2,467 research outputs found
Regenerative tree growth: structural results and convergence
We introduce regenerative tree growth processes as consistent families of
random trees with n labelled leaves, n>=1, with a regenerative property at
branch points. This framework includes growth processes for exchangeably
labelled Markov branching trees, as well as non-exchangeable models such as the
alpha-theta model, the alpha-gamma model and all restricted exchangeable models
previously studied. Our main structural result is a representation of the
growth rule by a sigma-finite dislocation measure kappa on the set of
partitions of the natural numbers extending Bertoin's notion of exchangeable
dislocation measures from the setting of homogeneous fragmentations. We use
this representation to establish necessary and sufficient conditions on the
growth rule under which we can apply results by Haas and Miermont for
unlabelled and not necessarily consistent trees to establish self-similar
random trees and residual mass processes as scaling limits. While previous
studies exploited some form of exchangeability, our scaling limit results here
only require a regularity condition on the convergence of asymptotic
frequencies under kappa, in addition to a regular variation condition.Comment: 23 pages, new title, restructured, presentation improve
Leader election: A Markov chain approach
A well-studied randomized election algorithm proceeds as follows: In each
round the remaining candidates each toss a coin and leave the competition if
they obtain heads. Of interest is the number of rounds required and the number
of winners, both related to maxima of geometric random samples, as well as the
number of remaining participants as a function of the number of rounds. We
introduce two related Markov chains and use ideas and methods from discrete
potential theory to analyse the respective asymptotic behaviour as the initial
number of participants grows. One of the tools used is the approach via the
R\'enyi-Sukhatme representation of exponential order statistics, which was
first used in the leader election context by Bruss and Gr\"ubel in
\cite{BrGr03}
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