@article{ECP2390,
author = {Pascal Maillard},
title = {A note on stable point processes occurring in branching Brownian motion},
journal = {Electron. Commun. Probab.},
fjournal = {Electronic Communications in Probability},
volume = {18},
year = {2013},
keywords = {stable distribution ; point process ; random measure ; branching Brownian motion ; branching random walk},
abstract = {We call a point process $Z$ on $\mathbb R$ exp-1-stable if for every $\alpha,\beta\in\mathbb R$ with $e^\alpha+e^\beta=1$, $Z$ is equal in law to $T_\alpha Z+T_\beta Z'$, where $Z'$ is an independent copy of $Z$ and $T_x$ is the translation by $x$. Such processes appear in the study of the extremal particles of branching Brownian motion and branching random walk and several authors have proven in that setting the existence of a point process $D$ on $\mathbb R$ such that $Z$ is equal in law to $\sum_{i=1}^\infty T_{\xi_i} D_i$, where $(\xi_i)_{i\ge1}$ are the atoms of a Poisson process of intensity $e^{-x}\,\mathrm d x$ on $\mathbb R$ and $(D_i)_{i\ge 1}$ are independent copies of $D$ and independent of $(\xi_i)_{i\ge1}$. In this note, we show how this decomposition follows from the classic LePage decomposition of a (union)-stable point process. Moreover, we give a short proof of it in the general case of random measures on $\mathbb R$.},
pages = {no. 5, 1-9},
issn = {1083-589X},
doi = {10.1214/ECP.v18-2390},
url = {http://ecp.ejpecp.org/article/view/2390}}