Miguel González

This paper establishes general sufficient conditions for a sequence of controlled branching processes to converge weakly on the Skorokhod space. We focus on a class of control mechanisms that extend previous results by decomposing those random variables into the sum of two independent components: an immigration term, which depends on the current population size, and a size-divisible term, which can be expressed as the sum of random contributions from each individual. This extension allows us to capture a broad range of control functions including Poisson, binomial, and negative binomial distributions, commonly used in the literature. The assumptions are formulated in terms of probability generating functions of the offspring and control laws, distinguishing in this latter between the immigration and the size-divisible parts. The limit process is shown to be a continuous-state branching process with dependent immigration. The proof essentially relies on tightness arguments and the identification of a martingale problem. We also identify the special case in which the limit reduces to a classical Feller branching diffusion with immigration.

We establish general sufficient conditions for a sequence of controlled branching processes to converge weakly on the Skorokhod space. We focus on a class of controlled random variables that extends previous results by considering them as a sum of an immigration size-dependent term and a size-divisible term. Our assumptions are established in terms of the probability generating functions of the offspring and control distributions, distinguishing in this latter case between the immigration and the size-divisible parts. The limit process is a continuous-state process with dependent immigration.

Branching processes form an important family of stochastic processes that have been successfully applied in many fields. In this paper, we focus our attention on controlled multi-type branching processes (CMBPs). A Feller-type diffusion approximation is derived for some critical CMBPs. Namely, we consider a sequence of appropriately scaled random step functions formed from a critical CMBP with control distributions having expectations that satisfy a kind of linearity assumption. It is proved that such a sequence converges weakly toward a squared Bessel process supported by a ray determined by an eigenvector of a matrix related to the offspring mean matrix and the control distributions of the branching process in question. As applications, among others, we derive Feller-type diffusion approximations of critical, primitive multi-type branching processes with immigration and some two-sex branching processes. We also describe the asymptotic behaviour of the relative frequencies of distinct types of individuals for critical CMBPs.

The aim of this paper is to introduce a multitype branching process with random migration following the research initiated with the Galton-Watson process with migration introduced in [Yanev & Mitov (1980) C. R. Acad. Bulg. Sci. 33(4):473-475]. We focus our attention in what we call the critical case. Sufficient conditions are provided for the process to have unlimited growth or not. Furthermore, using suitable normalizing sequences, we study the asymptotic distribution of the process. Finally, we obtain a Feller-type diffusion approximation.

The aim of this paper is to introduce a multitype branching process with random migration following the research initiated with the Galton–Watson process with migration introduced in [N. M. Yanev and K. V. Mitov, Controlled branching processes: The case of random migration, C. R. Acad. Bulgare Sci. 33 1980, 4, 473–475]. We focus our attention in what we call the critical case. Sufficient conditions are provided for the process to have unlimited growth or not. Furthermore, using suitable normalizing sequences, we study the asymptotic distribution of the process. Finally, we obtain a Feller-type diffusion approximation.

Branching processes form an important family of stochastic processes that have been successfully applied in many fields. In this paper, we focus our attention on controlled multi-type branching processes (CMBPs). A Feller-type diffusion approximation is derived for some critical CMBPs. Namely, we consider a sequence of appropriately scaled random step functions formed from a critical CMBP with control distributions having expectations that satisfy a kind of linearity assumption. It is proved that such a sequence converges weakly toward a squared Bessel process supported by a ray determined by an eigenvector of a matrix related to the offspring mean matrix and the control distributions of the branching process in question. As applications, among others, we derive Feller-type diffusion approximations of critical, primitive multi-type branching processes with immigration and some two-sex branching processes. We also describe the asymptotic behaviour of the relative frequencies of distinct types of individuals for critical CMBPs.

The 5th International Workshop on Branching Processes and their Applications (IWBPA2021) was held on 6th, 8th, 13th, 15th, 20th and 22nd April 2021. It was the last in a series of meetings held eve...

A controlled branching process (CBP) is a modification of the standard Bienayme-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.

Our purpose is to estimate the posterior distribution of the parameters of interest for controlled branching processes (CBPs) without prior knowledge of the maximum number of offspring that an individual can give birth to and without explicit likelihood calculations. We consider that only the population sizes at each generation and at least the number of progenitors of the last generation are observed, but the number of offspring produced by any individual at any generation is unknown. The proposed approach is twofold. Firstly, to estimate the maximum progeny per individual we make use of an approximate Bayesian computation (ABC) algorithm for model choice and based on sequential importance sampling with the raw data. Secondly, given such an estimate and taking advantage of the simulated values of the previous stage, we approximate the posterior distribution of the main parameters of a CBP by applying the rejection ABC algorithm with an appropriate summary statistic and a post-processing adjustment. The accuracy of the proposed method is illustrated by means of simulated examples developed with the statistical software R. Moreover, we apply the methodology to two real datasets describing populations with logistic growth. To this end, different population growth models based on CBPs are proposed for the first time.