A Cluster Limit Theorem for Infinitely Divisible Point Processes

Raluca Balan, Sana Louhichi

Published 2009-11-29, updated 2010-11-16Version 3

In this article, we consider a sequence $(N_n)_{n \geq 1}$ of point processes, whose points lie in a subset $E$ of $\bR \verb2\2 \{0\}$, and satisfy an asymptotic independence condition. Our main result gives some necessary and sufficient conditions for the convergence in distribution of $(N_n)_{n \geq 1}$ to an infinitely divisible point process $N$. As applications, we discuss the exceedance processes and point processes based on regularly varying sequences.