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Term(s):Pitman Jim
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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

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