Fractional negative binomial and Pólya 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. It is shown that the one-dimensional distributions of the FPP and the FNBP are not infinitely divisible. Also, the space fractional Pólya process SFPP is defined by replacing the rate parameter λ 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 PDEs governing the density of the FNBP and the SFPP are also investigated.

On the inverse gamma subordinator

“…Recently, in [6] some new operators associated with Gamma subordinators appear whereas, in [7] the connection between parabolic and elliptic problems in case of Gamma (and inverse Gamma) time change is considered. We also recall an interesting connection between Gamma subordinator and (fractional) negative binomial processes, see for example [8].…”