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 Extinction Probabilities of Supercritical Decomposable Branching Processes

2012 ◽  
Vol 49 (03) ◽  
pp. 639-651 ◽  
Author(s):  
Sophie Hautphenne
Keyword(s):  
Nonnegative Solution ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Equivalence Classes ◽  
Specific Class ◽  
Initial Particle ◽  
Whole Process ◽  
Extinction Probabilities ◽  
The Minimal Nonnegative Solution ◽  
Multitype Branching Processes

We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints.

scholarly journals Extinction Probabilities of Supercritical Decomposable Branching Processes

2012 ◽  
Vol 49 (3) ◽  
pp. 639-651 ◽  
Author(s):  
Sophie Hautphenne
Keyword(s):  
Nonnegative Solution ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Equivalence Classes ◽  
Specific Class ◽  
Initial Particle ◽  
Whole Process ◽  
Extinction Probabilities ◽  
The Minimal Nonnegative Solution ◽  
Multitype Branching Processes

We focus on supercritical decomposable (reducible) multitype branching processes. Types are partitioned into irreducible equivalence classes. In this context, extinction of some classes is possible without the whole process becoming extinct. We derive criteria for the almost-sure extinction of the whole process, as well as of a specific class, conditionally given the class of the initial particle. We give sufficient conditions under which the extinction of a class implies the extinction of another class or of the whole process. Finally, we show that the extinction probability of a specific class is the minimal nonnegative solution of the usual extinction equation but with added constraints.

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.

Asymptotic behavior of near-critical multitype branching processes

1991 ◽  
Vol 28 (03) ◽  
pp. 512-519 ◽  
Author(s):  
Fima C. Klebaner
Keyword(s):  
Asymptotic Behavior ◽  
Population Size ◽  
Discrete Time ◽  
Growth Rates ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Limiting Distribution ◽  
Size Dependent ◽  
Generalized Gamma ◽  
Multitype Branching Processes

Sufficient conditions for survival and extinction of multitype population-size-dependent branching processes in discrete time are obtained. Growth rates are determined on the set of divergence to infinity. The limiting distribution of a properly normalized process can be generalized gamma, normal or degenerate.

Asymptotic behavior of near-critical multitype branching processes

1991 ◽  
Vol 28 (3) ◽  
pp. 512-519 ◽  
Author(s):  
Fima C. Klebaner
Keyword(s):  
Asymptotic Behavior ◽  
Population Size ◽  
Discrete Time ◽  
Growth Rates ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Limiting Distribution ◽  
Size Dependent ◽  
Generalized Gamma ◽  
Multitype Branching Processes

Sufficient conditions for survival and extinction of multitype population-size-dependent branching processes in discrete time are obtained. Growth rates are determined on the set of divergence to infinity. The limiting distribution of a properly normalized process can be generalized gamma, normal or degenerate.

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 (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.

scholarly journals Multitype branching processes in random environments

1971 ◽  
Vol 8 (1) ◽  
pp. 17-31 ◽  
Author(s):  
Edward W. Weissner
Keyword(s):  
Random Environment ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Random Environments ◽  
Restrictive Assumption ◽  
Multitype Branching Processes ◽  
Watson Process ◽  
Necessary And Sufficient

Consider the classical Galton-Watson process described by Harris ((1963), Chapter 1). Recently this model has been generalized in Smith (1968), Smith and Wilkinson (1969), and Wilkinson (1967). They removed the restrictive assumption that the particles always divide in accordance with the same p.g.f. Instead, they assumed that at each unit of time, Nature be allowed to choose a p.g.f. from a class of p.g.f.'s, independently of the population, past and present, and the previously selected p.g.f.'s, which would then be assigned to the present population. Each particle of the present population would then split, independently of the others, in accordance with the selected p.g.f. This process, called a branching process in a random environment (BPRE), is clearly more applicable than the Galton-Watson process. Moreover, Smith and Wilkinson have found necessary and sufficient conditions for almost certain extinction of the BPRE which are almost as easy to verify as those for the Galton-Watson process.

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