A generalization of Goldstein's comparison lemma and the exponential limit law in critical Crump-Mode-Jagers branching processes

Let Z(t) be the population size at time t in a general age-dependent branching process (as defined by Crump and Mode, or Jagers) in which the number N of offspring of a parent has expected value 1 (critical case). Assuming positivity and finiteness of the second moments of N, of the lifespan distribution and of the expected number of births per parent as a function of age (also assumed to be strongly non-lattice), the distribution of Z(t)/t conditioned on non-extinction at time t is asymptotically exponential. The main step in the proof is a comparison lemma for the probability generating functions of Z(t) and of the embedded generation process.

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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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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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Sums of Lifetimes in Age Dependent Branching Processes

Journal of Applied Probability ◽

10.1017/s0021900200032630 ◽

1969 ◽

Vol 6 (01) ◽

pp. 195-200 ◽

Cited By ~ 2

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.

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

Journal of Applied Probability ◽

10.2307/3212181 ◽

1971 ◽

Vol 8 (3) ◽

pp. 589-598 ◽

Cited By ~ 18

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.

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Coalescence in Subcritical Bellman-Harris Age-Dependent Branching Processes

Journal of Applied Probability ◽

10.1239/jap/1371648962 ◽

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→∞.

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Asymptotic behaviour of immigration-branching processes by general set of types. II: Supercritical branching part

Advances in Applied Probability ◽

10.1017/s000186780004074x ◽

1975 ◽

Vol 7 (03) ◽

pp. 468-494

Author(s):

H. Hering

Keyword(s):

Branching Process ◽

Branching Processes ◽

Almost Sure Convergence ◽

Transition Function ◽

Mean Square ◽

Second Moments ◽

Mean Square Convergence ◽

The Mean ◽

Parameter Dependent ◽

Dependent Probability

We construct an immigration-branching process from an inhomogeneous Poisson process, a parameter-dependent probability distribution of populations and a Markov branching process with homogeneous transition function. The set of types is arbitrary, and the parameter is allowed to be discrete or continuous. Assuming a supercritical branching part with primitive first moments and finite second moments, we prove propositions on the mean square convergence and the almost sure convergence of normalized averaging processes associated with the immigration-branching process.

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

Journal of Applied Probability ◽

10.1017/s002190020003566x ◽

1971 ◽

Vol 8 (03) ◽

pp. 589-598 ◽

Cited By ~ 6

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.

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

Journal of Applied Probability ◽

10.2307/3212920 ◽

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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A limit theorem for a class of supercritical branching processes

Journal of Applied Probability ◽

10.1017/s002190020003610x ◽

1972 ◽

Vol 9 (04) ◽

pp. 707-724 ◽

Cited By ~ 17

Author(s):

R. A. Doney

Keyword(s):

General Class ◽

Branching Process ◽

Branching Processes ◽

Generating Functional ◽

Number Of Children ◽

Supercritical Case ◽

Age Dependent ◽

Watson Process ◽

Supercritical Branching Processes ◽

Convergence In Mean

In the Bellman-Harris (B-H) age-dependent branching process, the birth of a child can occur only at the time of its parent's death. A general class of branching process in which births can occur throughout the lifetime of a parent has been introduced by Crump and Mode. This class shares with the B-H process the property that the generation sizes {ξn } form a Galton-Watson process, and so may be classified into subcritical, critical or supercritical according to the value of m = E{ξ 1}. Crump and Mode showed that, as regards extinction probability, asymptotic behaviour, and for the supercritical case, convergence in mean square of Z(t)/E[Z(t)], as t → ∞, where Z(t) is the population size at time t given one ancestor at t = 0, properties of the B-H process can be extended to this general class. In this paper conditions are found for the convergence in distribution of Z(t)/E{Z(t)} in the supercritical case to a non-degenerate limit distribution. In contrast to the B-H process, these conditions are not the same as those for ξn /mn to have a non-degenerate limit. An integral equation is established for the generating function of Z(t), which is more complicated than the corresponding one for the B-H process and involves the conditional probability generating functional of N(x), x 0, ≧ the number of children born to an individual in the age interval [0, x].

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