Limit theorems for critical age-dependent nonhomogeneous branching processes

1978 ◽  
Vol 18 (5) ◽  
pp. 835-844
Author(s):  
V. A. Topchii
Keyword(s):  
Limit Theorems ◽  
Branching Processes ◽  
Age Dependent ◽  
Critical Age

Extinction probability for critical age-dependent branching processes with generation dependence

1980 ◽  
Vol 17 (1) ◽  
pp. 16-24
Author(s):  
Dean H. Fearn
Keyword(s):  
Branching Processes ◽  
Extinction Probability ◽  
Regularity Conditions ◽  
Limiting Behavior ◽  
Age Dependent ◽  
Mean And Variance ◽  
Critical Age ◽  
The Mean ◽  
Probability Of Extinction ◽  
Lifespan Distribution

The limiting behavior of the probability of extinction of critical age-dependent branching processes with generation dependence is obtained using Goldstein's methods. Regularity conditions on the mean and variance of the birth distributions are assumed. Also the lifespan distribution is assumed to satisfy suitable regularity conditions.

LIMIT THEOREMS FOR SUBCRITICAL AGE-DEPENDENT BRANCHING PROCESSES WITH TWO TYPES OF IMMIGRATION

2005 ◽  
Vol 21 (1) ◽  
pp. 133-147 ◽  
Author(s):  
Gerold Alsmeyer ◽  
Maroussia Slavtchova-Bojkova
Keyword(s):  
Limit Theorems ◽  
Branching Processes ◽  
Age Dependent

scholarly journals Limit theorems for supercritical age-dependent branching processes with neutral immigration

2011 ◽  
Vol 43 (1) ◽  
pp. 276-300 ◽  
Author(s):  
M. Richard
Keyword(s):  
Continuous Spectrum ◽  
Limit Theorems ◽  
Branching Process ◽  
Total Population ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Structured Populations ◽  
Total Population Size ◽  
Constant Rate ◽  
Age Dependent

We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate θ, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have independent and identically distributed lifetime durations (nonnecessarily exponential) during which they give birth independently at a constant rateb. First, using spine decomposition, we relax previously known assumptions required for almost-sure convergence of the total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P1,P2,…) of relative abundances of surviving families converges almost surely. In the first model, the limit is the GEM distribution with parameter θ /b.

scholarly journals Limit theorems for supercritical age-dependent branching processes with neutral immigration

2011 ◽  
Vol 43 (01) ◽  
pp. 276-300 ◽  
Author(s):  
M. Richard
Keyword(s):  
Continuous Spectrum ◽  
Limit Theorems ◽  
Branching Process ◽  
Total Population ◽  
Branching Processes ◽  
Almost Sure Convergence ◽  
Structured Populations ◽  
Total Population Size ◽  
Constant Rate ◽  
Age Dependent

We consider a branching process with Poissonian immigration where individuals have inheritable types. At rate θ, new individuals singly enter the total population and start a new population which evolves like a supercritical, homogeneous, binary Crump-Mode-Jagers process: individuals have independent and identically distributed lifetime durations (nonnecessarily exponential) during which they give birth independently at a constant rateb. First, using spine decomposition, we relax previously known assumptions required for almost-sure convergence of the total population size. Then, we consider three models of structured populations: either all immigrants have a different type, or types are drawn in a discrete spectrum or in a continuous spectrum. In each model, the vector (P1,P2,…) of relative abundances of surviving families converges almost surely. In the first model, the limit is the GEM distribution with parameter θ /b.

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