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 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.

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 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.

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.

Bellman–Harris branching processes with state-dependent immigration

1985 ◽  
Vol 22 (04) ◽  
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.

On population-size-dependent branching processes

1984 ◽  
Vol 16 (1) ◽  
pp. 30-55 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Stochastic Model ◽  
Asymptotic Behaviour ◽  
Rate Of Convergence ◽  
Population Size ◽  
Gamma Distribution ◽  
Branching Processes ◽  
Independent Random Variables ◽  
Size Dependent ◽  
Offspring Distribution ◽  
The Mean

We consider a stochastic model for the development in time of a population {Zn} where the law of offspring distribution depends on the population size. We are mainly concerned with the case when the mean mk and the variance of offspring distribution stabilize as the population size k grows to ∞, The process exhibits different asymptotic behaviour according to m < l, m = 1, m> l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m < 1 or m = 1 and mn approaches 1 not slower than n–2 then the process dies out with probability 1. If mn approaches 1 from above and the rate of convergence is n–1, then Zn/n converges in distribution to a gamma distribution, moreover a.s. both on a set of non-extinction and there are no constants an, such that Zn/an converges in probability to a non-degenerate limit. If mn approaches m > 1 not slower than n–α, α > 0, and do not grow to ∞ faster than nß, β <1 then Zn/mn converges almost surely and in L2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.

Asymptotic behaviour of population-size-dependent branching processes in Markovian random environments

1999 ◽  
Vol 36 (2) ◽  
pp. 611-619 ◽  
Author(s):  
Han-Xing Wang ◽  
Dafan Fang
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Process ◽  
Branching Processes ◽  
Random Environments ◽  
Real Numbers ◽  
Size Dependent ◽  
Recurrent Markov Chain ◽  
The Mean ◽  
Positive Recurrent

A population-size-dependent branching process {Zn} is considered where the population's evolution is controlled by a Markovian environment process {ξn}. For this model, let mk,θ and be the mean and the variance respectively of the offspring distribution when the population size is k and a environment θ is given. Let B = {ω : Zn(ω) = 0 for some n} and q = P(B). The asymptotic behaviour of limnZn and is studied in the case where supθ|mk,θ − mθ| → 0 for some real numbers {mθ} such that infθmθ > 1. When the environmental sequence {ξn} is a irreducible positive recurrent Markov chain (particularly, when its state space is finite), certain extinction (q = 1) and non-certain extinction (q < 1) are studied.

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.

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