Necessity of weak subordination for some strongly subordinated Lévy processes

Consider the strong subordination of a multivariate Levy process with a multivariate subordinator. If the subordinate is a stack of independent Levy processes and the components of the subordinator are indistinguishable within each stack, then strong subordination produces a Levy process; otherwise it may not. Weak subordination was introduced to extend strong subordination, always producing a Levy process even when strong subordination does not. Here we prove that strong and weak subordination are equal in law under the aforementioned condition. In addition, we prove that if strong subordination is a Levy process then it is necessarily equal in law to weak subordination in two cases: firstly when the subordinator is deterministic, and secondly when it is pure-jump with finite activity.