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

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.

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

Journal of Applied Probability ◽

10.2307/3212465 ◽

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.

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Attainable bounds for expectations

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s144678870001884x ◽

1982 ◽

Vol 33 (3) ◽

pp. 411-420 ◽

Cited By ~ 2

Author(s):

E. Seneta ◽

N. C. Weber

Keyword(s):

Lower Bound ◽

Generating Functions ◽

Branching Process ◽

Simple Technique ◽

Branching Processes ◽

Random Variables ◽

Random Variable ◽

Malthusian Parameter ◽

Age Dependent ◽

Coefficient Of Skewness

AbstractA simple technique for obtaining bounds in terms of means and variances for the expectations of certain functions of random variables in a given class is examined. The bounds given are sharp in the sense that they are attainable by at least one random variable in the class. This technique is applied to obtain bounds for moment generating functions, the coefficient of skewness and parameters associated with branching processes. In particular an improved lower bound for the Malthusian parameter in an age-dependent branching process is derived.

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

Journal of Applied Probability ◽

10.1017/s0021900200104000 ◽

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.

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Convergence of general branching processes and functionals thereof

Journal of Applied Probability ◽

10.1017/s002190020009625x ◽

1974 ◽

Vol 11 (03) ◽

pp. 471-478 ◽

Cited By ~ 5

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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Sur une procédure de branchement déterministe et ses dérivées aléatoires

Journal of Applied Probability ◽

10.1017/s0021900200044867 ◽

1994 ◽

Vol 31 (02) ◽

pp. 333-347

Author(s):

Thierry Huillet ◽

Andrzej Kłopotowski

Keyword(s):

Branching Processes ◽

Asymptotic Results ◽

Renewal Equations ◽

Age Dependent ◽

Probabilistic Version

This paper is concerned with the description of both a deterministic and stochastic branching procedure. The renewal equations for the deterministic branching population are first derived which allow for asymptotic results on the ‘number' and ‘generation' processes. A probabilistic version of these processes is then studied which presents some discrepancy with the standard Harris age-dependent branching processes.

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Labelling experiments and phase-age distributions for multiphase branching processes

Journal of Applied Probability ◽

10.2307/3212377 ◽

1973 ◽

Vol 10 (4) ◽

pp. 739-747 ◽

Cited By ~ 2

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.

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Supercritical age-dependent branching processes with immigration

Journal of Applied Probability ◽

10.2307/3212553 ◽

1974 ◽

Vol 11 (4) ◽

pp. 695-702 ◽

Cited By ~ 9

Author(s):

K. B. Athreya ◽

P. R. Parthasarathy ◽

G. Sankaranarayanan

Keyword(s):

Branching Process ◽

Random Number ◽

Branching Processes ◽

Life Time ◽

Random Variable ◽

One Dimensional ◽

Malthusian Parameter ◽

Age Dependent ◽

Branching Process With Immigration ◽

Mutually Independent

A branching process with immigration of the following type is considered. For every i, a random number Ni of particles join the system at time . These particles evolve according to a one-dimensional age-dependent branching process with offspring p.g.f. and life time distribution G(t). Assume . Then it is shown that Z(t) e–αt converges in distribution to an extended real-valued random variable Y where a is the Malthusian parameter. We do not require the sequences {τi} or {Ni} to be independent or identically distributed or even mutually independent.

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Sur une procédure de branchement déterministe et ses dérivées aléatoires

Journal of Applied Probability ◽

10.2307/3215027 ◽

1994 ◽

Vol 31 (2) ◽

pp. 333-347 ◽

Cited By ~ 1

Author(s):

Thierry Huillet ◽

Andrzej Kłopotowski

Keyword(s):

Branching Processes ◽

Asymptotic Results ◽

Renewal Equations ◽

Age Dependent ◽

Probabilistic Version

This paper is concerned with the description of both a deterministic and stochastic branching procedure. The renewal equations for the deterministic branching population are first derived which allow for asymptotic results on the ‘number' and ‘generation' processes. A probabilistic version of these processes is then studied which presents some discrepancy with the standard Harris age-dependent branching processes.

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Convergence of general branching processes and functionals thereof

Journal of Applied Probability ◽

10.2307/3212691 ◽

1974 ◽

Vol 11 (3) ◽

pp. 471-478 ◽

Cited By ~ 12

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.

Download Full-text