Results 1  10
of
27
The Standard Additive Coalescent
, 1997
"... Regard an element of the set \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; X i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent of Evans and Pitman (1997) is the \Deltavalued Markov process in which pairs of clusters of masses fx i ; x j g mer ..."
Abstract

Cited by 87 (21 self)
 Add to MetaCart
Regard an element of the set \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; X i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent of Evans and Pitman (1997) is the \Deltavalued Markov process in which pairs of clusters of masses fx i ; x j g merge into a cluster of mass x i +x j at rate x i +x j . They showed that a version (X 1 (t); \Gamma1 ! t ! 1) of this process arises as a n !1 weak limit of the process started at time \Gamma 1 2 log n with n clusters of mass 1=n. We show this standard additive coalescent may be constructed from the continuum random tree of Aldous (1991,1993) by Poisson splitting along the skeleton of the tree. We describe the distribution of X 1 (t) on \Delta at a fixed time t. We show that the size of the cluster containing a given atom, as a process in t, has a simple representation in terms of the stable subordinator of index 1=2. As t ! \Gamma1, we establish a Gaussian limit for (centered and norm...
Limit Distributions and Random Trees Derived From the Birthday Problem With Unequal Probabilities
, 1998
"... Given an arbitrary distribution on a countable set S consider the number of independent samples required until the first repeated value is seen. Exact and asymptotic formulae are derived for the distribution of this time and of the times until subsequent repeats. Asymptotic properties of the repeat ..."
Abstract

Cited by 37 (12 self)
 Add to MetaCart
(Show Context)
Given an arbitrary distribution on a countable set S consider the number of independent samples required until the first repeated value is seen. Exact and asymptotic formulae are derived for the distribution of this time and of the times until subsequent repeats. Asymptotic properties of the repeat times are derived by embedding in a Poisson process. In particular, necessary and sufficient conditions for convergence are given and the possible limits explicitly described. Under the same conditions the finite dimensional distributions of the repeat times converge to the arrival times of suitably modified Poisson processes, and random trees derived from the sequence of independent Research supported in part by N.S.F. Grants DMS 9224857, 9404345, 9224868 and 9703691 trials converge in distribution to an inhomogeneous continuum random tree. 1 Introduction Recall the classical birthday problem: given that each day of the year is equally likely as a possible birthday, and that birth...
Limits of normalized quadrangulations. The Brownian map
 Ann. Probab
, 2004
"... Consider qn a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces. In this paper, we show that, when n goes to +∞, qn suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name t ..."
Abstract

Cited by 32 (1 self)
 Add to MetaCart
(Show Context)
Consider qn a random pointed quadrangulation chosen equally likely among the pointed quadrangulations with n faces. In this paper, we show that, when n goes to +∞, qn suitably normalized converges weakly in a certain sense to a random limit object, which is continuous and compact, and that we name the Brownian map. The same result is shown for a model of rooted quadrangulations and for some models of rooted quadrangulations with random edge lengths. A metric space of rooted (resp. pointed) abstract maps that contains the model of discrete rooted (resp. pointed) quadrangulations and the model of Brownian map is defined. The weak convergences hold in these metric spaces. 1
The structure of the allelic partition of the total population for GaltonWatson processes with neutral mutations
"... We consider a (sub)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clonechildren and ..."
Abstract

Cited by 29 (4 self)
 Add to MetaCart
(Show Context)
We consider a (sub)critical Galton–Watson process with neutral mutations (infinite alleles model), and decompose the entire population into clusters of individuals carrying the same allele. We specify the law of this allelic partition in terms of the distribution of the number of clonechildren and the number of mutantchildren of a typical individual. The approach combines an extension of Harris representation of Galton–Watson processes and a version of the ballot theorem. Some limit theorems related to the distribution of the allelic partition are also given. 1. Introduction. We consider a Galton–Watson process, that is, a population model with asexual reproduction such that at every generation, each individual gives birth to a random number of children according to a fixed distribution and independently of the other individuals in the population. We are interested in the situation where a child can be either a clone, that
More uses of exchangeability: Representations of complex random structures
 Probability and Mathematical Genetics: Papers in Honour of Sir
, 2010
"... We review old and new uses of exchangeability, emphasizing the general theme of exchangeable representations of complex random structures. Illustrations of this theme include processes of stochastic coalescence and fragmentation; continuum random trees; secondorder limits of distances in random gra ..."
Abstract

Cited by 18 (2 self)
 Add to MetaCart
(Show Context)
We review old and new uses of exchangeability, emphasizing the general theme of exchangeable representations of complex random structures. Illustrations of this theme include processes of stochastic coalescence and fragmentation; continuum random trees; secondorder limits of distances in random graphs; isometry classes of metric spaces with probability measures; limits of dense random graphs; and more sophisticated uses in finitary combinatorics.
Random mappings, forests, and subsets associated with AbelCayleyHurwitz multinomial expansions
, 2001
"... ..."
A family of random trees with random edge lengths
, 1999
"... We introduce a family of probability distributions on the space of trees with I labeled vertices and possibly extra unlabeled vertices of degree 3, whose edges have positive real lengths. Formulas for distributions of quantities such asdegree sequence, shape, and total length are derived. An interpr ..."
Abstract

Cited by 12 (8 self)
 Add to MetaCart
(Show Context)
We introduce a family of probability distributions on the space of trees with I labeled vertices and possibly extra unlabeled vertices of degree 3, whose edges have positive real lengths. Formulas for distributions of quantities such asdegree sequence, shape, and total length are derived. An interpretation is given in terms of sampling from the inhomogeneous continuum random tree of Aldous and Pitman (1998). Key words and phrases. Continuum tree, enumeration, random tree, spanning tree, weighted tree, Cayley's multinomial expansion.
Invariance principles for nonuniform random mappings and trees
 ASYMPTOTIC COMBINATORICS WITH APPLICATIONS IN MATHEMATICAL PHYSICS
, 2002
"... In the context of uniform random mappings of an nelement set to itself, Aldous and Pitman (1994) established a functional invariance principle, showing that many n!1 limit distributions can be described as distributions of suitable functions of reflecting Brownian bridge. To study nonuniform cases ..."
Abstract

Cited by 11 (8 self)
 Add to MetaCart
(Show Context)
In the context of uniform random mappings of an nelement set to itself, Aldous and Pitman (1994) established a functional invariance principle, showing that many n!1 limit distributions can be described as distributions of suitable functions of reflecting Brownian bridge. To study nonuniform cases, in this paper we formulate a sampling invariance principle in terms of iterates of a fixed number of random elements. We show that the sampling invariance principle implies many, but not all, of the distributional limits implied by the functional invariance principle. We give direct verifications of the sampling invariance principle in two successive generalizations of the uniform case, to pmappings (where elements are mapped to i.i.d. nonuniform elements) and Pmappings (where elements are mapped according to a Markov matrix). We compare with parallel results in the simpler setting of random trees.
The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin’s local time identity
, 2004
"... ..."
Weak convergence of random pmappings and the exploration process of inhomogeneous continuum random trees
 PROBAB. THEORY RELAT. FIELDS
, 2005
"... ..."