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Abstract: This text surveys properties and applications of the exponential functional ∫ t exp(−ξs)ds of real-valued Lévy processes ξ = (ξt, t ≥ 0). 0

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 size-biased 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 two-parameter family of partition structures.

Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we obtain continuum random tree limits of Aldous’s beta-splitting models and Ford’s alpha models for phylogenetic trees. This confirms in a strong way that the whole trees grow at the same speed as the mean height of a randomly chosen leaf.

We develop some theory of spinal decompositions of discrete and continuous fragmentation trees. Specifically, we consider a coarse and a fine spinal integer partition derived from spinal tree decompositions. We prove that for a two-parameter Poisson–Dirichlet family of continuous fragmentation trees, including the stable trees of Duquesne and Le Gall, the fine partition is obtained from the coarse one by shattering each of its parts independently, according to the same law. As a second application of spinal decompositions, we prove that among the continuous fragmentation trees, stable trees are the only ones whose distribution is invariant under uniform re-rooting. 1. Introduction. Starting from a rooted combinatorial tree T[n] with n leaves labeled by [n] ={1,...,n}, we call the path from the root to the leaf labeled 1 the spine of T[n]. Deleting each edge along the spine of T[n] defines a graph whose connected components we call bushes. If, as well as cutting each edge on the spine, we cut each edge connected to a spinal vertex, each bush is further decomposed

Local limit theorems are derived for the number of occupied urns in general finite and infinite urn models under the minimum condition that the variance tends to infinity. Our results represent an optimal improvement over previous ones for normal approximation. 1. Introduction. A classical theorem of Rényi [29] for the number of empty boxes, denoted by μ0(n, M), in a sequence of n random allocations of indistinguishable balls into M boxes with equal probability 1/M, can be stated as follows: If the variance of μ0(n, M) tends to infinity with n, then μ0(n, M) is asymptotically normally distributed. This result, seldom stated in this form in the literature,

We use a natural ordered extension of the Chinese Restaurant Process to grow a two-parameter family of binary self-similar continuum fragmentation trees. We provide an explicit embedding of Ford’s sequence of alpha model trees in the continuum tree which we identified in a previous article as a distributional scaling limit of Ford’s trees. In general, the Markov branching trees induced by the two-parameter growth rule are not sampling consistent, so the existence of compact limiting trees cannot be deduced from previous work on the sampling consistent case. We develop here a new approach to establish such limits, based on regenerative interval partitions and the urnmodel description of sampling from Dirichlet random distributions. 1. Introduction. We

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.

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

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 non-lattice 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.