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.

The asymptotic shape of the branching random walk

1978 ◽  
Vol 10 (01) ◽  
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 R p , 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.

The first- and last-birth problems for a multitype age-dependent branching process

1976 ◽  
Vol 8 (03) ◽  
pp. 446-459 ◽  
Author(s):  
J. D. Biggins
Keyword(s):  
Random Walk ◽  
Branching Process ◽  
Branching Random Walk ◽  
First Birth ◽  
Age Dependent

IfBnis the time of the first birth in thenth generation in a supercritical irreducible multitype Crump–Mode process then when there are people in every generationBn/nconverges to a constant; ifDnis the time of the last birth in thenth generation thenDn/nalso converges to a constant on the survival set. Analogous results hold for the extreme members of thenth generation in a branching random walk.

The first- and last-birth problems for a multitype age-dependent branching process

1976 ◽  
Vol 8 (3) ◽  
pp. 446-459 ◽  
Author(s):  
J. D. Biggins
Keyword(s):  
Random Walk ◽  
Branching Process ◽  
Branching Random Walk ◽  
First Birth ◽  
Age Dependent

If Bn is the time of the first birth in the nth generation in a supercritical irreducible multitype Crump–Mode process then when there are people in every generation Bn/n converges to a constant; if Dn is the time of the last birth in the nth generation then Dn/n also converges to a constant on the survival set. Analogous results hold for the extreme members of the nth generation in a branching random walk.

Chernoff's theorem in the branching random walk

1977 ◽  
Vol 14 (3) ◽  
pp. 630-636 ◽  
Author(s):  
J. D. Biggins
Keyword(s):  
Random Walk ◽  
Analogous Result ◽  
Branching Random Walk ◽  
Limiting Behaviour ◽  
Increasing Function ◽  
Chernoff’S Theorem

If Fn∗ is the n-fold Stieltjes convolution of the increasing function F, then a version of Chernoff's theorem, on the limiting behaviour of (Fn∗(na))1/n, is established for Fn∗. If Z(n)(t) is the number of the nth-generation people to the left of t in a supercritical branching random walk then an analogous result is proved for Z(n).

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.

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

1971 ◽  
Vol 8 (03) ◽  
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 &lt; m = h'(1 –) &lt; ∞ and where F(t) is a non-decreasing right continuous function with F(0 –) = 0, F(0 +) &lt; 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.

A limit theorem for a class of supercritical branching processes

1972 ◽  
Vol 9 (04) ◽  
pp. 707-724 ◽  
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].

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

2004 ◽  
Vol 41 (A) ◽  
pp. 273-280 ◽  
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.

Chernoff's theorem in the branching random walk

1977 ◽  
Vol 14 (03) ◽  
pp. 630-636 ◽  
Author(s):  
J. D. Biggins
Keyword(s):  
Random Walk ◽  
Analogous Result ◽  
Branching Random Walk ◽  
Limiting Behaviour ◽  
Increasing Function ◽  
Chernoff’S Theorem

If Fn∗ is the n-fold Stieltjes convolution of the increasing function F, then a version of Chernoff's theorem, on the limiting behaviour of (Fn∗ (na))1/n , is established for Fn∗ . If Z (n)(t) is the number of the nth-generation people to the left of t in a supercritical branching random walk then an analogous result is proved for Z (n).

An age dependent branching process with variable lifetime distribution: The generation size

1974 ◽  
Vol 6 (02) ◽  
pp. 291-308 ◽  
Author(s):  
Robert Fildes
Keyword(s):  
Branching Process ◽  
Random Variable ◽  
Lifetime Distribution ◽  
Mean Square ◽  
Initial Particle ◽  
Age Dependent ◽  
The Mean ◽  
Watson Process ◽  
Number Of Particles

In a branching process with variable lifetime, introduced by Fildes (1972) define Yjk (t) as the number of particles alive in generation k at time t when the initial particle is born in generation j. A limit law similar to that derived in the Bellman-Harris process is proved where it is shown that Yjk (t) suitably normalised converges in mean square to a random variable which is the limit random variable of Znm–n in the Galton-Watson process (m is the mean number of particles born).

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