scholarly journals Weak convergence of random processes with immigration at random times

2020 ◽  
Vol 57 (1) ◽  
pp. 250-265
Author(s):  
Congzao Dong ◽  
Alexander Iksanov
Keyword(s):  
Weak Convergence ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Processes ◽  
Branching Random Walk ◽  
General Point ◽  
Shot Noise Process ◽  
Finite Dimensional ◽  
Strong Resemblance ◽  
Random Times

AbstractBy a random process with immigration at random times we mean a shot noise process with a random response function (response process) in which shots occur at arbitrary random times. Such random processes generalize random processes with immigration at the epochs of a renewal process which were introduced in Iksanov et al. (2017) and bear a strong resemblance to a random characteristic in general branching processes and the counting process in a fixed generation of a branching random walk generated by a general point process. We provide sufficient conditions which ensure weak convergence of finite-dimensional distributions of these processes to certain Gaussian processes. Our main result is specialised to several particular instances of random times and response processes.

scholarly journals A Limit Theorem for Discrete Galton-Watson Branching Processes with Immigration

2006 ◽  
Vol 43 (01) ◽  
pp. 289-295 ◽  
Author(s):  
Zenghu Li
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Discrete Time ◽  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Discrete State ◽  
Simple Set ◽  
Continuous State

We provide a simple set of sufficient conditions for the weak convergence of discrete-time, discrete-state Galton-Watson branching processes with immigration to continuous-time, continuous-state branching processes with immigration.

scholarly journals Measure change in multitype branching

2004 ◽  
Vol 36 (2) ◽  
pp. 544-581 ◽  
Author(s):  
J. D. Biggins ◽  
A. E. Kyprianou
Keyword(s):  
Branching Processes ◽  
Sufficient Conditions ◽  
Branching Random Walk ◽  
Stochastic Domination ◽  
Moment Conditions ◽  
Branching Brownian Motion ◽  
Mean Convergence ◽  
Convergence Results ◽  
Watson Process ◽  
The One

The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its expectation. Here, the approach to this theorem pioneered by Lyons, Pemantle and Peres (1995) is extended to certain kinds of martingales defined for Galton-Watson processes with a general type space. Many examples satisfy stochastic domination conditions on the offspring distributions and suitable domination conditions combine nicely with general conditions for mean convergence to produce moment conditions, like the X log X condition of the Kesten-Stigum theorem. A general treatment of this phenomenon is given. The application of the approach to various branching processes is indicated. However, the main reason for developing the theory is to obtain martingale convergence results in a branching random walk that do not seem readily accessible with other techniques. These results, which are natural extensions of known results for martingales associated with binary branching Brownian motion, form the main application.

Strong ergodicity for single-birth processes

2001 ◽  
Vol 38 (01) ◽  
pp. 270-277 ◽  
Author(s):  
Yu-Hui Zhang
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Strong Ergodicity ◽  
Finite Dimensional ◽  
Comparison Methods ◽  
Schlögl Model ◽  
Birth Processes ◽  
Single Birth ◽  
Continuous Time Branching Processes

An explicit and computable criterion for strong ergodicity of single-birth processes is presented. As an application, some sufficient conditions are obtained for strong ergodicity of an extended class of continuous-time branching processes and multi-dimensional Q-processes by comparison methods respectively. Consequently strong ergodicity of the Q-process corresponding to the finite-dimensional Schlögl model is proven.

Weak convergence of sequences of semimartingales with applications to multitype branching processes

1986 ◽  
Vol 18 (1) ◽  
pp. 20-65 ◽  
Author(s):  
A. Joffe ◽  
M. Metivier
Keyword(s):  
Weak Convergence ◽  
Stochastic Processes ◽  
Special Class ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Convergence Of Sequences ◽  
Multitype Branching Processes

The paper is devoted to a systematic discussion of recently developed techniques for the study of weak convergence of sequences of stochastic processes. The methods described make essential use of the semimartingale structure of the processes. Sufficient conditions for tightness including the results of Rebolledo are derived. The techniques are applied to a special class of processes, namely the D-semimartingales. Applications to multitype branching processes are given.

