Results 1  10
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14
Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws
, 2008
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Asymptotic laws for compositions derived from transformed subordinators
 ANN. PROBAB
, 2006
"... A random composition of n appears when the points of a random closed set ˜ R ⊂ [0, 1] are used to separate into blocks n points sampled from the uniform distribution. We study the number of parts Kn of this composition and other related functionals under the assumption that ˜ R = φ(S•) where (St, t ..."
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Cited by 25 (10 self)
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A random composition of n appears when the points of a random closed set ˜ R ⊂ [0, 1] are used to separate into blocks n points sampled from the uniform distribution. We study the number of parts Kn of this composition and other related functionals under the assumption that ˜ R = φ(S•) where (St, t ≥ 0) is a subordinator and φ: [0, ∞] → [0, 1] is a diffeomorphism. We derive the asymptotics of Kn when the Lévy measure of the subordinator is regularly varying at 0 with positive index. Specialising to the case of exponential function φ(x) = 1 −e −x we establish a connection between the asymptotics of Kn and the exponential functional of the subordinator.
Regenerative partition structures
 Electron. J. Combin. 11 Research Paper
"... We consider Kingman’s partition structures which are regenerative with respect to a general operation of random deletion of some part. Prototypes of this class are the Ewens partition structures which Kingman characterised by regeneration after deletion of a part chosen by sizebiased sampling. We a ..."
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Cited by 14 (7 self)
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We consider Kingman’s partition structures which are regenerative with respect to a general operation of random deletion of some part. Prototypes of this class are the Ewens partition structures which Kingman characterised by regeneration after deletion of a part chosen by sizebiased sampling. We associate each regenerative partition structure with a corresponding regenerative composition structure, which (as we showed in a previous paper) can be associated in turn with a regenerative random subset of the positive halfline, that is the closed range of a subordinator. A general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator. We also analyse deletion properties characteristic of the twoparameter family of partition structures.
Poisson calculus for spatial neutral to the right processes
, 2003
"... In this paper we consider classes of nonparametric priors on spaces of distribution functions and cumulative hazard measures that are based on extensions of the neutral to the right (NTR) concept. In particular, spatial neutral to the right processes that extend the NTR concept from priors on the cl ..."
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Cited by 10 (1 self)
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In this paper we consider classes of nonparametric priors on spaces of distribution functions and cumulative hazard measures that are based on extensions of the neutral to the right (NTR) concept. In particular, spatial neutral to the right processes that extend the NTR concept from priors on the class of distributions on the real line to classes of distributions on general spaces are discussed. Representations of the posterior distribution of the spatial NTR processes are given. A different type of calculus than traditionally employed in the Bayesian literature, based on Poisson process partition calculus methods described in James (2002), is provided which offers a streamlined proof of posterior results for NTR models and its spatial extension. The techniques are applied to progressively more complex models ranging from the complete data case to semiparametric multiplicative intensity models. Refinements are then given which describes the underlying properties of spatial NTR processes analogous to those developed for the Dirichlet process. The analysis yields accessible moment formulae and characterizations of the posterior distribution and relevant marginal distributions. An EPPF formula and additionally a distribution related to the risk and death sets is computed. In the homogeneous case, these distributions turn out to be connected and overlap with recent work on regenerative compositions defined by suitable discretisation of subordinators. The formulae we develop for the marginal distribution of spatial NTR models provide clues on how to sample posterior distributions in complex settings. In addition the spatial NTR is further extended to the mixture model setting which allows for applicability of such processes to much more complex data structures. A description of a species sampling model derived from a spatial NTR model is also given.
Selfsimilar and Markov compositions structures
 Metody
, 2005
"... Abstract The bijection between composition structures and random closed subsets of the unit interval implies that the composition structures associated with S ∩[0, 1] for a selfsimilar random set S ⊂ R+ are those which are consistent with respect to a simple truncation operation. Using the standard ..."
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Cited by 9 (5 self)
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Abstract The bijection between composition structures and random closed subsets of the unit interval implies that the composition structures associated with S ∩[0, 1] for a selfsimilar random set S ⊂ R+ are those which are consistent with respect to a simple truncation operation. Using the standard coding of compositions by finite strings of binary digits starting with a 1, the random composition of n is defined by the first n terms of a random binary sequence of infinite length. The locations of 1s in the sequence are the places visited by an increasing timehomogeneous Markov chain on the positive integers if and only if S = exp(−W) for some stationary regenerative random subset W of the real line. Complementing our study in previous papers, we identify selfsimilar Markovian composition structures associated with the twoparameter family of partition structures. 1
Asymptotics of the allele frequency spectrum associated with the BolthausenSznitman coalescent
, 2007
"... We work in the context of the infinitely many alleles model. The allelic partition associated with a coalescent process started from n individuals is obtained by placing mutations along the skeleton of the coalescent tree; for each individual, we trace back to the most recent mutation affecting it a ..."
