Convergence of branching processes to the local time of a Bessel process (original) (raw)
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Limiting distributions of Galton-Watson branching processes with immigration
Communications on Stochastic Analysis, 2012
In this paper, we introduce a decomposition of the transition probability matrix of Galton-Watson (G-W) branching processes with immigration and applying the decomposition, we give a limiting distribution of the process. Moreover, we extend the definition of this process to that of the multiple G-W branching process, and give an explicit form of a limiting distribution of the multiple G-W branching process.
Asymptotic Behavior of Critical Primitive Multi-Type Branching Processes with Immigration
Stochastic Analysis and Applications, 2014
Under natural assumptions a Feller type diffusion approximation is derived for critical multi-type branching processes with immigration when the offspring mean matrix is primitive (in other words, positively regular). Namely, it is proved that a sequence of appropriately scaled random step functions formed from a sequence of critical primitive multi-type branching processes with immigration converges weakly towards a squared Bessel process supported by a ray determined by the Perron vector of the offspring mean matrix.
Probability and Mathematical Statistics
We investigate limit properties of discrete time branching processes with application of the theory of regularly varying functions in the sense of Karamata. In the critical situation we suppose that the offspring probability generating function has an infinite second moment but its tail regularly varies. In the noncritical case, the finite moment of type Ε[x ln x] is required. The lemma on the asymptotic representation of the generating function of the process and its differential analogue will underlie our conclusions.
ASYMPTOTIC BEHAVIOR OF THE MEASURE VALUED BRANCHING PROCESS WITH IMMIGRATION
1993
The measure-valued branching process with immigration is defined as Y t = X t + I t , t ≥ 0, where X t satisfies the branching property and I t with I 0 = 0 is independent of X t . This formulation leads to the model of . We prove a large number law for Y t . Equilibrium distributions and spatial transformations are also studied.
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.
Limit Theorems for Branching Processes with Immigration in a Random Environment
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.
A note on the asymptotic behaviour of a periodic multitype Galton-Watson branching process
2004
In this work, the problem of the limiting behaviour of an irreducible Multitype Galton-Watson Branching Process with period d greater than 1 is considered. More specifically, almost sure convergence of some linear functionals depending on d consecutive generations is studied under hypothesis of non extinction. As consequence the main parameters of the model are given a convenient interpretation from a practical point of view. For a better understanding of the theoretical results, an illustrative example is provided. 1. Introduction. The Multitype Galton-Watson Branching Process (MP) is a modification of standard Galton-Watson Branching process, in which 2000 Mathematics Subject Classification: 60J80.
Some asymptotic results for near critical branching processes
Near critical single type Bienaymé-Galton-Watson (BGW) processes are considered. It is shown that, under appropriate conditions, Yaglom distributions of suitably scaled BGW processes converge to that of the corresponding diffusion approximation. Convergences of stationary distributions for Q-processes and models with immigration to the corresponding distributions of the associated diffusion approximations are established as well. Although most of the work is concerned with the single type case, similar results for multitype settings can be obtained. As an illustration, convergence of Yaglom distributions of suitably scaled multitype subcritical BGW processes to that of the associated diffusion model is established.
Critical Galton-Watson branching processes with a countable set of types and infinite second moments
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.