A note on stable point processes occurring in branching Brownian motion

We call a point process Z on R exp-1-stable if for every α, β ∈ R with e + e = 1, Z is equal in law to TαZ + TβZ , where Z ′ is an independent copy of Z and Tx is the translation by x. Such processes appear in the study of the extremal particles of branching Brownian motion and branching random walk and several authors have proven in that setting the existence of a point process D on R such that Z is equal in law to ∑∞ i=1 TξiDi, where (ξi)i≥1 are the atoms of a Poisson process of intensity e dx on R and (Di)i≥1 are independent copies of D and independent of (ξi)i≥1. 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 R.