Exchangeable random measures1

Tim Austin

doi:10.1214/13-AIHP584

Exchangeable random measures1

Tim Austin

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

Let A be a standard Borel space, and consider the space Aℕ^(k) 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 language	English (US)
Pages (from-to)	842-861
Number of pages	20
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	51
Issue number	3
DOIs	https://doi.org/10.1214/13-AIHP584
State	Published - Aug 1 2015

Keywords

Dilute spin glass models
Dovbysh-Sudakov Theorem
Exchangeability
Random measures

ASJC Scopus subject areas

Statistics and Probability
Statistics, Probability and Uncertainty

Access to Document

10.1214/13-AIHP584

Fingerprint Dive into the research topics of 'Exchangeable random measures1'. Together they form a unique fingerprint.

Cite this

@article{82a94b86b67c4ed7986d95f4a9353885,

title = "Exchangeable random measures1",

abstract = "Let A be a standard Borel space, and consider the space Aℕ(k) 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.",

keywords = "Dilute spin glass models, Dovbysh-Sudakov Theorem, Exchangeability, Random measures",

author = "Tim Austin",

note = "Publisher Copyright: {\textcopyright} 2015 Association des Publications de l'Institut Henri Poincar{\'e}.",

year = "2015",

month = aug,

day = "1",

doi = "10.1214/13-AIHP584",

language = "English (US)",

volume = "51",

pages = "842--861",

journal = "Annales de l'institut Henri Poincare (B) Probability and Statistics",

issn = "0246-0203",

publisher = "Institute of Mathematical Statistics",

number = "3",

}

TY - JOUR

T1 - Exchangeable random measures1

AU - Austin, Tim

PY - 2015/8/1

Y1 - 2015/8/1

N2 - Let A be a standard Borel space, and consider the space Aℕ(k) 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.

AB - Let A be a standard Borel space, and consider the space Aℕ(k) 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.

KW - Dilute spin glass models

KW - Dovbysh-Sudakov Theorem

KW - Exchangeability

KW - Random measures

UR - http://www.scopus.com/inward/record.url?scp=84936931360&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84936931360&partnerID=8YFLogxK

U2 - 10.1214/13-AIHP584

DO - 10.1214/13-AIHP584

M3 - Article

AN - SCOPUS:84936931360

VL - 51

SP - 842

EP - 861

JO - Annales de l'institut Henri Poincare (B) Probability and Statistics

JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

SN - 0246-0203

IS - 3

ER -

Exchangeable random measures1

Abstract

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Cite this