Asymptotic probabilities in a critical age-dependent branching process

1976 ◽  
Vol 13 (3) ◽  
pp. 476-485 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Branching Process ◽  
Comparison Method ◽  
Lifetime Distribution ◽  
Order Moment ◽  
Moment Condition ◽  
Age Dependent ◽  
Total Progeny ◽  
Critical Age ◽  
The Mean ◽  
Critical Process

Let Z(t) denote the number of cells alive at time t in a critical Bellman-Harris age-dependent branching process, that is, where the mean number of offspring per parent is one. A comparison method is used to show for k ≧ 1, and a high-order moment condition on G(t), where G(t) is the cell lifetime distribution, that lim t→∞t2P[Z(t) = k] = ak > 0, where {ak} are constants.The method is also applied to the total progeny in the critical process.

Asymptotic probabilities in a critical age-dependent branching process

1976 ◽  
Vol 13 (03) ◽  
pp. 476-485
Author(s):  
Howard J. Weiner
Keyword(s):  
Branching Process ◽  
Comparison Method ◽  
Lifetime Distribution ◽  
Order Moment ◽  
Moment Condition ◽  
Age Dependent ◽  
Total Progeny ◽  
Critical Age ◽  
The Mean ◽  
Critical Process

Let Z(t) denote the number of cells alive at time t in a critical Bellman-Harris age-dependent branching process, that is, where the mean number of offspring per parent is one. A comparison method is used to show for k ≧ 1, and a high-order moment condition on G(t), where G(t) is the cell lifetime distribution, that lim t→∞ t 2 P[Z(t) = k] = ak > 0, where {ak } are constants. The method is also applied to the total progeny in the critical process.

Total progeny in a multitype critical age dependent branching process with immigration

1974 ◽  
Vol 11 (3) ◽  
pp. 458-470 ◽  
Author(s):  
Howard J. Weiner
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Cell Types ◽  
Dimensional Case ◽  
One Dimensional ◽  
Age Dependent ◽  
Total Progeny ◽  
Critical Age ◽  
The One ◽  
Branching Process With Immigration

In a multitype critical age dependent branching process with immigration, the numbers of cell types born by t, divided by t2, tends in law to a one-dimensional (degenerate) law whose Laplace transform is explicitily given. The method of proof makes a correspondence between the moments in the m-dimensional case and the one-dimensional case, for which the corresponding limit theorem is known. Other applications are given, a possible relaxation of moment assumptions, and extensions are indicated.

An age dependent branching process with variable lifetime distribution: The generation size

1974 ◽  
Vol 6 (02) ◽  
pp. 291-308 ◽  
Author(s):  
Robert Fildes
Keyword(s):  
Branching Process ◽  
Random Variable ◽  
Lifetime Distribution ◽  
Mean Square ◽  
Initial Particle ◽  
Age Dependent ◽  
The Mean ◽  
Watson Process ◽  
Number Of Particles

In a branching process with variable lifetime, introduced by Fildes (1972) define Yjk (t) as the number of particles alive in generation k at time t when the initial particle is born in generation j. A limit law similar to that derived in the Bellman-Harris process is proved where it is shown that Yjk (t) suitably normalised converges in mean square to a random variable which is the limit random variable of Znm–n in the Galton-Watson process (m is the mean number of particles born).

An age dependent branching process with variable lifetime distribution: The generation size

1974 ◽  
Vol 6 (2) ◽  
pp. 291-308 ◽  
Author(s):  
Robert Fildes
Keyword(s):  
Branching Process ◽  
Random Variable ◽  
Lifetime Distribution ◽  
Mean Square ◽  
Initial Particle ◽  
Age Dependent ◽  
The Mean ◽  
Watson Process ◽  
Number Of Particles

In a branching process with variable lifetime, introduced by Fildes (1972) define Yjk(t) as the number of particles alive in generation k at time t when the initial particle is born in generation j. A limit law similar to that derived in the Bellman-Harris process is proved where it is shown that Yjk(t) suitably normalised converges in mean square to a random variable which is the limit random variable of Znm–n in the Galton-Watson process (m is the mean number of particles born).

