Galton-Watson Research Articles

Parameterised branching processes: A functional version of Kesten & Stigum theorem

Let (Zn,n≥0) be a supercritical Galton–Watson process whose offspring distribution μ has mean λ>1 and is such that ∫xlog+(x)dμ(x)<+∞. According to the famous Kesten & Stigum theorem, (Zn/λn) converges almost surely, as n→+∞. The limiting random variable has mean 1, and its distribution is characterised as the solution of a fixed point equation.In this paper, we consider a family of Galton–Watson processes (Zn(λ),n≥0) defined for λ ranging in an interval I⊂(1,∞), and where we interpret λ as the time (when n is the generation). The number of children of an individual at time λ is given by X(λ), where (X(λ))λ∈I is a càdlàg integer-valued process which is assumed to be almost surely non-decreasing and such that E(X(λ))=λ>1 for all λ∈I. This allows us to define Zn(λ) the number of elements in the nth generation at time λ.Set Wn(λ)=Zn(λ)/λn for all n≥0 and λ∈I. We prove that, under some moment conditions on the process X, the sequence of processes (Wn(λ),λ∈I)n≥0 converges in probability as n tends to infinity in the space of càdlàg processes equipped with the Skorokhod topology to a process, which we characterise as the solution of a fixed point equation.

Diffusion approximation of controlled branching processes using limit theorems for random step processes

A controlled branching process (CBP) is a modification of the standard Bienaymé–Galton–Watson process in which the number of progenitors in each generation is determined by a random mechanism. We consider a CBP starting from a random number of initial individuals. The main aim of this article is to provide a Feller diffusion approximation for critical CBPs. A similar result by considering a fixed number of initial individuals by using operator semigroup convergence theorems has been previously proved by Sriram et al. (Stochastic Processes Appl. 2007;117:928–946). An alternative proof is now provided making use of limit theorems for random step processes.

Invariant Galton–Watson branching process for earthquake occurrence

SUMMARY We propose a theoretical modelling framework for earthquake occurrence and clustering based on a family of invariant Galton–Watson (IGW) stochastic branching processes. The IGW process is a rigorously defined approximation to imprecisely observed or incorrectly estimated earthquake clusters modelled by Galton–Watson branching processes, including the Epidemic Type Aftershock Sequence (ETAS) model. The theory of IGW processes yields explicit distributions for multiple cluster attributes, including magnitude-dependent and magnitude-independent offspring number, cluster size and cluster combinatorial depth. Analysis of the observed seismicity in southern California demonstrates that the IGW model provides a close fit to the observed earthquake clusters. The estimated IGW parameters and derived statistics are robust with respect to the catalogue lower cut-off magnitude. The proposed model facilitates analyses of multiple quantities of seismicity based on self-similar tree attributes, and may be used to assess the proximity of seismicity to criticality.

Convergence of partial sum processes to stable processes with application for aggregation of branching processes

We provide a generalization of Theorem 1 in Bartkiewicz et al. (2011) in the sense that we give sufficient conditions for weak convergence of finite dimensional distributions of the partial sum processes of a strongly stationary sequence to the corresponding finite dimensional distributions of a non-Gaussian stable process instead of weak convergence of the partial sums themselves to a non-Gaussian stable distribution. As an application, we describe the asymptotic behaviour of finite dimensional distributions of aggregation of independent copies of a strongly stationary subcritical Galton–Watson branching process with regularly varying immigration having index in (0,1)∪(1,4/3) in a so-called iterated case, namely when first taking the limit as the time scale and then the number of copies tend to infinity.

Capacity of the range of tree-indexed random walk

By introducing a new measure for the infinite Galton–Watson process and providing estimates for (discrete) Green’s functions on trees, we establish the asymptotic behavior of the capacity of critical branching random walks: in high dimensions d≥7, the capacity grows linearly; and in the critical dimension d=6, it grows asymptotically proportional to nlogn.

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−fn(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 N0∪{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.

