An infinite alleles version of the Markov branching process

Pakes, Anthony G.

doi:10.1017/s1446788700030445

Cited by 11 publications

(24 citation statements)

References 7 publications

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“…In continuous time, Pakes [29] studied Markovian branching processes and gave the counterpart in the time-continuous setting, of properties found in the previously cited paper [14]. In particular, his results about the frequency spectrum and the "limiting frequency spectrum" are similar to ours, stated in Section 3.…”

Section: Introductionsupporting

confidence: 64%

Birth and Death Processes with Neutral Mutations

Champagnat

Lambert

Richard

2012

International Journal of Stochastic Analysis

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We review recent results of ours concerning branching processes with general lifetimes and neutral mutations, under the infinitely many alleles model, where mutations can occur either at the birth of particles or at a constant rate during their lives. In both models, we study the allelic partition of the population at time t. We give closed-form formulae for the expected frequency spectrum at t and prove a pathwise convergence to an explicit limit, as t → ∞, of the relative numbers of types younger than some given age and carried by a given number of particles small families . We also provide the convergences in distribution of the sizes or ages of the largest families and of the oldest families. In the case of exponential lifetimes, population dynamics are given by linear birth and death processes, and we can most of the time provide general formulations of our results unifying both models.

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Section: Introductionsupporting

confidence: 64%

Birth and Death Processes with Neutral Mutations

Champagnat

Lambert

Richard

2012

International Journal of Stochastic Analysis

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“…In [18], Griffiths and Pakes study the case of a Bienaymé-Galton-Watson process where children can independently be mutants with a given probability: the authors obtained asymptotic results about the number of alleles and the frequency spectrum at generation n as n → ∞. In [30], Pakes gets analogous properties concerning continuous-time Markov branching processes. In particular, his formula of the expected frequency spectrum can be seen as a counterpart of ours, stated in Section 3.1.…”

Section: Introductionmentioning

confidence: 99%

“…In particular, his formula of the expected frequency spectrum can be seen as a counterpart of ours, stated in Section 3.1. These two works [18,30] have recently been used by Kimmel and coworkers in several articles. In [24], the authors are interested in the evolution of parts of DNA called Alu sequences.…”

Section: Introductionmentioning

confidence: 99%

“…They model the evolution of these sequences using the infinite-alleles branching process of [18] with a linear-fractional offspring distribution. In [37], the authors consider the model of Pakes [30] and they especially obtain an explicit expression of the limiting mean frequency spectrum in the particular case of birth and death processes and they also study the variance frequency spectrum.…”

Section: Introductionmentioning

confidence: 99%

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Splitting trees with neutral mutations at birth

Richard

2014

Stochastic Processes and their Applications

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26 pagesInternational audienceWe consider a population model where individuals behave independently from each other and whose genealogy is described by a chronological tree called splitting tree. The individuals have i.i.d. (non-exponential) lifetime durations and give birth at constant rate to clonal or mutant children in an infinitely many alleles model with neutral mutations. First, to study the allelic partition of the population, we are interested in its frequency spectrum, which, at a fixed time, describes the number of alleles carried by a given number of individuals and with a given age. We compute the expected value of this spectrum and obtain some almost sure convergence results thanks to classical properties of Crump-Mode-Jagers (CMJ) processes counted by random characteristics. Then, by using multitype CMJ-processes, we get asymptotic properties about the number of alleles that have undergone a fixed number of mutations with respect to the ancestral allele of the population

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“…In a biological context it is more natural to study the corresponding continuous-time version of the model. Such studies were done recently in [6,7] and earlier in [14]. In this framework one would attach to each clique two exponential clocks, ringing at rate kλ and kµ respectively, where k is the size of the clique and λ, µ > 0 two parameters.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Degree Distribution of a Duplication–Deletion Random Graph Model

Thörnblad¹

2015

Internet Mathematics

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We study a discrete-time duplication-deletion random graph model and analyse its asymptotic degree distribution. The random graphs consists of disjoint cliques. In each time step either a new vertex is brought in with probability 0 < p < 1 and attached to an existing clique, chosen with probability proportional to the clique size, or all the edges of a random vertex are deleted with probability 1 − p. We prove almost sure convergence of the asymptotic degree distribution and find its exact values in terms of a hypergeometric integral, expressed in terms of the parameter p. In the regime 0 < p < 1 2 we show that the degree sequence decays exponentially at rate p 1−p , whereas it satisfies a power-law with exponent p 2p−1 if 1 2 < p < 1. At the threshold p = 1 2 the degree sequence lies between a power-law and exponential decay.

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An infinite alleles version of the Markov branching process

Cited by 11 publications

References 7 publications

Birth and Death Processes with Neutral Mutations

Birth and Death Processes with Neutral Mutations

Splitting trees with neutral mutations at birth

Asymptotic Degree Distribution of a Duplication–Deletion Random Graph Model

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