Jim Pitman

About

Jim Pitman is a Professor of the Graduate School in the Department of Statistics at the University of California, Berkeley. His research interests encompass a broad range of topics within probability, including fragmentation, statistics, mathematics, Brownian motion, distribution theory, path transformations, stochastic processes, local time, excursions, random trees, random partitions, and processes of coalescence. He has been interested in interfaces between the traditional theory of stochastic processes and other areas of mathematics, especially combinatorics. His work involves studying various random combinatorial objects such as permutations, partitions, and trees, and analyzing their asymptotic behavior over large numbers of elements in probabilistic terms, often involving Brownian motion and related processes. Pitman's research has led to the development of measure-valued and partition-valued Markov processes, with behavior understood through combinatorial constructions involving random trees. Currently, he is engaged in developing ideas related to random partitions, random trees, irreversible processes of coalescence, and their time reversals, which serve as models for random splitting or fragmentation. His approach views this line of research as pure mathematics that is often motivated and influenced by applications, recognizing that stochastic models with natural probabilistic structures frequently appear in diverse fields. The mathematical structure of these…

Research topics

• Mathematics
• Combinatorics
• Statistical physics
• Mathematical analysis
• Discrete mathematics

Selected publications

• Extreme order statistics of random walks

  Annales de l Institut Henri Poincaré Probabilités et Statistiques · 2023-01-16 · 15 citations

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  Cet article traite de la théorie limite des statistiques d’ordre extrêmes provenant des marches aléatoires. Nous établissons la convergence conjointe des statistiques d’ordre près du minimum d’une marche aléatoire en termes des chaînes de Feller. Des descriptions détaillées du processus limite sont données dans le cas de marches simples symétriques et des marches gaussiennes.

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• Hidden symmetries and limit laws in the extreme order statistics of the Laplace random walk

  The Annals of Probability · 2022-05-12 · 15 citations

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  This paper is concerned with the limit laws of the extreme order statistics derived from a symmetric Laplace walk. We provide two different descriptions of the point process of the limiting extreme order statistics: a branching representation and a squared Bessel representation. These complementary descriptions expose various hidden symmetries in branching processes and Brownian motion which lie behind some striking formulas found by Schehr and Majumdar (Phys. Rev. Lett. 108 (2012) 040601). In p…

  PublisherDOI

• Squared Bessel processes of positive and negative dimension embedded in\n Brownian local times

  Oxford University Research Archive (ORA) (University of Oxford) · 2018-01-01 · 14 citations

  articleOpen access1st authorCorresponding

  The Ray--Knight theorems show that the local time processes of various path fragments derived from a one-dimensional Brownian motion $B$ are squared Bessel processes of dimensions $0$, $2$, and $4$. It is also known that for various singular perturbations $X= |B| + \\mu \\ell$ of a reflecting Brownian motion $|B|$ by a multiple $\\mu$ of its local time process $\\ell$ at $0$, corresponding local time processes of $X$ are squared Bessel with other real dimension parameters, both positive and nega…

  Publisher

• A representation of exchangeable hierarchies by sampling from random real trees

  Probability Theory and Related Fields · 2017-11-08 · 12 citations

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• Random weighted averages, partition structures and generalized arcsine laws

  arXiv (Cornell University) · 2018-04-21 · 11 citations

  preprintOpen access1st authorCorresponding

  This article offers a simplified approach to the distribution theory of randomly weighted averages or $P$-means $M_P(X):= \sum_{j} X_j P_j$, for a sequence of i.i.d.random variables $X, X_1, X_2, \ldots$, and independent random weights $P:= (P_j)$ with $P_j \ge 0$ and $\sum_{j} P_j = 1$. The collection of distributions of $M_P(X)$, indexed by distributions of $X$, is shown to encode Kingman's partition structure derived from $P$. For instance, if $X_p$ has Bernoulli$(p)$ distribution on $\{0,1\}…

  PublisherOA PDFDOI

Recent grants

• CDI-Type II: Collaborative Research: Bibliographic Knowledge Network

  NSF · $167k · 2008–2011

• Combinatorial Stochastic Processes

  NSF · $240k · 2008–2013

• Brownian Motion and Combinatorial Stochastic Processes

  NSF · $315k · 2004–2008

Frequent coauthors

• Marc Yor

  42 shared

• Alexander Gnedin

  25 shared

• Matthias Winkel

  University of Oxford

  24 shared

• Wenpin Tang

  15 shared

• David Aldous

  13 shared

• Yu. V. Yakubovich

  11 shared

• Grégory Miermont

  Unité de Mathématiques Pures et Appliquées

  10 shared

• Marc Yor

  9 shared

Education

• Ph.D., Probability and Statistics

  University of Sheffield

  1974

Similar researchers at University of California, Berkeley

• Michael I. Jordanh-175
• Jennifer Chayesh-74
• Matei Zahariah-64