, while in the third model polarisation is possible. Our main results are functional laws of large numbers for the densities of the two opinions, functional laws of large numbers for the dynamic random graphs in the space of graphons, and a characterisation of the limiting densities in terms of Beta-distributions. Our results are supported by simulations. To prove our results we develop a novel method that involves coupling the co-evolutionary process to a mimicking process with one-way feedback. We expect that this method can be extended to other dense co-evolutionary models., Comment: 39 pages

Population-size dependent branching processes (PSDBP) and controlled branching processes (CBP) are two classes of branching processes widely used to model biological populations that exhibit logistic growth. In this paper we develop connections between the two, with the ultimate goal of determining when a population is more appropriately modelled with a PSDBP or a CBP. In particular, we state conditions for the existence of equivalent PSDBPs and CBPs, we then consider the subclass of CBPs with deterministic control functions (DCBPs), stating a necessary and sufficient condition for DCBP-PSDBP equivalence. Finally, we derive an upper bound on the total variation distance between non-equivalent DCBPs and PSDBPs with matching first and second moments and equal initial population size, and show that under certain conditions this bound tends to zero as the initial population size becomes large.

We derive the first conditionally consistent estimators for a class of parametric Markov population models with logistic growth, which are suitable for modelling endangered populations in restricted habitats with a carrying capacity. We focus on discrete-time parametric population-size-dependent branching processes, for which we propose a new class of weighted least-squares estimators based on a single trajectory of population size counts. We establish the consistency and asymptotic normality of our estimators, conditional on non-extinction up to time $n$, as $n\to\infty$. Since Markov population models with a carrying capacity become extinct almost surely under general conditions, our proofs rely on arguments distinct from those in the existing literature. Our results are motivated by conservation biology, where endangered populations are often studied precisely because they are still alive, leading to an observation bias. Through simulated examples, we show that our conditionally consistent estimators generally reduce this bias for key quantities such as a habitat's carrying capacity. We apply our methodology to estimate the carrying capacity of the Chatham Island black robin, a population reduced to a single breeding female in the 1970's, which has since recovered but has yet to reach the island's carrying capacity.

We consider a class of graph-valued stochastic processes in which each vertex has a type that fluctuates randomly over time. Collectively, the paths of the vertex types up to a given time determine the probabilities that the edges are active or inactive at that time. Our focus is on the evolution of the associated empirical graphon in the limit as the number of vertices tends to infinity, in the setting where fluctuations in the graph-valued process are more likely to be caused by fluctuations in the vertex types than by fluctuations in the states of the edges given these types. We derive both sample-path large deviation principles and convergence of stochastic processes. We demonstrate the flexibility of our approach by treating a class of stochastic processes where the edge probabilities depend not only on the fluctuations in the vertex types but also on the state of the graph itself., Comment: 41 pages, 4 figures

This paper considers the Cram\'er-Lundberg model, with the additional feature that the number of clients can fluctuate over time. Clients arrive according to a Poisson process, where the times they spend in the system form a sequence of independent and identically distributed non-negative random variables. While in the system, every client generates claims and pays premiums. In order to describe the model's rare-event behaviour, we establish a sample-path large-deviation principle. This describes the joint rare-event behaviour of the reserve-level process and the client-population size process. The large-deviation principle can be used to determine the decay rate of the time-dependent ruin probability as well as the most likely path to ruin. Our results allow us to determine whether the chance of ruin is greater with more or with fewer clients and, more generally, to determine to what extent a large deviation in the reserve-level process can be attributed to an unusual outcome of the client-population size process., Comment: 28 pages, 2 figures; v2-minor revision

We consider Galton-Watson branching processes with countable typeset $\mathcal{X}$. We study the vectors ${\bf q}(A)=(q_x(A))_{x\in\mathcal{X}}$ recording the conditional probabilities of extinction in subsets of types $A\subseteq \mathcal{X}$, given that the type of the initial individual is $x$. We first investigate the location of the vectors ${\bf q}(A)$ in the set of fixed points of the progeny generating vector and prove that $q_x(\{x\})$ is larger than or equal to the $x$th entry of any fixed point, whenever it is different from 1. Next, we present equivalent conditions for $q_x(A)< q_x (B)$ for any initial type $x$ and $A,B\subseteq \mathcal{X}$. Finally, we develop a general framework to characterise all \emph{distinct} extinction probability vectors, and thereby to determine whether there are finitely many, countably many, or uncountably many distinct vectors. We illustrate our results with examples, and conclude with open questions., Comment: 33 pages, 7 figures

