We describe the CGMY and Meixner processes as time changed Brownian motions. The CGMY uses a time change absolutely continuous with respect to the one-sided stable $(Y/2)$ subordinator while the Meixner time change is absolutely continuous with respect to the one sided stable $(1/2)$ subordinator$.$ The required time changes may be generated by simulating the requisite one-sided stable subordinator and throwing away some of the jumps as described in Rosinski (2001).