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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)
 
  • hal-00564630, version 1
  • http://hal.archives-ouvertes.fr/hal-00564630
  • oai:hal.archives-ouvertes.fr:hal-00564630
  • From: Pascal Maillard <>
  • Submitted on: Wednesday, 9 February 2011 14:48:46
  • Updated on: Wednesday, 9 February 2011 16:49:26