Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees

Chen, Bo; Winkel, Matthias

Annales de l'I.H.P. Probabilités et statistiques, Tome 49 (2013) no. 3, pp. 839-872.

Nous introduisons la notion d’une partition restreinte échangeable de $\mathbb{N}$. Nous obtenons des représentations intégrales, nous considérons les fragmentations associées, des plongements dans des arbres aléatoires continus et la convergence vers de tels arbres limites. En particulier, nous déduisons de la théorie générale développée içi un résultat limite formulé en conjecture dans un travail précédent. Ce résultat particulier concerne les arbres alpha de Ford et leurs généralisations, les arbres alpha-gamma, deux exemples où l’échangeabilité restreinte arrive de manière naturelle.

We introduce the notion of a restricted exchangeable partition of $\mathbb{N}$. We obtain integral representations, consider associated fragmentations, embeddings into continuum random trees and convergence to such limit trees. In particular, we deduce from the general theory developed here a limit result conjectured previously for Ford’s alpha model and its extension, the alpha-gamma model, where restricted exchangeability arises naturally.

Classification : 60G09, 60J80
Mots clés : exchangeability, hierarchy, coalescent, fragmentation, continuum random tree, renewal theory

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