57 research outputs found
Persisting randomness in randomly growing discrete structures: graphs and search trees
The successive discrete structures generated by a sequential algorithm from
random input constitute a Markov chain that may exhibit long term dependence on
its first few input values. Using examples from random graph theory and search
algorithms we show how such persistence of randomness can be detected and
quantified with techniques from discrete potential theory. We also show that
this approach can be used to obtain strong limit theorems in cases where
previously only distributional convergence was known.Comment: Official journal fil
Random recursive trees: A boundary theory approach
We show that an algorithmic construction of sequences of recursive trees
leads to a direct proof of the convergence of random recursive trees in an
associated Doob-Martin compactification; it also gives a representation of the
limit in terms of the input sequence of the algorithm. We further show that
this approach can be used to obtain strong limit theorems for various tree
functionals, such as path length or the Wiener index
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}
A note on limits of sequences of binary trees
We discuss a notion of convergence for binary trees that is based on subtree
sizes. In analogy to recent developments in the theory of graphs, posets and
permutations we investigate some general aspects of the topology, such as a
characterization of the set of possible limits and its structure as a metric
space. For random trees the subtree size topology arises in the context of
algorithms for searching and sorting when applied to random input, resulting in
a sequence of nested trees. For these we obtain a structural result based on a
local version of exchangeability. This in turn leads to a central limit
theorem, with possibly mixed asymptotic normality
Ranks, copulas, and permutons
We review a recent development at the interface between discrete mathematics
on one hand and probability theory and statistics on the other, specifically
the use of Markov chains and their boundary theory in connection with the
asymptotics of randomly growing permutations. Permutations connect total orders
on a finite set, which leads to the use of a pattern frequencies. This view is
closely related to classical concepts of nonparametric statistics. We give
several applications and discuss related topics and research areas, in
particular the treatment of other combinatorial families, the cycle view of
permutations, and an approach via exchangeability.Comment: Corrected a statement in Section 6.
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