Fractional Negative Binomial and Polya Processes

Abstract

In this paper, we define a fractional negative binomial process (FNBP) by replacing the Poisson process by a fractional Poisson process (FPP) in the gamma subordinated form of the negative binomial process. First, it is shown that the one-dimensional distributions of the FPP are not infinitely divisible. The long-range dependence of the FNBP, the short-range dependence of its increments and the infinite divisibility of the FPP and the FNBP are investigated. Also, the space fractional Polya process (SFPP) is defined by replacing the rate parameter $\lambda$ by a gamma random variable in the definition of the space fractional Poisson process. The properties of the FNBP and the SFPP and the connections to $pde$'$s$ governing the density of the FNBP and the SFPP are also investigated.