Title:Random discrete distributions invariant under size-biased permutation
Author(s):Pitman, J. W.;
Date issued:June 1992
Date issued:September 2003
Date issued:July 1986
Date modified:revised November 1986
Title:Combinatorial Stochastic Processes
Date issued:Aug 2002
Abstract:This is a preliminary set of lecture notes for a course to be given at the St. Flour summer school in July 2002. The theme of the course is the study of various combinatorial models of random partitions and random trees, and the asymptotics of these models related to continuous parameter stochastic processes. Following is a list of the main topics treated: models for random combinatorial structures, such as trees, forests, permutations, mappings, and partitions; probabilistic interpretations of various combinatorial notions e.g. Bell polynomials, Stirling numbers, polynomials of binomial type, Lagrange inversion; Kingman's theory of exchangeable random partitions and random discrete distributions; connections between random combinatorial structures and processes with independent increments: Poisson-Dirichlet limits; random partitions derived from subordinators; asymptotics of random trees, graphs and mappings related to excursions of Brownian motion; continuum random trees embedded in Brownian motion; Brownian local times and squares of Bessel processes; various processes of fragmentation and coagulation, including Kingman's coalescent, the additive and multiplicative coalescents
Title:Poisson-Dirichlet and GEM invariant distributions for split-and-merge transformations of an interval partition
Date issued:Jun 2001
Abstract:This paper introduces a split-and-merge transformation of interval partitions which combines some features of one model studied by Gnedin and Kerov and another studied by Tsilevich and by Mayer-Wolf, Zeitouni and Zerner. 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 Mayer-Wolf, Zeitouni and Zerner 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.
Title:Cyclically Stationary Brownian Local Time Processes
Date issued:Jun 1995
Abstract: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 such processes are typically non-Markovian, their Laplace functionals are expressed by series formulae related to similar formulae for the Markovian local time processes subject to the Ray-Knight theorems for BM on the line, and for squares of Bessel processes and their bridges. For $T$ the time that BM on the circle first returns to its starting point after a complete loop around the circle, the local time process is cyclically stationary, with same two-dimensional distributions, but not the same three-dimensional distributions, as the sum of squares of two i.i.d. cyclically stationary Gaussian processes. This local time process is the infinitely divisible sum of a Poisson point process of local time processes derived from Brownian excursions. The corresponding intensity measure on path space, and similar L\'evy measures derived from squares of Bessel processes, are described in terms of a 4-dimensional Bessel bridge by Williams' decomposition of It\^o's law of Brownian excursions.
Pub info:Prob. Th. Rel. Fields. 106 299-329, 1996
Title:Kac's moment formula for additive functionals of a Markov process
Date issued:Jul 1995
Abstract:Mark Kac introduced a method for calculating the distribution of the integral $A_v = \int_0^T v(X_t) dt$ for a function $v$ of a Markov process $(X_t, t \ge 0 )$, and suitable random $T$, which yields the Feynman-Kac formula for the moment-generating function of $A_v$. This paper reviews Kac's method, with emphasis on an aspect often overlooked. This is Kac's formula for moments of $A_v$ which may be stated as follows. For any random time $T$ such that the killed process $(X_t, 0 \le t <T)$ is Markov with some substochastic semi-group $K_t(x,dy) = P_x(X_t \in dy, T > t)$, and any non-negative measurable $v$, for $X$ with initial distribution $\lambda$ the $n$th moment of $A_v$ is $P_\lambda A_v^n = n! \lambda (G M_v) ^n 1 $ where $G = \int_(0)^\infty K_t dt$ is the Green's operator of the killed process, $M_v$ is the operator of multiplication by $v$, and $1$ is the function that is identically 1.
Title:Partition structures derived from Brownian motion and stable subordinators
Date issued:May 1992
Date modified:revised July 1995
Abstract:Explicit formulae are obtained for the distribution of various random partitions of a positive integer $n$, both ordered and unordered, derived from the zero set $M$ of a Brownian motion by the following scheme: pick $n$ points uniformly at random from $[0,1]$, and classify them by whether they fall in the same or different component intervals of the complement of $M$. Corresponding results are obtained for $M$ the range of a stable subordinator and for bridges defined by conditioning on $1 \in M$. These formulae are related to discrete renewal theory by a general method of discretizing a subordinator using the points of an independent homogeneous Poisson process.
Pub info:Bernoulli 3, 79-96, 1997
Title:Some probabilistic aspects of set partitions
Date issued:Mar 1996
Abstract:Some classical combinatorial identities related to partitions of a finite set, involving the Stirling numbers of the second kind, and the Bell numbers, are viewed from a probabilistic perspective.
Pub info:Amer. Math. Monthly, 104, 201-209, 1997.