Total progeny in a multitype critical age dependent branching process with immigration

1974 ◽  
Vol 11 (03) ◽  
pp. 458-470
Author(s):  
Howard J. Weiner
Keyword(s):  
Limit Theorem ◽  
Branching Process ◽  
Cell Types ◽  
Dimensional Case ◽  
One Dimensional ◽  
Age Dependent ◽  
Total Progeny ◽  
Critical Age ◽  
The One ◽  
Branching Process With Immigration

In a multitype critical age dependent branching process with immigration, the numbers of cell types born by t, divided by t 2, tends in law to a one-dimensional (degenerate) law whose Laplace transform is explicitily given. The method of proof makes a correspondence between the moments in the m-dimensional case and the one-dimensional case, for which the corresponding limit theorem is known. Other applications are given, a possible relaxation of moment assumptions, and extensions are indicated.

A note on the subcritical generalized age-dependent branching process

1976 ◽  
Vol 13 (4) ◽  
pp. 798-803 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Branching Process ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
The Mean ◽  
Mean Function ◽  
Necessary And Sufficient

For a subcritical Bellman-Harris process for which the Malthusian parameter α exists and the mean function M(t)∼ aeat as t → ∞, a necessary and sufficient condition for e–at (1 –F(s, t)) to have a non-zero limit is known. The corresponding condition is given for the generalized branching process.

Sums of Lifetimes in Age Dependent Branching Processes

1969 ◽  
Vol 6 (01) ◽  
pp. 195-200 ◽  
Author(s):  
J. Howard Weiner
Keyword(s):  
Distribution Function ◽  
Branching Process ◽  
Past History ◽  
Branching Processes ◽  
Lifetime Distribution ◽  
Parent Cell ◽  
Absolutely Continuous ◽  
Age Dependent ◽  
A Cell ◽  
Additional Assumptions

Consider a Bellman-Harris [1] age dependent branching process. At t = 0, a cell is born, has lifetime distribution function G(t), G(0) = 0, assumed to be absolutely continuous with density g(t). At the death of the cell, k new cells are born, each proceeding independently and identically as the parent cell, and independent of past history. Denote by h(s) = Σ k=0 ∞ pk s k and suppose h(1) ≡ m, and assume h”(1) < ∞. Additional assumptions will be added as required.

Distribution of the branching-process population among generations

1971 ◽  
Vol 8 (4) ◽  
pp. 655-667 ◽  
Author(s):  
M. L. Samuels
Keyword(s):  
Normal Form ◽  
Density Function ◽  
Branching Process ◽  
Random Variables ◽  
Normal Density ◽  
Asymptotically Linear ◽  
Random Functions ◽  
Normal Density Function ◽  
Age Dependent ◽  
The Mean

SummaryIn a standard age-dependent branching process, let Rn(t) denote the proportion of the population belonging to the nth generation at time t. It is shown that in the supercritical case the distribution {Rn(t); n = 0, 1, …} has asymptotically, for large t, a (non-random) normal form, and that the mean ΣnRn(t) is asymptotically linear in t. Further, it is found that, for large n, Rn(t) has the shape of a normal density function (of t).Two other random functions are also considered: (a) the proportion of the nth generation which is alive at time t, and (b) the proportion of the nth generation which has been born by time t. These functions are also found to have asymptotically a normal form, but with parameters different from those relevant for {Rn(t)}.For the critical and subcritical processes, analogous results hold with the random variables replaced by their expectations.

Extinction probability for critical age-dependent branching processes with generation dependence

1980 ◽  
Vol 17 (1) ◽  
pp. 16-24
Author(s):  
Dean H. Fearn
Keyword(s):  
Branching Processes ◽  
Extinction Probability ◽  
Regularity Conditions ◽  
Limiting Behavior ◽  
Age Dependent ◽  
Mean And Variance ◽  
Critical Age ◽  
The Mean ◽  
Probability Of Extinction ◽  
Lifespan Distribution

The limiting behavior of the probability of extinction of critical age-dependent branching processes with generation dependence is obtained using Goldstein's methods. Regularity conditions on the mean and variance of the birth distributions are assumed. Also the lifespan distribution is assumed to satisfy suitable regularity conditions.

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