A recursive distribution equation for the stable tree

2018/12/20 by Chee, Nicholas, Rembart, Franz, Winkel, Matthias
#60 J 05 #60 J 80 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.1812.08636

Abstract

We provide a new characterisation of Duquesne and Le Gall's α-stable tree, α∈(1,2], as the solution of a recursive distribution equation (RDE) of the form T\oversetd=g(ξ,T_i, i≥0), where g is a concatenation operator, ξ= (ξ_i, i≥ 0) a sequence of scaling factors, T_i, i ≥ 0, and T are i.i.d. trees independent of ξ. This generalises a version of the well-known characterisation of the Brownian Continuum Random Tree due to Aldous, Albenque and Goldschmidt. By relating to previous results on a rather different class of RDE, we explore the present RDE and obtain for a large class of similar RDEs that the fixpoint is unique (up to multiplication by a constant) and attractive.

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