Several New Proofs Of The Nature Of Mutual Exclusion

Mutual exclusivity was first put forward and studied by Dhaene and Denuit (1999). It is an extreme negative dependence structure. Following that, generalized mutual ex-clusivity and characterizations were recapitulated in Cheung and Lo (2014), including the minimal convex sum property, the distributional representation and the character-istic function of the sum. Among those, the property of pairwise counter-monotonic of mutually exclusive random vectors plays a significant role in generalizing counter-monotonic as the strongest negative dependence structure in multidimensional setting. In this paper, we first restate some notions and lemmas mentioned in the above two articles. We will give the most crucial content of this thesis that some new proofs concerning properties of mutually exclusive random vectors and some new equivalen-t conditions for a random vector which is mutually exclusive based on the paper of Cheung and Lo (2014).The paper is organized as follows. In Section 2, we first give the notions about Frechet lower bound, stochastic order, comonotonicity, counter-monotonicity, the cor-relation coefficient, the monotone coefficient and several lemmas used later. In Section 3, we revisit the properties of mutually exclusive and generalized sense concerned by Dhaene and Denuit (1999), Cheung and Lo (2014) respectively. And we transfor-m the theorem 4.1 of Cheung et al.(2014). We use four methods to prove that the equivalent condition between mutually exclusive and the minimal convex sum prop-erty in Section 4. In Section 5, we give the new proof method about some results of Cheung and Lo (2014). We present the relation between mutually exclusive and the correlation coefficient rp, the monotone coefficient pm. The two sections are the core content of this paper. Finally, we prove the conclusion which used in the thesis, that is, X1*+X2*+…+Xn*(?)X1M+X2M…+XnM(?)X1*+X2*+…+Xn-1*(?) X1M+X2M+…Xn-1M.