Ordering Functions of Random Vectors, with Application to Partial Sums

Abstract

It is known that the sums of the components of two random vectors (X1,X2,...,Xn) and (Y1,Y2,...,Yn) ordered in the multivariate (s1,s2,...,sn)-increasing convex order are ordered in the univariate (s1+s2+...+sn)-increasing convex order. More generally, real-valued functions of (X1,X2,...,Xn) and (Y1,Y2,...,Yn) are ordered in the same sense as long as these functions possess some specified non-negative cross derivatives. This note extends these results to multivariate functions. In particular, we consider vectors of partial sums (S1,S2,...,Sn) and (T1,T2,...,Tn) where Sj = X1+...+Xj and Tj = Y1 +...+Yj and we show that these random vectors are ordered in the multivariate (s1,s1+s2,...,s1+...+sn)-increasing convex order. The consequences of these general results for the upper orthant order and the orthant convex order are discussed.