Large deviations of Markov chains indexed by random trees

Abstract

Given a finite typed rooted tree T with n vertices, the empirical subtree measure is the uniform measure on the n typed subtrees of T formed by taking all descendants of a single vertex. We prove a large deviation principle in n, with explicit rate function, for the empirical subtree measures of multitype Galton–Watson trees conditioned to have exactly n vertices. In the process, we extend the notions of shift-invariance and specific relative entropy—as typically understood for Markov fields on deterministic graphs such as Zd—to Markov fields on random trees. We also develop single-generation empirical measure large deviation principles for a more general class of random trees including trees sampled uniformly from the set of all trees with n vertices. Etant donné un arbre enraciné T à n sommets, on appelle mesure empirique des sous arbres la mesure uniforme sur les sous arbres obtenus en prenant les descendants des n sommets. On demande notamment un principe explicite de grandes déviations en n pour la mesure empirique des sous arbres des arbres de Galton–Watson multitypes conditionnés à avoir n sommets.

Locations

• Annales de l Institut Henri Poincaré Probabilités et Statistiques - View
• French digital mathematics library (Numdam) - View - PDF