Supercritical Markov branching processes with general set of types

This paper concerns the asymptotic behaviour of normalized averaging processes associated with a supercritical, indecomposable Markov branching process. Results wellknown in case of a finite set of types are extended to processes with an arbitrary set of types. The process parameter is allowed to be discrete or continuous.Convergence in the quadratic mean is proved on the basis of a weak form of positive regularity. In this setting, limit variables corresponding to different averaging functions are proportional almost everywhere. The rate of convergence is such that process skeletons, defined by uniform partitions of the parameter set, converge with probability 1. Starting from the almost sure convergence of skeletons, we obtain almost sure convergence of the processes themselves. The final sections deals with properties of the limiting distribution functions, in particular with the possible jump at the origin and the existence of a continuous density everywhere else.Other investigations of supercritical Markov branching processes with an infinite or arbitrary set of types are to be found in [2, 3, 4, 6, 7, 13, 15, 19, 21, 23], and [24].The author was supported by DFG grant HE 678/1.     

Shift Self-Similar Additive Random Sequences Associated with Supercritical Branching Processes

Natural examples of increasing shift self-similar additive random sequences are constructed, which are associated with supercritical branching processes. The rate of growth and the distributional properties of them are studied in terms of the offspring distributions of the supercritical branching processes. The results are applied to two types of laws of the iterated logarithm for a Brownian motion on the unbounded Sierpinski gasket. An extension of the Bingham–Doney–de Meyer theorem on the limits of supercritical branching processes is also proved.     

Fluctuation limits of site-dependent branching systems in critical and large dimensions

In this paper, some limit processes of occupation time fluctuations of the branching particle systems with varied branching laws from site to site are obtained. The results show that the varied branching laws can not affect the limit processes and the scaling parameters in the case of large dimensions, but in the case of critical dimensions, under suitable assumptions, it changes the limit processes with simple and isotropic spatial structures to those with complicated and anisotropic spatial structures and gives log corrections in the scaling parameters.     

Continuous branching processes and spectral positivity

Certain properties of continuous-state branching processes are studied via the random time-change linking them with spectrally positive Lévy processes. The results are compared and contrasted with those for simple branching processes.     

Large scale localization of a spatial version of Neveu’s branching process

Recently a spatial version of Neveu’s (1992) continuous-state branching process was constructed by Fleischmann and Sturm (2004). This superprocess with infinite mean branching behaves quite differently from usual supercritical spatial branching processes. In fact, at macroscopic scales, the mass renormalized to a (random) probability measure is concentrated in a single space point which randomly fluctuates according to the underlying symmetric stable motion process.     

Catalytic Discrete State Branching Models and Related Limit Theorems

Catalytic discrete state branching processes with immigration are defined as strong solutions of stochastic integral equations. We provide main limit theorems of those processes using different scalings. The class of limit processes of the theorems includes essentially all continuous state catalytic branching processes and spectrally positive regular affine processes.      

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首先, 当$Q$是一个拟单调的q矩阵的时候, 我们找出最小的$Q$函数是一个Feller的转移函数的准则. 然后我们把这个结论应用于生成分支q矩阵并得到相应的生成分支过程的Feller准则. 特别地, 设$\theta$是分支q矩阵中的非线性数, 总是存在一个分点$\theta_0$满足$1\leq\theta_0\leq2$或$\theta_0<+\infty$使得 生成分支过程是否是Feller的要依据$\theta<\theta_0$或者$\theta>\theta_0$.     

Occupation Time Fluctuations of Weakly Degenerate Branching Systems

We establish limit theorems for rescaled occupation time fluctuations of a sequence of branching particle systems in ? ^d with anisotropic space motion and weakly degenerate splitting ability. In the case of large dimensions, our limit processes lead to a new class of operator-scaling Gaussian random fields with nonstationary increments. In the intermediate and critical dimensions, the limit processes have spatial structures analogous to (but more complicated than) those arising from the critical branching particle system without degeneration considered by Bojdecki et?al. (Stoch. Process. Appl. 116:1?C18 and 19?C35, 2006). Due to the weakly degenerate branching ability, temporal structures of the limit processes in all three cases are different from those obtained by Bojdecki et?al. (Stoch. Process. Appl. 116:1?C18 and 19?C35, 2006).     

