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.

On calculating extinction probabilities for branching processes in random environments

1969 ◽  
Vol 6 (3) ◽  
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 Z0 = k, a finite positive integer.

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.

On the extinction time distribution of a branching process in varying environments

1980 ◽  
Vol 12 (2) ◽  
pp. 350-366 ◽  
Author(s):  
Tetsuo Fujimagari
Keyword(s):  
Generating Functions ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Tail Probability ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Extinction Time ◽  
Asymptotic Formulae ◽  
Varying Environments

The extinction time distributions of a class of branching processes in varying environments are considered. We obtain (i) sufficient conditions for the extinction probability q = 1 or q < 1; (ii) asymptotic formulae for the tail probability of the extinction time if q = 1; and (iii) upper bounds for 1 – q if q < 1. To derive these results, we give upper and lower bounds for the tail probability of the extinction time. For the proofs, we use a method that compares probability generating functions with fractional linear generating functions.

On the extinction time distribution of a branching process in varying environments

1980 ◽  
Vol 12 (02) ◽  
pp. 350-366 ◽  
Author(s):  
Tetsuo Fujimagari
Keyword(s):  
Generating Functions ◽  
Branching Process ◽  
Branching Processes ◽  
Sufficient Conditions ◽  
Tail Probability ◽  
Upper Bounds ◽  
Upper And Lower Bounds ◽  
Extinction Time ◽  
Asymptotic Formulae ◽  
Varying Environments

The extinction time distributions of a class of branching processes in varying environments are considered. We obtain (i) sufficient conditions for the extinction probability q = 1 or q &lt; 1; (ii) asymptotic formulae for the tail probability of the extinction time if q = 1; and (iii) upper bounds for 1 – q if q &lt; 1. To derive these results, we give upper and lower bounds for the tail probability of the extinction time. For the proofs, we use a method that compares probability generating functions with fractional linear generating functions.

A theorem on compositions of random probability generating functions and applications to branching processes with random environments

1972 ◽  
Vol 9 (1) ◽  
pp. 1-12 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Generating Functions ◽  
Limit Theorems ◽  
Branching Processes ◽  
Random Environments ◽  
Probability Generating Functions ◽  
Random Probability

The results of this paper deal directly with the behavior of compositions of random probability generating functions. These results are then applied to the theory of branching processes with random environments. In particular, it is shown that it is not possible to strengthen many of the existing limit theorems on this subject.

A theorem on compositions of random probability generating functions and applications to branching processes with random environments

1972 ◽  
Vol 9 (01) ◽  
pp. 1-12 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Generating Functions ◽  
Limit Theorems ◽  
Branching Processes ◽  
Random Environments ◽  
Probability Generating Functions ◽  
Random Probability

The results of this paper deal directly with the behavior of compositions of random probability generating functions. These results are then applied to the theory of branching processes with random environments. In particular, it is shown that it is not possible to strengthen many of the existing limit theorems on this subject.

Sign in / Sign up

Export Citation Format

Share Document