The asymptotic behaviour of extinction probability in the Smith–Wilkinson branching process

1993 ◽  
Vol 25 (2) ◽  
pp. 263-289 ◽  
Author(s):  
D. R. Grey ◽  
Lu Zhunwei
Keyword(s):  
Asymptotic Behaviour ◽  
Rate Of Convergence ◽  
Population Size ◽  
Random Environment ◽  
Critical State ◽  
Branching Process ◽  
Extinction Probability ◽  
Subcritical State ◽  
Regularity Conditions

Under some regularity conditions, in the supercritical Smith–Wilkinson branching process it is shown that as k, the starting population size, tends to infinity, the rate of convergence of qk, the corresponding extinction probability, to zero is similar to that of:k–θ, if there exists at least one subcritical state in the random environment space; xkk–α, if there exist only supercritical states in the random environment space; exp , if there exists at least one critical state and the others are supercritical in the random environment space.Here θ, x, α and c are positive constants determined by the process.

The asymptotic behaviour of extinction probability in the Smith–Wilkinson branching process

1993 ◽  
Vol 25 (02) ◽  
pp. 263-289
Author(s):  
D. R. Grey ◽  
Lu Zhunwei
Keyword(s):  
Asymptotic Behaviour ◽  
Rate Of Convergence ◽  
Population Size ◽  
Random Environment ◽  
Critical State ◽  
Branching Process ◽  
Extinction Probability ◽  
Subcritical State ◽  
Regularity Conditions

Under some regularity conditions, in the supercritical Smith–Wilkinson branching process it is shown that as k, the starting population size, tends to infinity, the rate of convergence of qk, the corresponding extinction probability, to zero is similar to that of: k–θ, if there exists at least one subcritical state in the random environment space; xkk–α , if there exist only supercritical states in the random environment space; exp , if there exists at least one critical state and the others are supercritical in the random environment space. Here θ, x, α and c are positive constants determined by the process.

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.

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.

Estimation of the parameters of a branching process from migrating binomial observations

1998 ◽  
Vol 30 (4) ◽  
pp. 948-967 ◽  
Author(s):  
C. Jacob ◽  
J. Peccoud
Keyword(s):  
Confidence Interval ◽  
Asymptotic Behaviour ◽  
Population Size ◽  
Branching Process ◽  
Total Population ◽  
Initial Population ◽  
Asymptotic Confidence Interval ◽  
Total Population Size ◽  
Initial Population Size ◽  
Watson Process

This paper considers a branching process generated by an offspring distribution F with mean m < ∞ and variance σ2 < ∞ and such that, at each generation n, there is an observed δ-migration, according to a binomial law Bpvn*Nnbef which depends on the total population size Nnbef. The δ-migration is defined as an emigration, an immigration or a null migration, depending on the value of δ, which is assumed constant throughout the different generations. The process with δ-migration is a generation-dependent Galton-Watson process, whereas the observed process is not in general a martingale. Under the assumption that the process with δ-migration is supercritical, we generalize for the observed migrating process the results relative to the Galton-Watson supercritical case that concern the asymptotic behaviour of the process and the estimation of m and σ2, as n → ∞. Moreover, an asymptotic confidence interval of the initial population size is given.

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.

On higher moments of the population size in a subcritical branching process

1988 ◽  
Vol 25 (2) ◽  
pp. 413-417 ◽  
Author(s):  
Eric Willekens
Keyword(s):  
Rate Of Convergence ◽  
Population Size ◽  
Branching Process ◽  
Convergence Result ◽  
Lifetime Distribution ◽  
Higher Moments ◽  
Age Dependent

Let {Z(t), t ≧ 0} be an age-dependent subcritical branching process. In this paper we show that if the lifetime distribution is subexponential, EZα (t) ~ EZ(t) (t →∞) for every α ≧ 1. If furthermore the lifetime distribution has a subexponential density, a rate of convergence result in the above relation is established.

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