A Cluster Limit Theorem for Infinitely Divisible Point Processes

TL;DR

This paper establishes conditions under which sequences of point processes converge to infinitely divisible processes, with applications to exceedance processes and regularly varying sequences.

Contribution

It provides necessary and sufficient conditions for the convergence of point processes to infinitely divisible limits under asymptotic independence.

Findings

Derived convergence criteria for point processes

Applied results to exceedance processes

Analyzed processes based on regularly varying sequences

Abstract

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.