A cluster-limit theorem for infinitely divisible point processes

In this article, we consider a sequence (N n ) n≥1 of point processes, whose points lie in a subset E of ℝ \{0} and satisfy an asymptotic independence condition. Our main result gives some necessary and sufficient conditions for the convergence in the distribution of (N n ) n≥1 to an infinitely divisible point process N. As applications, we discuss the exceedance processes and point processes based on regularly varying sequences.