Necessity of weak subordination for some strongly subordinated Lévy processes

Consider the strong subordination of a multivariate Lévy process with a multivariate subordinator. If the subordinate is a stack of independent Lévy processes and the components of the subordinator are indistinguishable within each stack, then strong subordination produces a Lévy process; otherwise it may not. Weak subordination was introduced to extend strong subordination, always producing a Lévy 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 Lévy 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