Abstract
Assume that there is a random number K of positive integer random variables S1, …, SK that are conditionally independent given K and all have identical distributions. A random integer partition N = S1 + S2 + … + SK arises, and we denote by PN the conditional distribution of this partition for a fixed value of N. We prove that the distributions {PN} ∞N=1 form a partition structure in the sense of Kingman if and only if they are governed by the Ewens-Pitman Formula. The latter generalizes the celebrated Ewens sampling formula, which has numerous applications in pure and applied mathematics. The distributions of the random variables K and Sj belong to a family of integer distributions with two real parameters, which we call quasi-binomial. Hence every Ewens-Pitman distribution arises as a result of a two-stage random procedure based on this simple class of integer distributions. Bibliography: 25 titles.
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Additional information
This paper is an edited and actualized version of the unpublished PDMI preprint 21/1995. Further development of the ideas of this work can be found in [21, 25]. A number of detected misprints was fixed without notice, the bibliography was extended beyond the original 19 references, and a few comments were added as footnotes. (Comments by Alexander Gnedin.)
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 325, 2005, pp. 127–145.
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Kerov, S.V. Coherent random allocations, and the Ewens-Pitman formula. J Math Sci 138, 5699–5710 (2006). https://doi.org/10.1007/s10958-006-0338-9
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DOI: https://doi.org/10.1007/s10958-006-0338-9