Selfdecomposability of Weak Variance Generalised Gamma Convolutions

TL;DR

This paper investigates the self-decomposability of weak variance generalized gamma convolution processes, extending known results from strong to weak subordination and establishing necessary conditions under moment assumptions.

Contribution

It extends the theory of self-decomposability to weakly subordinated processes and clarifies the role of moment conditions on the Thorin measure.

Findings

Driftless Brownian motion leads to self-decomposable processes under weak subordination.

Moment conditions on the Thorin measure are necessary for self-decomposability.

Application to weak variance alpha-gamma process demonstrates the theoretical results.

Abstract

Weak variance generalised gamma convolution processes are multivariate Brownian motions weakly subordinated by multivariate Thorin subordinators. Within this class, we extend a result from strong to weak subordination that a driftless Brownian motion gives rise to a self-decomposable process. Under moment conditions on the underlying Thorin measure, we show that this condition is also necessary. We apply our results to some prominent processes such as the weak variance alpha-gamma process, and illustrate the necessity of our moment conditions in some cases.