Sums of Lifetimes in Age Dependent Branching Processes

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) = Σk=0∞pksk and suppose h(1) ≡ m, and assume h”(1) < ∞. Additional assumptions will be added as required.

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On the convergence of the supercritical branching processes with immigration

Journal of Applied Probability ◽

10.2307/3213010 ◽

1977 ◽

Vol 14 (2) ◽

pp. 387-390 ◽

Cited By ~ 3

Author(s):

Harry Cohn

Keyword(s):

Distribution Function ◽

Limit Distribution ◽

Branching Process ◽

Branching Processes ◽

Sufficient Condition ◽

Necessary And Sufficient Condition ◽

Limit Distribution Function ◽

Branching Process With Immigration ◽

Necessary And Sufficient ◽

Supercritical Branching Processes

It is shown for a supercritical branching process with immigration that if the log moment of the immigration distribution is infinite, then no sequence of positive constants {cn} exists such that {Xn/cn} converges in law to a proper limit distribution function F, except for the case F(0 +) = 1. Seneta's result [1] combined with the above-mentioned one imply that if 1 < m < ∞ then the finiteness of the log moment of the immigration distribution is a necessary and sufficient condition for the existence of some constants {cn} such that {Xn/cn} converges in law to a proper limit distribution function F, with F(0 +) < 1.

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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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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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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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Asymptotic probabilities in a critical age-dependent branching process

Journal of Applied Probability ◽

10.2307/3212467 ◽

1976 ◽

Vol 13 (3) ◽

pp. 476-485 ◽

Cited By ~ 2

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.

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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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On finite exponential moments for branching processes and busy periods for queues

Journal of Applied Probability ◽

10.1239/jap/1082552204 ◽

2004 ◽

Vol 41 (A) ◽

pp. 273-280 ◽

Cited By ~ 4

Author(s):

Marvin K. Nakayama ◽

Perwez Shahabuddin ◽

Karl Sigman

Keyword(s):

Random Walk ◽

Busy Period ◽

Branching Process ◽

Branching Processes ◽

Exponential Moment ◽

Exponential Moments ◽

Simulation Based ◽

Age Dependent ◽

Semi Markov Processes ◽

Number Of Customers

Using a known fact that a Galton–Watson branching process can be represented as an embedded random walk, together with a result of Heyde (1964), we first derive finite exponential moment results for the total number of descendants of an individual. We use this basic and simple result to prove analogous results for the population size at time t and the total number of descendants by time t in an age-dependent branching process. This has applications in justifying the interchange of expectation and derivative operators in simulation-based derivative estimation for generalized semi-Markov processes. Next, using the result of Heyde (1964), we show that, in a stable GI/GI/1 queue, the length of a busy period and the number of customers served in a busy period have finite exponential moments if and only if the service time does.

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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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