On the correlation order

Recently a new multiple dimensional stochastic order named correlation order was proposed to examine how the dependence among the individual risks effects riskness of the portfolios. This paper aims to discuss the relationship between correlation order and the supermodular order in general. Moreover, this paper also makes a comparison in the closeness of distributions of order statistics of random vectors ordered by correlation order. Such a comparison leads to a result more general than that in Hu and Hu [1997. Statist. Probab. Lett. 37, 1-6].