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...
Moment problems and boundaries of number triangles (2008)
Gnedin, Alexander, Pitman, Jim
The boundary problem for graphs like Pascal's but with general multiplicities of edges is related to a `backward' problem of moments of the Hausdorff type.
Spinal partitions and invariance under re-rooting of continuum random trees (2007)
Haas, Bénédicte, Pitman, Jim, Winkel, Matthias
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...
Spinal partitions and invariance under re-rooting of continuum random trees (2007)
Haas, Bénédicte, Pitman, Jim, Winkel, Matthias
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...
Spinal partitions and invariance under re-rooting of continuum random trees (2007)
Haas, Bénédicte, Pitman, Jim, Winkel, Matthias
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...
Gibbs fragmentation trees (2007)
McCullagh, Peter, Pitman, Jim, Winkel, Matthias
We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbs type fragmentation tree with Aldous's beta-splitting model, which has an extended parameter range...
One-dimensional Brownian particle systems with rank dependent drifts (2007)
We study interacting systems of linear Brownian motions whose drift vector at every time point is determined by the relative ranks of the coordinate processes at that time. Our main objective has...
Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws (2007)
Gnedin, Alexander, Hansen, Ben, Pitman, Jim
This paper collects facts about the number of occupied boxes in the classical balls-in-boxes occupancy scheme with infinitely many positive frequencies: equivalently, about the number of species...
Colored loop-erased random walk on the complete graph (2006)
Starting from a sequence regarded as a walk through some set of values, we consider the associated loop-erased walk as a sequence of directed edges, with an edge from $i$ to $j$ if the loop erased...
Gibbs distributions for random partitions generated by a fragmentation process (2006)
Berestycki, Nathanael, Pitman, Jim
In this paper we study random partitions of 1,...n, where every cluster of size j can be in any of w_j possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly among...
Gibbs distributions for random partitions generated by a fragmentation process (2006)
Berestycki, Nathanael, Pitman, Jim
In this paper we study random partitions of 1,...n, where every cluster of size j can be in any of w_j possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly among...
Poisson representation of a Ewens fragmentation process (2006)
Gnedin, Alexander, Pitman, Jim
A simple explicit construction is provided of a partition-valued fragmentation process whose distribution on partitions of $[n]=\{1,...,n\}$ at time $\theta \ge 0$ is governed by the Ewens sampling...
Continuum tree asymptotics of discrete fragmentations and applications to phylogenetic models (2006)
Haas, Bénédicte, Miermont, Grégory, Pitman, Jim, Winkel, Matthias
Given any regularly varying dislocation measure, we identify a natural self-similar fragmentation tree as scaling limit of discrete fragmentation trees with unit edge lengths. As an application, we...
Exchangeable partitions derived from Markovian coalescents (2006)
Dong, Rui, Gnedin, Alexander, Pitman, Jim
Kingman derived the Ewens sampling formula for random partitions describing the genetic variation in a neutral mutation model defined by a Poisson process of mutations along lines of descent governed...
Aldous, David, Miermont, Grégory, Pitman, Jim
We study the asymptotics of the $p$-mapping model of random mappings on $[n]$ as $n$ gets large, under a large class of asymptotic regimes for the underlying distribution $p$. We encode these random...
Aldous, David, Miermont, Grégory, Pitman, Jim
We study the asymptotics of the $p$-mapping model of random mappings on $[n]$ as $n$ gets large, under a large class of asymptotic regimes for the underlying distribution $p$. We encode these random...
Asymptotic laws for compositions derived from transformed subordinators (2006)
Gnedin, Alexander, Pitman, Jim, Yor, Marc
A random composition of n appears when the points of a random closed set ℛ̃⊂[0,1] are used to separate into blocks n points sampled from the uniform distribution. We study the number of parts Kn...
Moments of convex distribution functions and completely alternating sequences (2006)
Gnedin, Alexander, Pitman, Jim
We solve the moment problem for convex distribution functions on $[0,1]$ in terms of completely alternating sequences. This complements a recent solution of this problem by Diaconis and Freedman, and...
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Gibbs distributions for random partitions generated by a fragmentation process (2005)
Berestycki, Nathanael, Pitman, Jim
In this paper we study random partitions of {1,...,n} where every cluster of size j can be in any of w(j) possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly...
Gibbs distributions for random partitions generated by a fragmentation process (2005)
Berestycki, Nathanael, Pitman, Jim
In this paper we study random partitions of {1,...,n} where every cluster of size j can be in any of w(j) possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly...
Gibbs distributions for random partitions generated by a fragmentation process (2005)
Berestycki, Nathanael, Pitman, Jim
In this paper we study random partitions of {1,...,n} where every cluster of size j can be in any of w(j) possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly...
