scholarly journals Extremes of multitype branching random walks: heaviest tail wins

2019 ◽  
Vol 51 (2) ◽  
pp. 514-540
Author(s):  
Ayan Bhattacharya ◽  
Krishanu Maulik ◽  
Zbigniew Palmowski ◽  
Parthanil Roy
Keyword(s):  
Random Walk ◽  
Random Walks ◽  
Asymptotic Distribution ◽  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Weak Limit ◽  
The Other ◽  
Branching Random Walk ◽  
Branching Random Walks

AbstractWe consider a branching random walk on a multitype (with Q types of particles), supercritical Galton–Watson tree which satisfies the Kesten–Stigum condition. We assume that the displacements associated with the particles of type Q have regularly varying tails of index $\alpha$ , while the other types of particles have lighter tails than the particles of type Q. In this paper we derive the weak limit of the sequence of point processes associated with the positions of the particles in the nth generation. We verify that the limiting point process is a randomly scaled scale-decorated Poisson point process using the tools developed by Bhattacharya, Hazra, and Roy (2018). As a consequence, we obtain the asymptotic distribution of the position of the rightmost particle in the nth generation.

scholarly journals Persistence and Equilibria of Branching Populations with Exponential Intensity

2012 ◽  
Vol 49 (1) ◽  
pp. 226-244
Author(s):  
Zakhar Kabluchko
Keyword(s):  
Laplace Transform ◽  
Random Walks ◽  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Local Extinction ◽  
Branching Random Walks ◽  
Cluster Distribution

We consider a system of independent branching random walks on R which start from a Poisson point process with intensity of the form eλ(du) = e-λudu, where λ ∈ R is chosen in such a way that the overall intensity of particles is preserved. Denote by χ the cluster distribution, and let φ be the log-Laplace transform of the intensity of χ. If λφ'(λ) > 0, we show that the system is persistent, meaning that the point process formed by the particles in the nth generation converges as n → ∞ to a non-trivial point process Πeλχ with intensity eλ. If λφ'(λ) < 0 then the branching population suffers local extinction, meaning that the limiting point process is empty. We characterize point processes on R which are cluster invariant with respect to the cluster distribution χ as mixtures of the point processes Πceλχ over c > 0 and λ ∈ Kst, where Kst = {λ ∈ R: φ(λ) = 0, λφ'(λ) > 0}.

scholarly journals Persistence and Equilibria of Branching Populations with Exponential Intensity

2012 ◽  
Vol 49 (01) ◽  
pp. 226-244
Author(s):  
Zakhar Kabluchko
Keyword(s):  
Laplace Transform ◽  
Random Walks ◽  
Point Process ◽  
Poisson Point Process ◽  
Point Processes ◽  
Local Extinction ◽  
Branching Random Walks ◽  
Cluster Distribution

We consider a system of independent branching random walks on R which start from a Poisson point process with intensity of the form e λ(du) = e-λu du, where λ ∈ R is chosen in such a way that the overall intensity of particles is preserved. Denote by χ the cluster distribution, and let φ be the log-Laplace transform of the intensity of χ. If λφ'(λ) &gt; 0, we show that the system is persistent, meaning that the point process formed by the particles in the nth generation converges as n → ∞ to a non-trivial point process Πeλ χ with intensity e λ. If λφ'(λ) &lt; 0 then the branching population suffers local extinction, meaning that the limiting point process is empty. We characterize point processes on R which are cluster invariant with respect to the cluster distribution χ as mixtures of the point processes Π ce λ χ over c &gt; 0 and λ ∈ K st, where K st = {λ ∈ R: φ(λ) = 0, λφ'(λ) &gt; 0}.

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.

Extreme values of independent stochastic processes

1977 ◽  
Vol 14 (4) ◽  
pp. 732-739 ◽  
Author(s):  
Bruce M. Brown ◽  
Sidney I. Resnick
Keyword(s):  
Stochastic Processes ◽  
Point Process ◽  
Time Scales ◽  
Poisson Point Process ◽  
Point Processes ◽  
Extreme Values ◽  
Weak Limit ◽  
One Dimensional ◽  
Limit Process ◽  
Related Point

The maxima of independent Weiner processes spatially normalized with time scales compressed is considered and it is shown that a weak limit process exists. This limit process is stationary, and its one-dimensional distributions are of standard extreme-value type. The method of proof involves showing convergence of related point processes to a limit Poisson point process. The method is extended to handle the maxima of independent Ornstein–Uhlenbeck processes.

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.

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