  In this paper, we study the fractional Poisson process (FPP) time-changed by
an independent L\'evy subordinator and the inverse of the L\'evy subordinator,
which we call TCFPP-I and TCFPP-II, respectively. Various distributional
properties of these processes are established. We show that, under certain
conditions, the TCFPP-I has the long-range dependence property and also its law
of iterated logarithm is proved. It is shown that the TCFPP-II is a renewal
process and its waiting time distribution is identified. Its bivariate
distributions and also the governing difference-differential equation are
derived. Some specific examples for both the processes are discussed. Finally,
we present the simulations of the sample paths of these processes.
