On the asymptotic distribution of certain bivariate reinsurance treaties

Abstract

Let { ( X n , Y n ) , 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 ( X 1 , Y 1 ) , ( X 2 , Y 2 ) , … , ( X N ( t ) , Y N ( 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.