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Construction Of Markovian Coalescents
 Ann. Inst. Henri Poincar'e
, 1997
"... Partitionvalued and measurevalued coalescent Markov processes are constructed whose state describes the decomposition of a finite total mass m into a finite or countably infinite number of masses with sum m, and whose evolution is determined by the following intuitive prescription: each pair of ma ..."
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Cited by 44 (20 self)
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Partitionvalued and measurevalued coalescent Markov processes are constructed whose state describes the decomposition of a finite total mass m into a finite or countably infinite number of masses with sum m, and whose evolution is determined by the following intuitive prescription: each pair of masses of magnitudes x and y runs the risk of a binary collision to form a single mass of magnitude x+y at rate (x; y), for some nonnegative, symmetric collision rate kernel (x; y). Such processes with finitely many masses have been used to model polymerization, coagulation, condensation, and the evolution of galactic clusters by gravitational attraction. With a suitable choice of state space, and under appropriate restrictions on and the initial distribution of mass, it is shown that such processes can be constructed as Feller or Fellerlike processes. A number of further results are obtained for the additive coalescent with collision kernel (x; y) = x + y. This process, which arises fro...
Coalescent Random Forests
 J. COMBINATORIAL THEORY A
, 1998
"... Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ..."
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Cited by 38 (18 self)
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Various enumerations of labeled trees and forests, including Cayley's formula n n\Gamma2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from the following coalescent construction of a sequence of random forests (R n ; R n\Gamma1 ; : : : ; R 1 ) such that R k has uniform distribution over the set of all forests of k rooted trees labeled by [n]. Let R n be the trivial forest with n root vertices and no edges. For n k 2, given that R n ; : : : ; R k have been defined so that R k is a rooted forest of k trees, define R k\Gamma1 by addition to R k of a single edge picked uniformly at random from the set of n(k \Gamma 1) edges which when added to R k yield a rooted forest of k \Gamma 1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, the additive coalescent in which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitude...
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 ..."
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Cited by 26 (14 self)
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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...
The SDE solved by local times of a Brownian excursion or bridge derived from the height profile of a random tree or forest
, 1997
"... Let B be a standard onedimensional Brownian motion started at 0. Let L t;v (jBj) be the occupation density of jBj at level v up to time t. The distribution of the process of local times (L t;v (jBj); v 0) conditionally given B t = 0 and L t;0 (jBj) = ` is shown to be that of the unique strong solu ..."
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Cited by 25 (7 self)
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Let B be a standard onedimensional Brownian motion started at 0. Let L t;v (jBj) be the occupation density of jBj at level v up to time t. The distribution of the process of local times (L t;v (jBj); v 0) conditionally given B t = 0 and L t;0 (jBj) = ` is shown to be that of the unique strong solution X of the Ito SDE dXv = n 4 \Gamma X 2 v \Gamma t \Gamma R v 0 Xudu \Delta \Gamma1 o dv + 2 p XvdBv on the interval [0; V t (X)), where V t (X) := inffv : R v 0 Xudu = tg, and Xv = 0 for all v V t (X). This conditioned form of the RayKnight description of Brownian local times arises from study of the asymptotic distribution as n !1 and 2k= p n ! ` of the height profile of a uniform rooted random forest of k trees labeled by a set of n elements, as obtained by conditioning a uniform random mapping of the set to itself to have k cyclic points. The SDE is the continuous analog of a simple description of a GaltonWatson branching process conditioned on its total progeny....
Inhomogeneous Continuum Random Trees and the Entrance Boundary of the Additive Coalescent
 PROBAB. TH. REL. FIELDS
, 1998
"... Regard an element of the set of ranked discrete distributions \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; P i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent is the \Deltavalued Markov process in which pairs of clusters of masses fx i ; ..."
