Study of a risk model based on the entrance process

In this paper, we construct a new insurance risk model based on the entrance process and consider the asymptotic behavior of the risk process under the ultimate stability condition on the intensity of the entrance process. We find when the claim size has finite second moment the risk process is asymptotically normal and when the moment condition is dropped the risk process is asymptotically α-stable under additional restriction on the tail probability of the claim size. Our result provides a case in which the weak convergence property of the random sum of dependent and differently distributioned r.v.'s can be tackled.