On single- and multi-type general age-dependent branching processes

1976 ◽  
Vol 13 (2) ◽  
pp. 239-246 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Limiting Distribution ◽  
Age Dependent ◽  
Number Of Individuals ◽  
Random Times

We consider a p-type general age-dependent branching process (g.a.d.b.p) in which each individual lives a random length of time and at random times during its life and possibly after its death gives birth to offspring. If this process starts with one individual of type i then it turns out that the number of individuals of type i alive at time t forms a one-type g.a.d.b.p. This fact is exploited to establish a NASC for a properly normalized, supercritical p-type g.a.d.b.p. to have a limiting distribution which is not degenerate at zero.

On single- and multi-type general age-dependent branching processes

1976 ◽  
Vol 13 (02) ◽  
pp. 239-246 ◽  
Author(s):  
R. A. Doney
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Limiting Distribution ◽  
Age Dependent ◽  
Number Of Individuals ◽  
Random Times

We consider a p-type general age-dependent branching process (g.a.d.b.p) in which each individual lives a random length of time and at random times during its life and possibly after its death gives birth to offspring. If this process starts with one individual of type i then it turns out that the number of individuals of type i alive at time t forms a one-type g.a.d.b.p. This fact is exploited to establish a NASC for a properly normalized, supercritical p-type g.a.d.b.p. to have a limiting distribution which is not degenerate at zero.

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.

Labelling experiments and phase-age distributions for multiphase branching processes

1973 ◽  
Vol 10 (4) ◽  
pp. 739-747 ◽  
Author(s):  
P. J. Brockwell ◽  
W. H. Kuo
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Joint Probability ◽  
Random Variable ◽  
Cell Labelling ◽  
Single Individual ◽  
Age Dependent ◽  
The Individual ◽  
Number Of Individuals ◽  
Joint Probability Density

A supercritical age-dependent branching process is considered in which the lifespan of each individual is composed of four phases whose durations have joint probability density f(x1, x2, x3, x4). Starting with a single individual of age zero at time zero we consider the asymptotic behaviour as t→ ∞ of the random variable Z(4) (a0,…, an, t) defined as the number of individuals in phase 4 at time t for which the elapsed phase durations Y01,…, Y04,…, Yi1,…, Yi4,…, Yn4 of the individual itself and its first n ancestors satisfy the inequalities Yij ≦ aij, i = 0,…, n, j = 1,…, 4. The application of the results to the analysis of cell-labelling experiments is described. Finally we state an analogous result which defines (conditional on eventual non-extinction of the population) the asymptotic joint distribution of the phase and elapsed phase durations of an individual drawn at random from the population and the phase durations of its ancestors.

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.

Labelling experiments and phase-age distributions for multiphase branching processes

1973 ◽  
Vol 10 (04) ◽  
pp. 739-747 ◽  
Author(s):  
P. J. Brockwell ◽  
W. H. Kuo
Keyword(s):  
Branching Process ◽  
Branching Processes ◽  
Joint Probability ◽  
Random Variable ◽  
Cell Labelling ◽  
Single Individual ◽  
Age Dependent ◽  
The Individual ◽  
Number Of Individuals ◽  
Joint Probability Density

A supercritical age-dependent branching process is considered in which the lifespan of each individual is composed of four phases whose durations have joint probability density f(x 1, x 2, x 3, x 4). Starting with a single individual of age zero at time zero we consider the asymptotic behaviour as t→ ∞ of the random variable Z (4) (a 0,…, a n , t) defined as the number of individuals in phase 4 at time t for which the elapsed phase durations Y 01,…, Y 04,…, Yi 1,…, Yi 4,…, Yn 4 of the individual itself and its first n ancestors satisfy the inequalities Yij ≦ aij , i = 0,…, n, j = 1,…, 4. The application of the results to the analysis of cell-labelling experiments is described. Finally we state an analogous result which defines (conditional on eventual non-extinction of the population) the asymptotic joint distribution of the phase and elapsed phase durations of an individual drawn at random from the population and the phase durations of its ancestors.

