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...
The distribution of the maximal difference between Brownian bridge and its concave majorant (2009)
Balabdaoui, Fadoua, Pitman, Jim
We provide a representation of the maximal difference between a standard Brownian bridge and its concave majorant on the unit interval, from which we deduce expressions for the distribution and...
Gnedin, Alexander, Haulk, Chris, Pitman, Jim
We prove a long-standing conjecture which characterises the Ewens-Pitman two-parameter family of exchangeable random partitions, plus a short list of limit and exceptional cases, by the following...
David Aldous, Grégory Miermont, Jim Pitman
Weak convergence of random p-mappings and the exploration process of inhomogeneous continuum random trees
David Aldous, Grégory Miermont, Jim Pitman
The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin’s local time identity
A partition structure is a sequence of probability distributions for πn, a random partition of n, such that if πn is regarded as a random allocation of n unlabeled balls into some random number of...
Nathanaël Berestycki, Jim Pitman
distributions for random partitions generated by a
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.
IMS Lecture Notes Monograph (2008)
Some properties of the arc-sine law related to its invariance under a family of rational maps ∗
COALESCENTS WITH MULTIPLE COLLISIONS 1 (2008)
state space the compact set of all partitions of the set � of positive integers, is constructed so the restriction of the partition to each finite subset of � is a Markov chain with the following...
Inhomogeneous continuum random trees and the entrance boundary of the additive coalescent ⋆
THE STANDARD ADDITIVE COALESCENT 1 (2008)
Regard an element of the set � � = �x1�x2����� � x1 ≥ x2 ≥···≥0 � � xi = 1
THE TWO-PARAMETER POISSON–DIRICHLET DISTRIBUTION DERIVED FROM (2008)
A Stable Subordinator, Jim Pitman, Marc Yor
The two-parameter Poisson–Dirichlet distribution, denoted PD�α � θ�, is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson–Dirichlet...
This article appeared in Itô’s Stochastic Calculus and Probability (2008)
Jim Pitman, Marc Yor, Edited N. Ikeda, S. Watanabe, M. Fukushima
reprinted from the original tex source, August 3, 2004
Revised version. Accepted for publication in (2007)
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...
Jean Bertoin, Jim Pitman, Juan Ruiz, De Chavez
AMS subject classification: 60J65 Conditioned Brownian motion, path transformations We construct a Brownian path conditioned on its minimum value over a fixed time interval by simple transformations...
Path decompositions of a Brownian bridge related to the ratio of its maximum and amplitude (2007)
We give two new proofs of Cs'aki's formula for the law of the ratio 1 \Gamma Q of the maximum relative to the amplitude (i.e. the maximum minus minimum) for a standard Brownian bridge. The...
Running head: Stationary Markov Processes (2007)
We consider some classes of stationary, counting--measure--valued Markov processes and their companions under time--reversal. Examples arise in the L'evy--Ito decomposition of stable...
COMMUNICATIONS in PROBABILITY CONSTRUCTIONS OF A BROWNIAN PATH WITH A GIVEN MINIMUM (2007)
Jean Bertoin, Jim Pitman, Juan Ruiz, De Chavez
AMS subject classification: 60J65 Conditioned Brownian motion, path transformations We construct a Brownian path conditioned on its minimum value over a fixed time interval by simple transformations...
By Jim Pitman, Jim Pitman, Richard Stanley
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 )...
Richard P. Stanley, Jim Pitman
polytope related to empirical distributions, plane trees, parking functions, and the associahedron
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...
and the times spent by X above and below level y up to time (2007)
Hitting, occupation, and inverse local times of one-dimensional diffusions: martingale and
Forest volume decompositions and Abel-Cayley-Hurwitz multinomial expansions (2007)
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...
Smoluchowski's coagulation equation describes the growth of clustering in a system in which clusters coalesce irreversibly by binary mergers. If the rate with which two objects merge is...
trees, parking functions (2007)
polytope related to empirical distributions, plane
Where Did The Brownian Particle Go? (2007)
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...
the relative lengths of excursions derived from a stable subordinator
Fakultat fur Mathematik (2007)
Holger Dette, Jim Pitman, James Allen Fill, William J. Studden
For a birth and death chain on the nonnegative integers with birth and death probabilities p i and q i j 1 \Gamma p i and reflecting barrier at 0, it is shown that the right tails of the probability...
