Fractional negative binomial and Pólya processes

• Autorzy: Vellaisamy P. , Maheshwari A.

Identyfikatory

DOI

10.19195/0208-4147.38.1.5

• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

fractional negative binomial process; fractional Pólya process; fractional Poisson process; infinite divisibility; Lévy process; PDEs

• Wydawca: Wrocław University of Science and Technology
• Czasopismo: Probability and Mathematical Statistics
• Rocznik: 2018
• Tom: Vol. 38, Fasc. 1
• Strony: 77--101
• Opis fizyczny: Bibliogr. 26 poz., rys.

Twórcy

autor

Vellaisamy P.

• Indian Institute of Technology Bombay, Department of Mathematics, Powai, Mumbai 400076, India

autor

Maheshwari A.

• Indian Institute of Management Indore, Operations Management and Quantitative Techniques Area, Indore-453556, Madhya Pradesh, India

Bibliografia

• [1] G. E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, Cambridge 1999.
• [2] D. Applebaum, Lévy Processes and Stochastic Calculus, second edition, Cambridge University Press, Cambridge 2009.
• [3] L. Beghin, Fractional gamma processes and fractional gamma-subordinated processes, Stoch. Anal. Appl. 33 (2015), pp. 903-926.
• [4] L. Beghin and C. Macci, Fractional discrete processes: Compound and mixed Poisson representations, J. Appl. Probab. 51(1) (2014), pp. 9-36.
• [5] L. Beghin and E. Orsingher, Fractional Poisson processes and related planar random motions, Electron. J. Probab. 14 (2009), pp. 1790-1827.
• [6] R. Biard and B. Saussereau, Fractional Poisson process: Long-range dependence, applications in ruin theory, J. Appl. Probab. 51 (3) (2014), pp. 727-740.
• [7] N. H. Bingham, Limit theorems for occupation times of Markov processes, Z. Wahrsch. Verw. Gebiete 17 (1971), pp. 1-22.
• [8] A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vol. 3, McGraw-Hill Book Company Inc., New York-Toronto-London 1955.
• [9] W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 2, second edition, Wiley, New York 1971.
• [10] R. Gorenflo and F. Mainardi, On the fractional Poisson process and the discretized stable subordinator, Axioms 4 (2015), pp. 321-344.
• [11] A. A. Kilbas, M. Saigo, and J. J. Trujillo, On the generalized Wright function, Fract. Calc. Appl. Anal. 5 (2002), pp. 437-460.
• [12] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, Elsevier Science, Amsterdam 2006.
• [13] T. J. Kozubowski and K. Podgórski, Distributional properties of the negative binomial Lévy process, Probab. Math. Statist. 29 (2009), pp. 43-71.
• [14] N. Laskin, Fractional Poisson process, Commun. Nonlinear Sci. Numer. Simul. 8 (2003), pp. 201-213.
• [15] N. Laskin, Some applications of the fractional Poisson probability distribution, J. Math. Phys. 50 (2009), 113513.
• [16] N. N. Leonenko, M. M. Meerschaert, R. L. Schilling, and A. Sikorskii, Correlation structure of time-changed Lévy processes, Commun. Appl. Ind. Math. 6 (1) (2014), e-483.
• [17] F. Mainardi, Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London 2010.
• [18] F. Mainardi, R. Gorenflo, and E. Scalas, A fractional generalization of the Poisson processes, Vietnam J. Math. 32 (2004), pp. 53-64.
• [19] A. M. Mathai, R. K. Saxena, and H. J. Haubold, The H-Function: Theory and Applications, Springer, New York 2010.
• [20] M. M. Meerschaert, E. Nane, and P. Vellaisamy, The fractional Poisson process and the inverse stable subordinator, Electron. J. Probab. 16 (2009), pp. 1600-1620.
• [21] M. M. Meerschaert and P. Straka, Inverse stable subordinators, Math. Model. Nat. Phenom. 8 (2013), pp. 1-16.
• [22] E. Orsingher and F. Polito, The space-fractional Poisson process, Statist. Probab. Lett. 82 (2012), pp. 852-858.
• [23] K. Sato, Lévy Processes and Infinitely Divisible Distributions, Cambridge University Press, Cambridge 1999.
• [24] N. M. Temme, Special Functions: An Introduction to the Classical Functions of Mathematical Physics, Wiley, New York 1996.
• [25] P. Vellaisamy and A. Kumar, First-exit times of an inverse Gaussian process, Stochastics 90 (2018), pp. 29-48.
• [26] P. Vellaisamy and M. Sreehari, Some intrinsic properties of the gamma distribution, J. Japan Statist. Soc. 40 (2010), pp. 133-144.

Uwagi

Opracowanie rekordu w ramach umowy 509/P-DUN/2018 ze środków MNiSW przeznaczonych na działalność upowszechniającą naukę (2019).