scholarly journals Fluctuation limit theorems for age-dependent critical binary branching systems

2011 ◽  
Vol 31 ◽  
pp. 55-72
Author(s):  
José Alfredo López-Mimbela ◽  
Antonio Murillo-Salas
Keyword(s):  
Limit Theorems ◽  
Fluctuation Limit ◽  
Age Dependent

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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