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 ( X i , Y i ) , i = 1 , 2 , … , n , be independent pairs of random assets. It is shown that if X i is less than Y i in the location independent risk order for each i and the X i and Y i have log-concave density or probability functions, or if X i is less than Y i in the dispersive order and the X i and Y i have log-concave distribution functions, then ∑ i = 1 n X i is less than ∑ i = 1 n Y i in the location independent risk order. Similar results also hold for the excess wealth order.