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A hierarchy on a set S, also called a total partition of S, is a collection $$\mathcal {H}$$ H of subsets of S such that $$S \in \mathcal {H}$$ S∈H , each singleton subset of S belongs to $$\mathcal {H}$$ H , and if $$A, B \in \mathcal {H}$$ A,B∈H then $$A \cap B$$ A∩B equals either A or B or $$\varnothing $$ ∅ . Every exchangeable random hierarchy of positive integers has the same distribution...
The work of Kingman and others shows how the theory of exchangeable random partitions and associated random discrete distributions provides a natural mathematical framework for the analysis of coagulation processes (also called coalescents) and fragmentation processes.
This chapter is inspired by the following quotation from Harris’s 1952 paper [193, §6]: Walks and trees. Random walks and branching processes are both objects of considerable interest in probability theory. We may consider a random walk as a probability measure on sequences of steps-that is, on “walks”. A branching process is a probability measure on “trees”. The purpose of the present section...
The Harris correspondence between random walks and random trees, reviewed in Section 6.3, suggests that a continuous path be regarded as encoding some kind of infinite tree, with each upward excursion of the path corresponding to a subtree. This idea has been developed and applied in various ways by Neveu- Pitman [324, 323], Aldous [5, 6, 7] and Le Gall [271, 272, 273, 275]. This chapter...
This chapter is a review of various constructions of random partitions from Poisson point processes of random lengths, based on the work of Kingman and subsequent authors [249, 341, 371, 351, 362]. The lengths can be interpreted as the jumps of a subordinator, or as the lengths of excursions of some Markov process. The treatment is organized by sections as follows:
It is well known that the random occupation measure induced by the sample path of a Brownian motion B = (Bt, t ≥ 0) admits a jointly continuous local time process (Lxt (B); x ∈ ℝ, t ≥ 0) such that
This chapter reviews Brownian bridge asymptotics for random mappings, first described in 1994 by Aldous and Pitman. The limit distributions as n→∞, of various functionals of a uniformly distributed random mapping from an n element set to itself, are those of corresponding functionals of a Brownian bridge. Similar results known to hold for various non-uniform models of random mappings, according...
Results are obtained concerning the distribution of ranked relative lengths of excursions of a recurrent Markov process from a point in its state space whose inverse local time process is a stable subordinator. It is shown that for a large class of random times T the distribution of relative excursion lengths prior to T is the same as if T were a fixed time. It follows that the generalized arc-sine...
This chapter is a review of basic ideas from Kingman’s theory of exchangeable random partitions [253], as further developed in [14, 347, 350]. This theory turns out to be of interest in a number of contexts, for instance in the study of population genetics, Bayesian statistics, and models for processes of coagulation and fragmentation. The chapter is arranged as follows.
This is a collection of expository articles about various topics at the interface between enumerative combinatorics and stochastic processes. These articles expand on a course of lectures given at the Ecole d’Eté de Probabilités de St. Flour in July 2002. The articles are also called ‘chapters’. Each chapter is fairly self-contained, so readers with adequate background can start reading any...
Results are obtained regarding the distribution of the ranked lengths of component intervals in the complement of the random set of times when a recurrent Markov process returns to its starting point. Various martingales are described in terms of the Lévy measure of the Poisson point process of interval lengths on the local time scale. The martingales derived from the zero set of a one-dimensional...
This chapter introduces a basic sequential construction of random partitions, motivated at first by consideration of uniform random permutations of [n] which are consistent in a certain sense as n varies. This leads to consideration of a particular two-parameter family of exchangeable random partition structures, which can be characterized in various ways, and which is closely related to...
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