Gibbs distributions for random partitions generated by a fragmentation process (2005)
Berestycki, Nathanael, Pitman, Jim
In this paper we study random partitions of 1,...n, where every cluster of size j can be in any of w\_j possible internal states. The Gibbs (n,k,w) distribution is obtained by sampling uniformly...
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...
Self-similar and Markov composition structures (2005)
Gnedin, Alexander, Pitman, Jim
The bijection between composition structures and random closed subsets of the unit interval implies that the composition structures associated with $S \cap [0,1]$ for a self-similar random set...
Regenerative composition structures (2005)
Gnedin, Alexander, Pitman, Jim
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...
Aldous, David, Miermont, Grégory, Pitman, Jim
We study the asymptotics of the $p$-mapping model of random mappings on $[n]$ as $n$ gets large, under a large class of asymptotic regimes for the underlying distribution $p$. We encode these random...
Aldous, David, Miermont, Grégory, Pitman, Jim
We study the asymptotics of the $p$-mapping model of random mappings on $[n]$ as $n$ gets large, under a large class of asymptotic regimes for the underlying distribution $p$. We encode these random...
Exchangeable Gibbs partitions and Stirling triangles (2004)
Gnedin, Alexander, Pitman, Jim
For two collections of nonnegative and suitably normalised weights $\W=(\W_j)$ and $\V=(\V_{n,k})$, a probability distribution on the set of partitions of the set $\{1,...,n\}$ is defined by...
Regenerative partition structures (2004)
Gnedin, Alexander, Pitman, Jim
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...
Growth of the Brownian forest (2004)
De Probabilit, J. Pitman, M. Winkel, Jim Pitman, Matthias Winkel
Trees in Brownian excursions have been studied since the late 1980s.
Growth of the Brownian forest (2004)
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...
Where did the Brownian particle go? (2004)
Pemantle, Robin, Peres, Yuval, Pitman, Jim, Yor, Marc
Consider the radial projection onto the unit sphere of the path a d-dimensional Brownian motion W, started at the center of the sphere and run for unit time. Given the occupation measure mu of this...
Asymptotic laws for compositions derived from transformed subordinators (2004)
Gnedin, Alexander, Pitman, Jim, Yor, Marc
A random composition of $n$ appears when the points of a random closed set $\widetilde{\mathcal{R}}\subset[0,1]$ are used to separate into blocks $n$ points sampled from the uniform distribution. We...
Aldous, David, Miermont, Grégory, Pitman, Jim
We study the inhomogeneous continuum random trees (ICRT) that arise as weak limits of birthday trees. We give a description of the exploration process, a function defined on [0,1] that encodes the...
Aldous, David J, Miermont, Gregory, Pitman, Jim
We study the inhomogeneous continuum random trees (ICRT) that arise as weak limits of birthday trees. We give a description of the exploration process, a function defined on [0,1] that encodes the...
Aldous, David, Miermont, Grégory, Pitman, Jim
We study the inhomogeneous continuum random trees (ICRT) that arise as weak limits of birthday trees. We give a description of the exploration process, a function defined on [0,1] that encodes the...
Two recursive decompositions of Brownian bridge (2004)
Aldous and Pitman (1994) studied asymptotic distributions, as n tends to infinity, of various functionals of a uniform random mapping of a set of n elements, by constructing a mapping-walk and...
Rayleigh processes, real trees, and root growth with re-grafting (2004)
Evans, Steven N., Pitman, Jim, Winter, Anita
The real trees form a class of metric spaces that extends the class of trees with edge lengths by allowing behavior such as infinite total edge length and vertices with infinite branching degree....
Brownian Bridge Asymptotics for Random p-Mappings (2004)
Aldous, David; University Of California, Berkeley; Aldous@stat.berkeley.edu, Miermont, Gregory; Ecole Normale Superieure; Miermont@dma.ens.fr, Pitman, Jim; University Of California, Berkeley; Pitman@stat.berkeley.edu
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...
Aldous, David J., Miermont, Gregory, Pitman, Jim
We study the asymptotics of the $p$-mapping model of random mappings on $[n]$ as $n$ gets large, under a large class of asymptotic regimes for the underlying distribution $p$. We encode these random...
Aldous, David, Miermont, Grégory, Pitman, Jim
We study the inhomogeneous continuum random trees (ICRT) that arise as weak limits of birthday trees. We give a description of the exploration process, a function defined on [0,1] that encodes the...
Aldous, David, Miermont, Grégory, Pitman, Jim
We study the inhomogeneous continuum random trees (ICRT) that arise as weak limits of birthday trees. We give a description of the exploration process, a function defined on [0,1] that encodes the...
