Galton-Watson Research Articles

Perron–Frobenius theory for kernels and Crump–Mode–Jagers processes with macro-individuals

AbstractPerron–Frobenius theory developed for irreducible non-negative kernels deals with so-called R-positive recurrent kernels. If the kernel M is R-positive recurrent, then the main result determines the limit of the scaled kernel iterations $R^nM^n$ as $n\to\infty$ . In Nummelin (1984) this important result is proven using a regeneration method whose major focus is on M having an atom. In the special case when $M=P$ is a stochastic kernel with an atom, the regeneration method has an elegant explanation in terms of an associated split chain. In this paper we give a new probabilistic interpretation of the general regeneration method in terms of multi-type Galton–Watson processes producing clusters of particles. Treating clusters as macro-individuals, we arrive at a single-type Crump–Mode–Jagers process with a naturally embedded renewal structure.

Some Dissimilarity Measures of Branching Processes and Optimal Decision Making in the Presence of Potential Pandemics.

We compute exact values respectively bounds of dissimilarity/distinguishability measures–in the sense of the Kullback-Leibler information distance (relative entropy) and some transforms of more general power divergences and Renyi divergences–between two competing discrete-time Galton-Watson branching processes with immigration GWI for which the offspring as well as the immigration (importation) is arbitrarily Poisson-distributed; especially, we allow for arbitrary type of extinction-concerning criticality and thus for non-stationarity. We apply this to optimal decision making in the context of the spread of potentially pandemic infectious diseases (such as e.g., the current COVID-19 pandemic), e.g., covering different levels of dangerousness and different kinds of intervention/mitigation strategies. Asymptotic distinguishability behaviour and diffusion limits are investigated, too.

Galton–Watson processes in varying environment and accessibility percolation

This paper deals with branching processes in varying environment with selection, where the offspring distribution depends on the generation and every particle has a random fitness which can only increase along genealogical lineages (descendants with small fitness do not survive). We view the branching process in varying environment (BPVE) as a particular example of branching random walk. We obtain conditions for the survival or extinction of a BPVE (with or without selection), using fixed point techniques for branching random walks. These conditions rely only on the first and second moments of the offspring distributions. Our results can be interpreted in terms of accessibility percolation on Galton-Watson trees. In particular, we obtain that there is no accessibility percolation on almost every Galton-Watson tree where the expected number of offspring grows sublinearly in time, while superlinear growths allows percolation. This result is in agreement with what was found for deterministic trees in Nowak and Krug (Europhysics Letters 101 (2013) 66004).

Determining the effects of transplacental and direct transmission on the probability of persistence in a bluetongue virus model in temperate and tropical regions

Bluetongue virus (BTV) is an important concern for the ruminant livestock industry owing to its high morbidity rate. It is mainly transmitted by a midge to a host (vectorial transmission) but even though its replication in the midge ceases during winter in temperate countries, outbreaks re-emerge in the next season. Transplacental and direct transmission have been proposed as mechanisms that facilitate its overwintering (persistence), but to date, their effects on the probability of BTV persistence are not fully understood. In addition, the effects of these mechanisms on the probability of BTV persistence in tropical regions where the virus is endemic are not clear. A deterministic model that accounts for BTV transmission to cattle is formulated and analysed. Using the next generation approach the basic reproduction number (R0) is determined. When R0<1, the model exhibits a backward bifurcation indicating that the virus persists. When R0>1, a continuous-time Markov chain (CTMC) model derived from the deterministic model is used to estimate the probability of BTV persistence. By approximating the CTMC model with a multitype Galton–Watson branching process, it is shown that both mechanisms can have a high and a low positive effect on the probability of BTV persistence in temperate and tropical regions, respectively. These results indicate the importance of transplacental and direct transmission for BTV persistence in temperate regions, but not in tropical regions where vectorial transmission is the dominant mechanism as expected.

The evaluation of a weighted sum of Gauss hypergeometric functions and its connection with Galton–Watson processes

We evaluate a weighted sum of Gauss hypergeometric functions for certain ranges of the argument, weights and parameters. We establish the domain of absolute convergence of this series by determining the growth of the hypergeometric function for large summation index. We present an application to Galton–Watson branching processes arising in the theory of stochastic processes. We introduce a new class of positive integer-valued distributions with power tails.

Critical Galton-Watson branching processes with a countable set of types and infinite second moments

We consider an indecomposable Galton-Watson branching process with a countable set of types. Assuming that the process is critical and may have infinite variance of the offspring sizes of some (or all) types of particles we describe the asymptotic behaviour of the survival probability of the process and establish a Yaglom-type conditional limit theorem for the infinite-dimensional vector of the number of particles of all types. Bibliography: 20 titles.

