Infinite Divisibility and Max-Infinite Divisibility with Random Sample Size

TL;DR

This paper explores generalized forms of infinitely divisible and max-infinitely divisible laws as limits of random sums and maximums, providing conditions for convergence and extending previous foundational work.

Contribution

It introduces new classes of ID and MID laws based on random sample sizes, with necessary and sufficient conditions for their convergence, expanding the theoretical framework of these distributions.

Findings

Identifies classes of probability generating functions for random sample sizes.

Provides necessary and sufficient conditions for convergence to generalized ID and MID laws.

Extends previous results on ID and MID laws, including geometric MID laws.

Abstract

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.