Discontinuous Measure-Valued Branching Processes and Generalized Stochastic Equations

We study a class of integrable and discontinuous measure-valued branching processes. They are constructed as limits of renormalized spatial branching processes, the underlying branching distribution belonging to the domain of attraction of a stable law. These processes, computed on a test function f, are semimartingales whose martingale terms are identified with integrals of f with respect to a martingale measure. According to a representation theorem of continuous (respectively purely discontinuous) martingale measures as stochastic integrals with respect to a white noise (resp. to a POISSON process), we prove that the measure-valued processes that we consider are solutions of stochastic differential equations in the space of L² (Ω)-valued vector measures.     

Limit theorems for sums determined by branching and other exponentially growing processes

A branching process counted by a random characteristic has been defined as a process which at time t is the superposition of individual stochastic processes evaluated at the actual ages of the individuals of a branching population. Now characteristics which may depend not only on age but also on absolute time are considered. For supercritical processes a distributional limit theorem is proved, which implies that classical limit theorems for sums of characteristics evaluated at a fixed age point transfer into limit theorems for branching processes counted by these characteristics. A point is that, though characteristics of different individuals should be independent, the characteristics of an individual may well interplay with the reproduction of the latter. The result requires a sort of L^p-continuity for some 1 ? p ? 2. Its proof turns out to be valid for a wider class of processes than branching ones.For the case p = 1 a number of Poisson type limits follow and for p = 2 some normality approximations are concluded. For example results are obtained for processes for rare events, the age of the oldest individual, and the error of population predictions.This work has been supported by a grant from the Swedish Natural Science Research Council.     

Conditional log-Laplace functional for a class of branching processes in random environments

A conditional log-Laplace functional(CLLF) for a class of branching processes in random environments is derived. The basic idea is the decomposition of a dependent branching dynamic into a no-interacting branching and an interacting dynamic generated by the random environments.CLLF will play an important role in the investigation of branching processes and superprocesses with interaction.     

Law of Large Numbers for Branching Symmetric Hunt Processes with Measure-Valued Branching Rates

We establish weak and strong laws of large numbers for a class of branching symmetric Hunt processes with the branching rate being a smooth measure with respect to the underlying Hunt process, and the branching mechanism being general and state dependent. Our work is motivated by recent work on the strong law of large numbers for branching symmetric Markov processes by Chen and Shiozawa (J Funct Anal 250:374–399, 2007) and for branching diffusions by Engländer et al. (Ann Inst Henri Poincaré Probab Stat 46:279–298, 2010). Our results can be applied to some interesting examples that are covered by neither of these papers.     

Weak Convergence Results for Multiple Generations of a Branching Process

We establish limit theorems involving weak convergence of multiple generations of critical and supercritical branching processes. These results arise naturally when dealing with the joint asymptotic behavior of functionals defined in terms of several generations of such processes. Applications of our main result include a functional central limit theorem (CLT), a Darling–Erdös result, and an extremal process result. The limiting process for our functional CLT is an infinite dimensional Brownian motion with sample paths in the infinite product space (C ₀[0,1])^∞, with the product topology, or in Banach subspaces of (C ₀[0,1])^∞ determined by norms related to the distribution of the population size of the branching process. As an application of this CLT we obtain a central limit theorem for ratios of weighted sums of generations of a branching processes, and also to various maximums of these generations. The Darling–Erdös result and the application to extremal distributions also include infinite-dimensional limit laws. Some branching process examples where the CLT fails are also included.     

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.     

Applications of Feller–Reuter–Riley Transition Functions

A Feller–Reuter–Riley function is a Markov transition function whose corresponding semigroup maps the set of the real-valued continuous functions vanishing at infinity into itself. The aim of this paper is to investigate applications of such functions in the dual problem, Markov branching processes, and the Williams-matrix. The remarkable property of a Feller–Reuter–Riley function is that it is a Feller minimal transition function with a stable q-matrix. By using this property we are able to prove that, in the theory of branching processes, the branching property is equivalent to the requirement that the corresponding transition function satisfies the Kolmogorov forward equations associated with a stable q-matrix. It follows that the probabilistic definition and the analytic definition for Markov branching processes are actually equivalent. Also, by using this property, together with the Resolvent Decomposition Theorem, a simple analytical proof of the Williams' existence theorem with respect to the Williams-matrix is obtained. The close link between the dual problem and the Feller–Reuter–Riley transition functions is revealed. It enables us to prove that a dual transition function must satisfy the Kolmogorov forward equations. A necessary and sufficient condition for a dual transition function satisfying the Kolmogorov backward equations is also provided.     

