  In this paper, we introduce and study a convoluted version of the time
fractional Poisson process by taking the discrete convolution with respect to
space variable in the system of fractional differential equations that governs
its state probabilities. We call the introduced process as the convoluted
fractional Poisson process (CFPP). The explicit expression for the Laplace
transform of its state probabilities are obtained whose inversion yields its
one-dimensional distribution. Some of its statistical properties such as
probability generating function, moment generating function, moments etc. are
obtained. A special case of CFPP, namely, the convoluted Poisson process (CPP)
is studied and its time-changed subordination relationships with CFPP are
discussed. It is shown that the CPP is a L\'evy process using which the
long-range dependence property of CFPP is established. Moreover, we show that
the increments of CFPP exhibits short-range dependence property.
