On the weak convergence of subordinated systems

General sufficient conditions are given for the convergence of a sequence of subordinated continuous or càdlàg functions on [0,∞). Based on this result, we consider the weak convergence of continuous processes and càdlàg processes under random time change, e.g., subordination. Quenched, annealed, and joint weak convergence of subordinated sequence of continuous processes or càdlàg processes are proved under the condition that both the basic processes and the subordinators converge weakly, and the condition that the basic processes have no fixed discontinuity property if the subordinators are càdlàg processes. We give an example of marked birth and death process to illustrate the application of these convergence theorems.