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

**Title:**A lattice path model for the Bessel polynomials**Author(s):**Pitman, Jim; **Date issued:**Mar 1998

http://nma.berkeley.edu/ark:/28722/bk0000n225b (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n226w (PostScript) **Abstract:**The (n-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 +1 or -1 which starts and ends at 0.**Keyword note:**Pitman__Jim**Report ID:**551**Relevance:**100

**Title:**Probabilistic bounds on the coefficients of polynomials with only real zeros**Author(s):**Pitman, Jim; **Date issued:**Mar 1996

http://nma.berkeley.edu/ark:/28722/bk0000n235v (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n236d (PostScript) **Abstract:**The work of Harper and subsequent authors has shown that finite sequences $(a_0, \cdots , a_n)$ arising from combinatorial
problems are often such that the polynomial $A(z):= \sum_(k=0)^n a_k z^k$ has only real zeros. Basic examples include rows
from the arrays of binomial coefficients, Stirling numbers of the first and second kinds, and Eulerian numbers. Assuming the
$a_k$ are non-negative, $A(1) > 0$ and that $A(z)$ is not constant, it is known that $A(z)$ has only real zeros iff the normalized
sequence $(a_0/A(1), \cdots , a_n/A(1))$ is the probability distribution of the number of successes in $n$ independent trials
for some sequence of success probabilities. Such sequences $(a_0, \cdots , a_n)$ are also known to be characterized by total
positivity of the infinite matrix $(a_(i-j))$ indexed by non-negative integers $i$ and $j$. This papers reviews inequalities
and approximations for such sequences, called (\em P(\'o)lya frequency sequences) which follow from their probabilistic representation.
In combinatorial examples these inequalities yield a number of improvements of known estimates.**Pub info:**J. Comb. Theory A, 77, 279-303 (1997)**Keyword note:**Pitman__Jim**Report ID:**453**Relevance:**100

**Title:**The distribution of local times of a Brownian bridge**Author(s):**Pitman, Jim; **Date issued:**Nov 1998

http://nma.berkeley.edu/ark:/28722/bk0000n278k (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n2794 (PostScript) **Abstract:**L\'evy's approach to Brownian local times is used to give a simple derivation of a formula of Borodin which determines the
distribution of the local time at level x up to time 1 for a Brownian bridge of length 1 from 0 to b. A number of identities
in distribution involving functionals of the bridge are derived from this formula. A stationarity property of the bridge local
times is derived by a simple path transformation, and related to Ray's description of the local time process of Brownian motion
stopped at an independent exponential time.**Pub info:**S\'{e}minaire de Probabilit\'{e}s XXXIII, 388-394, Lecture Notes in Math. 1709, Springer, 1999**Keyword note:**Pitman__Jim**Report ID:**539**Relevance:**100

**Title:**Forest volume decompositions and Abel-Cayley-Hurwitz multinomial expansions**Author(s):**Pitman, Jim; **Date issued:**Apr 2001

http://nma.berkeley.edu/ark:/28722/bk0000n2b4x (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n2b5g (PostScript) **Abstract:**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 all subsets of a finite set. The identities are interpreted
as decompositions of forest volumes defined by the enumerator polynomials of sets of rooted labeled forests. These decompositions
involve the following basic forest volume formula, which is a refinement of Cayley's multinomial expansion: for R a subset
of S the polynomial enumerating out-degrees of vertices of rooted forests labeled by S whose set of roots is R, with edges
directed away from the roots, is (\sum_(r \in R) x_r ) (\sum_(s \in S) x_s )^(|S|-|R|-1))**Pub info:**J. Comb. Theory A. 98,175-191 (2002)**Keyword note:**Pitman__Jim**Report ID:**591**Relevance:**100

**Title:**Poisson-Kingman partitions**Author(s):**Pitman, Jim; **Date issued:**Oct 2002

http://nma.berkeley.edu/ark:/28722/bk0000n2j1c (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n2j2x (PostScript) **Abstract:**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.**Keyword note:**Pitman__Jim**Report ID:**625**Relevance:**100

**Title:**The two-parameter generalization of Ewens' random partition structure**Author(s):**Pitman, Jim; **Date issued:**Mar 1992**Date modified:**reprinted with an appendix and updated references

http://nma.berkeley.edu/ark:/28722/bk0000n2p0w (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n2p1f (PostScript) **Abstract:**A two-parameter generalization of Ewens' random partition structure is obtained by sampling from a random discrete distribution
derived by Perman, Pitman and Yor from the lengths of excursions of a recurrent Bessel process.**Keyword note:**Pitman__Jim**Report ID:**345**Relevance:**100

**Title:**Coalescent random forests.**Author(s):**Pitman, Jim; **Date issued:**Sep 1996

http://nma.berkeley.edu/ark:/28722/bk0000n366f (PDF) **Abstract:**Various enumerations of labeled trees and forests, due to Cayley, Moon, and other authors, are consequences of the following
(\em coalescent algorithm) for construction of a sequence of random forests $(R_n, R_(n-1), \cdots, R_1)$ such that $R_k$
has uniform distribution over the set of all forests of $k$ rooted trees labeled by $\INn := \(1, \cdots , n\)$. Let $R_n$
be the trivial forest with $n$ root vertices and no edges. For $n \ge k \ge 2$, given that $R_n, \cdots, R_k$ have been defined
so that $R_k$ is a rooted forest of $k$ trees, define $R_(k-1)$ by addition to $R_k$ of a single directed edge picked uniformly
at random from the set of $n(k-1)$ directed edges which when added to $R_k$ yield a rooted forest of $k-1$ trees labeled by
$\INn$. Variations of this coalescent algorithm are described, and related to the literature of physical processes of clustering
and polymerization.**Keyword note:**Pitman__Jim**Report ID:**457**Relevance:**100

