  The space-time fractional Poisson process (STFPP), defined by Orsingher and
Poilto in \cite{sfpp}, is a generalization of the time fractional Poisson
process (TFPP) and the space fractional Poisson process (SFPP). We study the
fractional generalization of the non-homogeneous Poisson process and call it
the non-homogeneous space-time fractional Poisson process (NSTFPP). We compute
their {\it pmf} and generating function and investigate the associated
differential equation. The limit theorems and the law of iterated logarithm for
the NSTFPP process are studied. We study the distributional properties, the
asymptotic expansion of the correlation function of the non-homogeneous time
fractional Poisson process (NTFPP) and subsequently investigate the long-range
dependence (LRD) property of a special NTFPP. We investigate the limit theorem
and the LRD property for the fractional non-homogeneous Poisson process (FNPP),
studied by Leonenko et. al. (2016). Finally, we present some simulated sample
paths of the NSTFPP process.
