Age-dependent birth and death processes

Sbornik: Mathematics, 2021

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.

arXiv (Cornell University), 2020

Our principal aim is to observe the Markov discrete-time process of population growth with long-living trajectory. First we study asymptotical decay of generating function of Galton-Watson process for all cases as the Basic Lemma. Afterwards we get a Differential analogue of the Basic Lemma. This Lemma plays main role in our discussions throughout the paper. Hereupon we improve and supplement classical results concerning Galton-Watson process. Further we investigate properties of the population process so called Q-process. In particular we obtain a joint limit law of Q-process and its total state. And also we prove the analogue of Law of large numbers and the Central limit theorem for total state of Q-process.

Journal of Siberian Federal University. Mathematics & Physics, 2019

We investigate an application of slowly varying functions (in sense of Karamata) in the theory of Galton-Watson branching processes. Consider the critical case so that the generating function of the per-capita offspring distribution has the infinite second moment, but its tail is regularly varying with remainder. We improve the Basic Lemma of the theory of critical Galton-Watson branching processes and refine some well-known limit results.

Stochastic Processes and their Applications, 1999

Let (Z(t): t¿0) be a supercritical age-dependent branching process and let {Yn} be the natural martingale arising in a homogeneous branching random walk. Let Z be the almost sure limit of Z(t)=EZ(t)(t → ∞) or that of Yn (n → ∞). We study the following problems: (a) the absolute continuity of the distribution of Z and the regularity of the density function; (b) the decay rate (polynomial or exponential) of the left tail probability P(Z6x) as x → 0, and that of the characteristic function Ee itZ and its derivative as |t| → ∞; (c) the moments and decay rate (polynomial or exponential) of the right tail probability P(Z ¿ x) as x → ∞, the analyticity of the characteristic function (t) = Ee itZ and its growth rate as an entire characteristic function. The results are established for non-trivial solutions of an associated functional equation, and are therefore also applicable for other limit variables arising in age-dependent branching processes and in homogeneous branching random walks.

Electronic Journal of Probability, 2015

We establish a general sufficient condition for a sequence of Galton-Watson branching processes in varying environments to converge weakly. This condition extends previous results by allowing offspring distributions to have infinite variance. Our assumptions are stated in terms of pointwise convergence of a triplet of two realvalued functions and a measure. The limiting process is characterized by a backwards integro-differential equation satisfied by its Laplace exponent, which generalizes the branching equation satisfied by continuous state branching processes. Several examples are discussed, namely branching processes in random environment, Feller diffusion in varying environments and branching processes with catastrophes.

Comptes Rendus Mathematique, 2013

Let (Z n ) be a supercritical Galton-Watson process, and let W be the limit of the normalized population size Z n /m n , where m = EZ 1 > 1 is the mean of the offspring distribution. Let be a positive function slowly varying at ∞. [4] showed that for α > 1 not an integer,

Stochastic Models, 2005

For the classical subcritical age-dependent branching process the effect of the following two-type immigration pattern is studied. At a sequence of renewal epochs a random number of immigrants enters the population. Each subpopulation stemming from one of these immigrants or one of the ancestors is revived by new immigrants and their offspring whenever it dies out, possibly after an additional delay period. All individuals have the same lifetime distribution and produce offspring according to the same reproduction law. This is the Bellman-Harris process with immigration at zero and immigration of renewal type (BHPIOR). We prove a strong law of large numbers and a central limit theorem for such processes. Similar conclusions are obtained for their discrete-time counterparts (lifetime per individual equals one), called Galton-Watson processes with immigration at zero and immigration of renewal type (GWPIOR). Our approach is based on the theory of regenerative processes, renewal theory and occupation measures and is quite different from those in earlier related work using analytic tools.

2019

We observe the Galton-Watson Branching Processes. Limit properties of transition functions and their convergence to invariant measures are investigated.

Stochastic Models

Coalescence processes have received a lot of attention in the context of conditional branching processes with fixed population size and nonoverlapping generations. Here we focus on similar problems in the context of the standard unconditional Bienaymé-Galton-Watson branching processes, either (sub)-critical or supercritical. Using an analytical tool, we derive the structure of some counting aspects of the ancestral genealogy of such processes, including: the transition matrix of the ancestral count process and an integral representation of various coalescence times distributions, such as the time to most recent common ancestor of a random sample of arbitrary size, including full size. We illustrate our results on two important examples of branching mechanisms displaying either finite or infinite reproduction mean, their main interest being to offer a closed form expression for their probability generating functions at all times. Large time behaviors are investigated.

A multitype Galton-Watson process describes populations of particles that live one season and are then replaced by a random number of children of possibly different types. Biological interpretation of the event that the daughter's type differs form the mother's type is that a mutation has occurred. We study a situation when mutations are rare and, among the types connected in a network, there is a supercritical type allowing the system to escape from extinction. We establish a neat asymptotic structure for the Galton-Watson process escaping extinction due to a sequence of mutations towards the supercritical type. The conditional limit process is a GW process with a multitype immigration stopped after a sequence of geometric times.

Problems of Information Transmission, 2018

2019

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 ...

2011

Branching processes are mathematical models which are applied to the physical and biological sciences. The most famous branching process is a GaltonWatson branching process. In this paper we consider a discrete time GaltonWatson branching process with immigration. It is known that this process is a Markov chain whose state space is a countably infinite set. Discrete time GaltonWatson branching processes with immigration have been described in, for instance, [1, 5, 6, 7]. In the preceding studies, the concrete structure of the limiting distribution and the stationary distribution of the general Galton-Watson branching process with immigration has not been fully investigated. The goal of this study is to find these distributions. In this paper, we find these distributions for the Bernoulli type Galton-Watson branching process with immigration. This is the simplest case. However, it seems that even in this simplest case, to find these distributions is complicated because the transition...

arXiv (Cornell University), 2022

The paper considers the well-known Galton-Watson stochastic branching process. We are dealing with a non-critical case. In the subcritical case, when the mean of the direct descendants of one particle per generation of the time step is less than 1, the population mean of the number of particles on the positive trajectories of the process stabilizes and approaches 1 K, where K is the so-called Kolmogorov constant. The paper is devoted to the search for an explicit expression of this constant depending on the structural parameters of the process. Our reasoning is essentially based on the Basic Lemma, which describes the asymptotic expansion of the generating function of the distribution of the number of particles. An important role is also played by the asymptotic properties of the transition probabilities of the so-called Q-process and their property convergence to invariant measures.

2020

We investigate subcritical Galton-Watson branching processes with immigration in a random environment. Using Goldie's implicit renewal theory we show that under general Cramer condition the stationary distribution has a power law tail. We determine the tail process of the stationary Markov chain, prove point process convergence, and convergence of the partial sums. The original motivation comes from Kesten, Kozlov and Spitzer seminal 1975 paper, which connects a random walk in a random environment model to a special Galton-Watson process with immigration in a random environment. We obtain new results even in this very special setting.