Critical Galton–Watson processes by decreasing state-dependent immigration

1984 ◽  
Vol 21 (1) ◽  
pp. 22-39 ◽  
Author(s):  
K. V. Mitov ◽  
N. M. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Limit Theorems ◽  
Critical Case ◽  
Branching Processes ◽  
State Dependent ◽  
Different Types ◽  
Number Of Particles

This paper deals with the Foster–Pakes model for Galton–Watson branching processes allowing immigration whenever the number of particles is 0. In the critical case we investigate the asymptotic behaviour of the probability of non-extinction, of the expectation and of the variance, and obtain different types of limit theorems depending on the temporally-decreasing sizes of the immigrants.

Critical Galton–Watson processes by decreasing state-dependent immigration

1984 ◽  
Vol 21 (01) ◽  
pp. 22-39
Author(s):  
K. V. Mitov ◽  
N. M. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Limit Theorems ◽  
Critical Case ◽  
Branching Processes ◽  
State Dependent ◽  
Different Types ◽  
Number Of Particles

This paper deals with the Foster–Pakes model for Galton–Watson branching processes allowing immigration whenever the number of particles is 0. In the critical case we investigate the asymptotic behaviour of the probability of non-extinction, of the expectation and of the variance, and obtain different types of limit theorems depending on the temporally-decreasing sizes of the immigrants.

Continuous-time branching processes with decreasing state-dependent immigration

1984 ◽  
Vol 16 (4) ◽  
pp. 697-714 ◽  
Author(s):  
K. V. Mitov ◽  
V. A. Vatutin ◽  
N. M. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Continuous Time ◽  
Limit Theorems ◽  
Critical Case ◽  
Branching Processes ◽  
State Dependent ◽  
Different Types ◽  
Continuous Time Branching Processes

This paper deals with continuous-time branching processes which allow a temporally-decreasing immigration whenever the population size is 0. In the critical case the asymptotic behaviour of the probability of non-extinction and of the first two moments is investigated and different types of limit theorems are also proved.

Continuous-time branching processes with decreasing state-dependent immigration

1984 ◽  
Vol 16 (04) ◽  
pp. 697-714 ◽  
Author(s):  
K. V. Mitov ◽  
V. A. Vatutin ◽  
N. M. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Continuous Time ◽  
Limit Theorems ◽  
Critical Case ◽  
Branching Processes ◽  
State Dependent ◽  
Different Types ◽  
Continuous Time Branching Processes

This paper deals with continuous-time branching processes which allow a temporally-decreasing immigration whenever the population size is 0. In the critical case the asymptotic behaviour of the probability of non-extinction and of the first two moments is investigated and different types of limit theorems are also proved.

Bellman-Harris branching processes with a special type of state-dependent immigration

1989 ◽  
Vol 21 (02) ◽  
pp. 270-283 ◽  
Author(s):  
K. V. Mitov ◽  
N. M. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Limit Theorems ◽  
Branching Processes ◽  
State Dependent ◽  
Probability Of Extinction ◽  
New Particles

We investigate critical Bellman-Harris processes which allow immigration of new particles whenever the population size is 0. Under some special conditions on the immigration component the asymptotic behaviour of the probability of extinction is obtained and limit theorems are also proved.

Bellman-Harris branching processes with a special type of state-dependent immigration

1989 ◽  
Vol 21 (2) ◽  
pp. 270-283 ◽  
Author(s):  
K. V. Mitov ◽  
N. M. Yanev
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Limit Theorems ◽  
Branching Processes ◽  
State Dependent ◽  
Probability Of Extinction ◽  
New Particles

We investigate critical Bellman-Harris processes which allow immigration of new particles whenever the population size is 0. Under some special conditions on the immigration component the asymptotic behaviour of the probability of extinction is obtained and limit theorems are also proved.

Bellman–Harris branching processes with state-dependent immigration

1985 ◽  
Vol 22 (4) ◽  
pp. 757-765 ◽  
Author(s):  
K. V. Mitov ◽  
N. M. Yanev
Keyword(s):  
Limit Theorem ◽  
Asymptotic Behaviour ◽  
Branching Processes ◽  
The State ◽  
State Dependent ◽  
Probability Of Extinction

We consider critical Bellman-Harris processes which admit an immigration component only in the state 0. The asymptotic behaviour of the probability of extinction and of the first two moments is investigated and a limit theorem is also proved.

scholarly journals Asymptotic Behaviour of Continuous Time and State Branching Processes

2000 ◽  
Vol 68 (1) ◽  
pp. 68-84 ◽  
Author(s):  
Zeng-Hu Li
Keyword(s):  
Asymptotic Behaviour ◽  
Continuous Time ◽  
Limit Theorems ◽  
Branching Processes ◽  
Limit Laws ◽  
Classical Setting

AbstractWe prove some limit theorems for contiunous time and state branching processes. The non-degenerate limit laws are obtained in critical and non-critical cases by conditioning or introducing immigration processes. The limit laws in non-critical cases are characterized in terms of the cononical measure of the cumulant semigroup. The proofs are based on estimates of the cumulant semigroup derived from the forward and backward equations, which are easier than the proffs in the classical setting.

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.

scholarly journals Model biological population by branching processes

2017 ◽  
Vol 3 (2) ◽  
Author(s):  
Antoanela Terzieva
Keyword(s):  
Population Dynamics ◽  
Cell Division ◽  
Stochastic Processes ◽  
End Of Life ◽  
Branching Processes ◽  
Expected Number ◽  
Biological Population ◽  
Different Types ◽  
Unicellular Organisms ◽  
Number Of Particles

Consider a population of two or more different types of cells that at the end of life create two new cells through cell division. We model the population dynamics using a multitype branching stochastic processes. Under consideration are processes of Bieneme-Galton-Watson and of Bellman-Harris for the Markovian case.   drawn Conclusions about the expected number of particles of each type after a random time are drawn. The proposed models could be applicable not only for populations of a unicellular organisms, but also for sets of objects which operate a certain period of time and then split into two new objects or change their type.

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