Ben Hansen, Ben Hansen, Jim Pitman, Jim Pitman
rules for exchangeable sequences related to species sampling 1
The (n \Gamma 1)th Bessel polynomial is represented by an exponential generating function derived from the number of returns to 0 of a sequence with 2n increments of \Sigma1 which starts and ends at...
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...
Forest volume decompositions and Abel-Cayley-Hurwitz multinomial expansions (2007)
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...
Where Did The Brownian Particle Go? (2007)
Robin Pemantle Yuval, 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.
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' 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...
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...
Arcsine Laws and Interval Partitions Derived from a Stable Subordinator (2006)
Lévy discovered that the fraction of time a standard one-dimensional Brownian motion B spends positive before time t has arcsine distribution, both for t a fixed time when Bt ≠ 0 almost surely,...
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...
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...
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...
Incluye bibliografía e índice
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...
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...
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 among...
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 among...
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)
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 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...
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...
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...
Growth of the Brownian forest (2004)
Paris Cnrs (umr, J. Pitman, M. Winkel, Jim Pitman, Matthias Winkel
Trees in Brownian excursions have been studied since the late 1980s.
Rayleigh processes, real trees, and root growth with re-grafting (2004)
Steven N. Evans, Steven N. Evans, Jim Pitman, Jim Pitman, Anita Winter, Anita Winter
Abstract. 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...
Brownian Bridge Asymptotics for Random p-Mappings (2004)
David Aldous, Grégory Miermont, Jim Pitman
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...
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...
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 (2003)
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...
Path Transformations of First Passage Bridges (2003)
Paris Cnrs (umr, J. Bertoin, L. Chaumont, J. Pitman, Jean Bertoin, ...
this paper, it follows that conditionally on {B 1 = #}, the process (B 1) has the law of the first passage bridge F # . The next lemma is obtained by following the same arguments as in the proofs of...
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, Grégory 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...
Regenerative composition structures (2003)
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...
Juan Dubra, Federico Echenique, Jel Classification C, David Levine, Jim Pitman, Maxwell Stinchcombe, ...
We present a simple example where the use of σ-algebras as a model of information leads to a paradoxical conclusion: a decision maker prefers less information to more. We then explain that the...
Path transformations of first passage bridges (2003)
Jean Bertoin, Loïc Chaumont, Jim Pitman
Summary. We define the first passage bridge from 0 to λ as the Brownian motion on the time interval [0,1] conditioned to first hit λ at time 1. We show that this process may be related to the...
Poisson-Kingman partitions (2003)
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...
Brownian Bridge Asymptotics for Random p-Mappings (2002)
David Aldous, Grégory Miermont, Jim Pitman
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...
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!1limit distributions can be...
Let Fnbeauniformlydistributedrandommappingfromtheset[n]:= (2002)
asymptotic distribution of the diameter of
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...
Forest volume decompositions and Abel-Cayley-Hurwitz multinomial expansions (2001)
This paper presents a systematic approach to the discovery, inter-pretation and veri cation of various extensions of Hurwitz's multino-mial identities, involving polynomials de ned by sumsover...
To appearinCombinatorics, Probability and Computing (2001)
distributions for split-and-merge transformations of an interval partition
Two coalescents derived from the ranges of stable subordinators (2000)
Let M ff be the closure of the range of a stable subordinator of index ff 2]0; 1[. There are two natural constructions of the M ff 's simultaneously for all ff 2]0; 1[, so that M ff ` M fi for...
E l e c t r o n i J o u r n a l o f
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...
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...
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...
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 Law of the Maximum of a Bessel Bridge (1999)
Pitman, Jim; University Of California, Berkeley; Pitman@stat.berkeley.edu, Yor, Marc; Université Pierre Et Marie Curie; Pitman@stat.berkeley.edu
Let $M_d$ be the maximum of a standard Bessel bridge of dimension $d$. A series formula for $P(M_d le a)$ due to Gikhman and Kiefer for $d = 1,2, ldots$ is shown to be valid for all real $d >0$....
