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Combinatorial Stochastic Processes
Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.- Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random
The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator
The two-parameter Poisson-Dirichlet distribution, denoted PD(α,θ), is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with
Exchangeable and partially exchangeable random partitions
SummaryCall a random partition of the positive integerspartially exchangeable if for each finite sequence of positive integersn1,...,nk, the probability that the partition breaks the firstn1+...+nk
Coalescents with multiple collisions
k−2 � 1 − xb−k � � dx� . Call this process a � -coalescent. Discrete measure-valued processes derived from the � -coalescent model a system of masses undergoing coalescent collisions. Kingman's
Exchangeable Gibbs partitions and Stirling triangles
AbstractFor two collections of nonnegative and suitably normalized weights W = (Wj) and V = (Vn,k), a probability distribution on the set of partitions of the set {1, …, n} is defined by assigning to
Some developments of the Blackwell-MacQueen urn scheme
The Blackwell-MacQueen description of sampling from a Dirichlet random distribution on an abstract space is reviewed, and extended to a general family of random discrete distributions. Results are
The standard additive coalescent
Regard an element of the set Δ := {(x 1 , x 2 , . . .): x 1 ≥ x 2 ≥ ⋯ ≥ 0, ∑ i x i = 1} as a fragmentation of unit mass into clusters of masses x i . The additive coalescent of Evans and Pitman is
Size-biased sampling of Poisson point processes and excursions
SummarySome general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function.
Poisson-Kingman partitions
This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are