Necessity of weak subordination for some strongly subordinated L\\'evy\n processes

Consider the strong subordination of a multivariate L\\'evy process with a\nmultivariate subordinator. If the subordinate is a stack of independent L\\'evy\nprocesses and the components of the subordinator are indistinguishable within\neach stack, then strong subordination produces a L\\'evy process, otherwise it\nmay not. Weak subordination was introduced to extend strong subordination,\nalways producing a L\\'evy process even when strong subordination does not.\nHere, we prove that strong and weak subordination are equal in law under the\naforementioned condition. In addition, we prove that if strong subordination is\na L\\'evy process, then it is necessarily equal in law to weak subordination in\ntwo cases: firstly, when the subordinator is deterministic and secondly, when\nit is pure-jump with finite activity.\n