Pitman, Jim; University Of California, Berkeley; Pitman@stat.berkeley.edu
For a random process $X$ consider the random vector defined by the values of $X$ at times $0 < U_{n,1} < ... < U_{n,n} < 1$ and the minimal values of $X$ on each of the intervals between consecutive...
Laplace transforms related to excursions of a one-dimensional diffusion (1999)
Various known expressions in terms of hyperbolic functions for the Laplace transforms of random times related to one-dimensional Brownian motion are derived in a unified way by excursion theory and...
Let $B$ be a standard one-dimensional Brownian motion started at 0. Let $L_{t,v}(|B|)$ be the occupation density of $|B|$ at level $v$ up to time $t$. The distribution of the process of local times...
Coalescents with multiple collisions (1999)
For each finite measure on [0; 1], a coalescent Markov process, with state space the compact set of all partitions of the set N of positive integers, is constructed so the restriction of the...
Laplace transforms related to excursions of a one-dimensional diffusion (1999)
Various known expressions in terms of hyperbolic functions for the Laplace transforms of random times related to one-dimensional Brownian motion are derived in a unified way by excursion theory and...
E l e c t r o n i J o u r n a l o f
Philippe Biane, Jim Pitman, Marc Yor
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...
For a random process X consider the random vector defined by the values of X at times 0 < U n,1 < ... < U n,n < 1 and the minimal values of X on each of the intervals between consecutive...
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 run for unit time. Given the occupation measure ¯ of this projected path, what can be said about the...
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...
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...
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...
For a random process X consider the random vector defined by the values of X at times 0 ! U n;1 ! ::: ! U n;n ! 1 and the minimal values of X on each of the intervals between consecutive pairs of...
Let B be a standard one-dimensional Brownian motion started at 0. Let L Ž�B�. be the occupation density of �B � t, v at level v up to time t. The distribution of the process of local times...
Philippe Biane, Jim Pitman, Marc Yor
Abstract. 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...
Kac's moment formula and the Feynman-Kac formula for additive functionals of a Markov process (1999)
Process by
and the associahedron
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...
Coalescents with multiple collisions (1999)
For each nite measure on [0; 1], a coalescent Markov process, with state space the compact set of all partitions of the set N of positive integers, is constructed so the restriction of the partition...
A lattice path model for the Bessel polynomials (1999)
The (n, 1)th Bessel polynomial is represented by anexponential generating function derived from the number of returns to 0 of a sequence with 2n increments of 1 which starts and ends at 0. AMS 1991...
Explicit evaluations of the symmetric Euler integral u (1,u) f(u)du are obtained 0 for some particular functions f.Theseevaluations are related to duplication formulae for Appell's...
Laplace transforms related to excursions of a one-dimensional diffusion (1999)
Various known expressions in terms of hyperbolic functions for the Laplace transforms of random times related to one-dimensional Brownian motion are derived in a uni ed way by excursion theory and...
Prediction rules for exchangeable sequences related to species sampling � (1999)
Suppose an exchangable sequence with values in a nice measurable space S admits a prediction rule of the following form: given the rst n terms of the sequence, the next term equals the jth distinct...
A family of random trees with random edge lengths (1999)
ABSTRACT: 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....
Random Brownian scaling identities and splicing of Bessel processes (1998)
An identity in distribution due to Knight for Brownian motion is extended in two different ways: first by replacing the supremum of a reflecting Brownian motion by the range of an unreflected...
The standard additive coalescent (1998)
Regard an element of the set $$\Delta := {(x_1, x_2, \dots): x_1 \geq x_2 \geq \dots \geq 0, \Sigma_i x_i = 1}$$ as a fragmentation of unit mass into clusters of masses $x_i$. The additive coalescent...
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Enumerations of Trees and Forests related to Branching Processes and Random Walks (1998)
In a Galton-Watson branching process with offspring distribution (p 0; p 1; : ::) started with k individuals, the distribution of the total progeny is identical to the distribution of the first...
