On the asymptotic distribution of certain bivariate reinsurance treaties

Let {(Xn,Yn),n≥1} be bivariate random claim sizes with common distribution function F and let {N(t),t≥0} be a stochastic process which counts the number of claims that occur in the time interval [0,t],t≥0. In this paper we derive the joint asymptotic distribution of randomly indexed order statistics of the random sample (X1,Y1),(X2,Y2),…,(XN(t),YN(t)) which is then used to obtain asymptotic representations for the joint distribution of two generalised largest claims reinsurance treaties available under specific insurance settings. As a by-product we obtain a stochastic representation of a m-dimensional Λ-extremal variate in terms of iid unit exponential random variables.