arXiv subjects

Jim Pitman

Publications and source records attributed to Jim Pitman.

At least 73 records · Page 4Linked to original sources

arXiv2008

One-dimensional Brownian particle systems with rank dependent drifts

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 been to study the long range behavior of the spacings between the Brownian motions arranged in increasing order. For finitely many Brownian motions interacting in this manner, we characterize drifts for which the family of laws of the vector of spacings is tight, and show its convergence to a unique stationary joint distribution given by independent exponential distributions with varying means. We also study one particular countably infinite system, where only the minimum Brownian particle gets a constant upward drift, and prove that independent and identically distributed exponential spacings remain stationary under the dynamics of such a process. Some related conjectures in this direction have also been discussed.

math.PR↗

arXiv2007

Notes on the occupancy problem with infinitely many boxes: general asymptotics and power laws

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 represented in samples from populations with infinitely many species. We present moments of this random variable, discuss asymptotic relations among them and with related random variables, and draw connections with regular variation, which appears in various manifestations.

math.PR↗

arXiv2006

Colored loop-erased random walk on the complete graph

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 walk makes a step from $i$ to $j$. We introduce a coloring of these edges by painting edges with a fixed color as long as the walk does not loop back on itself, then switching to a new color whenever a loop is erased, with each new color distinct from all previous colors. The pattern of colors along the edges of the loop-erased walk then displays stretches of consecutive steps of the walk left untouched by the loop-erasure process. Assuming that the underlying sequence generating the loop-erased walk is a sequence of independent random variables, each uniform on $[N]:=\{1, 2, ..., N\}$, we condition the walk to start at $N$ and stop the walk when it first reaches the subset $[k]$, for some $1 \leq k \leq N-1$. We relate the distribution of the random length of this loop-erased walk to the distribution of the length of the first loop of the walk, via Cayley's enumerations of trees, and via Wilson's algorithm. For fixed $N$ and $k$, and $i = 1,2, ...$, let $B_i$ denote the event that the loop-erased walk from $N$ to $[k]$ has $i +1$ or more edges, and the $i^{th}$ and $(i+1)^{th}$ of these edges are colored differently. We show that given that the loop-erased random walk has $j$ edges for some $1\leq j \leq N-k$, the events $B_i$ for $1 \leq i \leq j-1$ are independent, with the probability of $B_i$ equal to $1/(k+i+1)$. This determines the distribution of the sequence of random lengths of differently colored segments of the loop-erased walk, and yields asymptotic descriptions of these random lengths as $N \to \infty$.

math.PR↗

arXiv2006

Gibbs distributions for random partitions generated by a fragmentation process

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 such partitions with k clusters. We provide conditions on the weight sequence w allowing construction of a partition valued random process where at step k the state has the Gibbs (n,k,w) distribution, so the partition is subject to irreversible fragmentation as time evolves. For a particular one-parameter family of weight sequences w\_j, the time-reversed process is the discrete Marcus-Lushnikov coalescent process with affine collision rate K\_{i,j}=a+b(i+j) for some real numbers a and b. Under further restrictions on a and b, the fragmentation process can be realized by conditioning a Galton-Watson tree with suitable offspring distribution to have n nodes, and cutting the edges of this tree by random sampling of edges without replacement, to partition the tree into a collection of subtrees. Suitable offspring distributions include the binomial, negative binomial and Poisson distributions.

math.PR↗

arXiv2006

Poisson representation of a Ewens fragmentation process

A simple explicit construction is provided of a partition-valued fragmentation process whose distribution on partitions of $[n]=\{1,...,n\}$ at time $θ\ge 0$ is governed by the Ewens sampling formula with parameter $θ$. These partition-valued processes are exchangeable and consistent, as $n$ varies. They can be derived by uniform sampling from a corresponding mass fragmentation process defined by cutting a unit interval at the points of a Poisson process with intensity $θx^{-1} \diff x$ on ${\mathbb R}_+$, arranged to be intensifying as $θ$ increases.

math.PR↗

arXiv2006

Asymptotic laws for compositions derived from transformed subordinators

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 study the number of parts $K_n$ of this composition and other related functionals under the assumption that $\widetilde{\mathcal{R}}=ϕ(S_{\bullet})$, where $(S_t,t\geq0)$ is a subordinator and $ϕ:[0,\infty]\to[0,1]$ is a diffeomorphism. We derive the asymptotics of $K_n$ when the Lévy measure of the subordinator is regularly varying at 0 with positive index. Specializing to the case of exponential function $ϕ(x)=1-e^{-x}$, we establish a connection between the asymptotics of $K_n$ and the exponential functional of the subordinator.

math.PR↗

arXiv2006

Exchangeable partitions derived from Markovian coalescents

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 by a simple coalescent process, and observed that similar methods could be applied to more complex models. M{ö}hle described the recursion which determines the generalization of the Ewens sampling formula in the situation when the lines of descent are governed by a $Λ$-coalescent, which allows multiple mergers. Here we show that the basic integral representation of transition rates for the $Λ$-coalescent is forced by sampling consistency under more general assumptions on the coalescent process. Exploiting an analogy with the theory of regenerative partition structures, we provide various characterizations of the associated partition structures in terms of discrete-time Markov chains.

math.PR↗

arXiv2006

Growth of the Brownian forest

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 forest-valued 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.

