Jim Pitman

Size-biased sampling of Poisson point processes and excursions

SummarySome general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function. These formulae explain why in certain cases (gamma and stable) the size-biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking (residual allocation) scheme defined by independent beta random variables. An application is made to length biased sampling of excursions of a Markov process away from a recurrent point of its statespace, with emphasis on the Brownian and Bessel cases when the associated inverse local time is a stable subordinator. Results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by Brownian motion.

ONE-DIMENSIONAL BROWNIAN MOTION AND THE THREE-DIMENSIONAL BESSEL PROCESS

A simple path transformation is described which connects one-dimensional Brownian motion with the radial part of three-dimensional Brownian motion. This provides simple proofs of various path decompositions for these processes described by David Williams.

Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions

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 governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws.

Markovian Bridges: Construction, Palm Interpretation, and Splicing

By a Markovian bridge we mean a process obtained by conditioning a Markov process X to start in some state x at time 0 and arrive at some state z at time t. Once the definition is made precise, we call this process the (x, t, z)-bridge derived from X. Important examples are provided by Brownian and Bessel bridges, which have been extensively studied and find numerous applications. See for example [PY1,SW,Sa,H,EL,AP,BP].It is part of Markovian folklore that the right way to define bridges in some generality is by a suitable Doob h -transform of the space-time process. This method was used by Getoor and Sharpe [GS4] for excursion bridges, and by Salminen [Sa] for one-dimensional diffusions, but the idea of using h-transforms to construct bridges seems to be much older. Our first object in this paper is to make this definition of bridges precise in a suitable degree of generality, with the aim of dispelling all doubts about the existence of clearly defined bridges for nice Markov processes. This we undertake in Section 2. In Section 3 we establish a conditioning formula involving bridges and continuous additive functionals of the Markov process. This formula can be found in [RY, Ex. (1.16) of Ch. X, p.378] under rather stringent continuity conditions. One of our goals here is to prove the formula in its “natural” setting. We apply the conditioning formula in Section 4 to show how Markovian bridges are involved in a family of Palm distributions associated with continuous additive functionals of the Markov process. This generalizes an approach to bridges suggested in a particular case by Kallenberg [K1], and connects this approach to the more conventional definition of bridges adopted here.

Random discrete distributions invariant under size-biased permutation

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 characterize residual allocation models with independent factors that are invariant under size-biased permutation. Apart from some exceptional cases and minor modifications, such models form a two-parameter family of generalized Dirichlet distributions.

A Polytope Related to Empirical Distributions, Plane Trees, Parking Functions, and the Associahedron

AbstractThe volume of the n -dimensional polytope Πn(x):= {y ∈ Rn : yi ≥ 0 and y1 + · · · + yi ≤ x1 + · · ·+ xi for all 1 ≤ i ≤ n } for arbitrary x:=(x1, . . ., xn) with xi >0 for all i defines a polynomial in variables xi which admits a number of interpretations, in terms of empirical distributions, plane partitions, and parking functions. We interpret the terms of this polynomial as the volumes of chambers in two different polytopal subdivisions of Πn(x) . The first of these subdivisions generalizes to a class of polytopes called sections of order cones. In the second subdivision the chambers are indexed in a natural way by rooted binary trees with n+1 vertices, and the configuration of these chambers provides a representation of another polytope with many applications, the associahedron .

Fluctuation identities for Levy processes and splitting at the maximum

It6s notion of a Poisson point process of excursions is used to give a unified approach to a number of results in the fluctuation theory of LUvy processes, including identities of Pecherskii, Rogozin and Fristedt, and Millars path decomposition at the maximum. LEVY PROCESS; SPLITTING TIME; FLUCTUATION THEORY; POINT PROCESS OF EXCURSIONS

Poisson–Dirichlet and GEM Invariant Distributions for Split-and-Merge Transformations of an Interval Partition

This paper introduces a split-and-merge transformation of interval partitions which combines some features of one model studied by Gnedin and Kerov [12, 11] and another studied by Tsilevich [30, 31] and Mayer-Wolf, Zeitouni and Zerner [21]. The invariance under this split-and-merge transformation of the interval partition generated by a suitable Poisson process yields a simple proof of the recent result of [21] that a Poisson–Dirichlet distribution is invariant for a closely related fragmentation–coagulation process. Uniqueness and convergence to the invariant measure are established for the split-and-merge transformation of interval partitions, but the corresponding problems for the fragmentation–coagulation process remain open.

Tree-valued Markov chains derived from Galton-Watson processes

Abstract 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 process ( G u∗, 0 ≤ u ≤ 1) in which G 1∗ is a critical or subcritical Galton-Watson tree conditioned to be infinite. Results simplify and are further developed in the special case of Poisson offspring distribution.

Coalescent Random Forests

Various enumerations of labeled trees and forests, including Cayleys formulann?2for the number of trees labeled by [n], and Cayleys multinomial expansion over trees, are derived from the followingcoalescent constructionof a sequence of random forests (Rn,Rn?1,?,R1) such thatRkhas uniform distribution over the set of all forests ofkrooted trees labeled by [n]. LetRnbe the trivial forest withnroot vertices and no edges. Forn?k?2, given thatRn,?,Rkhave been defined so thatRkis a rooted forest ofktrees, defineRk?1by addition toRkof a single edge picked uniformly at random from the set ofn(k?1) edges which when added toRkyield a rooted forest ofk?1 trees. This coalescent construction is related to a model for a physical process of clustering or coagulation, theadditive coalescentin which a system of masses is subject to binary coalescent collisions, with each pair of masses of magnitudesxandyrunning a risk at ratex+yof a coalescent collision resulting in a mass of magnitudex+y. The transition semigroup of the additive coalescent is shown to involve probability distributions associated with a multinomial expansion over rooted forests.