The stable trees are nested

Abstract

We show that we can construct simultaneously all the stable trees as a nested family. More precisely, if \(1 < \alpha < \alpha ^{\prime } \le 2\) we prove that hidden inside any \(\alpha \)-stable tree we can find a version of an \(\alpha ^{\prime }\)-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.