The Aggregate Discounted Claims Process Under Multiple and Terminable Arrivals Renewal Processes

We study the aggregate discounted claims process when the claim arrivals follow a renewal process, allowing for multiple arrivals or terminable arrivals. Under both cases, their Laplace transforms are derived, which consequently give the moments formulas up to any orders. Then distributions of the aggregate discounted claims process under particular arrival processes and claim amount distributions are discussed, with closed form formulas of the defective density functions and numerical illustrations. We also prove that the central limit theorem works for multiple arrivals but not for the terminable arrivals. Finally, the connection between multiple or terminable arrivals and mixture of exponentials distributions are discussed.