  Continuing the study reported in Satheesh (2001),(math.PR/0304499 dated 01
May 2003) and Satheesh (2002)(math.PR/0305030 dated 02May 2003), here we study
generalizations of infinitely divisible (ID) and max-infinitely divisible (MID)
laws. We show that these generalizations appear as limits of random sums and
random maximums respectively. For the random sample size N, we identify a class
of probability generating functions. Necessary and sufficient conditions that
implies the convergence to an ID (MID) law by the convergence to these
generalizations and vise versa are given. The results generalize those on ID
and random ID laws studied previously in Satheesh (2001b, 2002) and those on
geometric MID laws studies in Rachev and Resnick (1991). We discuss attraction
and partial attraction in this generalization of ID and MID laws.
