Critical Galton-Watson Process Research Articles

Limit Theorems for Critical Galton-Watson Processes with Immigration Stopped at Zero

Limit Theorems for Critical Galton-Watson Processes with Immigration Stopped at Zero

A Galton–Watson process with a threshold at 1 and an immigration at 0

In this paper we consider a special class of population-size-dependent branching processes, which can also be seen as an extension of Galton–Watson processes with state-dependent immigration (GWPSDI). The model is formulated as follows. Let Zn be the size of individuals belonging to the nth generation of a population. If Zn>1, the population evolves as a critical Galton–Watson process with finite variance; if Zn=1, the population evolves as another Galton–Watson process; if Zn=0, Zn+1 is drawn from a fixed immigration distribution. Based on the technical routes in Foster (1971) and Pakes (1971), some asymptotic results identical with those of GWPSDI are obtained by detailed computations.

Critical branching as a pure death process coming down from infinity

AbstractWe consider the critical Galton–Watson process with overlapping generations stemming from a single founder. Assuming that both the variance of the offspring number and the average generation length are finite, we establish the convergence of the finite-dimensional distributions, conditioned on non-extinction at a remote time of observation. The limiting process is identified as a pure death process coming down from infinity.This result brings a new perspective on Vatutin’s dichotomy, claiming that in the critical regime of age-dependent reproduction, an extant population either contains a large number of short-living individuals or consists of few long-living individuals.

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 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.

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 .

Yaglom’s limit for critical Galton–Watson processes in varying environment: A probabilistic approach

A Galton–Watson process in varying environment is a discrete time branching process where the offspring distributions vary among generations. Based on a two-spine decomposition technique, we provide a probabilistic argument of a Yaglom-type limit for this family processes. The result states that, in the critical case, a suitable normalisation of the process conditioned on non-extinction converges in distribution to a standard exponential random variable.

Harmonic moments and large deviations for a critical Galton-Watson process with immigration

In this paper, a critical Galton-Watson branching process with immigration Zn is studied. We first obtain the convergence rate of the harmonic moment of Zn. Then the large deviation of $${S_{{Z_n}}}≔ \sum\nolimits_{i = 1}^{{Z_n}} {{X_i}} $$ is obtained, where {Xi} is a sequence of independent and identically distributed zero-mean random variables with the tail index α > 2. We shall see that the converging rate is determined by the immigration mean, the variance of reproducing and the tail index of X 1 + , compared with the previous result for the supercritical case, where the rate depends on the Schroder constant and the tail index.

ОБ АСИМПТОТИЧЕСКИХ СООТНОШЕНИЯХ ДЛЯ КРИТИЧЕСКОГО ПРОЦЕССА ГАЛЬТОНА – ВАТСОНА

Determining the asymptotics of the continuation probability for a Galton–Watson branching process is one of the most important problems in the theory of branching processes. This problem was solved by A.N. Kolmogorov (1938) in the case when the process starts with a single particle, and the classical result is obtained. A similar result for continuous branching processes was proved by B.A. Sevastyanov (1951). The next term in the expansion for continuous branching processes was obtained by V.M. Zolotarev (1957). The next term in the expansion for continuous branching processes in the critical case was obtained by V.P. Chistyakov (1957); the asymptotic expansion in the subcritical case under the condition of finiteness of the k-factorial moment was obtained by R. Mukhamedkhanova (1966). Asymptotic expansions for discrete branching processes in the subcritical and supercritical cases, provided that any m-factorial moment is finite, were obtained by S.V. Nagaev and R. Mukhamedkhanova (1966). In the critical case, the weak convergence of the conditional distribution of the quantity P(Z(n) > 0)Z(n) under the condition Z(n) > 0 to the exponential distribution was proved by A.M. Yaglom (1947) for processes starting with a single particle in the case of finiteness of the third moment of the number of generations. Subsequently, Spitzer, Kesten, and Ney (1966) proved this result under the condition that the second moment is finite. A similar result for branching processes with continuous parameters was established by V.M. Zolotarev (1957). In this paper, we study the asymptotics of the probability of continuation of the critical Galton-Watson process, starting with η particles. In addition, we prove an analogue of Yaglom’s theorem for critical Galton – Watson processes starting with a random number of particles.

