Exchangeable random measures

Austin, Tim

Annales de l'I.H.P. Probabilités et statistiques, Tome 51 (2015) no. 3, pp. 842-861.

Soit $A$ un espace de Borel standard, et soit $A^{{\mathbb{N}}^{\left(k\right)}}$ l’ensemble des tableaux à valeurs dans $A$ indexés par les sous-ensembles de $\mathbb{N}$ de taille $k$. On s’intéresse aux mesures aléatoires sur un tel espace dont la loi est invariante par l’action naturelle des permutations de $\mathbb{N}$. Le résultat principal est une représentation de ces mesures aléatoires « échangeables », obtenue à partir des théorèmes de représentation classiques de de Finetti, Hoover, Aldous et Kallenberg pour des tableaux échangeables. Après avoir prouvé cette représentation, on en donne deux applications. La première est une nouvelle courte preuve du théorème de représentation de Dovbysh–Sudakov pour des matrices définies semi-positives échangeables. La seconde concerne la formulation d’une classe naturelle d’objets limites pour des modèles de champ moyen dilués pour des verres de spins qui capture plus d’information que la seule matrice limite de Gram–de Finetti qui est notamment utilisée dans l’étude du modèle de Sherrington–Kirkpatrick.

Let $A$ be a standard Borel space, and consider the space $A^{{\mathbb{N}}^{\left(k\right)}}$ of $A$-valued arrays indexed by all size-$k$ subsets of $\mathbb{N}$. This paper concerns random measures on such a space whose laws are invariant under the natural action of permutations of $\mathbb{N}$. 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.

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