Preservation of the location independent risk order under convolution

The location independent risk order has been used to compare different random assets in risk analysis without the requirement of equal means. Let (Xi,Yi),i=1,2,…,n, be independent pairs of random assets. It is shown that if Xi is less than Yi in the location independent risk order for each i and the Xi and Yi have log-concave density or probability functions, or if Xi is less than Yi in the dispersive order and the Xi and Yi have log-concave distribution functions, then ∑i=1nXi is less than ∑i=1nYi in the location independent risk order. Similar results also hold for the excess wealth order.