Path transformations of first passage bridges (2003)
Bertoin, Jean; Universite Pierre Et Marie Curie; Jbe@ccr.jussieu.fr, Chaumont, Loic; Universite Pierre Et Marie Curie; Chaumont@ccr.jussieu.fr, Pitman, Jim; University Of California At Berkeley; Pitman@stat.berkeley.edu
We define the first passage bridge from 0 to $lambda$ as the Brownian motion on the time interval [0,1] conditioned to first hit $lambda$ at time 1. We show that this process may be related to the...
Regenerative Composition Structures (2003)
Gnedin, Alexander, Pitman, Jim
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...
David Aldous, Gregory Miermont, Jim Pitman
We study the inhomogeneous continuum random trees (ICRT) that arise as weak limits of birthday trees. We give a description of the exploration process, a function defined on [0, 1] that encodes the...
David Aldous, Gregory Miermont, Jim Pitman
We study the inhomogeneous continuum random trees (ICRT) that arise as weak limits of birthday trees. We give a description of the exploration process, a function defined on [0, 1] that encodes the...
Basic relations between the distributions of hitting, occupation and inverse local times of a one-dimensional diffusion process $X$, first discussed by It\^o and McKean, are reviewed from the...
Poisson-Kingman partitions (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...
Poisson-Kingman Partitions (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...
Combinatorial Stochastic Processes (2002)
CONTENTS Contents 0 Preliminaries 3 0.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 0.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 0.3...
Brownian Bridge Asymptotics for Random p-Mappings (2002)
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...
Where Did The Brownian Particle Go? (2002)
Robin Pemantle, Yuval Peres, Jim Pitman, Marc Yor
Consider the radial projection onto the unit sphere of the path a d-dimensional Brownian motion W , started at the center of the sphere and run for unit time.
Brownian Bridge Asymptotics for Random (2002)
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...
Two recursive decompositions of Brownian bridge related to the asymptotics of random mappings (2002)
Aldous and Pitman (1994) studied asymptotic distributions as n -> infty, of various functionals of a uniform random mapping of the set {1, ..., n}, by constructing a mapping-walk and showing these...
The Asymptotic Distribution of the Diameter of a Random Mapping (2002)
The asymptotic distribution of the diameter of the digraph of a uniformly distributed random mapping of an n-element set to itself is represented as the distribution of a functional of a reflecting...
Invariance Principles for Non-uniform Random Mappings and Trees (2002)
In the context of uniform random mappings of an n-element set to itself, Aldous and Pitman (1994) established a functional invariance principle, showing that many n ! 1 limit distributions can be...
Poisson-Kingman Partitions (2001)
Contents 1 Introduction 2 2 Preliminaries 2 3 The Poisson-Kingman Model 5 4 Operations 11 4.1 Scaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 4.2 Exponential tilting . . . ....
Basic relations between the distributions of hitting, occupation, and inverse local times of a one-dimensional diffusion process X , first discussed by Ito-McKean, are reviewed from the perspectives...
Forest volume decompositions and Abel-Cayley-Hurwitz 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...
Various random combinatorial objects, such as mappings, trees, forests, and subsets of a finite set, are constructed with probability distributions related to the binomial and multinomial expansions...
This paper introduces a split-and-merge transformation of interval partitions which combines some features of one model studied by Gnedin and Kerov [10, 11] and another studied by Tsilevich [30, 29]...
Richard P. Stanley, Jim Pitman
The volume of the n-dimensional polytope for arbitrary x := (x 1 ; : : : ; x n ) with x i > 0 for all i de nes a polynomial in variables x i which admits a number of interpretations, in terms of...
Forest volume decompositions and Abel-Cayley-Hurwitz 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...
On the distribution of ranked heights of excursions of a Brownian bridge (2001)
The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge $(B^{br}_t, 0 \le t \le 1)$ is described. The height $M^{br +}_j$of the...
Where Did the Brownian Particle Go? (2001)
Pemantle, Robin; Ohio State University; Pemantle@math.ohio_state.edu, Peres, Yuval; University Of California, Berkeley; Peres@stat.berkeley.edu, Pitman, Jim; University Of California, Berkeley; Pitman@stat.berkeley.edu, Yor, Marc; Université Pierre Et Marie Curie; Pitman@stat.berkeley.edu
Consider the radial projection onto the unit sphere of the path a $d$-dimensional Brownian motion $W$, started at the center of the sphere and run for unit time. Given the occupation measure $mu$ of...
Infinitely Divisible Laws Associated With Hyperbolic Functions (2000)
The infinitely divisible distributions on R + of random variables C t , S t and T t with Laplace transforms ` 1 cosh p 2 ' t ; / p 2 sinh p 2 ! t ; and / tanh p 2 p 2 ! t respectively are...
