# Conditionally monotone independence II: Multiplicative convolutions and infinite divisibility

## TL;DR

This paper explores the properties of the c-monotone multiplicative convolution, unifying various convolutions, characterizing semigroups, and discussing infinite divisibility and embeddings, with connections to Boolean convolution.

## Contribution

It introduces a unified framework for multiplicative convolutions under c-monotone independence and characterizes convolution semigroups and infinite divisibility on the unit circle.

## Key findings

- Characterization of convolution semigroups for c-monotone multiplicative convolution
- Proof that infinitely divisible distributions can be embedded in convolution semigroups
- Discussion of the (non)-uniqueness of embeddings, including the monotone case

## Abstract

We study the multiplicative convolution for c-monotone independence. This convolution unifies the monotone, Boolean and orthogonal multiplicative convolutions. We characterize convolution semigroups for the c-monotone multiplicative convolution on the unit circle. We also prove that an infinitely divisible distribution can always be embedded in a convolution semigroup. We furthermore discuss the (non)-uniqueness of such embeddings including the monotone case. Finally connections to the multiplicative Boolean convolution are discussed.