Asymptotic properties of the ratio of order statistics

Let X1,n⩽X2,n⩽⋯⩽Xn,nX1,n⩽X2,n⩽⋯⩽Xn,n be the order statistics obtained from a sample from a continuous distribution and Zi,j,n=Xj,n/Xi,nZi,j,n=Xj,n/Xi,n be the ratio of order statistics. In this paper, asymptotic properties of Zi,j,nZi,j,n are discussed. Specifically, we make use of the theory of regular variation to derive weak and strong limit results for Zi,j,nZi,j,n.