2012/07/23 by Nicolas Curien, Curien, Nicolas, Bénédicte Haas +1
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR
paper · pdf · doi:10.48550/arxiv.1207.5418
arxiv created 2012/07/23 · arxiv updated 2012/07/24
We show that we can construct simultaneously all the stable trees as a nested family. More precisely, if 1 < a < a' ≤ 2 we prove that hidden inside any a-stable we can find a version of an a'-stable tree rescaled by an independent Mittag-Leffler type distribution. This tree can be explicitly constructed by a pruning procedure of the underlying stable tree or by a modification of the fragmentation associated with it. Our proofs are based on a recursive construction due to Marchal which is proved to converge almost surely towards a stable tree.