# On total claim amount for marked Poisson cluster models

## TL;DR

This paper investigates the asymptotic behavior of total claim amounts in marked Poisson cluster models, establishing conditions for normal or stable limit distributions, with a focus on marked Hawkes processes.

## Contribution

It provides new theoretical results on the limit distributions of total claim amounts, including conditions for CLT and stable laws, especially for marked Hawkes processes.

## Key findings

- Conditions for CLT and stable law convergence identified.
- Detailed analysis of marked Hawkes processes.
- Application to insurance claim modeling.

## Abstract

We study the asymptotic distribution of the total claim amount for marked Poisson cluster models. The marks determine the size and other characteristics of the individual claims and potentially influence arrival rate of the future claims. We find sufficient conditions under which the total claim amount satisfies the central limit theorem or alternatively tends in distribution to an infinite variance stable random variable. We discuss several Poisson cluster models in detail, paying special attention to the marked Hawkes processes as our key example.