On the number of near-maximum insurance claims

A near-maximum claim size is one falling within a fixed distance a of the current record claim size. In this paper we study the statistic Kt(a) counting the number of near-maximum insurance claims in the time period [0,t]. Some distributional results of Kt(a) are considered, as well as asymptotic behaviours as t→∞. Subject to a limitation on the tail thickness of the claim-size law, these results provide an estimate of the long-term total value of near-maximum insurance claims.