A note on stable point processes occurring in branching Brownian motion

Abstract

We call a point process $Z$ on $\mathbb{R}$ \emph{exp-1-stable} if for every $\alpha ,\beta \in \mathbb{R}$ with $e^{\alpha }+e^{\beta }=1$, $Z$ is equal in law to $T_{\alpha }Z+T_{\beta }{Z}^{\prime }$, where $Z^{\prime }$ 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 $({\xi }_i{)}_{i\ge 1}$ 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 $({\xi }_i{)}_{i\ge 1}$. In this note, we show how this decomposition follows from the classic \emph{LePage decomposition} of a (union)-stable point process. Moreover, we give a short proof of it in the general case of random measures on $\mathbb{R}$.

References in corpus (2)

• Stationary max-stable fields associated to negative definite functions
• A branching random walk seen from the tip

Cited by in corpus (2)

• Genealogy of the extremal process of the branching random walk
• On the branching convolution equation $\mathcal{E}=\mathcal{Z}⊛\mathcal{E}$