A Cluster Limit Theorem for Infinitely Divisible Point Processes

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