The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin's local time identity

David Aldous (1) , Grégory Miermont (2, 3) , Jim Pitman (1)

1 Department of Statistics [Berkeley]
2 LPMA - Laboratoire de Probabilités et Modèles Aléatoires
3 DMA - Département de Mathématiques et Applications - ENS Paris

We study the inhomogeneous continuum random trees (ICRT) that arise as weak limits of birthday trees. We give a description of the exploration process, a function defined on [0,1] that encodes the structure of an ICRT, and also of its width process, determining the size of layers in order of height. These processes turn out to be transformations of bridges with exchangeable increments, which have already appeared in other ICRT related topics such as stochastic additive coalescence.The results rely on two different constructions of birthday trees from processes with exchangeable increments, on weak convergence arguments, and on general theory on continuum random trees.

Mots clés: Continuum random tree; exchangeable increments; exploration process; Levy process; weak convergence

Domaines: Probabilités [math.PR]

David Aldous, Grégory Miermont, Jim Pitman. The exploration process of inhomogeneous continuum random trees, and an extension of Jeulin's local time identity. Probability Theory and Related Fields, 2004, 129, pp.182-218. ⟨10.1007/s00440-003-0334-7⟩. ⟨hal-00001253⟩