A Scaling Limit Theorem for Galton–Watson Processes in Varying Environments

We prove a scaling limit theorem for discrete Galton-Watson processes in varying environments. A simple sufficient condition for the weak convergence in the Skorokhod space is given in terms of probability generating functions. The limit theorem gives rise to the continuous-state branching processes in varying environments studied recently by several authors.

A Scaling Limit Theorem for Galton-Watson Processes in Varying Environments

Доказана теорема о предельном распределении нормированных дискретных процессов Гальтона-Ватсона, эволюционирующих в меняющихся средах. В терминах вероятностных производящих функций указано простое достаточное условие, обеспечивающее слабую сходимость таких процессов в пространстве Скорохода. Предельная теорема приводит к ветвящемуся процессу в меняющейся среде с непрерывным множеством состояний, который исследовался в последнее время несколькими авторами.

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).

A Comparison of Deterministic and Stochastic Plant-Vector-Virus Models Based on Probability of Disease Extinction and Outbreak

In this investigation, we formulate and analyse a stochastic epidemic model using the continuous-time Markov chain model for the propagation of a vector-borne cassava mosaic disease in a single population. The stochastic model is based upon a pre-existing deterministic plant-vector-virus model. To see how demographic stochasticity affects the vector-borne cassava mosaic disease dynamics, we compare the disease dynamics of both deterministic and stochastic models through disease extinction process. The probability of disease extinction and therefore the major outbreak are estimated analytically using the multitype Galton-Watson branching process (GWbp) approximation. Also, we have found the approximate probabilities of disease extinction numerically based on 30000 sample paths, and it is shown to be good estimate with the calculated probabilities from GWbp approximation. In particular, it is observed that there is a very high probability of disease extinction when the disease is introduced via the infected vectors rather than through infected plants.

Two-type linear-fractional branching processes in varying environments with asymptotically constant mean matrices

AbstractConsider two-type linear-fractional branching processes in varying environments with asymptotically constant mean matrices. Let $\nu$ be the extinction time. Under certain conditions, we show that both $\mathbb{P}(\nu=n)$ and $\mathbb{P}(\nu&gt;n)$ are asymptotically the same as some functions of the products of spectral radii of the mean matrices. We also give an example for which $\mathbb{P}(\nu=n)$ decays with various speeds such as ${c}/({n^{1/2}\log n)^2}$ , ${c}/{n^\beta}$ , $\beta &gt;1$ , which are very different from those of homogeneous multitype Galton–Watson processes.

Parameter estimation in branching processes with almost sure extinction

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≥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→∞; 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.

On asymptotic structure of critical Galton-Watson branching processes allowing immigration with infinite variance

We consider the Galton-Watson branching process allowing immigration. We are dealing with the critical case, in which the immigration law has infinite mean and the offspring law have an infinite variance. An explicit-integral form of the generating function of a stationary measure for the process without immigration is found. We study the asymptotic properties of transition probabilities and their convergence to stationary measures in the case of processes with immigration, when the process is ergodic. And also we define a rate of speed of this convergence.

Extinction probabilities in branching processes with countably many types: a general framework

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.

Galton-Watson processes and their role as building blocks for branching processes

Эта статья - обзор, одновременно объясняющий и обосновывающий роль процессов Гальтона-Ватсона как инструмента в теории ветвящихся процессов. Для автора статья явилась возможностью отметить вклад в эту область двух выдающихся ученых-специалистов из Российской академии наук и поздравить их с юбилеями. Основной темой статьи является роль, которую процессы Гальтона-Ватсона сыграли в исследованиях автора. Мы начинаем статью с управляемых процессов Гальтона-Ватсона. Затем мы переходим к случайным процессам с поглощениями и рассматриваем связанную с ними задачу, возникающую в медицине. Далее после обсуждения вариантов леммы Бореля-Кантелли мы переходим к $\varphi$-ветвящимся процессам и их обобщениям. В качестве дальнейших обобщений рассматриваются процессы Гальтона-Ватсона с двумя полами. Наконец, кратко упоминаются относительно более сложные ветвящиеся процессы с ресурсными ограничениями, чтобы показать еще раз, что использование (когда это имеет смысл) тех же схем размножения, что и в процессах Гальтона-Ватсона, может быть удобным способом упростить исследование новых процессов и получать новые интересные результаты.