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Cited by 7 (0 self)
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We work in the context of the infinitely many alleles model. The allelic partition associated with a coalescent process started from n individuals is obtained by placing mutations along the skeleton of the coalescent tree; for each individual, we trace back to the most recent mutation affecting it and group together individuals whose most recent mutations are the same. The number of blocks of each of the different possible sizes in this partition is the allele frequency spectrum. The celebrated Ewens sampling formula gives precise probabilities for the allele frequency spectrum associated with Kingman’s coalescent. This (and the degenerate starshaped coalescent) are the only Λcoalescents for which explicit probabilities are known, although they are known to satisfy a recursion due to Möhle. Recently, Berestycki, Berestycki and Schweinsberg have proved asymptotic results for the allele frequency spectra of the Beta(2 − α,α) coalescents with α ∈ (1,2). In this paper, we prove full asymptotics for the case of the BolthausenSznitman coalescent.
Regenerative compositions in the case of slow variation
 Stoch. Process. Appl
, 2006
"... For S a subordinator and Πn an independent Poisson process of intensity ne −x,x> 0, we are interested in the number Kn of gaps in the range of S that are hit by at least one point of Πn. Extending previous studies in [7, 10, 11] we focus on the case when the tail of the Lévy measure of S is slowly v ..."
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Cited by 7 (4 self)
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For S a subordinator and Πn an independent Poisson process of intensity ne −x,x> 0, we are interested in the number Kn of gaps in the range of S that are hit by at least one point of Πn. Extending previous studies in [7, 10, 11] we focus on the case when the tail of the Lévy measure of S is slowly varying. We view Kn as the terminal value of a random process Kn, and provide an asymptotic analysis of the fluctuations of Kn, as n → ∞, for a wide spectrum of situations. 1
The Bernoulli sieve revisited
, 2008
"... We consider an occupancy scheme in which ‘balls’ are identified with n points sampled from the standard exponential distribution, while the role of ‘boxes’ is played by the spacings induced by an independent random walk with positive and nonlattice steps. We discuss the asymptotic behaviour of five ..."
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Cited by 3 (2 self)
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We consider an occupancy scheme in which ‘balls’ are identified with n points sampled from the standard exponential distribution, while the role of ‘boxes’ is played by the spacings induced by an independent random walk with positive and nonlattice steps. We discuss the asymptotic behaviour of five quantities: the index K ∗ n of the last occupied box, the number Kn of occupied boxes, the number Kn,0 of empty boxes whose index is at most K ∗ n, the index Wn of the first empty box and the number of balls Zn in the last occupied box. It is shown that the limiting distribution of properly scaled and centered K ∗ n coincides with that of the number of renewals not exceeding log n. A similar result is shown for Kn and Wn under a side condition that prevents occurrence of very small boxes. The condition also ensures that Kn,0 converges in distribution. Limiting results for Zn are established under an assumption of regular variation.
Regeneration in Random Combinatorial Structures
, 2009
"... Theory of Kingman’s partition structures has two culminating points • the general paintbox representation, relating finite partitions to hypothetical infinite populations via a natural sampling procedure, • a central example of the theory: the EwensPitman twoparameter partitions. In these notes we ..."
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Cited by 2 (2 self)
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Theory of Kingman’s partition structures has two culminating points • the general paintbox representation, relating finite partitions to hypothetical infinite populations via a natural sampling procedure, • a central example of the theory: the EwensPitman twoparameter partitions. In these notes we further develop the theory by • passing to structures enriched by the order on the collection of categories, • extending the class of tractable models by exploring the idea of regeneration, • analysing regenerative properties of the EwensPitman partitions, • studying asymptotic features of the regenerative compositions.
On a random recursion related to absorption times of death Markov chains
, 2007
"... Let X1, X2,... be a sequence of random variables satisfying the d distributional recursion X1 = 0 and Xn = Xn−In + 1 for n = 2, 3,..., where In is a random variable with values in {1,..., n − 1} which is independent of X2,..., Xn−1. The random variable Xn can be interpreted as the absorption time of ..."
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Cited by 2 (0 self)
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Let X1, X2,... be a sequence of random variables satisfying the d distributional recursion X1 = 0 and Xn = Xn−In + 1 for n = 2, 3,..., where In is a random variable with values in {1,..., n − 1} which is independent of X2,..., Xn−1. The random variable Xn can be interpreted as the absorption time of a suitable death Markov chain with state space N: = {1, 2,...} and absorbing state 1, conditioned that the chain starts in the initial state n. This paper focuses on the asymptotics of Xn as n tends to infinity under the particular but important assumption that the distribution of In satisfies P{In = k} = pk/(p1+ · · ·+pn−1) for some given probability distribution pk = P{ξ = k}, k ∈ N. Depending on the tail behaviour of the distribution of ξ, several scalings for Xn and corresponding limiting distributions come into play, among them stable distributions and distributions of exponential integrals of subordinators. The methods used in this paper are mainly probabilistic. The key tool is a coupling technique which relates the distribution of Xn to a random walk, which explains, for example, the appearance of the MittagLeffler distribution in this context.