Selfdecomposability and semi-selfdecomposability in subordination of cone-parameter convolution semigroups

TL;DR

This paper extends known results about how selfdecomposability and semi-selfdecomposability properties are preserved under subordination from simple cases like Brownian motion to more complex cone-parameter convolution semigroups.

Contribution

It generalizes the inheritance of selfdecomposability and semi-selfdecomposability properties to cone-parameter convolution semigroups, broadening the scope of subordination theory.

Findings

Selfdecomposability is inherited in cone-parameter convolution semigroups.

Semi-selfdecomposability is inherited in stable cone-parameter convolution semigroups.

Extensions apply from Brownian motion to more complex semigroup structures.

Abstract

Extension of two known facts concerning subordination is made. The first fact is that, in subordination of 1-dimensional Brownian motion with drift, selfdecomposability is inherited from subordinator to subordinated. This is extended to subordination of cone-parameter convolution semigroups. The second fact is that, in subordination of strictly stable cone-parameter convolution semigroups on $\mathbb{R}^d$, selfdecomposability is inherited from subordinator to subordinated. This is extended to semi-selfdecomposability.