We consider a dynamic Erd\H{o}s-R\'enyi random graph (ERRG) on $n$ vertices in which each edge switches on at rate $\lambda$ and switches off at rate $\mu$, independently of other edges. The focus is on the analysis of the evolution of the associated empirical graphon in the limit as $n\to\infty$. Our main result is a large deviation principle (LDP) for the sample path of the empirical graphon observed until a fixed time horizon. The rate is $\binom{n}{2}$, the rate function is a specific action integral on the space of graphon trajectories. We apply the LDP to identify (i) the most likely path that starting from a constant graphon creates a graphon with an atypically large density of $d$-regular subgraphs, and (ii) the mostly likely path between two given graphons. It turns out that bifurcations may occur in the solutions of associated variational problems.

We consider population-size-dependent branching processes (PSDBPs) which eventually become extinct with probability one. For these processes, we derive maximum likelihood estimators for the mean number of offspring born to individuals when the current population size is $z\geq 1$. As is standard in branching process theory, an asymptotic analysis of the estimators requires us to condition on non-extinction up to a finite generation $n$ and let $n\to\infty$; however, because the processes become extinct with probability one, we are able to demonstrate that our estimators do not satisfy the classical consistency property ($C$-consistency). This leads us to define the concept of $Q$-consistency, and we prove that our estimators are $Q$-consistent and asymptotically normal. To investigate the circumstances in which a $C$-consistent estimator is preferable to a $Q$-consistent estimator, we then provide two $C$-consistent estimators for subcritical Galton-Watson branching processes. Our results rely on a combination of linear operator theory, coupling arguments, and martingale methods.

We present a rather general method for proving local limit theorems, with a good rate of convergence, for sums of dependent random variables. The method is applicable when a Stein coupling can be exhibited. Our approach involves both Stein's method for distributional approximation and Stein's method for concentration. As applications, we prove local central limit theorems with rate of convergence for the number of germs with $d$ neighbours in a germ--grain model, and the number of degree-$d$ vertices in an Erd\H{o}s--R\'enyi random graph. In both cases, the error rate is optimal, up to logarithmic factors., Comment: Ver2: 33 pages, minor revision; Ver 1: 32 pages

We consider a class of multitype Galton-Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, the probability $\boldsymbol{q}(A)$ of extinction in subsets of types $A\subseteq \mathcal{X}_d$ may differ from the global extinction probability $\boldsymbol{q}$ and the partial extinction probability $\tilde{\boldsymbol{q}}$. After deriving partial and global extinction criteria, we develop conditions for $\boldsymbol{q}<\boldsymbol{q}(A)<\tilde{\boldsymbol{q}}$. We then present an iterative method to compute the vector $\boldsymbol{q}(A)$ for any set $A$. Finally, we investigate the location of the vectors $\boldsymbol{q}(A)$ in the set of fixed points of the progeny generating vector.

We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton-Watson processes with typeset $\mathcal{X}=\{0,1,2,\dots\}$, in which individuals of type $i$ may give birth to offspring of type $j\leq i+1$ only. For this class of processes, we study the set $S$ of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector $\boldsymbol{q}$ and whose maximum is the partial extinction probability vector $\boldsymbol{\tilde{q}}$. In the case where $\boldsymbol{\tilde{q}}=\boldsymbol{1}$, we derive a global extinction criterion which holds under second moment conditions, and when $\boldsymbol{\tilde{q}}<\boldsymbol{1}$ we develop necessary and sufficient conditions for $\boldsymbol{q}=\boldsymbol{\tilde{q}}$.

We consider the extinction events of Galton-Watson processes with countably infinitely many types. In particular, we construct truncated and augmented Galton-Watson processes with finite but increasing sets of types. A pathwise approach is then used to show that, under some sufficient conditions, the corresponding sequence of extinction probability vectors converges to the global extinction probability vector of the Galton-Watson processes with countably infinitely many types. This gives rise to a number of iterative methods for the computation of the global extinction probability vector., Comment: 35 pages, 11 figures; References 4 and 6 updated