285 research outputs found
Pruning Galton-Watson Trees and Tree-valued Markov Processes
We present a new pruning procedure on discrete trees by adding marks on the
nodes of trees. This procedure allows us to construct and study a tree-valued
Markov process by pruning Galton-Watson trees and an
analogous process by pruning a critical or subcritical
Galton-Watson tree conditioned to be infinite. Under a mild condition on
offspring distributions, we show that the process run until
its ascension time has a representation in terms of . A
similar result was obtained by Aldous and Pitman (1998) in the special case of
Poisson offspring distributions where they considered uniform pruning of
Galton-Watson trees by adding marks on the edges of trees
Limit theorems for Markov processes indexed by continuous time Galton--Watson trees
We study the evolution of a particle system whose genealogy is given by a
supercritical continuous time Galton--Watson tree. The particles move
independently according to a Markov process and when a branching event occurs,
the offspring locations depend on the position of the mother and the number of
offspring. We prove a law of large numbers for the empirical measure of
individuals alive at time t. This relies on a probabilistic interpretation of
its intensity by mean of an auxiliary process. The latter has the same
generator as the Markov process along the branches plus additional jumps,
associated with branching events of accelerated rate and biased distribution.
This comes from the fact that choosing an individual uniformly at time t favors
lineages with more branching events and larger offspring number. The central
limit theorem is considered on a special case. Several examples are developed,
including applications to splitting diffusions, cellular aging, branching
L\'{e}vy processes.Comment: Published in at http://dx.doi.org/10.1214/10-AAP757 the Annals of
Applied Probability (http://www.imstat.org/aap/) by the Institute of
Mathematical Statistics (http://www.imstat.org
Routing on trees
We consider three different schemes for signal routing on a tree. The
vertices of the tree represent transceivers that can transmit and receive
signals, and are equipped with i.i.d. weights representing the strength of the
transceivers. The edges of the tree are also equipped with i.i.d. weights,
representing the costs for passing the edges. For each one of our schemes, we
derive sharp conditions on the distributions of the vertex weights and the edge
weights that determine when the root can transmit a signal over arbitrarily
large distances
A survey of max-type recursive distributional equations
In certain problems in a variety of applied probability settings (from
probabilistic analysis of algorithms to statistical physics), the central
requirement is to solve a recursive distributional equation of the form X =^d
g((\xi_i,X_i),i\geq 1). Here (\xi_i) and g(\cdot) are given and the X_i are
independent copies of the unknown distribution X. We survey this area,
emphasizing examples where the function g(\cdot) is essentially a ``maximum''
or ``minimum'' function. We draw attention to the theoretical question of
endogeny: in the associated recursive tree process X_i, are the X_i measurable
functions of the innovations process (\xi_i)?Comment: Published at http://dx.doi.org/10.1214/105051605000000142 in the
Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute
of Mathematical Statistics (http://www.imstat.org
Invariance principles for pruning processes of Galton-Watson trees
Pruning processes have been studied
separately for Galton-Watson trees and for L\'evy trees/forests. We establish
here a limit theory that strongly connects the two studies. This solves an open
problem by Abraham and Delmas, also formulated as a conjecture by L\"ohr,
Voisin and Winter. Specifically, we show that for any sequence of Galton-Watson
forests , , in the domain of attraction of a L\'evy
forest , suitably scaled pruning processes
converge in the Skorohod topology on
cadlag functions with values in the space of (isometry classes of) locally
compact real trees to limiting pruning processes. We separately treat pruning
at branch points and pruning at edges. We apply our results to study ascension
times and Kesten trees and forests.Comment: 33 page
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