Weak convergence of sequences of semimartingales with applications to multitype branching processes

1986 ◽  
Vol 18 (01) ◽  
pp. 20-65 ◽  
Author(s):  
A. Joffe ◽  
M. Metivier
Keyword(s):  
Weak Convergence ◽  
Stochastic Processes ◽  
Special Class ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Convergence Of Sequences ◽  
Multitype Branching Processes

The paper is devoted to a systematic discussion of recently developed techniques for the study of weak convergence of sequences of stochastic processes. The methods described make essential use of the semimartingale structure of the processes. Sufficient conditions for tightness including the results of Rebolledo are derived. The techniques are applied to a special class of processes, namely the D-semimartingales. Applications to multitype branching processes are given.

scholarly journals Measure change in multitype branching

2004 ◽  
Vol 36 (02) ◽  
pp. 544-581 ◽  
Author(s):  
J. D. Biggins ◽  
A. E. Kyprianou
Keyword(s):  
Branching Processes ◽  
Sufficient Conditions ◽  
Branching Random Walk ◽  
Stochastic Domination ◽  
Moment Conditions ◽  
Branching Brownian Motion ◽  
Mean Convergence ◽  
Convergence Results ◽  
Watson Process ◽  
The One

The Kesten-Stigum theorem for the one-type Galton-Watson process gives necessary and sufficient conditions for mean convergence of the martingale formed by the population size normed by its expectation. Here, the approach to this theorem pioneered by Lyons, Pemantle and Peres (1995) is extended to certain kinds of martingales defined for Galton-Watson processes with a general type space. Many examples satisfy stochastic domination conditions on the offspring distributions and suitable domination conditions combine nicely with general conditions for mean convergence to produce moment conditions, like theXlogXcondition of the Kesten-Stigum theorem. A general treatment of this phenomenon is given. The application of the approach to various branching processes is indicated. However, the main reason for developing the theory is to obtain martingale convergence results in a branching random walk that do not seem readily accessible with other techniques. These results, which are natural extensions of known results for martingales associated with binary branching Brownian motion, form the main application.

scholarly journals A Limit Theorem for Discrete Galton-Watson Branching Processes with Immigration

2006 ◽  
Vol 43 (1) ◽  
pp. 289-295 ◽  
Author(s):  
Zenghu Li
Keyword(s):  
Limit Theorem ◽  
Weak Convergence ◽  
Discrete Time ◽  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Discrete State ◽  
Simple Set ◽  
Continuous State

We provide a simple set of sufficient conditions for the weak convergence of discrete-time, discrete-state Galton-Watson branching processes with immigration to continuous-time, continuous-state branching processes with immigration.

Strong ergodicity for single-birth processes

2001 ◽  
Vol 38 (1) ◽  
pp. 270-277 ◽  
Author(s):  
Yu-Hui Zhang
Keyword(s):  
Continuous Time ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Strong Ergodicity ◽  
Finite Dimensional ◽  
Comparison Methods ◽  
Schlögl Model ◽  
Birth Processes ◽  
Single Birth ◽  
Continuous Time Branching Processes

An explicit and computable criterion for strong ergodicity of single-birth processes is presented. As an application, some sufficient conditions are obtained for strong ergodicity of an extended class of continuous-time branching processes and multi-dimensional Q-processes by comparison methods respectively. Consequently strong ergodicity of the Q-process corresponding to the finite-dimensional Schlögl model is proven.

scholarly journals Weak convergence of the number of vertices at intermediate levels of random recursive trees

2018 ◽  
Vol 55 (4) ◽  
pp. 1131-1142 ◽  
Author(s):  
Alexander Iksanov ◽  
Zakhar Kabluchko
Keyword(s):  
Asymptotic Behavior ◽  
Weak Convergence ◽  
Gaussian Process ◽  
Branching Processes ◽  
One Dimensional ◽  
Recursive Tree ◽  
Finite Dimensional ◽  
Recursive Trees

Abstract Let Xn(k) be the number of vertices at level k in a random recursive tree with n+1 vertices. We are interested in the asymptotic behavior of Xn(k) for intermediate levels k=kn satisfying kn→∞ and kn=o(logn) as n→∞. In particular, we prove weak convergence of finite-dimensional distributions for the process (Xn ([knu]))u>0, properly normalized and centered, as n→∞. The limit is a centered Gaussian process with covariance (u,v)↦(u+v)−1. One-dimensional distributional convergence of Xn(kn), properly normalized and centered, was obtained with the help of analytic tools by Fuchs et al. (2006). In contrast, our proofs, which are probabilistic in nature, exploit a connection of our model with certain Crump–Mode–Jagers branching processes.

Some Contributions to the Theory of Near-Critical Bisexual Branching Processes

2007 ◽  
Vol 44 (02) ◽  
pp. 492-505
Author(s):  
M. Molina ◽  
M. Mota ◽  
A. Ramos
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Positive Probability ◽  
Limit Laws ◽  
Bisexual Branching Processes ◽  
Probabilistic Evolution

We investigate the probabilistic evolution of a near-critical bisexual branching process with mating depending on the number of couples in the population. We determine sufficient conditions which guarantee either the almost sure extinction of such a process or its survival with positive probability. We also establish some limiting results concerning the sequences of couples, females, and males, suitably normalized. In particular, gamma, normal, and degenerate distributions are proved to be limit laws. The results also hold for bisexual Bienaymé–Galton–Watson processes, and can be adapted to other classes of near-critical bisexual branching processes.

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