For ˜ R = 1 −exp(−R) a random closed set obtained by exponential transformation of the closed range R of a subordinator, a regenerative composition of generic positive integer n is defined by recording the sizes of clusters of n uniform random points as they are separated by the points of ˜ R. We focus on the number of parts Kn of the composition when ˜ R is derived from a gamma subordinator. We prove logarithmic asymptotics of the moments and central limit theorems for Kn and other functionals of the composition such as the number of singletons, doubletons, etc. This study complements our previous work on asymptotics of these functionals when the tail of the Lévy measure is regularly varying at 0+.

The Joyal bijection between doubly-rooted trees and mappings can be lifted to a transformation on function space which takes tree-walks to mapping-walks. Applying known results on weak convergence of random tree walks to Brownian excursion, we give a conceptually simpler rederivation of the 1994 Aldous-Pitman result on convergence of uniform random mapping walks to reflecting Brownian bridge, and extend this result to random p-mappings.

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

Interpolated Kneser-Ney is one of the best smoothing methods for n-gram language models. Previous explanations for its superiority have been based on intuitive and empirical justifications of specific properties of the method. We propose a novel interpretation of interpolated Kneser-Ney as approximate inference in a hierarchical Bayesian model consisting of Pitman-Yor processes. As opposed to past explanations, our interpretation can recover exactly the formulation of interpolated Kneser-Ney, and performs better than interpolated Kneser-Ney when a better inference procedure is used. 1

Random planar maps are considered in the physics literature as the discrete counterpart of random surfaces. It is conjectured that properly rescaled random planar maps, when conditioned to have a large number of faces, should converge to a limiting surface whose law does not depend, up to scaling factors, on details of the class of maps that are sampled. Previous works on the topic, starting with Chassaing and Schaeffer, have shown that the radius of a random quadrangulation with n faces, that is, the maximal graph distance on such a quadrangulation to a fixed reference point, converges in distribution once rescaled by n 1/4 to the diameter of the Brownian snake, up to a scaling constant. Using a bijection due to Bouttier, Di Francesco and Guitter between bipartite planar maps and a family of labeled trees, we show the corresponding invariance principle for a class of random maps that follow a Boltzmann distribution putting weight qk on faces of degree 2k: the radius of such maps, conditioned to have n faces (or n vertices) and under a criticality assumption, converges in distribution once rescaled by n 1/4 to a scaled version of the diameter of the Brownian snake. Convergence results for the so-called profile of maps are also provided. The convergence of rescaled bipartite maps to the Brownian map, in the sense introduced by Marckert and Mokkadem, is also shown. The proofs of these results rely on a new invariance principle for two-type spatial Galton–Watson trees.

In this paper we give a new example of duality between fragmentation and coagulation operators. Consider the space of partitions of mass (i.e., decreasing sequences of nonnegative real numbers whose sum is 1) and the twoparameter family of Poisson–Dirichlet distributions PD(α, θ) that take values in this space. We introduce families of random fragmentation and coagulation operators Frag α and Coag α,θ, respectively, with the following property: if the input to Frag α has PD(α, θ) distribution, then the output has PD(α, θ +1) distribution, while the reverse is true for Coag α,θ. This result may be proved using a subordinator representation and it provides a companion set of relations to those of Pitman between PD(α, θ) and PD(αβ, θ). Repeated application of the Frag α operators gives rise to a family of fragmentation chains. We show that these Markov chains can be encoded naturally by certain random recursive trees, and use this representation to give an alternative and more concrete proof of the coagulation–fragmentation duality. 1. Introduction. The

This paper presents some general formulas for random partitions of a finite set derived by Kingman's model of random sampling from an interval partition generated by subintervals whose lengths are the points of a Poisson point process. These lengths can be also interpreted as the jumps of a subordinator, that is an increasing process with stationary independent increments. Examples include the two-parameter family of Poisson-Dirichlet models derived from the Poisson process of jumps of a stable subordinator. Applications are made to the random partition generated by the lengths of excursions of a Brownian motion or Brownian bridge conditioned on its local time at zero.

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.

In this work we describe a sequence compression method based on combining a Bayesian nonparametric sequence model with entropy encoding. The model, a hierarchy of Pitman-Yor processes of unbounded depth previously proposed by Wood et al. [2009] in the context of language modelling, allows modelling of long-range dependencies by allowing conditioning contexts of unbounded length. We show that incremental approximate inference can be performed in this model, thereby allowing it to be used in a text compression setting. The resulting compressor reliably outperforms several PPM variants on many types of data, but is particularly effective in compressing data that exhibits power law properties. 1