Barrier estimates for a critical Galton–Watson process and the cover time of the binary tree

Pour le processus critique de Galton–Watson avec loi de reproduction géométrique de la progéniture, nous fournissons des estimations fines de barrière pour des obstacles qui sont des (petites) perturbations de barrières linéaires. Les estimations sont utiles pour analyser le temps de recouvrement, par une marche aleatoire, de graphes finis dans le régime critique, et les temps de recouvrement brownien de variétés bidemensionelles compactes. Comme application des estimations de barrière, nous prouvons que si $C_{L}$ dénote le temps de recouvrement de l’arbre binaire de profondeur $L$ par une marche aleatoire simple, la suite $\sqrt{C_{L}/2^{L+1}}-\sqrt{2\log2}L+\log L/\sqrt{2\log2}$ est tendue. Ce dernier resultat améliore les résultats d’Aldous (J. Math. Anal. Appl. 157 (1991) 271–283), Bramson et Zeitouni (Ann. Probab. 37 (2009) 615–653) et Ding et Zeitouni (Stochastic Process. Appl. 122 (2012) 2117–2133). Dans un article compagnon, nous utilisons ces estimations de barrière pour prouver la tension du temps de recouvrement brownien pour des variétés riemanniennes compactes en deux dimensions.

Coagulation and universal scaling limits for critical Galton–Watson processes

AbstractThe basis of this paper is the elementary observation that then-step descendant distribution of any Galton–Watson process satisfies a discrete Smoluchowski coagulation equation with multiple coalescence. Using this we obtain simple necessary and sufficient criteria for the convergence of scaling limits of critical Galton–Watson processes in terms of scaled family-size distributions and a natural notion of convergence of Lévy triples. Our results provide a clear and natural interpretation, and an alternate proof, of the fact that the Lévy jump measure of certain continuous-state branching processes (CSBPs) satisfies a generalized Smoluchowski equation. (This result was previously proved by Bertoin and Le Gall (2006).) Our analysis shows that the nonlinear scaling dynamics of CSBPs becomelinear and purely dilatationalwhen expressed in terms of the Lévy triple associated with the branching mechanism. We prove a continuity theorem for CSBPs in terms of the associated Lévy triples, and use our scaling analysis to prove the existence ofuniversalcritical Galton–Watson processes and CSBPs analogous to Doeblin's `universal laws'. Namely, these universal processes generateallpossible critical and subcritical CSBPs as subsequential scaling limits. Our convergence results rely on a natural topology for Lévy triples and a continuity theorem for Bernstein transforms (Laplace exponents) which we develop in a self-contained appendix.

LOCAL LARGE DEVIATIONS: McMILLIAN THEOREM FOR MULTITYPE GALTON-WATSON PROCESS

In this article we prove a local large deviation principle (LLDP) for the critical multitype Galton-Watson process from spectral potential point. We define the so-called a spectral potential $U_{\skrik}(\,\cdot,\,\pi)$ for the Galton-Watson process, where $\pi$ is the normalized eigen vector corresponding to the leading \emph{Perron-Frobenius eigen value } $\1$ of the transition matrix $\skria(\cdot,\,\cdot)$ defined from ${\skrik},$ the transition kernel. We show that the Kullback action or the deviation function, $J(\pi,\rho),$ with respect to an empirical offspring measure, $\rho,$ is the Legendre dual of $U_{\skrik}(\,\cdot,\,\pi).$ From the LLDP we deduce a conditional large deviation principle and a weak variant of the classical McMillian Theorem for the multitype Galton-Watson process. To be specific, given any empirical offspring measure $\varpi,$ we show that the number of critical multitype Galton-Watson processes on $n$ vertices is approximately $e^{n\langle \skrih_{\varpi},\,\pi\rangle},$ where $\skrih_{\varpi}$ is a suitably defined entropy.

