Setwise independence for some dependence structures

Characterization problems of setwise independence are considered by introducing the setwise dependence class SMk(n). Two questions are answered: 1. (i) Under which conditions is a strongly positive orthant dependent random vector setwise independent.2. (ii) Under what kind of dependence structure does a block diagonal covariance matrix imply setwise independence. The techniques involved in dealing with the above questions allow us to answer some open problems involving the SPOD class raised in Newman (1984, Inequalities in Statistics and Probability. IMS Lecture Notes--Monograph Series 5, pp. 127-140. IMS, Hayward, CA).