A Supplement To The Bose-Dasgupta-Rubin (2002) Review Of Infinitely Divisible Laws And Processes

TL;DR

This paper extends the review of infinitely divisible laws by proving properties of discrete ID distributions, exploring their stability, and discussing recent developments and applications in probability theory.

Contribution

It provides new proofs and insights into discrete infinitely divisible laws, including their support, stability, and recent theoretical advancements not covered in the original review.

Findings

Discrete ID distributions with integer components have a mass at zero.

Discrete laws can be stable and have a domain of attraction.

The paper discusses recent developments in infinite divisibility, stability, and applications.

Abstract

This paper proves that if a discrete distribution is infinitely divisible (ID) with integer-valued components, then it has a mass at the origin, which also implies why certain ID discrete laws do not have gaps in its support. We argue that discrete laws also can be stable and such laws do have domain of attraction. Then we give certain recent developments and references not reported in the Bose Dasgupta Rubin (2002) review in Sankhya, and some examples in the topics; infinite divisibility and stability of discrete laws, random infinite divisibility, operator stable laws, class-L laws, Goldie-Steutel result, max-infinite divisibility and stability, simulation, alternate stable laws, applications and free probability theory.