Continuity Properties of Distributions with Some Decomposability

Absolute continuity and smoothness of distributions in the nested subclasses ~L m (B), m = 0, 1, 2,..., of the class of all Bdecomposable distributions are studied. All invertible matrices are classified into two types in terms of P.V. numbers. The minimum integer m for which all full distributions in ~L m (B) are absolutely continuous and the minimum integer m for which all absolutely continuous distributions in ~L m (B) have the densities of class C r for 0 <= r <= (infinity) are discussed according to the type of the matrix B related to P.V. numbers.