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Cited by 18 (13 self)
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Regard an element of the set of ranked discrete distributions \Delta := f(x 1 ; x 2 ; : : :) : x 1 x 2 : : : 0; P i x i = 1g as a fragmentation of unit mass into clusters of masses x i . The additive coalescent 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 . Aldous and Pitman (1998) showed that a version of this process starting from time \Gamma1 with infinitesimally small clusters can be constructed from the Brownian continuum random tree of Aldous (1991,1993) by Poisson splitting along the skeleton of the tree. In this paper it is shown that the general such process may be constructed analogously from a new family of inhomogeneous continuum random trees.
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 ..."
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Cited by 11 (9 self)
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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.
The Multinomial Distribution on Rooted Labeled Forests
, 1997
"... For a probability distribution (p s ; s 2 S) on a finite set S, call a random forest F of rooted trees labeled by S (with edges directed away from the roots) a pforest if given F has m edges the vector of outdegrees of vertices of F has a multinomial distribution with parameters m and (p s ; s 2 ..."
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Cited by 10 (10 self)
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For a probability distribution (p s ; s 2 S) on a finite set S, call a random forest F of rooted trees labeled by S (with edges directed away from the roots) a pforest if given F has m edges the vector of outdegrees of vertices of F has a multinomial distribution with parameters m and (p s ; s 2 S), and given also these outdegrees the distribution of F is uniform on all forests with the given outdegrees. The family of distributions of pforests is studied, and shown to be closed under various operations involving deletion of edges. Some related enumerations of rooted labeled forests are obtained as corollaries. 1 Introduction Let F(S) denote the set of all forests of rooted trees labeled by a finite set S of size jSj. Each f 2 F(S) is a directed graph labeled by S, that is a subset of S \Theta S, such that each Research supported in part by N.S.F. Grant DMS9703961 connected component of the graph is a tree with edges directed away from some root vertex. The notation v f ...
The asymptotic behavior of the Hurwitz binomial distribution
 Combinatorics, Probability and Computing
, 1998
"... Hurwitz's extension of Abel's binomial theorem defines a probability distribution on the set of integers from 0 to n. This is the distribution of the number of nonroot vertices of a fringe subtree of a suitably defined random tree with n + 2 vertices. The asymptotic behaviour of this distribution i ..."
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Cited by 5 (5 self)
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Hurwitz's extension of Abel's binomial theorem defines a probability distribution on the set of integers from 0 to n. This is the distribution of the number of nonroot vertices of a fringe subtree of a suitably defined random tree with n + 2 vertices. The asymptotic behaviour of this distribution is described in a limiting regime where the distribution of the delabeled fringe subtree approaches that of a GaltonWatson tree with a mixed Poisson offspring distribution. 1 Introduction and statement of results Hurwitz [10] discovered the following identity of polynomials in n + 2 variables x; y and z s ; s 2 [n] := f1; : : : ; ng, which reduces to the binomial expansion of (x + y) n when Research supported in part by N.S.F. Grant DMS9703961 z s j 0: X A`[n] x(x + z A ) jAj\Gamma1 (y + z ¯ A ) j ¯ Aj = (x + y + z [n] ) n (1) where the sum is over all 2 n subsets A of [n], with the notations z A := P s2A z s , and jAj for the number of elements of A, and ¯ A := [n] \...
Forest volume decompositions and AbelCayleyHurwitz multinomial expansions
, 2001
"... This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes define ..."
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Cited by 2 (0 self)
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This paper presents a systematic approach to the discovery, interpretation and verification of various extensions of Hurwitz's multinomial identities, involving polynomials defined by sums over all subsets of a finite set. The identities are interpreted as decompositions of forest volumes defined by the enumerator polynomials of sets of rooted labeled forests. These decompositions involve the following basic forest volume formula, which is a refinement of Cayley's multinomial expansion: for R ` S the polynomial enumerating outdegrees of vertices of rooted forests labeled by S whose set of roots is R, with edges directed away from the roots, is ( P r2R x r )( P s2S x s ) jS j\GammajRj\Gamma1