Bootstrap of the offspring mean in the critical process with a non-stationary immigration

In applications of branching processes, usually it is hard to obtain samples of a large size. Therefore, a bootstrap procedure allowing inference based on a small sample size is very useful. Unfortunately, in the critical branching process with stationary immigration the standard parametric bootstrap is invalid. In this paper, we consider a process with non-stationary immigration, whose mean and variance vary regularly with nonnegative exponents α and β, respectively. We prove that 1+2α is the threshold for the validity of the bootstrap in this model. If β<1+2α, the standard bootstrap is valid and if β>1+2α it is invalid. In the case β=1+2α, the validity of the bootstrap depends on the slowly varying parts of the immigration mean and variance. These results allow us to develop statistical inferences about the parameters of the process in its early stages.     

From Brownian-Time Brownian Sheet to a Fourth Order and a Kuramoto–Sivashinsky-Variant Interacting PDEs Systems

In this paper a bisexual Galton–Watson branching process allowing the immigration of mating units is introduced and its limit behaviour is investigated. For the supercritical case, i.e., asymptotic growth rate greater than one, necessary and sufficient conditions for the almost sure and L ¹ convergence of the suitably normed underlying sequences are given.

     

?-measure for continuous state branching processes and its application

In this paper, we first give a direct construction of the ℕ-measure of a continuous state branching process. Then we prove, with the help of this ℕ-measure, that any continuous state branching process with immigration can be constructed as the independent sum of a continuous state branching process (without immigration), and two immigration parts (jump immigration and continuum immigration). As an application of this construction of a continuous state branching process with immigration, we give a proof of a necessary and sufficient condition, first stated without proof by M. A. Pinsky [Bull. Amer. Math. Soc., 1972, 78: 242–244], for a continuous state branching process with immigration to a proper almost sure limit. As another application of the ℕ-measure, we give a “conceptual” proof of an L log L criterion for a continuous state branching process without immigration to have an L ¹-limit first proved by D. R. Grey [J. Appl. Prob., 1974, 11: 669–677].     

Sevastyanov branching processes with non-homogeneous Poisson immigration

Sevastyanov age-dependent branching processes allowing an immigration component are considered in the case when the moments of immigration form a non-homogeneous Poisson process with intensity r(t). The asymptotic behavior of the expectation and of the probability of non-extinction is investigated in the critical case depending on the asymptotic rate of r(t). Corresponding limit theorems are also proved using different types of normalization. Among them we obtained limiting distributions similar to the classical ones of Yaglom (1947) and Sevastyanov (1957) and also discovered new phenomena due to the non-homogeneity.     

Multitype branching processes with immigration in random environment,and polling systems

For multitype branching processes with immigration evolving in a random environment and producing a final product, we find the tail distribution of the size of the final product accumulated in the process for a life period. Using this result, we investigate the tail distributions of the busy periods of the queueing polling systems of branching type with random service disciplines and random positive switch-over times.     

Some new limit theorems for the critical branching process allowing immigration

Some limit theorems are obtained for the population size of a critical Bienaymé-Galton-Watson process allowing immigration and where the variance of the offspring distribution is infinite. An application is given to a limit theorem for the situation where the immigration does not occur but the population size is conditioned on non-extinction until the remote future. This complements a well-known result of Slack.     

The multitype branching random walk, II

Limit theorems for the multitype branching random walk as n → ∞ are given (n is the generation number) in the case in which the branching process has a mean matrix which is not positive regular. In particular, the existence of steady state distributions is proven in the subcritical case with immigration, and in the critical case with initial Poisson random fields of particles. In the supercritical case, analogues of the limit theorems of Kesten and Stigum are given.     

Self‐Intersection Local Time for 𝒮′(ℝd)‐Ornstein‐Uhlenbeck Processes Arising from Immigration Systems

Fluctuation limits of an immigration branching particle system and an immigration branching measure‐valued process yield different types of 𝒮′(ℝ^d)‐valued Ornstein‐Uhlenbeck processes whose covariances are given in terms of an excessive measure for the underlying motion in Rd, which is taken to be a symmetric α‐stable process. In this paper we prove existence and path continuity results for the self‐intersection local time of these Ornstein‐Uhlenbeck processes. The results depend on relationships between the dimension d and the parameter α.     

Quenouille-type theorem on autocorrelations

The central result is a limit theorem for not necessarily stationary processes resembling AR (p). Assumption of a vector limit distribution for standardized sample autocorrelations leads to the convergence of a vector limit distribution for ordinary sample partial autocorrelations, and to a clear relationship between the two limit distributions. The motivation is the study of the case p=1 by Mills and Seneta (1989, Stochastic Process Appl., 33, 151–161). The central result is used to explain the nature of the relationship between the two results of Quenouille in the classical stationary AR (p) setting.     

Critical Bellman-Harris branching processes with long-living particles

A critical indecomposable two-type Bellman-Harris branching process is considered in which the life-length of the first-type particles has finite variance while the tail of the life-length distribution of the second-type particles is regularly varying at infinity with parameter β ∈ (0, 1]. It is shown that, contrary to the critical indecomposable Bellman-Harris branching processes with finite variances of the life-lengths of particles of both types, the probability of observing first-type particles at a distant moment t is infinitesimally less than the survival probability of the whole process. In addition, a Yaglom-type limit theorem is proved for the distribution of the number of the first-type particles at moment t given that the population contains particles of the first type at this moment.     

