The falling apart of the tagged fragment and the asymptotic disintegration of the brownian height fragmentation
Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, p. 1130-1149
Une étude additionnelle de la fragmentation de hauteur brownienne est présentée. Plus précisément, une représentation de la masse du fragment marqué en termes d'une transformation de Doob du subordinateur stable d'indice 1/2 est décrite puis utilisée pour étudier les sauts du processus de masse; ceci nous renseigne sur la façon dans laquelle un fragment typique se casse. Ces résultats se généralisent au cadre des fragmentations de hauteur de l'arbre stable. Enfin, nous donnons un théorème limite de la fragmentation de l'excursion Brownienne par les hauteurs, centrée autour du dernier fragment qui se décompose en poussière.
We present a further analysis of the fragmentation at heights of the normalized brownian excursion. Specifically we study a representation for the mass of a tagged fragment in terms of a Doob transformation of the 1/2-stable subordinator and use it to study its jumps; this accounts for a description of how a typical fragment falls apart. These results carry over to the height fragmentation of the stable tree. Additionally, the sizes of the fragments in the brownian height fragmentation when it is about to reduce to dust are described in a limit theorem.
@article{AIHPB_2009__45_4_1130_0,
     author = {Uribe Bravo, Ger\'onimo},
     title = {The falling apart of the tagged fragment and the asymptotic disintegration of the brownian height fragmentation},
     journal = {Annales de l'I.H.P. Probabilit\'es et statistiques},
     publisher = {Gauthier-Villars},
     volume = {45},
     number = {4},
     year = {2009},
     pages = {1130-1149},
     doi = {10.1214/08-AIHP304},
     zbl = {1208.60036},
     mrnumber = {2572168},
     language = {en},
     url = {http://www.numdam.org/item/AIHPB_2009__45_4_1130_0}
}
Uribe Bravo, Gerónimo. The falling apart of the tagged fragment and the asymptotic disintegration of the brownian height fragmentation. Annales de l'I.H.P. Probabilités et statistiques, Tome 45 (2009) no. 4, pp. 1130-1149. doi : 10.1214/08-AIHP304. http://www.numdam.org/item/AIHPB_2009__45_4_1130_0/

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