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Contents of PMS, Vol. 38, Fasc. 5,
pages 77 - 101
DOI: 10.19195/0208-4147.38.1.5
 

FRACTIONAL NEGATIVE BINOMIAL AND PÓLYA PROCESSES

Palaniappan Vellaisamy
Aditya Maheshwari

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.

2010 AMS Mathematics Subject Classification: Primary: 60G22; Secondary: 60G51.

Keywords and phrases: Fractional negative binomial process, fractional Pólya process, fractional Poisson process, infinite divisibility, Lévy process, PDEs.

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