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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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