  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.