Parameter estimation for discretely observed linear birth-and-death processes.

Birth-and-death processes are widely used to model the development of biological populations. Although they are relatively simple models, their parameters can be challenging to estimate, as the likelihood can become numerically unstable when data arise from the most common sampling schemes, such as annual population censuses. A further difficulty arises when the discrete observations are not equi-spaced, for example, when census data are unavailable for some years. We present two approaches to estimating the birth, death, and growth rates of a discretely observed linear birth-and-death process: via an embedded Galton-Watson process and by maximizing a saddlepoint approximation to the likelihood. We study asymptotic properties of the estimators, compare them on numerical examples, and apply the methodology to data on monitoredpopulations.

La guerre de 1914-1918 et la disparition des noms de famille en France : un essai d'estimation à partir du modèle Galton-Watson

Aucune étude en France ne s’est intéressée à l’éventuelle réduction du stock patronymique imputable à la guerre de 1914-1918. L’objet de ce travail est d’estimer l’ampleur du phénomène en prenant en compte les effets aléatoires de la disparition des noms selon le modèle d’extinction de Galton-Watson, la distribution du nombre d’enfants par ménage et la proportion de patronymes selon le nombre de porteurs. Les estimations, sur la France entière et par département, sont établies sur la base du fichier des « Morts pour la France » et du fichier Insee des noms de familles. Si les pertes humaines imputables à la Grande Guerre sont inégalement distribuées sur le territoire, avec un nombre de morts plus élevé au nord d’une ligne La Rochelle-Mulhouse, les pertes patronymiques sont géographiquement différemment réparties et plus nombreuses dans les Pyrénées, en Corse et en Bretagne. Pour autant, les patronymes effectivement disparus ne représentent qu’une proportion faible (1,4%) de l’ensemble des patronymes des MPF. Par département, cette proportion est très variable et souvent en contradiction avec le modèle de Galton- Watson qui prévoit davantage de disparitions. Plusieurs explications sont avancées : elles tiennent au mode de constitution des deux fichiers et aux hypothèses démographiques formulées.

A NOTE ON BISEXUAL GALTON-WATSON BRANCHING PROCESSES IN RANDOM ENVIRONMENTS

A bisexual Galton-Watson branching process is a two-type branching model, in which matings in one generation give rise to random numbers of both males and females in the next. The mating function describes how many mating units are formed from given numbers of males and females. In this paper we consider the case that the distributions of the random numbers of males and females produced by the mating units depend on some fertility parameters evolving randomly in time. By means of a main stochastic comparison result, we show that the total population increases, in some stochastic sense, as the positive dependence between the fertility indexes increases. Simple examples of applications of this result are provided, together with other similar results for a different model of population growth.

A unifying approach to branching processes in a varying environment

AbstractBranching 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 almost sure extinction, square integrability of the martingale $(Z_n/\mathrm E[Z_n])_{n \ge 0}$, 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 $Z_n \gt 0$. The theorems generalize/unify diverse results from the literature and lead to a classification of the processes.

On tail behaviour of stationary second-order Galton–Watson processes with immigration

A second-order Galton-Watson process with immigration can be represented as a coordinate process of a 2-type Galton-Watson process with immigration. Sufficient conditions are derived on the offspring and immigration distributions of a second-order Galton-Watson process with immigration under which the corresponding 2-type Galton-Watson process with immigration has a unique stationary distribution such that its common marginals are regularly varying. In the course of the proof sufficient conditions are given under which the distribution of a second-order Galton-Watson process (without immigration) at any fixed time is regularly varying provided that the initial sizes of the population are independent and regularly varying.

Notes on large deviations for branching processes indexed by a Poisson process

Consider a continuous-time process {ZNt}, where {Zn} is a Galton–Watson process with offspring mean m, and {Nt} is a Poisson process independent of {Zn}. It turns out that Rt := ZNt+1/ZNt is an estimator of m. We deal with large deviation rates for the convergence of Rt to m for the supercritical and critical cases.

Deviations for Martingale Convergence of a Branching Process with Random Index

The purpose of this paper is to obtain large deviations, moderate deviations and normal deviations for the convergence of martingale generated by a supercritical Galton–Watson process with a Poisson process as its time index. Harmonic moments and Shroder index play an important role in the proofs.

