A New Compound Poisson Process and Its Fractional Versions

We consider a weighted sum of a series of independent Poisson random variables and show that it results in a new compound Poisson distribution which includes the Poisson distribution and Poisson distribution of order k. An explicit representation for its distribution is obtained in terms of Bell polynomials. We then extend it to a compound Poisson process and time fractional compound Poisson process (TFCPP). It is shown that the one-dimensional distributions of the TFCPP exhibit over-dispersion property, are not infinitely divisible and possess the long-range dependence property. Also, their moments and factorial moments are derived. Finally, the fractional differential equation associated with the TFCPP is also obtained.