Some results on population-size-dependent Galton-Watson processes

1986 ◽  
Vol 23 (2) ◽  
pp. 297-306 ◽  
Author(s):  
Reinhard Höpfner
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Limit Distributions ◽  
Probability Of Survival ◽  
Size Dependent

Some classes of population-size-dependent Galton-Watson processes are considered where extinction occurs with probability 1. Results on the asymptotic behaviour of the probability of survival up to time t, mean population size and conditioned limit distributions are found to hold. They correspond to those obtained in the study of Galton-Watson processes with immigration stopped at 0.

Some results on population-size-dependent Galton-Watson processes

1986 ◽  
Vol 23 (02) ◽  
pp. 297-306 ◽  
Author(s):  
Reinhard Höpfner
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Limit Distributions ◽  
Probability Of Survival ◽  
Size Dependent

Some classes of population-size-dependent Galton-Watson processes are considered where extinction occurs with probability 1. Results on the asymptotic behaviour of the probability of survival up to time t, mean population size and conditioned limit distributions are found to hold. They correspond to those obtained in the study of Galton-Watson processes with immigration stopped at 0.

On some classes of population-size-dependent Galton–Watson processes

1985 ◽  
Vol 22 (01) ◽  
pp. 25-36
Author(s):  
Reinhard Höpfner
Keyword(s):  
Population Size ◽  
Lower Bounds ◽  
Generating Functions ◽  
Transition Probability ◽  
Limit Distributions ◽  
Size Dependent ◽  
Probability Generating Functions ◽  
Iteration Methods ◽  
Functional Iteration

Some classes of population-size-dependent Galton–Watson processes {Z(t)} t=0,1, …, whose transition probability generating functions allow for certain upper or lower bounds, can be treated by means of functional iteration methods. Criteria for almost certain extinction are obtained as well as gammatype limit distributions for Z(t)/t as t → ∞ the results can be stated under conditions on moments of the reproduction distributions.

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.

On some classes of population-size-dependent Galton–Watson processes

1985 ◽  
Vol 22 (1) ◽  
pp. 25-36 ◽  
Author(s):  
Reinhard Höpfner
Keyword(s):  
Population Size ◽  
Lower Bounds ◽  
Generating Functions ◽  
Transition Probability ◽  
Limit Distributions ◽  
Size Dependent ◽  
Probability Generating Functions ◽  
Iteration Methods ◽  
Functional Iteration

Some classes of population-size-dependent Galton–Watson processes {Z(t)}t=0,1, …, whose transition probability generating functions allow for certain upper or lower bounds, can be treated by means of functional iteration methods. Criteria for almost certain extinction are obtained as well as gammatype limit distributions for Z(t)/t as t → ∞ the results can be stated under conditions on moments of the reproduction distributions.

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

1999 ◽  
Vol 36 (02) ◽  
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 {Z n } is considered where the population's evolution is controlled by a Markovian environment process {ξ n }. For this model, let m k,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 : Z n (ω) = 0 for some n} and q = P(B). The asymptotic behaviour of lim n Z n and is studied in the case where supθ|m k,θ − m θ| → 0 for some real numbers {m θ} such that infθ m θ &gt; 1. When the environmental sequence {ξ n } is a irreducible positive recurrent Markov chain (particularly, when its state space is finite), certain extinction (q = 1) and non-certain extinction (q &lt; 1) are studied.

On population-size-dependent branching processes

1984 ◽  
Vol 16 (01) ◽  
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 {Z n } 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 &lt; l, m = 1, m&gt; l; moreover, the rate of convergence of mk to m plays an important role. It is shown that if m &lt; 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 &gt; 1 not slower than n– α, α &gt; 0, and do not grow to ∞ faster than nß , β &lt;1 then Zn /mn converges almost surely and in L 2 to a non-degenerate limit. A number of general results concerning the behaviour of sums of independent random variables are also given.

A note on the asymptotic behaviour of the extinction probability in supercritical population-size-dependent branching processes with independent and identically distributed random environments

2004 ◽  
Vol 41 (1) ◽  
pp. 176-186 ◽  
Author(s):  
Lu Zhunwei ◽  
Peter Jagers
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Processes ◽  
Extinction Probability ◽  
Random Environments ◽  
Regularity Conditions ◽  
Size Dependent ◽  
Bounded Below ◽  
Independent And Identically Distributed

In supercritical population-size-dependent branching processes with independent and identically distributed random environments, it is shown that under certain regularity conditions there exist constants 0 < α1 ≤α0 < + ∞ and 0 < C1, C2 < + ∞ such that the extinction probability starting with k individuals is bounded below by C1k-α0 and above by C2k-α1 for sufficiently large k. Moreover, a similar conclusion, which follows from a result of Höpfner, is presented along with some remarks.

A note on the asymptotic behaviour of the extinction probability in supercritical population-size-dependent branching processes with independent and identically distributed random environments

2004 ◽  
Vol 41 (01) ◽  
pp. 176-186 ◽  
Author(s):  
Lu Zhunwei ◽  
Peter Jagers
Keyword(s):  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Processes ◽  
Extinction Probability ◽  
Random Environments ◽  
Regularity Conditions ◽  
Size Dependent ◽  
Bounded Below ◽  
Independent And Identically Distributed

In supercritical population-size-dependent branching processes with independent and identically distributed random environments, it is shown that under certain regularity conditions there exist constants 0 &lt; α 1 ≤α 0 &lt; + ∞ and 0 &lt; C 1, C 2 &lt; + ∞ such that the extinction probability starting with k individuals is bounded below by C 1 k -α 0 and above by C 2 k -α 1 for sufficiently large k. Moreover, a similar conclusion, which follows from a result of Höpfner, is presented along with some remarks.

Population-size-dependent branching process with linear rate of growth

1983 ◽  
Vol 20 (02) ◽  
pp. 242-250 ◽  
Author(s):  
F. C. Klebaner
Keyword(s):  
Population Size ◽  
Branching Process ◽  
Positive Probability ◽  
Linear Rate ◽  
Size Dependent ◽  
Rate Of Growth ◽  
Binary Splitting

The process we consider is a binary splitting, where the probability of division, , depends on the population size, 2i. We show that Zn converges to ∞ almost surely on a set of positive probability. Zn /n converges in distribution to a proper limit, diverges almost surely on converges almost surely on and there are no constants cn such that Zn /cn converges in probability to a non-degenerate limit.

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