321 research outputs found

    Linear-fractional branching processes with countably many types

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    We study multi-type Bienaym\'e-Galton-Watson processes with linear-fractional reproduction laws using various analytical tools like contour process, spinal representation, Perron-Frobenius theorem for countable matrices, renewal theory. For this special class of branching processes with countably many types we present a transparent criterion for RR-positive recurrence with respect to the type space. This criterion appeals to the Malthusian parameter and the mean age at childbearing of the associated linear-fractional Crump-Mode-Jagers process.Comment: 2nd version revised for SP

    General linear-fractional branching processes with discrete time

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    We study a linear-fractional Bienaym\'e-Galton-Watson process with a general type space. The corresponding tree contour process is described by an alternating random walk with the downward jumps having a geometric distribution. This leads to the linear-fractional distribution formula for an arbitrary observation time, which allows us to establish transparent limit theorems for the subcritical, critical and supercritical cases. Our results extend recent findings for the linear-fractional branching processes with countably many types

    Rank-dependent Galton-Watson processes and their pathwise duals

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    We introduce a modified Galton-Watson process using the framework of an infinite system of particles labeled by (x,t)(x,t), where xx is the rank of the particle born at time tt. The key assumption concerning the offspring numbers of different particles is that they are independent, but their distributions may depend on the particle label (x,t)(x,t). For the associated system of coupled monotone Markov chains, we address the issue of pathwise duality elucidated by a remarkable graphical representation, with the trajectories of the primary Markov chains and their duals coalescing together to form forest graphs on a two-dimensional grid

    The coalescent effective size of age-structured populations

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    We establish convergence to the Kingman coalescent for a class of age-structured population models with time-constant population size. Time is discrete with unit called a year. Offspring numbers in a year may depend on mother's age.Comment: Published at http://dx.doi.org/10.1214/105051605000000223 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

    A Consistent Estimator of the Evolutionary Rate

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    We consider a branching particle system where particles reproduce according to the pure birth Yule process with the birth rate L, conditioned on the observed number of particles to be equal n. Particles are assumed to move independently on the real line according to the Brownian motion with the local variance s2. In this paper we treat nn particles as a sample of related species. The spatial Brownian motion of a particle describes the development of a trait value of interest (e.g. log-body-size). We propose an unbiased estimator Rn2 of the evolutionary rate r2=s2/L. The estimator Rn2 is proportional to the sample variance Sn2 computed from n trait values. We find an approximate formula for the standard error of Rn2 based on a neat asymptotic relation for the variance of Sn2

    Reduced branching processes with very heavy tails

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    The reduced Markov branching process is a stochastic model for the genealogy of an unstructured biological population. Its limit behavior in the critical case is well studied for the Zolotarev-Slack regularity parameter α∈(0,1]\alpha\in(0,1]. We turn to the case of very heavy tailed reproduction distribution α=0\alpha=0 assuming Zubkov's regularity condition with parameter β∈(0,∞)\beta\in(0,\infty). Our main result gives a new asymptotic pattern for the reduced branching process conditioned on non-extinction during a long time interval.Comment: 15 pages, 1 figur

    Coalescent approximation for structured populations in a stationary random environment

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    We establish convergence to the Kingman coalescent for the genealogy of a geographically - or otherwise - structured version of the Wright-Fisher population model with fast migration. The new feature is that migration probabilities may change in a random fashion. This brings a novel formula for the coalescent effective population size (EPS). We call it a quenched EPS to emphasize the key feature of our model - random environment. The quenched EPS is compared with an annealed (mean-field) EPS which describes the case of constant migration probabilities obtained by averaging the random migration probabilities over possible environments

    A special family of Galton-Watson processes with explosions

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    The linear-fractional Galton-Watson processes is a well known case when many characteristics of a branching process can be computed explicitly. In this paper we extend the two-parameter linear-fractional family to a much richer four-parameter family of reproduction laws. The corresponding Galton-Watson processes also allow for explicit calculations, now with possibility for infinite mean, or even infinite number of offspring. We study the properties of this special family of branching processes, and show, in particular, that in some explosive cases the time to explosion can be approximated by the Gumbel distribution
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