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.