Branching processes

Athreya, Krishna B.; Vidyashankar, Anand N.

doi:10.1016/s0169-7161(01)19004-8

Cited by 594 publications

(1,309 citation statements)

References 49 publications

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“…Consider a discrete-time-t branching process whose reproduction law has probability mass P (M = m) = π m , m ≥ 0 for the number M of offspring per capita, see [9] and [1]. We avoid the trivial case π 1 = 1.…”

Section: Count Genealogies Of Bienaymé-galton-watson Branching Processesmentioning

confidence: 99%

On the genealogy and coalescence times of Bienaymé–Galton–Watson branching processes

Grosjean

Huillet

2017

Stochastic Models

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Abstract. Coalescence processes have received a lot of attention in the context of conditional branching processes with fixed population size and nonoverlapping generations. Here we focus on similar problems in the context of the standard unconditional Bienaymé-Galton-Watson branching processes, either (sub)-critical or supercritical. Using an analytical tool, we derive the structure of some counting aspects of the ancestral genealogy of such processes, including: the transition matrix of the ancestral count process and an integral representation of various coalescence times distributions, such as the time to most recent common ancestor of a random sample of arbitrary size, including full size.We illustrate our results on two important examples of branching mechanisms displaying either finite or infinite reproduction mean, their main interest being to offer a closed form expression for their probability generating functions at all times. Large time behaviors are investigated.

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Section: Count Genealogies Of Bienaymé-galton-watson Branching Processesmentioning

confidence: 99%

On the genealogy and coalescence times of Bienaymé–Galton–Watson branching processes

Grosjean

Huillet

2017

Stochastic Models

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“…where φ •n (z) is the n-th composition of φ (z) with itself, 1 . Similarly, if N n (i) is the number of individuals alive at generation n given there are N 0 = i independent founders, we clearly get…”

Section: The Pgf Approachmentioning

confidence: 99%

Additional aspects of the generalized linear-fractional branching process

Grosjean

Huillet

2016

Ann Inst Stat Math

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We derive some additional results on the Bienyamé-Galton-Watson branching process with θ−linear fractional branching mechanism, as studied in [16]. This includes: the explicit expression of the limit laws in both the sub-critical cases and the super-critical cases with finite mean, the long-run behavior of the population size in the critical case, limit laws in the super-critical cases with infinite mean when the θ-process is either regular or explosive, results regarding the time to absorption, an expression of the probability law of the θ-branching mechanism involving Bell polynomials, the explicit computation of the stochastic transition matrix of the θ−process, together with its powers.

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“…, X K (0)) of individuals of the different types {(k, 1), 1 ≤ k ≤ K}. The s-th individual (in some ordering) of type (k, 1) in Z(2r) = X(r) has offspring vector Y ks;r = (Y (1) ks;r , . .…”

Section: Intersection Graphs and Branching Processesmentioning

confidence: 99%

The shortest distance in random multi‐type intersection graphs

Barbour¹,

Reinert²

2010

Random Struct Algorithms

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Using an associated branching process as the basis of our approximation, we show that typical inter-point distances in a multi-type random intersection graph have a defective distribution, which is well described by a mixture of translated and scaled Gumbel distributions, the missing mass corresponding to the event that the vertices are not in the same component of the graph. © 2010 Wiley Periodicals, Inc. The shortest distance in random multi-type intersection graphsA. D. Barbour * and G. Reinert † Universität Zürich and University of Oxford AbstractUsing an associated branching process as the basis of our approximation, we show that typical inter-point distances in a multitype random intersection graph have a defective distribution, which is well described by a mixture of translated and scaled Gumbel distributions, the missing mass corresponding to the event that the vertices are not in the same component of the graph.

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Branching processes

Cited by 594 publications

References 49 publications

On the genealogy and coalescence times of Bienaymé–Galton–Watson branching processes

On the genealogy and coalescence times of Bienaymé–Galton–Watson branching processes

Additional aspects of the generalized linear-fractional branching process

The shortest distance in random multi‐type intersection graphs

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