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A note on stable point processes occurring in branching Brownian motion. (English) Zbl 1320.60116
Summary: 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 $$T_\alpha Z+T_\beta Z^{\prime}$$, 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 walks and several authors have proved 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 $$(\xi_i)_{i\geq1}$$ are the atoms of a Poisson process of intensity $$e^{-x}\,\text{d}x$$ on $$\mathbb R$$ and $$(D_i)_{i\geq 1}$$ are independent copies of $$D$$ and independent of $$(\xi_i)_{i\geq1}$$. 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 $$\mathbb R$$.

##### MSC:
 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G52 Stable stochastic processes 60J80 Branching processes (Galton-Watson, birth-and-death, etc.) 60J65 Brownian motion 60G50 Sums of independent random variables; random walks 60G57 Random measures
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