Estimation of the offspring mean in a controlled branching process with a random control function

Controlled branching processes (CBP) with a random control function provide a useful way to model generation sizes in population dynamics studies, where control on the growth of the population size is necessary at each generation. An important special case of this process is the well known branching process with immigration. Motivated by the work of Wei and Winnicki [C.Z. Wei, J. Winnicki, Estimation of the mean in the branching process with immigration, Ann. Statist. 18 (1990) 1757–1773], we develop a weighted conditional least squares estimator of the offspring mean of the CBP and derive the asymptotic limit distribution of the estimator when the process is subcritical, critical and supercritical. Moreover, we show the strong consistency of this estimator in all the cases. The results obtained here extend those of Wei and Winnicki [C.Z. Wei, J. Winnicki, Estimation of the mean in the branching process with immigration, Ann. Statist. 18 (1990) 1757–1773] for branching processes with immigration and provide a unified limit theory of estimation.     

随机环境中迁入分枝过程的极限性质

在方差和均值有限的条件下,得到了随机环境中迁入分枝过程对应的规范化过程的几乎处处收敛性和L^2收敛性．这对于刻画过程本身的发散速度,具有重要的意义．     

Analysis of a Recurrence Related to Critical Nonhomogeneous Branching Processes

Abstract

Some classes of controlled branching processes (with nonhomo-geneous migration or with nonhomo-geneous state-dependent immigration) lead in the critical case to a recurrence for the extinction probabilities. Under some additional conditions it is known that this recurrence depends on some parameter β and converges for 0 < β < 1. Now we show that the recurrence does converge for all positive values of the parameter β, which leads to an extension of some limit theorems for the corresponding branching processes. We also give a generalization of the recurrence and an asymptotic analysis of its behavior.     

A central limit theorem for branching Brownian motion with random immigration

We establish a central limit theorem for a branching Brownian motion with random immigration under the annealed law,where the immigration is determined by another branching Brownian motion.The limit is a Gaussian random measure and the normalization is t3/4for d=3 and t1/2for d≥4,where in the critical dimension d=4 both the immigration and the branching Brownian motion itself make contributions to the covariance of the limit.     

Estimation of the mean in partially observed branching processes with general immigration

In the paper we investigate asymptotic properties of the branching process with non-stationary immigration which are sufficient for a natural estimator of the offspring mean based on partial observations to be strongly consistent and asymptotically normal. The estimator uses only a binomially distributed subset of the population of each generation. This approach allows us to obtain results without conditions on the criticality of the process which makes possible to develop a unified estimation procedure without knowledge of the range of the offspring mean. These results are to be contrasted with the existing literature related to i.i.d. immigration case where the asymptotic normality depends on the criticality of the process and are new for the fully observed processes as well. Examples of applications in the process with immigration with regularly varying mean and variance and subcritical processes with i.i.d. immigration are also considered.

     

Simulation Study for the Clan of Ancestors in a Perfect Simulation Scheme of a Continuous One-Dimensional Loss Network

Perfect simulation of a one-dimensional loss network on ℝ with length distribution π and cable capacity C is performed using the clan of ancestors method. Previous works estimated the region of convergence of this scheme using a domination by a branching process. In this work, we show that the domination by the branching process is far from sharp and that there is room for improvement. Moreover, we derive an empirical relation concerning the critical value using simulation studies on the number of rectangles present in the clan of ancestors.      

Asymptotic distribution of the CLSE in a critical process with immigration

It is known that in the critical case the conditional least squares estimator (CLSE) of the offspring mean of a discrete time branching process with immigration is not asymptotically normal. If the offspring variance tends to zero, it is normal with normalization factor n^2/3$n^{2/3}$. We study a situation of its asymptotic normality in the case of non-degenerate offspring distribution for the process with time-dependent immigration, whose mean and variance vary regularly with non-negative exponents α$\alpha$ and β$\beta$, respectively. We prove that if β<1+2α$\beta <1+2\alpha$, the CLSE is asymptotically normal with two different normalization factors and if β>1+2α$\beta >1+2\alpha$, its limit distribution is not normal but can be expressed in terms of the distribution of certain functionals of the time-changed Wiener process. When β=1+2α$\beta =1+2\alpha$ the limit distribution depends on the behavior of the slowly varying parts of the mean and variance.     

Extinction properties of multi-type continuous-state branching processes

Recently in Barczy et al. (2015), the notion of a multi-type continuous-state branching process (with immigration) having $d$-types was introduced as a solution to an $d$-dimensional vector-valued SDE. Preceding that, work on affine processes, originally motivated by mathematical finance, in Duffie et al. (2003) also showed the existence of such processes. See also more recent contributions in this direction due to Gabrielli and Teichmann (2014) and Caballero and Pérez Garmendia (2017). Older work on multi-type continuous-state branching processes is more sparse but includes Watanabe (1969) and Ma (2013), where only two types are considered. In this paper we take a completely different approach and consider multi-type continuous-state branching process, now allowing for up to a countable infinity of types, defined instead as a super Markov chain with both local and non-local branching mechanisms. In the spirit of Engländer and Kypriano (2004) we explore their extinction properties and pose a number of open problems.     

Population Size Dependent Generalized Multitype Branching Processes

Abstract

In this paper, we introduce the population size dependent generalized multitype branching process. This is a Markovian model that allows us to study homogeneous multitype branching processes in a unified way. The basic properties for this model, transitions between its states, as well as the existence of a stationary limiting distribution, are investigated. Finally, we apply the obtained results to a new controlled multitype branching process.