The stable trees are nested

Haas, Bénédicte; Curien, Nicolas

Haas, Bénédicte; Curien, Nicolas (2013), The stable trees are nested, Probability Theory and Related Fields, 157, 3-4, p. 847-883. http://dx.doi.org/10.1007/s00440-012-0472-x

Type

Article accepté pour publication ou publié

External document link

http://fr.arxiv.org/abs/1207.5418

Date

2013

Journal name

Probability Theory and Related Fields

Volume

157

Number

3-4

Publisher

Springer

Pages

847-883

Abstract (EN)

We show that we can construct simultaneously all the stable trees as a nested family. More precisely, if $1 < a < a' \leq 2$ we prove that hidden inside any a-stable tree 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.

Subjects / Keywords

Marchal's algorithm; stable Levy trees; dissipative self-similar fragmentations; pruning