321 research outputs found
Linear-fractional branching processes with countably many types
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 -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
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
We introduce a modified Galton-Watson process using the framework of an
infinite system of particles labeled by , where is the rank of the
particle born at time . The key assumption concerning the offspring numbers
of different particles is that they are independent, but their distributions
may depend on the particle label . 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
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
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 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
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
. We turn to the case of very heavy tailed reproduction
distribution assuming Zubkov's regularity condition with parameter
. 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
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
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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