### 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 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 | |

State | Published - Aug 1 2015 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Statistics and Probability
- Statistics, Probability and Uncertainty

### Cite this

*Annales de l'institut Henri Poincare (B) Probability and Statistics*,

*51*(3), 842-861. https://doi.org/10.1214/13-AIHP584

**Exchangeable random measures1.** / Austin, Tim.

Research output: Contribution to journal › Article

*Annales de l'institut Henri Poincare (B) Probability and Statistics*, vol. 51, no. 3, pp. 842-861. https://doi.org/10.1214/13-AIHP584

}

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

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 -