Minimax functions on Galton–Watson trees

AbstractWe consider the behaviour of minimax recursions defined on random trees. Such recursions give the value of a general class of two-player combinatorial games. We examine in particular the case where the tree is given by a Galton–Watson branching process, truncated at some depth 2n, and the terminal values of the level 2n nodes are drawn independently from some common distribution. The case of a regular tree was previously considered by Pearl, who showed that as n → ∞ the value of the game converges to a constant, and by Ali Khan, Devroye and Neininger, who obtained a distributional limit under a suitable rescaling.For a general offspring distribution, there is a surprisingly rich variety of behaviour: the (unrescaled) value of the game may converge to a constant, or to a discrete limit with several atoms, or to a continuous distribution. We also give distributional limits under suitable rescalings in various cases.We also address questions of endogeny. Suppose the game is played on a tree with many levels, so that the terminal values are far from the root. To be confident of playing a good first move, do we need to see the whole tree and its terminal values, or can we play close to optimally by inspecting just the first few levels of the tree? The answers again depend in an interesting way on the offspring distribution.We also mention several open questions.

Skeletal stochastic differential equations for continuous-state branching processes

AbstractIt is well understood that a supercritical continuous-state branching process (CSBP) is equal in law to a discrete continuous-time Galton–Watson process (the skeleton of prolific individuals) whose edges are dressed in a Poissonian way with immigration which initiates subcritical CSBPs (non-prolific mass). Equally well understood in the setting of CSBPs and superprocesses is the notion of a spine or immortal particle dressed in a Poissonian way with immigration which initiates copies of the original CSBP, which emerges when conditioning the process to survive eternally. In this article we revisit these notions for CSBPs and put them in a common framework using the well-established language of (coupled) stochastic differential equations (SDEs). In this way we are able to deal simultaneously with all types of CSBPs (supercritical, critical, and subcritical) as well as understanding how the skeletal representation becomes, in the sense of weak convergence, a spinal decomposition when conditioning on survival. We have two principal motivations. The first is to prepare the way to expand the SDE approach to the spatial setting of superprocesses, where recent results have increasingly sought the use of skeletal decompositions to transfer results from the branching particle setting to the setting of measure valued processes. The second is to provide a pathwise decomposition of CSBPs in the spirit of genealogical coding of CSBPs via Lévy excursions, albeit precisely where the aforesaid coding fails to work because the underlying CSBP is supercritical.

Coalescence in the diffusion limit of a Bienaymé–Galton–Watson branching process

We consider the problem of estimating the elapsed time since the most recent common ancestor of a finite random sample drawn from a population which has evolved through a Bienaymé–Galton–Watson branching process. More specifically, we are interested in the diffusion limit appropriate to a supercritical process in the near-critical limit evolving over a large number of time steps. Our approach differs from earlier analyses in that we assume the only known information is the mean and variance of the number of offspring per parent, the observed total population size at the time of sampling, and the size of the sample. We obtain a formula for the probability that a finite random sample of the population is descended from a single ancestor in the initial population, and derive a confidence interval for the initial population size in terms of the final population size and the time since initiating the process. We also determine a joint likelihood surface from which confidence regions can be determined for simultaneously estimating two parameters, (1) the population size at the time of the most recent common ancestor, and (2) the time elapsed since the existence of the most recent common ancestor.

Extinction in lower Hessenberg branching processes with countably many types

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 $\mathcal{X}=\{0,1,2,\ldots\}$, 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 also correct a result in the literature on a sequence of finite extinction probability vectors that converge to the infinite vector $\boldsymbol{\tilde{q}}$.

Penalization of Galton–Watson processes

We apply the penalization technique introduced by Roynette, Vallois, Yor for Brownian motion to Galton–Watson processes with a penalizing function of the form P(x)sx where P is a polynomial of degree p and s∈[0,1]. We prove that the limiting martingales obtained by this method are most of the time classical ones, except in the super-critical case for s=1 (or s→1) where we obtain new martingales. If we make a change of probability measure with this martingale, we obtain a multi-type Galton–Watson tree with p distinguished infinite spines.

Resolving the Protein Folding Problem Using Clustering Partition and Random Walk Theory with Extended Applications in Alzheimer’s Research using Galton Watson Processes and Stochastic Theory

Resolving the Protein Folding Problem Using Clustering Partition and Random Walk Theory with Extended Applications in Alzheimer’s Research using Galton Watson Processes and Stochastic Theory

Moderate deviations for the total population arising from a nearly unstable sub-critical Galton-Watson process with immigration

In this paper, we consider the moderate deviation of the nearly unstable sub-critical Galton-Waston process with immigration for the centered total population arising. We discuss the main influence factors for the form of moderate deviation in different stochastic processes. Moreover, we compute the exact rate function in every different situation of MDP.