On the asymptotic distribution of certain bivariate reinsurance treaties

Abstract: 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 Λextremal variate in terms of iid unit exponential random variables.