Random Brownian scaling identities and splicing of Bessel processes (1998)
Jim Pitman, Jim Pitman, Marc Yor, Marc Yor
An identity in distribution due to F. Knight for Brownian motion is extended in two different ways: firstly by replacing the supremum of a reflecting Brownian motion by the range of an unreflected...
Construction of Markovian coalescents (1998)
Partition-valued and measure-valued 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...
On the laws of homogeneous functionals of the Brownian bridge (1998)
Philippe Carmona, Erique Petit, Jim Pitman, Marc Yor
Abstract. We develop a general and elementary method, which allows in particular to compute the distributions of a large number of interesting homogeneous functionals of the standard Brownian bridge....
Tree-valued Markov chains derived from Galton-Watson processes (1998)
Let G be a Galton-Watson tree, and for 0 u 1 let G u be the subtree of G obtained by retaining each edge with probability u. We study the tree-valued Markov process (G u; 0 u 1) and an analogous...
The law of the maximum of a Bessel bridge (1998)
Let M ffi be the maximum of a standard Bessel bridge of dimension ffi . A series formula for P (M ffi a) due to Gikhman and Kiefer for ffi = 1; 2; : : : is shown to be valid for all real ffi ? 0....
Inhomogeneous Continuum Random Trees and the Entrance Boundary of the Additive Coalescent (1998)
By David Aldous, David Aldous, Jim Pitman
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...
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...
Inhomogeneous Continuum Random Trees and the Entrance Boundary of the Additive Coalescent (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...
Abel-Cayley-Hurwitz multinomial expansions associated with random mappings, forests, and subsets
Revised version. Accepted for publication in (1998)
Various enumerations of labeled trees and forests, including Cayley's formula n n,2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived from...
Enumerations of Trees and Forests related to Branching Processes and Random Walks (1998)
In a Galton-Watson branching process with o spring distribution (p 0;p 1;:::) started with k individuals, the distribution of the total progeny is identical to the distribution of the rst passage...
Random Brownian scaling identities and splicing of Bessel processes (1998)
Jim Pitman, Jim Pitman, Marc Yor, Marc Yor
An identity in distribution due to F. Knight for Brownian motion is extended in two di erent ways: rstly by replacing the supremum of a re ecting Brownian motion by the range of an unre ected...
Prediction rules and exchangeable sequences related to species sampling (1998)
Ben Hansen, Ben Hansen, Jim Pitman, Jim Pitman
Prediction rules for exchangeable sequences related to species sampling 1 by
Path decompositions of a Brownian bridge related to the ratio of its maximum and amplitude (1998)
We give two new proofs of Csaki's formula for the law of the ratio 1, Q of the maximum relative to the amplitude (i.e. the maximum minus minimum) for a standard Brownian bridge. The second of...
The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator (1997)
The two-parameter Poisson-Dirichlet distribution, denoted $\mathsf{PD}(\alpha, \theta)$ is a probability distribution on the set of decreasing positive sequences with sum 1. The usual...
Stopped Markov chains with stationary occupation (1997)
Abstract: Let E be a finite set equipped with a group G of bijective transformations and suppose that X is an irreducible Markov chain on E that is equivariant under the action of G. In particular,...
The asymptotic behavior of the Hurwitz binomial distribution (1997)
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 non-root vertices of a fringe...
Some probabilistic aspects of set partitions (1997)
A partition of the set N n: = f1; 2; \Delta \Delta \Delta; ng is an unordered collection of nonempty subsets of N n. Let P n denote the set of all such partitions, and let
On the lengths of excursions of some Markov processes (1997)
Results are obtained regarding the distribution of the ranked lengths of component intervals in the complement of the random set of times when a recurrent Markov process returns to its starting...
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...
Coagulation and Branching Process Models of Gravitational Clustering (1997)
Smoluchowski's coagulation equation describes the growth of clustering in a system in which clusters coalesce irreversibly by binary mergers. If the rate with which two objects merge is...
