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.

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.

scholarly journals Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks

2010 ◽  
Vol 47 (2) ◽  
pp. 513-525 ◽  
Author(s):  
Alexander Iksanov ◽  
Matthias Meiners
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Sufficient Conditions ◽  
Almost Sure Convergence ◽  
Branching Random Walk ◽  
Exponential Rate ◽  
Branching Random Walks

We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. We include in particular the case of Galton-Watson processes so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering (1983). As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be of interest in themselves and which correct, generalize, and simplify some earlier works.

scholarly journals Exponential rate of almost-sure convergence of intrinsic martingales in supercritical branching random walks

2010 ◽  
Vol 47 (02) ◽  
pp. 513-525 ◽  
Author(s):  
Alexander Iksanov ◽  
Matthias Meiners
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Sufficient Conditions ◽  
Almost Sure Convergence ◽  
Branching Random Walk ◽  
Exponential Rate ◽  
Branching Random Walks

We provide sufficient conditions which ensure that the intrinsic martingale in the supercritical branching random walk converges exponentially fast to its limit. We include in particular the case of Galton-Watson processes so that our results can be seen as a generalization of a result given in the classical treatise by Asmussen and Hering (1983). As an auxiliary tool, we prove ultimate versions of two results concerning the exponential renewal measures which may be of interest in themselves and which correct, generalize, and simplify some earlier works.

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.

scholarly journals Large deviation estimates for branching random walks

2019 ◽  
Vol 23 ◽  
pp. 823-840
Author(s):  
Dariusz Buraczewski ◽  
Mariusz Maślanka
Keyword(s):  
Random Walk ◽  
Law Of Large Numbers ◽  
Large Deviation ◽  
First Passage Time ◽  
Passage Time ◽  
Branching Random Walk ◽  
First Passage ◽  
Branching Random Walks ◽  
The Central Limit Theorem ◽  
Large Numbers

For the branching random walk drifting to −∞ we study large deviations-type estimates for the first passage time. We prove the corresponding law of large numbers and the central limit theorem.

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