An ergodic theorem for the extremal process of branching Brownian motion
  In a previous paper, the authors proved a conjecture of Lalley and Sellke that the empirical (time-averaged) distribution function of the maximum of branching Brownian motion converges almost surely to a Gumbel distribution. The result is extended here to the entire system of particles that are extremal, i.e. close to the maximum. Namely, it is proved that the distribution of extremal particles under time-average converges to a Poisson cluster process. 
Louis-Pierre Arguin, Anton Bovier, Nicola Kistler
math.PR