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**11 - 20**of**20**### Elect. Comm. in Probab. 13 (2008), 461–474 ELECTRONIC COMMUNICATIONS in PROBABILITY FRAGMENTING RANDOM PERMUTATIONS

, 2007

"... process Problem 1.5.7 from Pitman’s Saint-Flour lecture notes [11]: Does there exist for each n a fragmentation process (Πn,k, 1 ≤ k ≤ n) such that Πn,k is distributed like the partition generated by cycles of a uniform random permutation of {1, 2,..., n} conditioned to have k cycles? We show that t ..."

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process Problem 1.5.7 from Pitman’s Saint-Flour lecture notes [11]: Does there exist for each n a fragmentation process (Πn,k, 1 ≤ k ≤ n) such that Πn,k is distributed like the partition generated by cycles of a uniform random permutation of {1, 2,..., n} conditioned to have k cycles? We show that the answer is yes. We also give a partial extension to general exchangeable Gibbs partitions. 1

### OCCUPANCY DISTRIBUTIONS ARISING IN SAMPLING FROM GIBBS-POISSON ABUNDANCE MODELS

, 2013

"... Abstract. Estimating the number n of unseen species from a k−sample displaying only p ≤ k distinct sampled species has received attention for long. It requires a model of species abundance together with a sampling model. We start with a discrete model of iid stochastic species abundances, each with ..."

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Abstract. Estimating the number n of unseen species from a k−sample displaying only p ≤ k distinct sampled species has received attention for long. It requires a model of species abundance together with a sampling model. We start with a discrete model of iid stochastic species abundances, each with Gibbs-Poisson distribution. A k−sample drawn from the n−species abundances vector is the one obtained while conditioning it on summing to k. We discuss the sampling formulae (species occupancy distributions, frequency of frequencies) in this context. We then develop some aspects of the estimation of n problem from the size k of the sample and the observed value of Pn,k, the number of distinct sampled species. It is shown that it always makes sense to study these occupancy problems from a Gibbs-Poisson abundance model in the context of a population with infinitely many species. From this extension, a parameter γ naturally appears, which is a measure of richness or diversity of species. We rederive the sampling formulae for a population with infinitely many species, together with the distribution of the number Pk of distinct sampled species. We investigate the estimation of γ problem from the sample size k and the observed value of Pk. We then exhibit a large special class of Gibbs-Poisson distributions having the property that sampling from a discrete abundance model may equivalently be viewed as a sampling problem from a random partition of unity, now in the continuum. When n is finite, this partition may be built upon normalizing n infinitely divisible iid positive random variables by its partial sum. It is shown that the sampling process in the continuum should generically be biased on the total length appearing in the latter normalization. A construction with sizebiased sampling from the ranked normalized jumps of a subordinator is also supplied, would the problem under study present infinitely many species. We illustrate our point of view with many examples, some of which being new ones.

### Manuscript submitted to: Electronic Journal of Probability Record indices and age-ordered frequencies in Exchangeable

, 2008

"... Abstract The frequencies X1, X2,... of an exchangeable Gibbs random partition Π of N = {1, 2,...} (Gnedin and Pitman (2006)) are considered in their age-order, i.e. their size-biased order. We study their dependence on the sequence i1, i2,... of least elements of the blocks of Π. In particular, cond ..."

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Abstract The frequencies X1, X2,... of an exchangeable Gibbs random partition Π of N = {1, 2,...} (Gnedin and Pitman (2006)) are considered in their age-order, i.e. their size-biased order. We study their dependence on the sequence i1, i2,... of least elements of the blocks of Π. In particular, conditioning on 1 = i1 < i2 <..., a representation is shown to be Xj = ξj−1 i=j (1 − ξi) j = 1, 2,... where {ξj: j = 1, 2,...} is a sequence of independent Beta random variables. Sequences with such a product form are called neutral to the left. We show that the property of conditional left-neutrality in fact characterizes the Gibbs family among all exchangeable partitions, and leads to further interesting results on: (i) the conditional Mellin transform of Xk, given ik, and (ii) the conditional distribution of the first k normalized frequencies, given ∑k j=1 Xj and ik; the latter turns out to be a mixture of Dirichlet distributions. Many of the mentioned representations are extensions of Griffiths and Lessard (2005) results on Ewens ’ partitions.

### unknown title

, 2007

"... Meinardus ’ theorem on weighted partitions: extensions and a probabilistic proof ..."

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Meinardus ’ theorem on weighted partitions: extensions and a probabilistic proof

### ELECTRONIC COMMUNICATIONS in PROBABILITY A SPECIES SAMPLING MODEL WITH FINITELY MANY TYPES

, 2010

"... A two-parameter family of exchangeable partitions with a simple updating rule is introduced. The partition is identified with a randomized version of a standard symmetric Dirichlet speciessampling model with finitely many types. A power-like distribution for the number of types is derived. 1 ..."

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A two-parameter family of exchangeable partitions with a simple updating rule is introduced. The partition is identified with a randomized version of a standard symmetric Dirichlet speciessampling model with finitely many types. A power-like distribution for the number of types is derived. 1

### Fragmenting random permutations

, 2007

"... Problem 1.5.7 from Pitman’s Saint-Flour lecture notes [9]: Does there exist for each n a Pn-valued fragmentation process (Πn,k, 1 ≤ k ≤ n) such that Πn,k is distributed like the partition generated by cycles of a uniform random permutation of [n] conditioned to have k cycles? We show that the answer ..."

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Problem 1.5.7 from Pitman’s Saint-Flour lecture notes [9]: Does there exist for each n a Pn-valued fragmentation process (Πn,k, 1 ≤ k ≤ n) such that Πn,k is distributed like the partition generated by cycles of a uniform random permutation of [n] conditioned to have k cycles? We show that the answer is yes. We also give a partial extension to general exchangeable Gibbs partitions. 1

### Fragmenting

"... This paper is made available online in accordance with publisher policies. Please scroll down to view the document itself. Please refer to the repository record for this item and our policy information available from the repository home page for further information. To see the final version of this ..."

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This paper is made available online in accordance with publisher policies. Please scroll down to view the document itself. Please refer to the repository record for this item and our policy information available from the repository home page for further information. To see the final version of this paper please visit the publisher’s website. Access to the published version may require a subscription. Author(s): C Goldschmidt, JB Martin and D Spano,