Selfdecomposability and Semi-selfdecomposability in Subordination of Cone-parameter Convolution Semigroups

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 $\mathbf{R}^d$, selfdecomposability is inherited from subordinator to subordinated. This is extended to semi-selfdecomposability.