Bivariate maximum insurance claim and related point processes

Abstract

Let X 1, X 2,… be independent bivariate claim sizes arising from an insurance portfolio. The number of claims occurring in the time interval [0, t] is denoted by N( t). We investigate in this paper distributional and asymptotic properties of the following point process: C t(B)≔ ∑ i=1 N(t) 1 ( X N(t):N(t)− X i∈B), B∈ B([0,∞) 2), t⩾0 with X N(t):N(t) , the bivariate maximum insurance claim occurring during [0, t]. We show that C t(·)/t, C t(·)/N(t) are strongly consistent estimators of a certain tail probability of the claim size distribution. Further, we investigate the connection between convergence in distribution of the bivariate maximum claim size and weak convergence of C t(·) . As a byproduct, a result for the ECOMOR reinsurance treaty is obtained.