Ordering Functions of Random Vectors, with Application to Partial Sums
(2013) Journal of Theoretical Probability — Vol. 26, n° 2, p. 474-479 (2013)
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- 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.
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Denuit, M., & Mesfioui, M. (2013). Ordering Functions of Random Vectors, with Application to Partial Sums. Journal of Theoretical Probability, 26(2), 474-479. https://doi.org/10.1007/s10959-012-0402-y (Original work published 2013)