**Title:**Coalescents with multiple collisions**Author(s):**Pitman, Jim; **Date issued:**Nov 1997

http://nma.berkeley.edu/ark:/28722/bk0000n381q (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n3828 (PostScript) **Abstract:**For each finite measure $\Lambda$ 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 to each finite subset of $N$ is a
Markov chain with the following transition rates: when the partition has $b$ blocks, each $k$-tuple of blocks is merging to
form a single block at rate $\int_0^1 x^(k-2) (1 - x) ^(b-k) \Lambda(dx)$. Call this process a (\em $\Lambda$-coalescent).
Discrete measure valued processes derived from the $\Lambda$-coalescent model a system of masses undergoing coalescent collisions.
Kingman's coalescent, which has numerous applications in population genetics, is the $\delta_0$-coalescent for $\delta_0$
a unit mass at 0. The coalescent recently derived by Bolthausen and Sznitman from Ruelle's probability cascades, in the context
of the Sherrington-Kirkpatrick spin glass model in mathematical physics, is the $U$-coalescent for $U$ uniform on $[0,1]$.
For $\Lambda=U$, and whenever an infinite number of masses are present, each collision in a $\Lambda$-coalescent involves
an infinite number of masses almost surely, and the proportion of masses involved exists almost surely and is distributed
proportionally to $\Lambda$. The two-parameter Poisson-Dirichlet family of random discrete distributions derived from a
stable subordinator, and corresponding exchangeable random partitions of $N$ governed by a generalization of the Ewens sampling
formula, are applied to describe transition mechanisms for processes of coalescence and fragmentation, including the $U$-coalescent
and its time reversal.**Keyword note:**Pitman__Jim**Report ID:**495**Relevance:**100

**Title:**The SDE solved by local times of a Brownian excursion or bridge derived from the height profile of a random tree or forest**Author(s):**Pitman, Jim; **Date issued:**Dec 1997

http://nma.berkeley.edu/ark:/28722/bk0000n3h0s (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n3h1b (PostScript) **Abstract:**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 $(L_(t,v)(|B|), v \ge 0 )$ conditionally given
$B_t = 0$ and $L_(t,0) (|B|)= \ell$ is shown to be that of the unique strong solution $X$ of the \Ito\ SDE $$ dX_v = \left\(
4 - X_v^2 \left( t - \mbox($\int_0^v X_u du$) \right) ^(-1) \right\) \, dv + 2 \sqrt(X_v) d B_v$$ on the interval $[0,V_t
(X))$,where $V_t(X):= \inf \( v: \int_0^v X_u du = t \)$and $X_v = 0$ for all $v \ge V_t(X)$. This conditioned form of the
Ray-Knight description of Brownian local times arises from study of the asymptotic distribution as $n \te \infty$ and $ 2
k/\sqrt(n) \te \ell$ of the height profile of a uniform rooted random forest of $k$ trees labeled by a set of $n$ elements,
as obtained by conditioning a uniform random mapping of the set to itself to have $k$ cyclic points. The SDE is the continuous
analog of a simple description of a Galton-Watson branching process conditioned on its total progeny. A result is obtained
regarding the weak convergence of normalizations of such conditioned Galton-Watson processes and height profiles of random
forests to a solution of the SDE. For $\ell = 0$, corresponding to asymptotics of a uniform random tree, the SDE gives a
new description of the process of local times of a Brownian excursion, implying Jeulin's description of these local times
as a time change of twice a Brownian excursion. Another corollary is the Biane-Yor description of the local times of a reflecting
Brownian bridge as a time changed reversal of twice a Brownian meander of the same length.**Pub info:**Ann. Prob. 27, 261-283 (1999)**Keyword note:**Pitman__Jim**Report ID:**503**Relevance:**100

**Title:**Enumerations of trees and forests related to branching processes and random walks**Author(s):**Pitman, Jim; **Date issued:**May 1997

http://nma.berkeley.edu/ark:/28722/bk0000n3p6p (PDF)

http://nma.berkeley.edu/ark:/28722/bk0000n3p77 (PostScript) **Abstract:**In a Galton-Watson branching process with offspring distribution $(p_0, p_1, \ldots )$ started with $k$ individuals, the
distribution of the total progeny is identical to the distribution of the first passage time to $-k$ for a random walk started
at 0 which takes steps of size $j$ with probability $p_(j+1)$ for $j \ge -1$. The formula for this distribution is a probabilistic
expression of the Lagrange inversion formula for the coefficients in the power series expansion of $f(z)^k$ in terms of those
of $g(z)$ for $f(z)$ defined implicitly by $f(z) = z g(f(z))$. The Lagrange inversion formula is the analytic counterpart
of various enumerations of trees and forests which generalize Cayley's formula $k n^(n-k-1)$ for the number of rooted forests
labeled by a set of size $n$ whose set of roots is a particular subset of size $k$. These known results are derived by elementary
combinatorial methods without appeal to the Lagrange formula, which is then obtained as a byproduct. This approach unifies
and extends a number of known identities involving the distributions of various kinds of random trees and random forests.**Pub info:**In Microsurveys in Discrete Probability edited by D. Aldous and J. Propp. DIMACS Ser. Discrete Math. Theoret. Comput. Sci..**Keyword note:**Pitman__Jim**Report ID:**482**Relevance:**100