1,034 research outputs found
Necessary and sufficient conditions for consistent root reconstruction in Markov models on trees
We establish necessary and sufficient conditions for consistent root
reconstruction in continuous-time Markov models with countable state space on
bounded-height trees. Here a root state estimator is said to be consistent if
the probability that it returns to the true root state converges to 1 as the
number of leaves tends to infinity. We also derive quantitative bounds on the
error of reconstruction. Our results answer a question of Gascuel and Steel and
have implications for ancestral sequence reconstruction in a classical
evolutionary model of nucleotide insertion and deletion.Comment: 30 pages, 3 figures, title of reference [FR] is update
Genealogies in Expanding Populations
The goal of this paper is to prove rigorous results for the behavior of
genealogies in a one-dimensional long range biased voter model introduced by
Hallatschek and Nelson [25]. The first step, which is easily accomplished using
results of Mueller and Tribe [38], is to show that when space and time are
rescaled correctly, our biased voter model converges to a Wright-Fisher SPDE. A
simple extension of a result of Durrett and Restrepo [18] then shows that the
dual branching coalescing random walk converges to a branching Brownian motion
in which particles coalesce after an exponentially distributed amount of
intersection local time. Brunet et al. [8] have conjectured that genealogies in
models of this type are described by the Bolthausen-Sznitman coalescent, see
[39]. However, in the model we study there are no simultaneous coalescences.
Our third and most significant result concerns "tracer dynamics" in which some
of the initial particles in the biased voter model are labeled. We show that
the joint distribution of the labeled and unlabeled particles converges to the
solution of a system of stochastic partial differential equations. A new
duality equation that generalizes the one Shiga [44] developed for the
Wright-Fisher SPDE is the key to the proof of that result.Comment: 40 pages, 1 figur
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