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.

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 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 Approximating Critical Parameters of Branching Random Walks

2009 ◽  
Vol 46 (02) ◽  
pp. 463-478 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Critical Parameter ◽  
Branching Random Walk ◽  
Critical Parameters ◽  
Bond Percolation ◽  
Branching Random Walks

Given a branching random walk on a graph, we consider two kinds of truncations: either by inhibiting the reproduction outside a subset of vertices or by allowing at most m particles per vertex. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.

scholarly journals A Note on Branching Random Walks on Finite Sets

2005 ◽  
Vol 42 (1) ◽  
pp. 287-294 ◽  
Author(s):  
Thomas Mountford ◽  
Rinaldo B. Schinazi
Keyword(s):  
Random Walk ◽  
Finite Number ◽  
Random Walks ◽  
Ecological Model ◽  
Critical Value ◽  
Strong Sense ◽  
Branching Random Walk ◽  
Finite Sets ◽  
Branching Random Walks ◽  
Number Of Species

We show that a branching random walk that is supercritical on is also supercritical, in a rather strong sense, when restricted to a large enough finite ball of This implies that the critical value of branching random walks on finite balls converges to the critical value of branching random walks on as the radius increases to infinity. Our main result also implies coexistence of an arbitrary finite number of species for an ecological model.

A note on the branching random walk

1982 ◽  
Vol 19 (2) ◽  
pp. 421-424 ◽  
Author(s):  
Norman Kaplan
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Branching Random Walk ◽  
Branching Random Walks

A well-known theorem in the theory of branching random walks is shown to hold when only Σj log jpj <∞. This result was asserted by Athreya and Kaplan (1978), but their proof was incorrect.

scholarly journals Non-intersection of transient branching random walks

2020 ◽  
Vol 178 (1-2) ◽  
pp. 1-23
Author(s):  
Tom Hutchcroft
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Cayley Graph ◽  
Spectral Radius ◽  
Branching Random Walk ◽  
Expected Number ◽  
Branching Random Walks ◽  
Offspring Distribution ◽  
The Future ◽  
Supercritical Phase

Abstract Let G be a Cayley graph of a nonamenable group with spectral radius $$\rho < 1$$ ρ < 1 . It is known that branching random walk on G with offspring distribution $$\mu $$ μ is transient, i.e., visits the origin at most finitely often almost surely, if and only if the expected number of offspring $${\overline{\mu }}$$ μ ¯ satisfies $$\overline{\mu }\le \rho ^{-1}$$ μ ¯ ≤ ρ - 1 . Benjamini and Müller (Groups Geom Dyn, 6:231–247, 2012) conjectured that throughout the transient supercritical phase $$1<\overline{\mu } \le \rho ^{-1}$$ 1 < μ ¯ ≤ ρ - 1 , and in particular at the recurrence threshold $${\overline{\mu }} = \rho ^{-1}$$ μ ¯ = ρ - 1 , the trace of the branching random walk is tree-like in the sense that it is infinitely-ended almost surely on the event that the walk survives forever. This is essentially equivalent to the assertion that two independent copies of the branching random walk intersect at most finitely often almost surely. We prove this conjecture, along with several other related conjectures made by the same authors. A central contribution of this work is the introduction of the notion of local unimodularity, which we expect to have several further applications in the future.

A note on the branching random walk

1982 ◽  
Vol 19 (02) ◽  
pp. 421-424
Author(s):  
Norman Kaplan
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Branching Random Walk ◽  
Branching Random Walks

A well-known theorem in the theory of branching random walks is shown to hold when only Σj log jpj &lt;∞. This result was asserted by Athreya and Kaplan (1978), but their proof was incorrect.

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 Approximating Critical Parameters of Branching Random Walks

2009 ◽  
Vol 46 (2) ◽  
pp. 463-478 ◽  
Author(s):  
Daniela Bertacchi ◽  
Fabio Zucca
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Critical Parameter ◽  
Branching Random Walk ◽  
Critical Parameters ◽  
Bond Percolation ◽  
Branching Random Walks

Given a branching random walk on a graph, we consider two kinds of truncations: either by inhibiting the reproduction outside a subset of vertices or by allowing at most m particles per vertex. We investigate the convergence of weak and strong critical parameters of these truncated branching random walks to the analogous parameters of the original branching random walk. As a corollary, we apply our results to the study of the strong critical parameter of a branching random walk restricted to the cluster of a Bernoulli bond percolation.

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