Stopped Markov chains with stationary occupation (1997)
Abstract: Let E be a nite set equipped with a group G of bijective transformations and suppose that X is an irreducible Markov chain on E that is equivariant under the action of G. In particular, if...
Probabilistic bounds on the coefficients of polynomials with only real zeros (1997)
The work of Harper and subsequent authors has shown that nite sequences (a 0;;an) arising from combinatorial problems are often such that the polynomial A(z): = P n k=0 akz k has only real zeros....
On the distribution of ranked heights of excursions of a Brownian bridge (1997)
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 sion of the bridge has...
On the distribution of ranked heights of excursions of a Brownian bridge (1997)
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...
Let G be a Galton-Watson tree, and for 0 u 1let Gu be the subtree of G obtained by retaining each edge with probability u. We study the tree-valued Markov process (Gu; 0 u 1) and an analogous process...
The two parameter Poisson–Dirichlet (1997)
distribution derived from a stable subordinator. by
: www.idealibrary.com on Coalescent Random Forests* (1997)
Various enumerations of labeled trees and forests, including Cayley's formula n n&2 for the number of trees labeled by [n], and Cayley's multinomial expansion over trees, are derived...
Random Discrete Distributions Derived from Self-Similar Random Sets (1996)
Pitman, Jim; University Of California, Berkeley; Pitman@stat.berkeley.edu, Yor, Marc; Université Pierre Et Marie Curie; Pitman@stat.berkeley.edu
A model is proposed for a decreasing sequence of random variables $(V_1, V_2, cdots)$ with $sum_n V_n = 1$, which generalizes the Poisson-Dirichlet distribution and the distribution of ranked lengths...
Random discrete distributions invariant under size-biased permutation (1996)
Invariance of a random discrete distribution under size-biased permutation is equivalent to a conjunction of symmetry conditions on its finite-dimensional distributions. This is applied to...
Some developments of the Blackwell-MacQueen urn scheme (1996)
The Blackwell-MacQueen description of sampling from a Dirichlet random distribution on an abstract space is reviewed, and extended to a general family of random discrete distributions. Results are...
Random Discrete Distributions Derived From Self-Similar Random Sets (1996)
: A model is proposed for a decreasing sequence of random variables (V 1 ; V 2 ; \Delta \Delta \Delta) with P n V n = 1, which generalizes the Poisson-Dirichlet distribution and the distribution of...
Communicated by the Managing Editors (1996)
The work of Harper and subsequent authors has shown that finite sequences (a0,..., an) arising from combinatorial problems are often such that the polynomial A(z): = n k=0 akz k has only real zeros....
To appearinSeminaire de Probabilites XXXI (1996)
On the relative lengths of excursions derived from a stable subordinator by
The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator. (1995)
The two-parameter Poisson-Dirichlet distribution, denoted pd(ff; `), is a distribution on the set of decreasing positive sequences with sum 1. The usual Poisson-Dirichlet distribution with a single...
Cyclically Stationary Brownian Local Time Processes (1995)
Local time processes parameterized by a circle, defined by the occupation density up to time T of Brownian motion with constant drift on the circle, are studied for various random times T . While...
Research supported by N.S.F. Grant DMS94-04345 (1995)
Local time processes parameterized by a circle, de ned by the occupation density up to time T of Brownian motion with constant drift on the circle, are studied for various random times T. While such...
Jim Pitman, William J. Studden, Holger Dette, William J. Studden, Jim Pitman
For the random walk on the nonnegative integers with re ecting barrier it is shown that the right tails of the probability of the rst return from state 0 to state 0 are simple transition...
Hitting, occupation, and inverse local times of one-dimensional di usions: martingale and
Prediction rules for exchangeable sequences related to species sampling
Suppose an exchangable sequence with values in a nice measurable space S admits a prediction rule of the following form: given the first n terms of the sequence, the next term equals the jth distinct...
Limit laws for Brownian motion conditioned to reach a high level
A functional limit theorem is presented for the behaviour of Brownian motion conditioned to reach a high level during a fixed time interval. The asymptotic behaviour of the conditioned path as the...