Space-Fractional Versions of the Negative Binomial and Polya-Type Processes
In this paper, we introduce a space fractional negative binomial process (SFNB) by time-changing the space fractional Poisson process by a gamma subordinator. Its one-dimensional distributions are derived in terms of generalized Wright functions and their governing equations are obtained. It is a Lévy process and the corresponding Lévy measure is given. Extensions to the case of distributed order SFNB, where the fractional index follows a two-point distribution, are investigated in detail. The relationship with space fractional Polya-type processes is also discussed. Moreover, we define and study multivariate versions, which we obtain by time-changing a d-dimensional space-fractional Poisson process by a common independent gamma subordinator. Some applications to population’s growth and epidemiology models are explored. Finally, we discuss algorithms for the simulation of the SFNB process.
KeywordsFractional negative binomial process Stable subordinator Wright function Polya-type process Governing equations
Mathematics Subject Classification (2010)Primary: 60G22 Secondary: 60G51 60E05
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The authors thank Mr. A. Maheshwari for his simulations and computational help. They are also grateful to Prof. C. Macci for his careful reading of the paper and suggesting some useful comments. The authors thank also the referee for some helpful comments.
- Avramidis A, Lecuyer P, Tremblay PA (2003) Efficient simulation of gamma and variance-gamma processes, Simulation Conference, 2003. Proceedings of the 2003 Winter, 1, 319–326Google Scholar
- Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and Applications of Fractional Differential Equations, vol 204 of North-Holland Mathematics Studies. Elsevier Science B.V., AmsterdamGoogle Scholar
- Sato KI (1999) Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Adv. Math. Cambridge Univ. PressGoogle Scholar
- Vellaisamy P, Maheshwari A (2015) Fractional negative binomial and Polya processes, Probab. Math. Statist., to appearGoogle Scholar