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).

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).

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.

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.

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

2014 ◽  
Vol 46 (2) ◽  
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 function G and a maximum principle which, we prove, is satisfied by every fixed point of G. 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.

scholarly journals Branching random walk with selection at critical rate

Bernoulli ◽  
2017 ◽  
Vol 23 (3) ◽  
pp. 1784-1821
Author(s):  
Bastien Mallein
Keyword(s):  
Random Walk ◽  
Branching Random Walk ◽  
Critical Rate
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