Fractional Negative Binomial and Polya Processes

2013/06/11 by P. Vellaisamy, Aditya Maheshwari, Vellaisamy, P. +1
Mathematics · #60E07 #60G51 #FOS: Mathematics #Fractional Differential Equations Solutions #Mathematical functions and polynomials #Primay : 60G22 #Probability (math.PR) #Secondary : 60G55 #Statistical Distribution Estimation and Applications

paper · pdf · doi:10.48550/arxiv.1306.2493

openalex publication_date 2013/06/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 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 pde's governing the density of the FNBP and the SFPP are also investigated.

Citations

• The space-fractional Poisson process

Related

• Space-fractional versions of the negative binomial and Polya-type processes
• Generalized Fractional Negative Binomial Process
• Fractional discrete processes: compound and mixed Poisson representations
• Fractional Generalizations of the Compound Poisson Process
• On the long-range dependence of fractional Poisson and negative binomial processes