SUPERMODULAR ORDERING OF POISSON AND BINOMIAL RANDOM
VECTORS BY TREE-BASED CORRELATIONS
Bünyamin Kızıldemir
Nicolas Privault
Abstract: We construct a dependence structure for binomial, Poisson and Gaussian random
vectors, based on partially ordered binary trees and sums of independent random variables.
Using this construction, we characterize the supermodular ordering of such random vectors
via the componentwise ordering of their covariance matrices. For this, we apply Möbius
inversion techniques on partially ordered trees, which allow us to connect the Lévy measures
of Poisson random vectors on the discrete -dimensional hypercube to their covariance
matrices.
2010 AMS Mathematics Subject Classification: Primary: 60E15; Secondary: 62H20,
05C05, 06A11, 60E07.
Keywords and phrases: Stochastic ordering, supermodular functions, Möbius
transform, Möbius inversion, binary trees, Poisson random vectors, binomial random
vectors.