Asymptotic correlation structure of discounted Incurred But Not Reported claims under fractional Poisson arrival process

•The Incurred But Not Reported claims in the insurance context are considered.•The number of such claims is also the number of customers in infinite server queue.•Insurance claims are assumed to arrive according to a fractional Poisson process.•The reporting delays are assumed to follow exponential or Pareto distribution.•Asymptotic correlation structures and implication in long-range dependence are given. This paper studies the joint moments of a compound discounted renewal process observed at different times with each arrival removed from the system after a random delay. This process can be used to describe the aggregate (discounted) Incurred But Not Reported claims in insurance and also the total number of customers in an infinite server queue. It is shown that the joint moments can be obtained recursively in terms of the renewal density, from which the covariance and correlation structures are derived. In particular, the fractional Poisson process defined via the renewal approach is also considered. Furthermore, the asymptotic behaviour of covariance and correlation coefficient of the aforementioned quantities is analyzed as the time horizon goes to infinity. Special attention is paid to the cases of exponential and Pareto delays. Some numerical examples in relation to our theoretical results are also presented.