A characterisation of superposable random measures
Résumé
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.
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