This paper reviews known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws...
Constructions Of A Brownian Pathwith A Given Minimum (2000)
Jean Bertoin, Jim Pitman, Juan Ruiz, De Chavez
We construct a Brownian path conditioned on its minimum value over a fixed time interval by simple transformations of a Brownian bridge. Path transformations have proved useful in the study of...
This paper reviews known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws...
This paper reviews known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws...
This paper reviews known results which connect Riemann's integral representations of his zeta function, involving Jacobi's theta function and its derivatives, to some particular probability laws...
The volume of the n-dimensional polytope Pi n (x) := fy 2 R n : y i 0 and y 1 + Delta Delta Delta + y i x 1 + Delta Delta Delta + x i for all 1 i ng for arbitrary x := (x 1 ; : : : ; x n ) with x i ?...
Pitman, Jim; University Of California, Berkeley; Pitman@stat.berkeley.edu
Given an arbitrary distribution on a countable set, consider the number of independent samples required until the first repeated value is seen. Exact and asymptotic formulae are derived for the...
Two Coalescents Derived from the Ranges of Stable Subordinators (1999)
Bertoin, Jean; Université Paris VI; Jbe@ccr.jussieu.fr, Pitman, Jim; University Of California, Berkeley; Pitman@stat.berkeley.edu
Let $M_alpha$ be the closure of the range of a stable subordinator of index $alphain ]0,1[$. There are two natural constructions of the $M_{alpha}$'s simultaneously for all $alphain ]0,1[$, so that...
Explicit evaluations of the symmetric Euler integral R 1 0 u ff (1 Gamma u) ff f(u)du are obtained for some particular functions f . These evaluations are related to duplication formulae for Appell's...
Coalescents With Multiple Collisions (1999)
For each finite measure $\Lambda$ on [0,1] a coalescent Markov process, with state space the compact set of all partitions of the set $\mathbbN$of positive integers, is constructed so the restriction...
Where Did The Brownian Particle Go? (1999)
Robin Pemantle, Yuval Peres, Jim Pitman, Marc Yor
Consider the radial projection onto the unit sphere of the path a d-dimensional Brownian motion W , started at the center of the sphere and run for unit time. Given the occupation measure of this...
On the distribution of ranked heights of excursions of a Brownian bridge (1999)
The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (B br t ; 0 t 1) is described. The height M br+ j of the jth highest...
Where Did The Brownian Particle Go? (1999)
Robin Pemantle, Yuval Peres, Jim Pitman, Marc Yor
Consider the radial projection onto the unit sphere of the path a d-dimensional Brownian motion W , started at the center of the sphere and run for unit time. Given the occupation measure ¯ of this...
Where Did The Brownian Particle Go? (1999)
Robin Pemantle, Yuval Peres, Jim Pitman, Marc Yor
Consider the radial projection onto the unit sphere of the path a d-dimensional Brownian motion W , started at the center of the sphere and run for unit time. Given the occupation measure ¯ of this...
On the distribution of ranked heights of excursions of a Brownian bridge (1999)
The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge (B br t ; 0 t 1) is described. The height M br+ j of the jth highest...
Where Did The Brownian Particle Go? (1999)
Robin Pemantle, Yuval Peres, Jim Pitman, Marc Yor
Consider the radial projection onto the unit sphere of the path a d-dimensional Brownian motion W , started at the center of the sphere and run for unit time. Given the occupation measure of this...
We define an n-dimensional polytope Pi_n(x), depending on parameters x_i>0, whose combinatorial properties are closely connected with empirical distributions, plane trees, plane partitions, parking...
Constructions Of A Brownian Pathwith A Given Minimum (1999)
Jean Bertoin, Jim Pitman, Juan Ruiz, De Chavez
We construct a Brownian path conditioned on its minimum value over a fixed time interval by simple transformations of a Brownian bridge. Path transformations have proved useful in the study of...
Construction of a Brownian Path With a Given Minimum (1999)
Bertoin, Jean; Universite Pierre Et Marie Curie; Jbe@ccr.jussieu.fr, Pitman, Jim; University Of California, Berkeley; Pitman@stat.Berkeley.EDU, Chavez, Juan Ruiz De; UAM-I; Jrch@xanum.uam.mx
We construct a Brownian path conditioned on its minimum value over a fixed time interval by a simple transformation of a Brownian bridge.
The volume of the n-dimensional polytope n (x) := fy 2 R n : y i 0 and y 1 + + y i x 1 + + x i for all 1 i ng for arbitrary x := (x 1 ; : : : ; x n ) with x i > 0 for all i denes a polynomial in...