Exchangeable Gibbs partitions and Stirling triangles |
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Authors: | A Gnedin J Pitman |
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Affiliation: | (1) Utrecht University, The Netherlands;(2) University of California at Berkeley, USA |
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Abstract: | For 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 a generic partition
{Aj, j ≤ k} the probability Vn,k
, where |Aj| is the number of elements of Aj. We impose constraints on the weights by assuming that the resulting random partitions Π n of n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers.
This implies that the weights W must be of a very special form depending on a single parameter α ∈ − ∞, 1]. The case α =
1 is trivial, and for each value of α ≠ = 1 the set of possible V-weights is an infinite-dimensional simplex. We identify
the extreme points of the simplex by solving the boundary problem for a generalized Stirling triangle. In particular, we show
that the boundary is discrete for − ∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members
of the Ewens-Pitman family of random partitions indexed by (α,θ), while for 0 < α < 1 the extremes are obtained by conditioning
an (α,θ)-partition on the asymptotics of the number of blocks of Πn as n tends to infinity. Bibliography: 29 titles.
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 83–102. |
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