A new class of random composition structures (the ordered analog of Kingman’s partition structures) is defined by a regenerative description of component sizes. Each regenerative composition structure is represented by a process of random sampling of points from an exponential distribution on the positive halfline, and separating the points into clusters by an independent regenerative random set. Examples are composition structures derived from residual allocation models, including one associated with the Ewens sampling formula, and composition structures derived from the zero set of a Brownian motion or Bessel process. We provide characterisation results and formulas relating the distribution of the regenerative composition to the Lévy parameters of a subordinator whose range is the corresponding regenerative set. In particular, the only reversible regenerative composition structures are those associated with the interval partition of [0, 1] generated by excursions of a standard Bessel bridge of dimension 2 − 2α for some α ∈ [0, 1].

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.

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For two collections of nonnegative and suitably normalised weights W = (Wj) and V = (Vn,k), a probability distribution on the set of partitions of the set {1,...,n} is defined by assigning to a generic partition {Aj, j ≤ k} the probability Vn,k W |A1 | · · ·W |Ak|, where |Aj | is the number of elements of Aj. We impose constraints on the weights by assuming that the resulting random partitions Πn of [n] are consistent as n varies, meaning that they define an exchangeable partition of the set of all natural numbers. This implies that the weights W must be of a very special form depending on a single parameter α ∈ [−∞, 1]. The case α = 1 is trivial, and for each value of α ̸ = 1 the set of possible V-weights is an infinite-dimensional simplex. We identify the extreme points of the simplex by solving the boundary problem for a generalised Stirling triangle. In particular, we show that the boundary is discrete for − ∞ ≤ α < 0 and continuous for 0 ≤ α < 1. For α ≤ 0 the extremes correspond to the members of the Ewens-Pitman family of random partitions indexed by (α, θ), while for 0 < α < 1 the extremes are obtained by conditioning an (α, θ)-partition on the asymptotics of the number of blocks of Πn as n tends to infinity.

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.

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+.

continuum random tree

Abstract. Bernoulli sieve is a recursive construction of a random composition (ordered partition) of integer n. This composition can be induced by sampling from a random discrete distribution which has frequencies equal to the sizes of component intervals of a stick-breaking interval partition of [0, 1]. We exploit Markov property of the composition and its renewal representation to derive asymptotics of the moments and to prove a central limit theorem for the number of parts. 1. The Bernoulli sieve can be seen as a generalisation of the ‘game ’ found in [3]. The first round of the game starts with n players and amounts to tossing a coin with probability X1 for tails. Each of the players tosses one time and the players flipping tails must drop out. If all n get heads the trial is disqualified and must be repeated completely with all n players, as many times as necessary until some players do quit. If at least one player remains after the first round, the second round continues with the remaining players, who must toss another coin with probability X2 for tails. The game lasts with probabilities X3, X4,... for tails until all players are sorted out. It is assumed that the probabilities X1, X2,... are independent random variables with a given distribution ω on]0, 1 [ , and that given Xj the individual outcomes at

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 self-similar 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 time-homogeneous 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 self-similar Markovian composition structures associated with the two-parameter family of partition structures. 1

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