A pathwise approach to the extinction of branching processes with countably many types

Braunsteins, Peter; Decrouez, Geoffrey; Hautphenne, Sophie

doi:10.1016/j.spa.2018.03.013

Cited by 13 publications

(9 citation statements)

References 29 publications

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“…For each ω ∈ Ω the family tree of {Y k } is then given by f (X(ω)), where X(ω) is defined in (2.1). Variants of {Y k } (which do not permit explosion) can be found in [12] and [19].…”

Section: An Embedded Gwpve With Explosionsmentioning

confidence: 99%

“…To resolve the problem in the infinite-type setting we should give both a partial and a global extinction criterion. A number of authors have progressed in this direction [8,12,21,22,30,36,37]. In the infinite-type case, the analogue of the Perron-Frobenius eigenvalue is the convergence norm ν(M ) of M defined in (2.4), which gives a partial extinction criterion:q = 1 if and only if ν(M ) ≤ 1, see [37,Theorem 4.1].…”

Section: Introductionmentioning

confidence: 99%

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Extinction in lower Hessenberg branching processes with countably many types

Braunsteins¹,

Hautphenne²

2019

Ann. Appl. Probab.

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We consider a class of branching processes with countably many types which we refer to as Lower Hessenberg branching processes. These are multitype Galton-Watson processes with typeset X = {0, 1, 2, . . . }, in which individuals of type i may give birth to offspring of type j ≤ i + 1 only. For this class of processes, we study the set S of fixed points of the progeny generating function. In particular, we highlight the existence of a continuum of fixed points whose minimum is the global extinction probability vector q and whose maximum is the partial extinction probability vectorq. In the case whereq = 1, we derive a global extinction criterion which holds under second moment conditions, and whenq < 1 we develop necessary and sufficient conditions for q =q.

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Section: An Embedded Gwpve With Explosionsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Extinction in lower Hessenberg branching processes with countably many types

Braunsteins¹,

Hautphenne²

2019

Ann. Appl. Probab.

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“…For an introduction to infinite type branching processes, we recommend Braunsteins' exposition in [8,Chapter 2] and the references mentioned therein. This presentation also presents the rather wellunderstood results on the extinction of finite type branching processes.…”

Section: Solving This Recurrence Relation Yields the Representationmentioning

confidence: 99%

Boolean percolation on digraphs and random exchange processes

Braun¹

2021

Preprint

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We study, in a general graph-theoretic formulation, a long-range percolation model introduced by Lamperti in [27]. For various underlying directed graphs, we discuss connections between this model and random exchange processes. We clarify, for n ∈ N, under which conditions the lattices N n 0 and Z n are essentially covered in this model. Moreover, for all n ≥ 2, we establish that it is impossible to cover the directed n-ary tree in our model.

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“…To generalise (i)-(iii) to the infinite type setting it is generally accepted that we should give the corresponding results for both q andq. That is, we aim to (i) derive a partial and a global extinction criterion, (ii) develop iterative methods to compute q andq when an algebraic expression cannot be found, and (iii) locate q andq in S. While open questions remain, a number of authors have made progress on (i) [7,9,15,18,20], (ii) [6,11,16], and (iii) [2,7,15] (to name a few).…”

Section: Introductionmentioning

confidence: 99%

The probabilities of extinction in a branching random walk on a strip

Braunsteins

Hautphenne

2020

J. Appl. Probab.

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We consider a class of multitype Galton–Watson branching processes with a countably infinite type set $\mathcal{X}_d$ whose mean progeny matrices have a block lower Hessenberg form. For these processes, we study the probabilities $\textbf{\textit{q}}(A)$ of extinction in sets of types $A\subseteq \mathcal{X}_d$ . We compare $\textbf{\textit{q}}(A)$ with the global extinction probability $\textbf{\textit{q}} = \textbf{\textit{q}}(\mathcal{X}_d)$ , that is, the probability that the population eventually becomes empty, and with the partial extinction probability $\tilde{\textbf{\textit{q}}}$ , that is, the probability that all types eventually disappear from the population. After deriving partial and global extinction criteria, we develop conditions for $\textbf{\textit{q}} < \textbf{\textit{q}}(A) < \tilde{\textbf{\textit{q}}}$ . We then present an iterative method to compute the vector $\textbf{\textit{q}}(A)$ for any set A. Finally, we investigate the location of the vectors $\textbf{\textit{q}}(A)$ in the set of fixed points of the progeny generating vector.

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A pathwise approach to the extinction of branching processes with countably many types

Cited by 13 publications

References 29 publications

Extinction in lower Hessenberg branching processes with countably many types

Extinction in lower Hessenberg branching processes with countably many types

Boolean percolation on digraphs and random exchange processes

The probabilities of extinction in a branching random walk on a strip

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