A unifying approach to branching processes in a varying environment

@article{Kersting2020AUA,
  title={A unifying approach to branching processes in a varying environment},
  author={G{\"o}tz Kersting},
  journal={Journal of Applied Probability},
  year={2020},
  volume={57},
  pages={196 - 220}
}
  • G. Kersting
  • Published 6 March 2017
  • Mathematics
  • Journal of Applied Probability
Abstract Branching processes $(Z_n)_{n \ge 0}$ in a varying environment generalize the Galton–Watson process, in that they allow time dependence of the offspring distribution. Our main results concern general criteria for almost sure extinction, square integrability of the martingale $(Z_n/\mathrm E[Z_n])_{n \ge 0}$, properties of the martingale limit W and a Yaglom-type result stating convergence to an exponential limit distribution of the suitably normalized population size $Z_n$, conditioned… 
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Background Citations
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Methods Citations
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Results Citations
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25 Citations

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References

SHOWING 1-10 OF 26 REFERENCES
On the scaling limits of Galton-Watson processes in varying environments
We establish a general sufficient condition for a sequence of Galton–Watson branching processes in varying environments to converge weakly. This condition extends previ- ous results by allowing…
The Rates of Growth of the Galton–Watson Process in Varying Environments
Let {Zn } be a supercritical Galton–Watson process in varying environments, and W be the limit of the non-negative martingale {Zn /EZ n }. Under a condition which ensures that W is not identically…
Branching processes in varying environment with generation-dependent immigration
Abstract A branching process in varying environment with generation-dependent immigration is a modification of the standard branching process in which immigration is allowed and the reproduction and…
Heavy traffic approximations for the Galton-Watson process
The behaviour of the Galton-Watson process in near critical conditions is discussed, both with and without immigration. Limit theorems are obtained which show that, suitably normalized, and…
Time-Inhomogeneous Branching Processes Conditioned on Non-Extinction
In this paper, we consider time-inhomogeneous branching processes and time-inhomogeneous birth-and-death processes, in which the offspring distribution and birth and death rates (respectively) vary…
The supercritical Galton- Watson process in varying environments
A GALTON-WATSON BRANCHING PROCESS IN VARYING ENVIRONMENTS WITH ESSENTIALLY CONSTANT OFFSPRING MEANS AND TWO RATES OF GROWTH1
Summary A Galton-Watson process in varying environments (Zn), with essentially constant offspring means, i.e. E(Zn)/mnα∈(0, ∞), and exactly two rates of growth is constructed. The underlying…
The Survival Probability of a Critical Branching Process in a Random Environment
In this paper we determine the asymptotic behavior of the survival probability of a critical branching process in a random environment. In the special case of independent identically distributed…
Rank-dependent Galton‒Watson processes and their pathwise duals
Abstract We introduce a modified Galton‒Watson process using the framework of an infinite system of particles labelled by (x,t), where x is the rank of the particle born at time t. The key assumption…
Transient Phenomena in branching stochastic processes
The following scheme for generating particles is considered. Each particle existing at a given time splits up into k particles with the probability $\delta _{k1} + p_k \Delta t + o(\Delta t),\delta…
...
...