Wright–Fisher-like models with constant population size on average

We first recall some basic facts from the theory of discrete-time Markov chains arising from two types neutral and non-neutral evolution models of population genetics with constant size. We then define and analyze a version of such models whose fluctuating total population size is conserved on average only. In our model, the population of interest is seen as being embedded in a frame process which is a critical Galton–Watson process. In this context, we address problems such as extinction, fixation, size of the population at fixation and survival probability to a bottleneck effect of the environment.

Statistical inference of 2-type critical Galton–Watson processes with immigration

In this paper the asymptotic behavior of the conditional least squares estimators of the offspring mean matrix for a 2-type critical positively regular Galton–Watson branching process with immigration is described. We also study this question for a natural estimator of the spectral radius of the offspring mean matrix, which we call criticality parameter. We discuss the subcritical case as well.

Asymptotic behavior of CLS estimators for 2-type doubly symmetric critical Galton–Watson processes with immigration

In this paper, the asymptotic behavior of the conditional least squares (CLS) estimators of the offspring means $(\alpha,\beta)$ and of the criticality parameter $\varrho:=\alpha+\beta$ for a $2$-type critical doubly symmetric positively regular Galton-Watson branching process with immigration is described.

On Largest Offspring in a Critical Branching Process with Finite Variance

Continuing the work in Bertoin (2011) we study the distribution of the maximal number X*k of offspring amongst all individuals in a critical Galton‒Watson process started with k ancestors, treating the case when the reproduction law has a regularly varying tail F̅ with index −α for α > 2 (and, hence, finite variance). We show that X*k suitably normalized converges in distribution to a Fréchet law with shape parameter α/2; this contrasts sharply with the case 1< α<2 when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.

On Largest Offspring in a Critical Branching Process with Finite Variance

Continuing the work in Bertoin (2011) we study the distribution of the maximal number X * k of offspring amongst all individuals in a critical Galton‒Watson process started with k ancestors, treating the case when the reproduction law has a regularly varying tail F̅ with index −α for α > 2 (and, hence, finite variance). We show that X * k suitably normalized converges in distribution to a Fréchet law with shape parameter α/2; this contrasts sharply with the case 1< α<2 when the variance is infinite. More generally, we obtain a weak limit theorem for the offspring sequence ranked in decreasing order, in terms of atoms of a certain doubly stochastic Poisson measure.

Total variation error bounds for geometric approximation

We develop a new formulation of Stein's method to obtain computable upper bounds on the total variation distance between the geometric distribution and a distribution of interest. Our framework reduces the problem to the construction of a coupling between the original distribution and the "discrete equilibrium" distribution from renewal theory. We illustrate the approach in four non-trivial examples: the geometric sum of independent, non-negative, integer-valued random variables having common mean, the generation size of the critical Galton-Watson process conditioned on non-extinction, the in-degree of a randomly chosen node in the uniform attachment random graph model and the total degree of both a fixed and randomly chosen node in the preferential attachment random graph model.

Multigraph models for causal quantum gravity and scale dependent spectral dimension

We study random walks on ensembles of a specific class of random multigraphs which provide an ‘effective graph ensemble’ for the causal dynamical triangulation (CDT) model of quantum gravity. In particular, we investigate the spectral dimension of the multigraph ensemble for recurrent as well as transient walks. We investigate the circumstances in which the spectral dimension and Hausdorff dimension are equal and show that this occurs when ρ, the exponent for anomalous behaviour of the resistance to infinity, is zero. The concept of scale dependent spectral dimension in these models is introduced. We apply this notion to a multigraph ensemble with a measure induced by a size biased critical Galton–Watson process which has a scale dependent spectral dimension of two at large scales and one at small scales. We conclude by discussing a specific model related to four dimensional CDT which has a spectral dimension of four at large scales and two at small scales.

Asymptotics of nearly critical Galton-Watson processes with immigration

Asymptotics of nearly critical Galton-Watson processes with immigration