Exchangeable random measures1

Tim Austin

Research output: Contribution to journalArticle

Abstract

Let A be a standard Borel space, and consider the space Aℕ<sup>(k)</sup> of A-valued arrays indexed by all size-k subsets of ℕ. This paper concerns random measures on such a space whose laws are invariant under the natural action of permutations of ℕ. The main result is a representation theorem for such "exchangeable" random measures, obtained using the classical representation theorems for exchangeable arrays due to de Finetti, Hoover, Aldous and Kallenberg. After proving this representation, two applications of exchangeable random measures are given. The first is a short new proof of the Dovbysh-Sudakov Representation Theorem for exchangeable positive semi-definite matrices. The second is in the formulation of a natural class of limit objects for dilute mean-field spin glass models, retaining more information than just the limiting Gram-de Finetti matrix used in the study of the Sherrington-Kirkpatrick model.

Original languageEnglish (US)
Pages (from-to)842-861
Number of pages20
JournalAnnales de l'institut Henri Poincare (B) Probability and Statistics
Volume51
Issue number3
DOIs
StatePublished - Aug 1 2015

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Random Measure
Representation Theorem
Positive Semidefinite Matrix
Spin Glass
Mean Field
Permutation
Limiting
Subset
Invariant
Representation theorem
Formulation
Model

Keywords

  • Dilute spin glass models
  • Dovbysh-Sudakov Theorem
  • Exchangeability
  • Random measures

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Exchangeable random measures1. / Austin, Tim.

In: Annales de l'institut Henri Poincare (B) Probability and Statistics, Vol. 51, No. 3, 01.08.2015, p. 842-861.

Research output: Contribution to journalArticle

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