math.PR↗

arXiv2005

Self-similar and Markov composition structures

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 $S\subset {\mathbb R}_+$ are those which are consistent with respect to a simple truncation operation. Using the standard coding of compositions by finite strings of binary digits starting with a 1, the random composition of $n$ is defined by the first $n$ terms of a random binary sequence of infinite length. The locations of 1s in the sequence are the places visited by an increasing time-homogeneous Markov chain on the positive integers if and only if $S = \exp(-W)$ for some stationary regenerative random subset $W$ of the real line. Complementing our study in previous papers, we identify self-similar Markovian composition structures associated with the two-parameter family of partition structures.

math.PR↗

arXiv2004

Exchangeable Gibbs partitions and Stirling triangles

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 assigning to a generic partition $\{A_j, j\leq k\}$ the probability $\V_{n,k} \W_{|A_1|}... \W_{|A_k|}$, where $|A_j|$ is the number of elements of $A_j$. We impose constraints on the weights by assuming that the resulting random partitions $Π_n$ 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 $α\in [-\infty,1]$. The case $α=1$ is trivial, and for each value of $α\neq 1$ the set of possible $\V$-weights is an infinite-dimensional 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 $-\infty\leqα<0$ and continuous for $0\leqα<1$. For $α\leq 0$ the extremes correspond to the members of the Ewens-Pitman 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 $Π_n$ as $n$ tends to infinity.

math.PR↗

arXiv2004

Rayleigh processes, real trees, and root growth with re-grafting

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. Aldous's Brownian continuum random tree, the random tree-like object naturally associated with a standard Brownian excursion, may be thought of as a random compact real tree. The continuum random tree is a scaling limit as N tends to infinity of both a critical Galton-Watson tree conditioned to have total population size N as well as a uniform random rooted combinatorial tree with N vertices. The Aldous--Broder algorithm is a Markov chain on the space of rooted combinatorial trees with N vertices that has the uniform tree as its stationary distribution. We construct and study a Markov process on the space of all rooted compact real trees that has the continuum random tree as its stationary distribution and arises as the scaling limit as N tends to infinity of the Aldous--Broder chain. A key technical ingredient in this work is the use of a pointed Gromov--Hausdorff distance to metrize the space of rooted compact real trees.

math.PR↗

arXiv2004

Regenerative partition structures

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 which Kingman characterised by regeneration after deletion of a part chosen by size-biased sampling. We associate each regenerative partition structure with a corresponding regenerative composition structure, which (as we showed in a previous paper) can be associated in turn with a regenerative random subset of the positive halfline, that is the closed range of a subordinator. A general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator. We also analyse deletion properties characteristic of the two-parameter family of partition structures.

math.PR↗

arXiv2004

Regenerative Composition Structures

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 structure is represented by a process of random sampling of points from an exponential distribution on the positive halfline, and separating the points into clusters by an independent regenerative random set. Examples are composition structures derived from residual allocation models, including one associated with the Ewens sampling formula, and composition structures derived from the zero set of a Brownian motion or Bessel process. We provide characterisation results and formulas relating the distribution of the regenerative composition to the L{é}vy parameters of a subordinator whose range is the corresponding regenerative set. In particular, the only reversible regenerative composition structures are those associated with the interval partition of $[0,1]$ generated by excursions of a standard Bessel bridge of dimension $2 - 2 α$ for some $α\in [0,1]$.

math.PR↗

arXiv2004

Where did the Brownian particle go?

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 projected path, what can be said about the terminal point W(1), or about the range of the original path? In any dimension, for each Borel set A subseteq S^{d-1}, the conditional probability that the projection of W(1) is in A given mu(A) is just mu(A). Nevertheless, in dimension d>=3, both the range and the terminal point of W can be recovered with probability 1 from mu. In particular, for d>=3 the conditional law of the projection of W(1) given mu is not mu. In dimension~2 we conjecture that the projection of W(1) cannot be recovered almost surely from mu, and show that the conditional law of the projection of W(1) given mu is not mu.

math.PR↗

arXiv2004

The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin's local time identity

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 structure of an ICRT, and also of its width process, determining the size of layers in order of height. These processes turn out to be transformations of bridges with exchangeable increments, which have already appeared in other ICRT related topics such as stochastic additive coalescence. The results rely on two different constructions of birthday trees from processes with exchangeable increments, on weak convergence arguments, and on general theory on continuum random trees.

math.PR↗

arXiv2004

Two recursive decompositions of Brownian bridge

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 showing these mapping-walks converge weakly to a reflecting Brownian bridge. Two different ways to encode a mapping as a walk lead to two different decompositions of the Brownian bridge, each defined by cutting the path of the bridge at an increasing sequence of recursively defined random times in the zero set of the bridge. The random mapping asymptotics entail some remarkable identities involving the random occupation measures of the bridge fragments defined by these decompositions. We derive various extensions of these identities for Brownian and Bessel bridges, and characterize the distributions of various path fragments involved, using the theory of Poisson processes of excursions for a self-similar Markov process whose zero set is the range of a stable subordinator of index between 0 and 1.

math.PR↗

arXiv2004

Weak convergence of random p-mappings and the exploration process of inhomogeneous continuum random trees

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 mappings in random walks which are shown to converge to a functional of the exploration process of inhomogeneous random trees, this exploration process being derived (Aldous-Miermont-Pitman 2003) from a bridge with exchangeable increments. Our setting generalizes previous results by allowing a finite number of ``attracting points'' to emerge.

math.PR↗

arXiv2002

Poisson-Kingman partitions

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 two-parameter family of Poisson-Dirichlet 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.

math.PR↗