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hal-00564630, version 1

## A characterisation of superposable random measures

Pascal Maillard () 1

(2011-02-09)

Abstract: Let $Z$ be a point process on $\R$ and $T_\alpha Z$ its translation by $\alpha\in\R$. Let $Z'$ be an independent copy of $Z$. We say that $Z$ is \emph{superposable}, if $T_\alpha Z + T_\beta Z'$ and $Z$ are equal in law for every $\alpha,\beta\in\R$, such that $\e^\alpha + \e^\beta = 1.$ We prove a characterisation of superposable point processes in terms of decorated Poisson processes, which was conjectured by Brunet and Derrida [A branching random walk seen from the tip, 2010, \url{http://arxiv.org/abs/1011.4864v1}]. We further prove a generalisation to random measures.

• 1:  Laboratoire de Probabilités et Modèles Aléatoires (LPMA)
• CNRS : UMR7599 – Université Pierre et Marie Curie [UPMC] - Paris VI – Université Paris VII - Paris Diderot
• Domain : Mathematics/Probability
• Keywords : Superposable random measures – superposable point processes – infinitely divisible random measures – branching Brownian motion
• Comment : 13 pages
• Available versions :  v1 (2011-02-09) v2 (2013-01-20)

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• Submitted on: Wednesday, 9 February 2011 14:48:46
• Updated on: Wednesday, 9 February 2011 16:49:26