Inequalities Via Majorization and Weak Majorization

The concept of majorization basically concerns the comparison of the degrees of diversity among the components between two vectors. For fixed k, two real vectors that are considered are a = (a1…, ak)' and b = (b1,…,bk)'. Let a[1]≥· ·≥a[k], b[1]≥ · ·≥b[k] denote the ordered values of the components of a and b. a is said to weakly majorize b or b is said to be weakly majorized by a, in symbols a ≫ b, if In this, the condition that the components of the vectors have a common total is dropped. Therefore, if two real vectors are not comparable through majorization, they might still be comparable through weak majorization. In particular, a> b holds if a>b or (and) a ≥ b, that is, a[i] ≥ b[i] for each i. This chapter discusses the probability inequalities in multivariate distributions, using majorization and weak majorization as a tool. If two parameter vectors or two sets can be partially ordered through majorization or weak majorization, then under suitable conditions, the corresponding probability contents can be ordered. It is in this capacity that majorization and weak majorization play an important role in deriving probability inequalities. The chapter presents several basic preservation theorems under integral transformations. The chapter discusses their implications in location and scale parameter families.