A note on asymptotic behavior of Galton-Watson branching processes in random environments

2006 ◽  
Vol 10 (2) ◽  
pp. 95-99 ◽  
Author(s):  
Han-xing Wang ◽  
Fei Zhao ◽  
Jn-yu Lu
Keyword(s):  
Asymptotic Behavior ◽  
Branching Processes ◽  
Random Environments

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.

Branching processes with varying and random geometric offspring distributions

1975 ◽  
Vol 12 (1) ◽  
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.

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 Interacting branching processes and linear file-sharing networks

2010 ◽  
Vol 42 (03) ◽  
pp. 834-854 ◽  
Author(s):  
Lasse Leskelä ◽  
Philippe Robert ◽  
Florian Simatos
Keyword(s):  
Communication Network ◽  
Asymptotic Behavior ◽  
Distributed Systems ◽  
Stochastic Model ◽  
Branching Processes ◽  
File Sharing ◽  
Independent Interest ◽  
Constant Flow

File-sharing networks are distributed systems used to disseminate files among nodes of a communication network. The general simple principle of these systems is that once a node has retrieved a file, it may become a server for this file. In this paper, the capacity of these networks is analyzed with a stochastic model when there is a constant flow of incoming requests for a given file. It is shown that the problem can be solved by analyzing the asymptotic behavior of a class of interacting branching processes. Several results of independent interest concerning these branching processes are derived and then used to study the file-sharing systems.

scholarly journals Branching processes with random environments

1970 ◽  
Vol 76 (4) ◽  
pp. 865-871 ◽  
Author(s):  
Krishna B. Athreya ◽  
Samuel Karlin
Keyword(s):  
Branching Processes ◽  
Random Environments

scholarly journals Modeling Y-Linked Pedigrees through Branching Processes

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 256
Author(s):  
Miguel González ◽  
Cristina Gutiérrez ◽  
Rodrigo Martínez
Keyword(s):  
Hearing Loss ◽  
Asymptotic Behavior ◽  
Gibbs Sampler ◽  
Mutant Allele ◽  
Branching Process ◽  
Branching Processes ◽  
Dominant Allele ◽  
Linked Gene ◽  
Different Types ◽  
The Mean

A multidimensional two-sex branching process is introduced to model the evolution of a pedigree originating from the mutation of an allele of a Y-linked gene in a monogamous population. The study of the extinction of the mutant allele and the analysis of the dominant allele in the pedigree is addressed on the basis of the classical theory of multi-type branching processes. The asymptotic behavior of the number of couples of different types in the pedigree is also derived. Finally, using the estimates of the mean growth rates of the allele and its mutation provided by a Gibbs sampler, a real Y-linked pedigree associated with hearing loss is analyzed, concluding that this mutation will persist in the population although without dominating the pedigree.

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 behaviour of population-size-dependent branching processes in Markovian random environments

1999 ◽  
Vol 36 (2) ◽  
pp. 611-619 ◽  
Author(s):  
Han-Xing Wang ◽  
Dafan Fang
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Process ◽  
Branching Processes ◽  
Random Environments ◽  
Real Numbers ◽  
Size Dependent ◽  
Recurrent Markov Chain ◽  
The Mean ◽  
Positive Recurrent

A population-size-dependent branching process {Zn} is considered where the population's evolution is controlled by a Markovian environment process {ξn}. For this model, let mk,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Zn(ω) = 0 for some n} and q = P(B). The asymptotic behaviour of limnZn and is studied in the case where supθ|mk,θ − mθ| → 0 for some real numbers {mθ} such that infθmθ > 1. When the environmental sequence {ξn} is a irreducible positive recurrent Markov chain (particularly, when its state space is finite), certain extinction (q = 1) and non-certain extinction (q < 1) are studied.

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