# 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} }

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…

## 25 Citations

Defective Galton-Watson processes in a varying environment

- MathematicsBernoulli
- 2022

We study an extension of the so-called defective Galton-Watson processes obtained by allowing the offspring distribution to change over the generations. Thus, in these processes, the individuals…

Yaglom’s limit for critical Galton–Watson processes in varying environment: A probabilistic approach

- Mathematics
- 2020

A Galton-Watson process in varying environment is a discrete time branching process where the offspring distributions vary among generations. Based on a two-spine decomposition technique, we provide…

Two-type linear-fractional branching processes in varying environments with asymptotically constant mean matrices

- MathematicsJournal of Applied Probability
- 2022

Abstract Consider two-type linear-fractional branching processes in varying environments with asymptotically constant mean matrices. Let
$\nu$
be the extinction time. Under certain conditions, we…

Two-type linear fractional branching processes in varying environments with asymptotically constant mean matrices

- Mathematics
- 2020

Consider two-type linear fractional branching processes in varying environments with asymptotically constant mean matrices. Let $\nu$ be the extinction time. Under certain conditions, we show that…

Regeneration of branching processes with immigration in varying environments

- Mathematics
- 2022

In this paper, we consider certain linear-fractional branching processes with immigration in varying environments. For n ≥ 0 , let Z n counts the number of individuals of the n -th generation, which…

Central limit theorem for a critical multitype branching process in random environments

- MathematicsTunisian Journal of Mathematics
- 2021

Let (Z n) n≥0 with Z n = (Z n (i, j)) 1≤i,j≤p be a p multi-type critical branching process in random environment, and let M n be the expectation of Z n given a fixed environment. We prove theorems on…

Branching processes in varying environment with generation-dependent immigration

- MathematicsStochastic Models
- 2019

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…

The genealogy of a nearly critical branching processes in varying environment

- Mathematics
- 2022

Building on the spinal decomposition technique in [10] we prove a Yaglom limit law for the rescaled size of a nearly critical branching process in varying environment conditional on survival. In…

Limit theorems for critical branching processes in a finite-state-space Markovian environment

- MathematicsAdvances in Applied Probability
- 2022

Abstract Let
$(Z_n)_{n\geq 0}$
be a critical branching process in a random environment defined by a Markov chain
$(X_n)_{n\geq 0}$
with values in a finite state space
$\mathbb{X}$
. Let
$ S_n…

Haldane's asymptotics for Supercritical Branching Processes in an iid Random Environment

- Mathematics
- 2021

Branching processes in a random environment are a natural generalisation of GaltonWatson processes. In this paper we analyse the asymptotic decay of the survival probability for a sequence of…

## References

SHOWING 1-10 OF 26 REFERENCES

On the scaling limits of Galton-Watson processes in varying environments

- Mathematics
- 2011

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

- MathematicsAdvances in Applied Probability
- 1994

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

- MathematicsStochastic Models
- 2019

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

- MathematicsAdvances in Applied Probability
- 1971

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

- Mathematics
- 2017

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…

A GALTON-WATSON BRANCHING PROCESS IN VARYING ENVIRONMENTS WITH ESSENTIALLY CONSTANT OFFSPRING MEANS AND TWO RATES OF GROWTH1

- Mathematics
- 1983

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

- Mathematics
- 2001

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

- MathematicsAdvances in Applied Probability
- 2018

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

- Mathematics
- 1959

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…