**Statistics Technical Reports:**Search | Browse by year

**Term(s):**Pitman Jim**Results:**77**Sorted by:**Relevance | Year**Page: 1 2 3 4 5 ... Next**

**Title:**Partially exchangeable random partitions**Author(s):**Pitman, J. W.; **Date issued:**May 1992

http://nma.berkeley.edu/ark:/28722/bk00047298w (PDF) **Keyword note:**Pitman__Jim**Report ID:**343**Relevance:**100

**Title:**Random discrete distributions invariant under size-biased permutation**Author(s):**Pitman, J. W.; **Date issued:**June 1992

http://nma.berkeley.edu/ark:/28722/bk00047299f (PDF) **Keyword note:**Pitman__Jim**Report ID:**344**Relevance:**100

**Title:**[Title unavailable]**Author(s):**Pitman, Jim; **Date issued:**September 2003**Keyword note:**Pitman__Jim**Report ID:**648**Relevance:**100

**Title:**Stationary Excursions**Author(s):**Pitman, J.; **Date issued:**July 1986**Date modified:**revised November 1986

http://nma.berkeley.edu/ark:/28722/bk000498b86 (PDF) **Keyword note:**Pitman__Jim**Report ID:**66**Relevance:**100

**Title:**Combinatorial Stochastic Processes**Author(s):**Pitman, Jim; **Date issued:**Aug 2002

http://nma.berkeley.edu/ark:/28722/bk0000n1q1g (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n1q21 (PostScript) **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**Keyword note:**Pitman__Jim**Report ID:**621**Relevance:**100

**Title:**Poisson-Dirichlet and GEM invariant distributions for split-and-merge transformations of an interval partition**Author(s):**Pitman, Jim; **Date issued:**Jun 2001

http://nma.berkeley.edu/ark:/28722/bk0000n1q44 (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n1q5p (PostScript) **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.**Keyword note:**Pitman__Jim**Report ID:**597**Relevance:**100

**Title:**Cyclically Stationary Brownian Local Time Processes**Author(s):**Pitman, Jim; **Date issued:**Jun 1995

http://nma.berkeley.edu/ark:/28722/bk0000n1x7w (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n1x8f (PostScript) **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**Keyword note:**Pitman__Jim**Report ID:**425**Relevance:**100

**Title:**Kac's moment formula for additive functionals of a Markov process**Author(s):**Pitman, Jim; **Date issued:**Jul 1995

http://nma.berkeley.edu/ark:/28722/bk0000n1z0j (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n1z13 (PostScript) **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.**Keyword note:**Pitman__Jim**Report ID:**427**Relevance:**100

**Title:**Partition structures derived from Brownian motion and stable subordinators**Author(s):**Pitman, Jim; **Date issued:**May 1992**Date modified:**revised July 1995

http://nma.berkeley.edu/ark:/28722/bk0000n1z36 (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n1z4r (PostScript) **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**Keyword note:**Pitman__Jim**Report ID:**346**Relevance:**100

**Title:**Some probabilistic aspects of set partitions**Author(s):**Pitman, Jim; **Date issued:**Mar 1996

http://nma.berkeley.edu/ark:/28722/bk0000n2148 (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n215t (PostScript) **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.**Keyword note:**Pitman__Jim**Report ID:**452**Relevance:**100