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.

A note on a functional equation arising in Galton-Watson branching processes

1971 ◽  
Vol 8 (3) ◽  
pp. 589-598 ◽  
Author(s):  
Krishna B. Athreya
Keyword(s):  
Functional Equation ◽  
Continuous Function ◽  
Generating Function ◽  
Branching Process ◽  
Branching Processes ◽  
Probability Generating Function ◽  
Age Dependent ◽  
Watson Process ◽  
Natural Way

The functional equation ϕ(mu) = h(ϕ(u)) where is a probability generating function with 1 < m = h'(1 –) < ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) < 1 and F(+ ∞) = 1 arises in a Galton-Watson process in a natural way. We prove here that for any if and only if This unifies several results in the literature on the supercritical Galton-Watson process. We generalize this to an age dependent branching process case as well.

scholarly journals Coalescence in Subcritical Bellman-Harris Age-Dependent Branching Processes

2013 ◽  
Vol 50 (2) ◽  
pp. 576-591
Author(s):  
Jyy-I Hong
Keyword(s):  
Limit Distribution ◽  
Branching Process ◽  
Branching Processes ◽  
Simple Random Sampling ◽  
Single Type ◽  
Last Common Ancestor ◽  
Death Time ◽  
Age Dependent ◽  
Lines Of Descent ◽  
First Time

We consider a continuous-time, single-type, age-dependent Bellman-Harris branching process. We investigate the limit distribution of the point process A(t)={at,i: 1≤ i≤ Z(t)}, where at,i is the age of the ith individual alive at time t, 1≤ i≤ Z(t), and Z(t) is the population size of individuals alive at time t. Also, if Z(t)≥ k, k≥2, is a positive integer, we pick k individuals from those who are alive at time t by simple random sampling without replacement and trace their lines of descent backward in time until they meet for the first time. Let Dk(t) be the coalescence time (the death time of the last common ancestor) of these k random chosen individuals. We study the distribution of Dk(t) and its limit distribution as t→∞.

Strict supercritical generation-dependent Crump–Mode–Jagers branching processes

1978 ◽  
Vol 10 (04) ◽  
pp. 744-763 ◽  
Author(s):  
L. Edler
Keyword(s):  
Generating Functions ◽  
Positive Real Number ◽  
Branching Processes ◽  
Positive Real ◽  
Quadratic Mean ◽  
Malthusian Parameter ◽  
Renewal Equations ◽  
Branching Model ◽  
Age Dependent ◽  
Number Of Individuals

The general age-dependent branching model of Crump, Mode and Jagers will be generalized towards generation-dependent varying lifespan and reproduction distributions. A system of integral and renewal equations is established for the generating functions and the first two moments of Zi (t) (the number of individuals alive at time t), if the population was initiated at time 0 by one ancestor of age 0 from generation i. Convergence in quadratic mean of Zi (t)/EZi (t) as t tends to infinity is obtained if the generation-dependent reproduction functions converge to a supercritical one. In particular, if this convergence is slow enough t γ exp (αt) is the asymptotic behavior of EZi (t) for t tending to infinity, where γ is a positive real number and α the Malthusian parameter of growth of the limiting reproduction function.

Convergence of general branching processes and functionals thereof

1974 ◽  
Vol 11 (03) ◽  
pp. 471-478 ◽  
Author(s):  
Peter Jagers
Keyword(s):  
Wide Class ◽  
Random Function ◽  
Branching Processes ◽  
Supercritical Case ◽  
Age Dependent ◽  
Dependent Property ◽  
Number Of Individuals

With each individual in a branching population associate a random function of the age. Count the population by the values of these functions. Different choices yield different processes. In the supercritical case a unified treatment of the asymptotics is possible for a wide class, including for example the number of individuals having some random age dependent property or integrals of branching processes. As an application, the demographic concept of average age at childbearing is given a rigorous interpretation.

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