A LIMIT THEOREM FOR INTERACTING MEASURE-VALUED BRANCHING PROCESSES

1IntroductionItiswellknownthatthe(classical)DW-superprocessiscloselyassociatedwiththe(c1assical)FV-superprocess.Tl1isconnectionisfiIStobservedbyKonnoandShiga(1988),rigorouslyprovedbyEtheridgeandMarch(1991),andgeneralizedbyPerkins(l992).Infact,intermofapproximationofparticlesystemsthisrelationshipisratherintuitive,buttoproveitisnoteasy.I11thispaper,weshallconsiderthesanequestionforthemeasure-valuedprocesseswithinteractions.Forthesakeofbrevity,weomittheintroductionoftl1iskindofmeasure-value…     

Interacting Fisher–Wright diffusions in a catalytic medium

We study the longtime behaviour of interacting systems in a randomly fluctuating (space–time) medium and focus on models from population genetics. There are two prototypes of spatial models in population genetics: spatial branching processes and interacting Fisher–Wright diffusions. Quite a bit is known on spatial branching processes where the local branching rate is proportional to a random environment (catalytic medium). Here we introduce a model of interacting Fisher–Wright diffusions where the local resampling rate (or genetic drift) is proportional to a catalytic medium. For a particular choice of the medium, we investigate the longtime behaviour in the case of nearest neighbour migration on the d-dimensional lattice. While in classical homogeneous systems the longtime behaviour exhibits a dichotomy along the transience/recurrence properties of the migration, now a more complicated behaviour arises. It turns out that resampling models in catalytic media show phenomena that are new even compared with branching in catalytic medium. Received: 15 November 1999 / Revised version: 16 June 2000 / Published online: 6 April 2001     

On Renewal Matrices Connected with Branching Processes with Tails of Distributions of Different Orders

We study irreducible renewal matrices generated by matrices whose rows are proportional to various distribution functions. Such matrices arise in studies of multi-dimensional critical Bellman–Harris branching processes. Proofs of limit theorems for such branching processes are based on asymptotic properties of a chosen family of renewal matrices. In the theory of branching processes, unsolved problems are known that correspond to the case in which the tails of some of the above mentioned distribution functions are integrable, while the other distributions lack this property.We assume that the heaviest tails are regularly varying at the infinity with parameter ?β ∈ [?1, 0) and asymptotically proportional, while the other tails are infinitesimal with respect to them. Under a series of additional conditions, we describe asymptotic properties of the first and second order increments for the renewal matrices.     

Decay parameter and related properties of n-type branching processes

We consider decay properties including the decay parameter, invariant measures, invariant vectors, and quasistationary distributions for n-type Markov branching processes on the basis of the 1-type Markov branching processes and 2-type Markov branching processes. Investigating such behavior is crucial in realizing life period of branching models. In this paper, some important properties of the generating functions for n-type Markov branching q-matrix are firstly investigated in detail. The exact value of the decay parameter λC of such model is given for the communicating class C = Zn+ \ 0. It is shown that this λC can be directly obtained from the generating functions of the corresponding q-matrix. Moreover, the λC -invariant measures/vectors and quasi-distributions of such processes are deeply considered. λC -invariant measures and quasi-stationary distributions for the process on C are presented.     

A continuous-state polynomial branching process

A continuous-state polynomial branching process is constructed as the pathwise unique solution of a stochastic integral equation with absorbing boundary condition. The process can also be obtained from a spectrally positive Lévy process through Lamperti type transformations. The extinction and explosion probabilities and the mean extinction and explosion times are computed explicitly. Some of those are also new for the classical linear branching process. We present necessary and sufficient conditions for the process to extinguish or explode in finite times. In the critical or subcritical case, we give a construction of the process coming down from infinity. Finally, it is shown that the continuous-state polynomial branching process arises naturally as the rescaled limit of a sequence of discrete-state processes.