Remarks on compositions of some random integral mappings

TL;DR

This paper explores the properties of compositions of random integral mappings related to Lévy processes, showing they can always be represented as a single integral mapping, with illustrative examples.

Contribution

It demonstrates that compositions of certain random integral mappings can be simplified into a single mapping, providing new insights and examples in the theory of infinitely divisible measures.

Findings

Compositions of random integral mappings can be expressed as a single integral mapping.

The paper provides both old and new examples illustrating this composition property.

Abstract

The random integral mappings (some type of functionals of L\'evy processes) are continuous homomorphisms between convolution subsemigroups of the semigroup of all infinitely divisible measures. Compositions of those random integrals (mappings) can be always expressed as another single random integral mapping. That fact is illustrated by some old and new examples.