Strong limit theorems for general supercritical branching processes with applications to branching diffusions

1976 ◽  
Vol 36 (3) ◽  
pp. 195-212 ◽  
Author(s):  
S�ren Asmussen ◽  
Heinrich Hering
Keyword(s):  
Limit Theorems ◽  
Branching Processes ◽  
Strong Limit ◽  
Branching Diffusions ◽  
Supercritical Branching Processes

Limit theorems for stochastic growth models. II

1972 ◽  
Vol 4 (3) ◽  
pp. 393-428 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Limit Theorems ◽  
Growth Models ◽  
Branching Processes ◽  
Random Variable ◽  
Present Situation ◽  
Stochastic Growth ◽  
Stochastic Growth Models ◽  
Zygotic Selection ◽  
Image Position ◽  
Supercritical Branching Processes

We consider d-dimensional stochastic processes which take values in (R+)d These processes generalize Galton-Watson branching processes, but the main assumption of branching processes, independence between particles, is dropped. Instead, we assume for some Here τ:(R+)d→R +, |x| = σ1d |x(i)|, A {x ∈(R+)d: |x| 1} and T: A→A. Under various assumptions on the maps τ and T it is shown that with probability one there exists a ρ > 1, a fixed point p ∈ A of T and a random variable w such that limn→∞Zn|ρnwp. This result is a generalization of the main limit theorem for supercritical branching processes; note, however, that in the present situation both ρ and ρ are random as well. The results are applied to a population genetical model for zygotic selection without mutation at one locus.

On maximum family size in branching processes

1999 ◽  
Vol 36 (3) ◽  
pp. 632-643 ◽  
Author(s):  
Ibrahim Rahimov ◽  
George P. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Sample Size ◽  
Random Sample ◽  
Family Size ◽  
Limit Theorems ◽  
Branching Processes ◽  
Random Sample Size ◽  
Convergence Results ◽  
Watson Process ◽  
Supercritical Branching Processes

The number Yn of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Yn as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Yn and EYn provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

On maximum family size in branching processes

1999 ◽  
Vol 36 (03) ◽  
pp. 632-643 ◽  
Author(s):  
Ibrahim Rahimov ◽  
George P. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Sample Size ◽  
Random Sample ◽  
Family Size ◽  
Limit Theorems ◽  
Branching Processes ◽  
Random Sample Size ◽  
Convergence Results ◽  
Watson Process ◽  
Supercritical Branching Processes

The number Y n of offspring of the most prolific individual in the nth generation of a Bienaymé–Galton–Watson process is studied. The asymptotic behaviour of Y n as n → ∞ may be viewed as an extreme value problem for i.i.d. random variables with random sample size. Limit theorems for both Y n and EY n provided that the offspring mean is finite are obtained using some convergence results for branching processes as well as a transfer limit lemma for maxima. Subcritical, critical and supercritical branching processes are considered separately.

Limit theorems for stochastic growth models. II

1972 ◽  
Vol 4 (03) ◽  
pp. 393-428 ◽  
Author(s):  
Harry Kesten
Keyword(s):  
Limit Theorems ◽  
Growth Models ◽  
Branching Processes ◽  
Random Variable ◽  
Present Situation ◽  
Stochastic Growth ◽  
Stochastic Growth Models ◽  
Zygotic Selection ◽  
Image Position ◽  
Supercritical Branching Processes

We consider d-dimensional stochastic processes which take values in (R+) d These processes generalize Galton-Watson branching processes, but the main assumption of branching processes, independence between particles, is dropped. Instead, we assume for some Here τ:(R+) d →R +, |x| = σ1 d |x(i)|, A {x ∈(R+)d: |x| 1} and T: A→A. Under various assumptions on the maps τ and T it is shown that with probability one there exists a ρ > 1, a fixed point p ∈ A of T and a random variable w such that lim n→∞ Z n |ρ n wp. This result is a generalization of the main limit theorem for supercritical branching processes; note, however, that in the present situation both ρ and ρ are random as well. The results are applied to a population genetical model for zygotic selection without mutation at one locus.

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.

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