scholarly journals Extinction probabilities in branching processes with countably many types: a general framework

2022 ◽  
Vol 19 (1) ◽  
pp. 311
Author(s):  
Daniela Bertacchi ◽  
Peter Braunsteins ◽  
Sophie Hautphenne ◽  
Fabio Zucca
Keyword(s):  
General Framework ◽  
Branching Processes ◽  
Extinction Probabilities

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (03) ◽  
pp. 478-492 ◽  
Author(s):  
William E. Wilkinson
Keyword(s):  
Generating Functions ◽  
Infinite Sequence ◽  
Branching Processes ◽  
Transition Probability ◽  
Transition Probability Matrix ◽  
Random Environments ◽  
Real Numbers ◽  
Discrete Time Markov Chain ◽  
Extinction Probabilities ◽  
Image Position

Consider a discrete time Markov chain {Zn } whose state space is the non-negative integers and whose transition probability matrix ║Pij ║ possesses the representation where {Pr }, r = 1,2,…, is a finite or denumerably infinite sequence of non-negative real numbers satisfying , and , is a corresponding sequence of probability generating functions. It is assumed that Z 0 = k, a finite positive integer.

scholarly journals Limit theorems for multi-type general branching processes with population dependence

2020 ◽  
Vol 52 (4) ◽  
pp. 1127-1163
Author(s):  
Jie Yen Fan ◽  
Kais Hamza ◽  
Peter Jagers ◽  
Fima C. Klebaner
Keyword(s):  
Carrying Capacity ◽  
Population Model ◽  
General Framework ◽  
Limit Theorems ◽  
Mating Systems ◽  
Law Of Large Numbers ◽  
Branching Processes ◽  
Special Section ◽  
Large Numbers ◽  
Type Population

AbstractA general multi-type population model is considered, where individuals live and reproduce according to their age and type, but also under the influence of the size and composition of the entire population. We describe the dynamics of the population as a measure-valued process and obtain its asymptotics as the population grows with the environmental carrying capacity. Thus, a deterministic approximation is given, in the form of a law of large numbers, as well as a central limit theorem. This general framework is then adapted to model sexual reproduction, with a special section on serial monogamic mating systems.

On determining absorption probabilities for Markov chains in random environments

1981 ◽  
Vol 13 (2) ◽  
pp. 369-387 ◽  
Author(s):  
Richard D. Bourgin ◽  
Robert Cogburn
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Random Environment ◽  
Efficient Method ◽  
General Framework ◽  
Random Environments ◽  
Extinction Probabilities ◽  
Environmental Process ◽  
Branching Chains ◽  
Absorption Probabilities

The general framework of a Markov chain in a random environment is presented and the problem of determining extinction probabilities is discussed. An efficient method for determining absorption probabilities and criteria for certain absorption are presented in the case that the environmental process is a two-state Markov chain. These results are then applied to birth and death, queueing and branching chains in random environments.

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.

Branching processes with varying and random geometric offspring distributions

1975 ◽  
Vol 12 (01) ◽  
pp. 135-141 ◽  
Author(s):  
Niels Keiding ◽  
John E. Nielsen
Keyword(s):  
Asymptotic Behavior ◽  
Generating Functions ◽  
Conditional Expectation ◽  
Elementary Proof ◽  
Branching Processes ◽  
Random Environments ◽  
Subcritical Case ◽  
Supercritical Case ◽  
Extinction Probabilities ◽  
Varying Environments

The class of fractional linear generating functions is used to illustrate various aspects of the theory of branching processes in varying and random environments. In particular, it is shown that Church's theorem on convergence of the varying environments process admits of an elementary proof in this particular case. For random environments, examples are given on the asymptotic behavior of extinction probabilities in the supercritical case and conditional expectation given non-extinction in the subcritical case.

Random environment branching processes with equal environmental extinction probabilities

1973 ◽  
Vol 10 (03) ◽  
pp. 659-665
Author(s):  
Donald C. Raffety
Keyword(s):  
Markov Chains ◽  
Population Size ◽  
Random Environment ◽  
Transition Matrix ◽  
Branching Processes ◽  
Random Variables ◽  
Conditional Probabilities ◽  
Left Eigenvector ◽  
Extinction Probabilities ◽  
Limiting Values

R-positivity theory for Markov chains is used to obtain results for random environment branching processes whose environment random variables are independent and identically distributed and whose environmental extinction probabilities are equal. For certain processes whose eventual extinction is almost sure, it is shown that the distribution of population size conditioned by non-extinction at time n tends to a left eigenvector of the transition matrix. Limiting values of other conditional probabilities are given in terms of this left eigenvector and it is shown that the probability of non-extinction at time n approaches zero geometrically as n approaches ∞. Analogous results are obtained for processes whose extinction is not almost sure.

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.

Bisexual Galton–Watson branching processes with superadditive mating functions

1986 ◽  
Vol 23 (03) ◽  
pp. 585-600 ◽  
Author(s):  
D. J. Daley ◽  
David M. Hull ◽  
James M. Taylor
Keyword(s):  
Growth Rate ◽  
Branching Process ◽  
Critical Case ◽  
Branching Processes ◽  
Upper And Lower Bounds ◽  
Simple Criterion ◽  
Asymptotic Growth ◽  
Extinction Probabilities ◽  
Mating Function ◽  
Asymptotic Growth Rate

For a bisexual Galton–Watson branching process with superadditive mating function there is a simple criterion for determining whether or not the process becomes extinct with probability 1, namely, that the asymptotic growth rate r should not exceed 1. When extinction is not certain (equivalently, r > 1), simple upper and lower bounds are established for the extinction probabilities. An example suggests that in the critical case that r = 1, some condition like superadditivity is essential for ultimate extinction to be certain. Some illustrative numerical comparisons of particular mating functions are made using a Poisson offspring distribution.

scholarly journals The Properties of Generalized Collision Branching Processes

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Juan Wang ◽  
Chunhao Cai
Keyword(s):  
Explicit Expression ◽  
Generating Functions ◽  
Branching Processes ◽  
Hitting Times ◽  
Conditional Mean ◽  
Basic Properties ◽  
Extinction Probabilities ◽  
The Mean

We consider basic properties regarding uniqueness, extinction, and explosivity for the Generalized Collision Branching Processes (GCBP). Firstly, we investigate some important properties of the generating functions for GCB q-matrix in detail. Then for any given GCB q-matrix, we prove that there always exists exactly one GCBP. Next, we devote to the study of extinction behavior and hitting times. Some elegant and important results regarding extinction probabilities, the mean extinction times, and the conditional mean extinction times are presented. Moreover, the explosivity is also investigated and an explicit expression for mean explosion time is established.

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