Abstract
We provide a new characterisation of Duquesne and Le Gall’s α-stable tree, , as the solution of a recursive distributional equation (RDE) of the form , where g is a concatenation operator, a sequence of scaling factors, , , and are i.i.d. trees independent of ξ. This generalises the characterisation of the Brownian Continuum Random Tree proved by Albenque and Goldschmidt, based on self-similarity observed by Aldous. 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.
Acknowledgements
We’d like to thank the referee for valuable suggestions that improved the legibility of the paper.
Citation
Nicholas Chee. Franz Rembart. Matthias Winkel. "A recursive distributional equation for the stable tree." Bernoulli 30 (2) 1029 - 1054, May 2024. https://doi.org/10.3150/23-BEJ1623
Information