Space-Fractional Versions of the Negative Binomial and Polya-Type Processes

  • L. Beghin
  • P. Vellaisamy


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.


Fractional negative binomial process Stable subordinator Wright function Polya-type process Governing equations 

Mathematics Subject Classification (2010)

Primary: 60G22 Secondary: 60G51 60E05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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.


  1. Anscombe FJ (1949) The statistical analysis of insect counts based on the negative binomial distribution. Biometrics 5:165–173CrossRefGoogle Scholar
  2. 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
  3. Beghin L (2012) Random-time processes governed by differential equations of fractional distributed order. Chaos Solitons Fractals 45:1314–1327MathSciNetCrossRefzbMATHGoogle Scholar
  4. Beghin L (2014) Geometric stable processes and related fractional differential equations, Electron. Commun Probab 19(13):1–14MathSciNetzbMATHGoogle Scholar
  5. Beghin L (2015) Fractional gamma process and fractional gamma-subordinated processes. Stoch Anal Appl 33(5):903–926MathSciNetCrossRefzbMATHGoogle Scholar
  6. Beghin L, Macci C (2014) Fractional discrete processes: compound and mixed Poisson representations. J Appl Probab 51.1:9–36MathSciNetzbMATHGoogle Scholar
  7. Beghin L, Macci C (2015) Multivariate fractional Poisson processes and compound sums. Adv Appl Probab 48:691–711MathSciNetCrossRefzbMATHGoogle Scholar
  8. Bliss CI, Fisher RA (1953) Fitting the negative binomial distribution to biological data. Biometrics 9(2):176–200MathSciNetCrossRefGoogle Scholar
  9. Chechkin AV, Gorenflo R, Sokolov IM (2002) Retarding subdiffusion and accelerating superdiffusion governed by distributed-order fractional diffusion equations. Phys Rev E 66:046129. /1–9/6CrossRefGoogle Scholar
  10. Kanter M (1975) Stable densities under change of scale and total variation inequalities. Ann Probab 34:697–707MathSciNetCrossRefzbMATHGoogle Scholar
  11. 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
  12. Kozubowski TJ, Podgórski K (2009) Distributional properties of the negative binomial Lévy process. Probab Math Stat 29:43–71zbMATHGoogle Scholar
  13. Laskin N (2003) Fractional Poisson process. Commun Nonlinear Sci Numer Simul 8(3-4):201–213. Chaotic transport and complexity in classical and quantum dynamicsMathSciNetCrossRefzbMATHGoogle Scholar
  14. Mainardi F, Mura A, Gorenflo R, Stojanovic M (2007) The two forms of fractional relaxation with distributed order. J Vib Control 9-10:1249–1268MathSciNetCrossRefzbMATHGoogle Scholar
  15. Mainardi F, Pagnini G (2007) The role of the Fox–Wright functions in fractional sub-diffusion of distributed order. J Comput Appl Math 207:245–257MathSciNetCrossRefzbMATHGoogle Scholar
  16. Meerschaert MM, Nane E, Vellaisamy P (2011) The fractional Poisson process and the inverse stable subordinator. Electron J Probab 16(59):1600–1620MathSciNetCrossRefzbMATHGoogle Scholar
  17. Orsingher E, Polito F (2012) The space-fractional Poisson process. Stat Probab Lett 82(4):852–858MathSciNetCrossRefzbMATHGoogle Scholar
  18. Sato KI (1999) Lévy Processes and Infinitely Divisible Distributions, Cambridge Studies in Adv. Math. Cambridge Univ. PressGoogle Scholar
  19. Srivastava R (2013) Some generalizations of Pochhammer’s symbol and their associated families of hypergeometric functions and hypergeometric polynomials. Appl Math Inf Sci 7(6):2195–2206MathSciNetCrossRefGoogle Scholar
  20. Upadhye NS, Vellaisamy P (2014) Compound Poisson approximation to convolutions of compound negative binomial variables. Method Comp Appl Probab 16 (4):951–968MathSciNetCrossRefzbMATHGoogle Scholar
  21. Valero J, Ginebra J, Pérez-Casany M (2012) Extended truncated Tweedie-Poisson model. Methodol Comput Appl Probab 14(3):811–829MathSciNetCrossRefzbMATHGoogle Scholar
  22. Vellaisamy P, Maheshwari A (2015) Fractional negative binomial and Polya processes, Probab. Math. Statist., to appearGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Department of Statistical SciencesSapienza University of RomeRomaItaly
  2. 2.Indian Institute of Technology BombayMumbaiIndia

Personalised recommendations