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.

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 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.

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.

Branching random walk in varying environments

1982 ◽  
Vol 14 (02) ◽  
pp. 359-367 ◽  
Author(s):  
C. F. Klebaner
Keyword(s):  
Random Walk ◽  
Asymptotic Theory ◽  
Sufficient Conditions ◽  
Almost Sure Convergence ◽  
Point Of View ◽  
Branching Random Walk ◽  
Time Interval ◽  
The Central Limit Theorem ◽  
Age Dependent ◽  
Varying Environments

The branching random walk model is generalized towards generation-dependent displacement and reproduction distributions. Asymptotic theory of branching random walk in varying environments from theL2point of view is given. IfZn(x) is the number of nth-generation particles to the left ofx, then under appropriate conditions for suitably chosenxn,Zn(xn)/Zn(+∞) converges inL2completely to a limiting distribution. Sufficient conditions for almost sure convergence are given. As a corollary an analogue of the central limit theorem for the proportion of particles of the nth generation in time intervalInin the age-dependent Crump–Mode–Jagers process is obtained.

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.

Branching random walk in varying environments

1982 ◽  
Vol 14 (2) ◽  
pp. 359-367 ◽  
Author(s):  
C. F. Klebaner
Keyword(s):  
Random Walk ◽  
Asymptotic Theory ◽  
Sufficient Conditions ◽  
Almost Sure Convergence ◽  
Point Of View ◽  
Branching Random Walk ◽  
Time Interval ◽  
The Central Limit Theorem ◽  
Age Dependent ◽  
Varying Environments

The branching random walk model is generalized towards generation-dependent displacement and reproduction distributions. Asymptotic theory of branching random walk in varying environments from the L2 point of view is given. If Zn(x) is the number of nth-generation particles to the left of x, then under appropriate conditions for suitably chosen xn, Zn (xn)/Zn (+∞) converges in L2 completely to a limiting distribution. Sufficient conditions for almost sure convergence are given. As a corollary an analogue of the central limit theorem for the proportion of particles of the nth generation in time interval In in the age-dependent Crump–Mode–Jagers process is obtained.

scholarly journals Strong Local Survival of Branching Random Walks is Not Monotone

2014 ◽  
Vol 46 (02) ◽  
pp. 400-421 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Generating Function ◽  
Continuous Time ◽  
Local Property ◽  
Branching Random Walk ◽  
Infinite Dimensional ◽  
Branching Random Walks ◽  
Local Survival ◽  
Branching Law

In this paper we study the strong local survival property for discrete-time and continuous-time branching random walks. We study this property by means of an infinite-dimensional generating functionGand a maximum principle which, we prove, is satisfied by every fixed point ofG. We give results for the existence of a strong local survival regime and we prove that, unlike local and global survival, in continuous time, strong local survival is not a monotone property in the general case (though it is monotone if the branching random walk is quasitransitive). We provide an example of an irreducible branching random walk where the strong local property depends on the starting site of the process. By means of other counterexamples, we show that the existence of a pure global phase is not equivalent to nonamenability of the process, and that even an irreducible branching random walk with the same branching law at each site may exhibit nonstrong local survival. Finally, we show that the generating function of an irreducible branching random walk can have more than two fixed points; this disproves a previously known result.

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