scholarly journals An age-dependent branching process with arbitrary state space I

1971 ◽  
Vol 36 (1) ◽  
pp. 41-59 ◽  
Author(s):  
John J Bircher ◽  
Charles J Mode
Keyword(s):  
State Space ◽  
Branching Process ◽  
Arbitrary State ◽  
Age Dependent

Malthusian behaviour of the critical and subcritical age-dependent branching processes with arbitrary state space

1976 ◽  
Vol 13 (3) ◽  
pp. 455-465
Author(s):  
D. I. Saunders
Keyword(s):  
State Space ◽  
Branching Process ◽  
Branching Processes ◽  
Rates Of Convergence ◽  
Expected Number ◽  
Subcritical Case ◽  
Arbitrary State ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Number Of Individuals

For the age-dependent branching process with arbitrary state space let M(x, t, A) be the expected number of individuals alive at time t with states in A given an initial individual at x. Subject to various conditions it is shown that M(x, t, A)e–at converges to a non-trivial limit where α is the Malthusian parameter (α = 0 for the critical case, and is negative in the subcritical case). The method of proof also yields rates of convergence.

Malthusian behaviour of the critical and subcritical age-dependent branching processes with arbitrary state space

1976 ◽  
Vol 13 (03) ◽  
pp. 455-465
Author(s):  
D. I. Saunders
Keyword(s):  
State Space ◽  
Branching Process ◽  
Branching Processes ◽  
Rates Of Convergence ◽  
Expected Number ◽  
Subcritical Case ◽  
Arbitrary State ◽  
Malthusian Parameter ◽  
Age Dependent ◽  
Number Of Individuals

For the age-dependent branching process with arbitrary state space let M(x, t, A) be the expected number of individuals alive at time t with states in A given an initial individual at x. Subject to various conditions it is shown that M(x, t, A)e–at converges to a non-trivial limit where α is the Malthusian parameter (α = 0 for the critical case, and is negative in the subcritical case). The method of proof also yields rates of convergence.

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.

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.

The asymptotic shape of the branching random walk

1978 ◽  
Vol 10 (1) ◽  
pp. 62-84 ◽  
Author(s):  
J. D. Biggins
Keyword(s):  
Random Walk ◽  
Explicit Formula ◽  
Branching Process ◽  
Analogous Result ◽  
Convex Set ◽  
Branching Random Walk ◽  
Asymptotic Shape ◽  
Age Dependent ◽  
Watson Process

In a supercritical branching random walk on Rp, a Galton–Watson process with the additional feature that people have positions, let be the set of positions of the nth-generation people, scaled by the factor n–1. It is shown that when the process survives looks like a convex set for large n. An analogous result is established for an age-dependent branching process in which people also have positions. In certain cases an explicit formula for the asymptotic shape is given.

A multitype decomposable age-dependent branching process and its applications

1995 ◽  
Vol 32 (3) ◽  
pp. 591-608 ◽  
Author(s):  
Chinsan Lee ◽  
Grace L. Yang
Keyword(s):  
Tumor Growth ◽  
Life Span ◽  
Growth Rates ◽  
Exponential Growth ◽  
Branching Process ◽  
Asymptotic Formulas ◽  
Stage Model ◽  
Two Stage ◽  
Markov Branching Process ◽  
Age Dependent

Asymptotic formulas for means and variances of a multitype decomposable age-dependent supercritical branching process are derived. This process is a generalization of the Kendall–Neyman–Scott two-stage model for tumor growth. Both means and variances have exponential growth rates as in the case of the Markov branching process. But unlike Markov branching, these asymptotic moments depend on the age of the original individual at the start of the process and the life span distribution of the progenies.

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