Large Deviation Rates for the Continuous‐Time Supercritical Branching Processes with Immigration

Let {Y(t); t ≥ 0} be a supercritical continuous‐time branching process with immigration; our focus is on the large deviation rates of Y(t) and thus extending the results of the discrete‐time Galton–Watson process to the continuous‐time case. Firstly, we prove that Z(t) = e−mt[Y(t) − ((em(t + 1) − 1)/(em − 1))ea+m] is a submartingale and converges to a random variable Z. Then, we study the decay rates of P(|Z(t) − Z| &gt; ε) as t⟶∞ and P(|(Y(t + v)/Y(t)) − emv| &gt; ε|Z ≥ α) as t⟶∞ for α &gt; 0 and ε &gt; 0 under various moment conditions on {bk; k ≥ 0} and {aj; j ≥ 0}. We conclude that the rates are supergeometric under the assumption of finite moment generation functions.

The asymptotic behavior of an SIR epidemic model: collective Reed-Frost process

The aim of this research is to provide a historical overview of the mathematical theory of epidemics and to study the asymptotic behavior of the final size of a collective Reed-Frost epidemic process with different types of infected people. This model was introduced by Picard and Lefevre [25] provides an extension of the model of Pettigrew and Weiss [24]. Under certain conditions, we show that when the number of the initial susceptible individuals is large and the number of the initial infected people is finite, the infection process is equivalent to a multitype Galton-Watson process. Our method is simple and based on Bernstein polynomials.

A note on asymptotic behavior of critical Galton–Watson processes with immigration

In this somewhat didactic note we give a detailed alternative proof of the known result due to Wei and Winnicki (1989) which states that under second order moment assumptions on the offspring and immigration distributions the sequence of appropriately scaled random step functions formed from a critical Galton-Watson process with immigration (starting from not necessarily zero) converges weakly towards a squared Bessel process. The proof of Wei and Winnicki (1989) is based on infinitesimal generators, while we use limit theorems for random step processes towards a diffusion process due to Ispany and Pap (2010). This technique was already used in Ispany (2008), where he proved functional limit theorems for a sequence of some appropriately normalized nearly critical Galton-Watson processes with immigration starting from zero, where the offspring means tend to its critical value 1. As a special case of Theorem 2.1 in Ispany (2008) one can get back the result of Wei and Winnicki (1989) in the case of zero initial value. In the present note we handle non-zero initial values with the technique used in Ispany (2008), and further, we simplify some of the arguments in the proof of Theorem 2.1 in Ispany (2008) as well.

Haldane\u2019s formula in Cannings models: the case of moderately strong selection

For a class of Cannings models we prove Haldane’s formula, pi (s_N) sim frac{2s_N}{rho ^2}, for the fixation probability of a single beneficial mutant in the limit of large population size N and in the regime of moderately strong selection, i.e. for s_N sim N^{-b} and 0< b<1/2. Here, s_N is the selective advantage of an individual carrying the beneficial type, and rho ^2 is the (asymptotic) offspring variance. Our assumptions on the reproduction mechanism allow for a coupling of the beneficial allele’s frequency process with slightly supercritical Galton–Watson processes in the early phase of fixation.

Critical Galton–Watson Processes with Overlapping Generations

Abstract A properly scaled critical Galton–Watson process converges to a continuous state critical branching process ξ ⁢ ( ⋅ ) \xi(\,{\cdot}\,) as the number of initial individuals tends to infinity. We extend this classical result by allowing for overlapping generations and considering a wide class of population counts. The main result of the paper establishes a convergence of the finite-dimensional distributions for a scaled vector of multiple population counts. The set of the limiting distributions is conveniently represented in terms of integrals ( ∫ 0 y ξ ⁢ ( y - u ) ⁢ d u γ \int_{0}^{y}\xi(y-u)\,du^{\gamma} , y ≥ 0 y\geq 0 ) with a pertinent γ ≥ 0 \gamma\geq 0 .