102 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
Critical branching as a pure death process coming down from infinity
We consider the critical Galton-Watson process with overlapping generations stemming from a single founder. Assuming that both the variance of the offspring number and the average generation length are finite, we establish the convergence of the finite-dimensional distributions, conditioned on non-extinction at a remote time of observation. The limiting process is identified as a pure death process coming down from infinity.This result brings a new perspective on Vatutin\u27s dichotomy, claiming that in the critical regime of age-dependent reproduction, an extant population either contains a large number of short-living individuals or consists of few long-living individuals
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
Critical branching as a pure death process coming down from infinity
We consider the critical Galton-Watson process with overlapping generations stemming from a single founder. Assuming that both the variance of the offspring number and the average generation length are finite, we establish the convergence of the finite-dimensional distributions, conditioned on non-extinction at a remote time of observation. The limiting process is identified as a pure death process coming down from infinity.This result brings a new perspective on Vatutin\u27s dichotomy, claiming that in the critical regime of age-dependent reproduction, an extant population either contains a large number of short-living individuals or consists of few long-living individuals
Undergraduate statistical inference
This text has grown from the lecture notes for the undergraduate course "Statistical Inference", given at the Chalmers University of Technology and the University of Gothenburg. The course material was originally based on the second edition of the book "Mathematical statistics and data analysis" by John Rice. A great number of examples and exercises included in this compendium are borrowed from Rice’s textbook
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