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12
Basic Analytic Combinatorics of Directed Lattice Paths
 Theoretical Computer Science
, 2001
"... This paper develops a unified enumerative and asymptotic theory of directed 2dimensional lattice paths in halfplanes and quarterplanes. The lattice paths are speci ed by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then ess ..."
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Cited by 59 (11 self)
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This paper develops a unified enumerative and asymptotic theory of directed 2dimensional lattice paths in halfplanes and quarterplanes. The lattice paths are speci ed by a finite set of rules that are both time and space homogeneous, and have a privileged direction of increase. (They are then essentially 1dimensional objects.) The theory relies on a specific "kernel method" that provides an important decomposition of the algebraic generating functions involved, as well as on a generic study of singularities of an associated algebraic curve. Consequences are precise computable estimates for the number of lattice paths of a given length under various constraints (bridges, excursions, meanders) as well as a characterization of the limit laws associated to several basic parameters of paths.
Betacoalescents and continuous stable random trees
, 2006
"... Coalescents with multiple collisions, also known as Λcoalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case t ..."
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Cited by 23 (8 self)
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Coalescents with multiple collisions, also known as Λcoalescents, were introduced by Pitman and Sagitov in 1999. These processes describe the evolution of particles that undergo stochastic coagulation in such a way that several blocks can merge at the same time to form a single block. In the case that the measure Λ is the Beta(2 − α, α) distribution, they are also known to describe the genealogies of large populations where a single individual can produce a large number of offspring. Here we use a recent result of Birkner et al. to prove that Betacoalescents can be embedded in continuous stable random trees, about which much is known due to recent progress of Duquesne and Le Gall. Our proof is based on a construction of the DonnellyKurtz lookdown process using continuous random trees which is of independent interest. This produces a number of results concerning the smalltime behavior of Betacoalescents. Most notably, we recover an almost sure limit theorem of the authors for the number of blocks at small times, and give the multifractal spectrum corresponding to the emergence of blocks with atypical size. Also, we are able to find exact asymptotics for sampling formulae corresponding to the site frequency spectrum and allele frequency spectrum associated with mutations in the context of population genetics.
Exchangeable Gibbs partitions and Stirling triangles
"... 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 ele ..."
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Cited by 22 (5 self)
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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 Vweights is an infinitedimensional 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 EwensPitman 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.
Recent Progress in Coalescent Theory
"... Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such ..."
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Cited by 17 (2 self)
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Coalescent theory is the study of random processes where particles may join each other to form clusters as time evolves. These notes provide an introduction to some aspects of the mathematics of coalescent processes and their applications to theoretical population genetics and in other fields such as spin glass models. The emphasis is on recent work concerning in particular the connection of these processes to continuum random trees and spatial models such as coalescing random walks.
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 ..."
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Cited by 11 (9 self)
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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.
PoissonKingman Partitions
 of Lecture NotesMonograph Series
, 2002
"... 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 subordin ..."
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Cited by 11 (3 self)
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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 twoparameter family of PoissonDirichlet 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.
Limit Laws for Basic Parameters of Lattice Paths with . . .
, 2002
"... This paper establishes the asymptotics of a class of random walks on N with regular but unbounded jumps and studies several basic parameters (returns to zero for meanders, bridges, excursions, nal altitude for meanders). All these results are generic (obtained by the kernel method for the combinator ..."
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Cited by 7 (4 self)
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This paper establishes the asymptotics of a class of random walks on N with regular but unbounded jumps and studies several basic parameters (returns to zero for meanders, bridges, excursions, nal altitude for meanders). All these results are generic (obtained by the kernel method for the combinatorial part and by singularity analysis for the asymptotic part). This paper
Growth of the Brownian forest
, 2005
"... Trees in Brownian excursions have been studied since the late 1980s. Forests in excursions of Brownian motion above its past minimum are a natural extension of this notion. In this paper we study a forestvalued Markov process which describes the growth of the Brownian forest. The key result is a co ..."
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Cited by 3 (1 self)
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Trees in Brownian excursions have been studied since the late 1980s. Forests in excursions of Brownian motion above its past minimum are a natural extension of this notion. In this paper we study a forestvalued Markov process which describes the growth of the Brownian forest. The key result is a composition rule for binary Galton–Watson forests with i.i.d. exponential branch lengths. We give elementary proofs of this composition rule and explain how it is intimately linked with Williams ’ decomposition for Brownian motion with drift. 1. Introduction. Given 0 ≤ λ<µ, the binary(λ, µ) forest is defined as a collection of independent binary Galton–Watson trees, with branching probability (µ − λ)/2µ, branch lengths independent exponential(2µ), planted into the positive real line at the points of a homogeneous Poisson process of rate µ − λ. We call the vertices on the positive real line roots. The unique branch connecting
A Bayesian Review of the PoissonDirichlet Process
, 2010
"... The two parameter PoissonDirichlet process is also known as the PitmanYor Process and related to the ChineseRestaurant Process, is a generalisation of the Dirichlet Process, and is increasingly being used for probabilistic modelling in discrete areas such as language and images. This article revie ..."
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Cited by 2 (1 self)
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The two parameter PoissonDirichlet process is also known as the PitmanYor Process and related to the ChineseRestaurant Process, is a generalisation of the Dirichlet Process, and is increasingly being used for probabilistic modelling in discrete areas such as language and images. This article reviews the theory of the PoissonDirichlet process in terms of its consistency for estimation, the convergence rates and the posteriors of data. This theory has been well developed for continuous distributions (more generally referred to as nonatomic distributions). This article then presents a Bayesian interpretation of the PoissonDirichlet process: it is a mixture using an improper and infinite dimensional Dirichlet distribution. This interpretation requires technicalities of priors, posteriors and Hilbert spaces, but conceptually, this means we can understand the process as just another Dirichlet and thus all its sampling properties fit naturally. Finally, this article also presents results for the discrete case which is the case seeing widespread use now in computer science, but which has received less attention in the literature.